text
stringlengths 1
2.55M
| id
stringlengths 21
25
| metadata
dict |
|---|---|---|
\section{Introduction}
The relation between the quantum and classical dynamics of nonlinear
systems includes a specific side in the correspondence between the
dynamical properties of systems treated in the mixed and fully
quantized descriptions. Various aspects of the correspondences between
classical nonlinear systems on the one side and their fully quantized
counterparts on the other have been intensively investigated in the
last decade (see e.\ g.\ \cite{Hel91,BTU,Rei}). In many systems
relevant for molecular and condensed matter physics the direct
quantization of the full system in one step is, however, not possible
from a practical point of view. As a rule such systems divide
naturally into interacting subsystems. Then a stepwise quantization
is applied resulting in a mixed description, in which one of the
subsystems is treated in the quantum and the other in the classical
context. Furthermore in complex systems the mixed description is
often necessary for understanding global dynamical properties, e.\ g.
the presence of bifurcations and separatrix structures dividing the
solution manifold into characteristic parts, before for a selected
energy region the full quantization can be performed.
This stepwise quantization is the basic idea on which the Born -
Oppenheimer approximation developed in the early days of quantum
mechanics for the quantization of systems dividing into subsystems is
based. As is well known this approximation can be complemented into a
rigorous scheme, if the nonadiabatic couplings are included \cite{BH}.
These couplings can be the source of nonintegrability and chaos of
systems treated in the mixed quantum-classical description
\cite{BK,BE}. Then the problem of the quantum-classical
correspondences arises on the level of the relation between the
dynamical properties of the mixed and fully quantized descriptions
\cite{BE}.
In this paper we consider this dynamic relation for the particular
model of a quasiparticle moving between two sites and coupled to
oscillators. This is an important model system with applications such
as excitons moving in molecular aggregates and coupled to vibrations,
see e.~g.\ \cite{W}. It has also attracted widespread attention in the
context of the spin-boson Hamiltonian and its classical-quantum
phase space behavior and correspondence (see e. g. \cite{GH,K,Sto} and
references therein). Hence it seems appropriate to use this system as
a model to analyze the relation between the mixed and fully quantized
descriptions. Treating the oscillators in the classical or quantum
contexts, whereas the quasiparticle moving between two sites is a
quantum object from the beginning, one arrives at mixed and fully
quantized levels of description. We have recently investigated the
dynamical properties of this model in the mixed description by
integrating the corresponding Bloch-oscillator equations and
demonstrated the presence of a phase space with an underlying
separatrix structure for overcritical coupling and chaos developing
from the region of the hyperbolic point at the center of this
structure. For increasing total energy chaos spreads over the product
phase space of the system constituted by the Bloch sphere and
oscillator plane, leaving only regular islands in the region of the
antibonding states \cite{ES}. Here we consider the problem of the
relation between the dynamics in the mixed and fully quantum levels of
description of the coupled quasiparticle-oscillator motion.
Investigating this relation we focus on the adiabatic parameter range,
where the closest correspondence between the classical and quantum
aspects of the oscillator dynamics can be expected. Although several
aspects of the dynamics of the system have been considered
\cite{GH,Sto}, there exists no systematic investigation in the
adiabatic parameter range. In particular, such an investigation
requires the numerical determination of a large number of eigenstates
for the fully quantized system. The stationary properties of these
states were reported in \cite{inc}. In this paper these states are
used to compute the dynamics of the fully quantized system and to
compare the quantum evolution with the dynamics of the mixed
description where the oscillator is treated classically. Performing
this comparison we use both the fixed and adiabatic basis sets in the
mixed description. The latter basis set is of particular importance to
clarify the role of the nonadiabatic couplings in the formation of the
dynamics. We demonstrate the effect of the separatrix structure of the
mixed description in the oscillator wave packet propagation of the
fully quantized version, of dynamical subsystem correlations deriving
from the separatrix structure and how the chaotic phase space regions
of the system in the mixed description show up in the nonstationary
properties of the time dependent full quantum state vector.
In section \ref{model} the model will be specified in detail. The
mixed quantum-classical description is discussed in section \ref{mqcd}
including the derivation of the equations of motion, the fixed point
structure and the dynamical properties of the system on this level of
description. In section \ref{qevol} the evolution of the fully
quantized system is presented and compared to the dynamics in the
mixed description.
\section{The Model}\label{model}
We consider a quasiparticle coupled to oscillator degrees of
freedom. The quasiparticle is specified as a molecular exciton in a tight
binding representation and can be substituted by any other quantum object
moving between discrete sites and described by a tight binding Hamiltonian
of the same structure. The system has the Hamiltonian
\begin{equation}\label{htot}
H^{{\rm (tot)}} = H^{{\rm (exc)}} + H^{{\rm (vib)}} + H^{{\rm (int)}},
\end{equation}
where $H^{{\rm (exc)}}$, $H^{{\rm (vib)}}$ and $H^{{\rm (int)}}$ are the excitonic, vibronic and interaction
parts, respectively. $H^{{\rm (exc)}}$ represents the quantum subsystem, which is taken
in the site representation
\begin{equation}\label{hex}
H^{{\rm (exc)}} = \sum_n \epsilon_n |c_n|^2 + \sum_{n \ne m} V_{nm} c_n^*\,c_m,
\end{equation}
where $c_n$ is the quantum probability amplitude of the exciton to
occupy the $n$-th molecule and $V_{nm}$ the transfer matrix element.
For the intramolecular vibrations coupling to the exciton we use the
harmonic approximation in $H^{{\rm (vib)}}$
\begin{equation}\label{hvi}
H^{{\rm (vib)}} = {1 \over 2} \sum_n (p_n^2 + \omega_n^2 q_n^2).
\end{equation}
Here $q_n$, $p_n$ and $\omega_n$ are the coordinate, the canonic conjugate momentum
and frequency of the intramolecular vibration of the $n$-th molecule,
respectively. The interaction Hamiltonian $H^{{\rm (int)}}$ represents the dependence
of the exciton energy on the intramolecular configuration for which we use
the first order expansion in $q_n$
\begin{equation}\label{int}
H^{{\rm (int)}} = \sum_{n} \gamma_n q_n |c_n|^2,
\end{equation}
where $\gamma_n$ are the coupling constants. The interaction is restricted to a
single oscillator at each molecule.
The case of a symmetric two site system $n=1,2$, e.~g.\ an exciton in a molecular
dimer constituted by two identical monomers, is considered in what follows.
For this case we set $\epsilon_1=\epsilon_2$, $\omega_1=\omega_2$, $\gamma_1=\gamma_2$ and
$V_{12}=V_{21}=-V$, $V>0$. Then by introducing for the vibronic subsystem
the new coordinates and momenta
\begin{equation}\label{qp}
q_\pm := {q_2 \pm q_1 \over \sqrt{2} }\hspace*{1cm}
p_\pm := {p_2 \pm p_1 \over \sqrt{2} },
\end{equation}
and for the excitonic subsystem the Bloch variables
\begin{equation}\label{bl}
x=\rho_{21}+\rho_{12},\hspace*{1cm}
y=i(\rho_{21}-\rho_{12}),\hspace*{1cm}
z=\rho_{22}-\rho_{11},
\end{equation}
where $\rho_{mn}$ is the density matrix of the excitonic subsystem
\begin{equation}\label{dm}
\rho_{mn}=c_n^*c_m,
\end{equation}
the relevant part of (\ref{htot})-(\ref{int}) connected with the exciton
coupled to the $q_-$ vibration is obtained in the form
\begin{equation}\label{hsymx}
H_- = -Vx + {1 \over 2} (p_-^2 + \omega^2 q_-^2) +
{\gamma q_- z \over \sqrt{2}}\,.
\end{equation}
The part corresponding to $q_+$ is not coupled to the exciton and omitted.
The Hamiltonian (\ref{hsymx}) can be represented as an operator in the
space of the two dimensional vectors $C=(c_1,c_2)$ constituted by the
excitonic amplitudes $c_n$ by using the standard Pauli spin matrices
$\sigma_i$ $(i=x,y,z)$. Passing in (\ref{hsymx}) to dimensionless
variables by measuring $H$ in units of $2V$ and replacing $q_-$, $p_-$
by
\begin{equation}\label{gqp}
Q := \sqrt{2V} q_- \hspace*{1cm} P := {1 \over \sqrt{2V}} p_-,
\end{equation}
one finally obtains
\begin{equation}\label{hsym}
H = -{\sigma_x\over2} + {1 \over 2} (P^2 + r^2 Q^2) +
\sqrt{p\over 2}r\,Q\sigma_z\, .
\end{equation}
Here
\begin{equation}\label{par}
p = {\gamma^2\over 2V\omega^2}
\end{equation}
represents the dimensionless excitonic-vibronic coupling
and
\begin{equation}\label{rad}
r = {\omega \over 2V}
\end{equation}
is the adiabatic parameter measuring the relative strength of quantum
effects in both subsystems. We focus on the adiabatic case $r\ll 1$
when the vibronic subsystem can be described in the classical
approximation. To make contact with the dynamical features following
from the adiabatic approximation we derive the basic equations in both
the fixed and the adiabatic bases.
\section{Mixed Quantum-Classical Description}\label{mqcd}
\subsection{Fixed Basis}
In this case the basis states are given by the fixed molecule sites
$|n\rangle$ . Representing the excitonic state by $|\psi\rangle = \sum_n c_n
|n\rangle$, inserting it into the time dependent Schroedinger equation for
(\ref{hsym}) and using (\ref{bl}) the quantum equations of motion for
the excitonic subsystem describing the transfer dynamics between the
two sites are obtained. The classical equations for the dynamics of
the oscillator are found by passing to the expectation values of $Q$
and $P$ and using (\ref{hsym}) as a classical Hamiltonian function
from which the canonical equations are derived. In this way one
obtains the coupled Bloch-oscillator equations representing the
dynamics of the system in the mixed description
\begin{eqnarray}\label{eommix}
\dot{x} &=& - \sqrt{2p}\,r\,Q\, y
\nonumber \\
\dot{y} &=& \sqrt{2p}\,r\,Q\, x + z
\nonumber \\
\dot{z} &=& -y
\\
\dot{Q} &=& P
\nonumber \\
\dot{P} &=& -\,r^2 Q - \sqrt{p\over2}\,r z
\nonumber
\end{eqnarray}
Besides the energy
\begin{equation}
E = -{x\over2} + {1\over2}(P^2 + r^2Q^2) + \sqrt{p\over2}rQz
\end{equation}
there is a second integral of the motion
restricting the flow associated with the quantum subsystem to the
surface of the unit radius Bloch sphere
\begin{equation}\label{bln}
R^2 = x^2 + y^2 + z^2 = 1\,.
\end{equation}
Sometimes it is advantageous to make use of this conserved quantity in
order to reduce the number of variables to four, e.\ g.\ when a
formulation in canonically conjugate variables is desired also for the
excitonic subsystem. One then introduces an angle $\phi$ by
\begin{equation}\label{defphi}
x = \sqrt{1-z^2}\cos\phi\,,\hspace*{1cm}y=\sqrt{1-z^2}\sin\phi\,.
\end{equation}
We shall replace the usual Bloch variables by these coordinates
where it is appropriate.
\subsection{ Adiabatic Basis}
In this case one first solves the eigenvalue problem of the part of
the Hamiltonian (\ref{hsym}) which contains excitonic operators
\begin{equation}\label{adham}
h^{{\rm (ad)}} = -{\sigma_x\over2} + \sqrt{p\over2}rQ\sigma_z \,,
\end{equation}
where $Q$ is considered as an adiabatic variable. The eigenvalues of
(\ref{adham}) are given by
\begin{equation}
\epsilon_\pm^{{\rm (ad)}}(Q)\, = \pm {1\over2}w(Q),
\end{equation}
where
\begin{equation}\label{wu}
w(Q)\, = \sqrt{1 + 2pr^2 Q^2}.
\end{equation}
The eigenvalues are part of the adiabatic potentials for the slow
subsystem
\begin{equation}\label{adpot}
U_\pm^{{\rm (ad)}}(Q) = {1\over 2} r^2Q^2 + \epsilon_\pm^{{\rm (ad)}}(Q) \, ,
\end{equation}
The two eigenstates $(\alpha=1,2)$ of (\ref{adham}) represented by
the fixed basis are given by
\begin{equation}\label{adbv+}
|\alpha=2,Q\rangle = {1\over\sqrt{2}}(-\sqrt{1 + c(Q)}\,|2\rangle+\sqrt{1 - c(Q)}\,|1\rangle)
\end{equation}
and
\begin{equation}\label{adbv-}
| \alpha=1,Q \rangle = {1\over\sqrt{2}}(+\sqrt{1 - c(Q)}\,|2\rangle+\sqrt{1 + c(Q)}\,|1\rangle)
\end{equation}
with
\begin{equation}
c(Q) := {\sqrt{2p}rQ\over w(Q)}\,.
\end{equation}
The state vector of the excitonic subsystem is expanded in the
adiabatic basis $|\psi\rangle = \sum_{\alpha}c^{{\rm (ad)}}_\alpha |\alpha,Q \rangle $ and
inserted into the time dependent Schr\"odinger equation. For obtaining
the complete evolution equations in the adiabatic basis one has to
take into account the time derivative of the expansion coefficients
$c^{{\rm (ad)}}_\alpha$ as well as the nonadiabatic couplings due to the time
dependence of the states $|\alpha,Q(t)\rangle$. The neglect of these
couplings would result in the adiabatic approximation. Using $(d/d t)
|\alpha,Q \rangle = \dot Q (d/d Q)|\alpha,Q\rangle$ the nonadiabatic coupling
function
\begin{equation}\label{ncg}
\varphi_{\alpha\beta} := \left\langle\alpha,Q\left|{\partial\over\partial Q}\right|\beta,Q \right\rangle,
\end{equation}
$(\varphi_{\alpha\beta}=-\varphi_{\beta\alpha})$ is found, which in
case of the eigenstates (\ref{adbv+},\ref{adbv-}) is explicitly given
by
\begin{equation}\label{nce}
\varphi_{12} = - \frac{\sqrt{p}r}{\sqrt{2}[w(Q)]^2}\,.
\end{equation}
Introducing now analogously to (\ref{bl}) the Bloch variables in the
adiabatic basis and treating the oscillator in the classical
approximation one obtains the coupled Bloch-oscillator equations in
the adiabatic basis
\begin{eqnarray}\label{eomad}
\dot x^{{\rm (ad)}} &=& 2P\, \varphi_{12}(Q)\, z^{{\rm (ad)}} - w(Q)\, y^{{\rm (ad)}}\nonumber \\
\dot y^{{\rm (ad)}} &=& w(Q)\, x^{{\rm (ad)}}\nonumber \\
\dot z^{{\rm (ad)}} &=& -2P\, \varphi_{12}(Q)\, x^{{\rm (ad)}}\\
\dot Q &=& P\nonumber \\
\dot P &=& -r^2Q + \sqrt{p}\,w(Q)\varphi_{12}\,x^{{\rm (ad)}} - \sqrt{p\over2}\,r\,c(Q)\,z^{{\rm (ad)}}\nonumber
\end{eqnarray}
The energy is now expressed by the adiabatic Bloch variable $z^{{\rm (ad)}}$
\begin{equation}\label{ad1dz}
E = w(Q){z^{{\rm (ad)}}\over2} + {1\over 2}\left(P^2 + r^2Q^2\right)
\end{equation}
and the flow is again located on the surface of the unit Bloch sphere.
Neglecting the nonadiabatic couplings $\varphi_{12}=0$ one obtains the
dynamics of the decoupled adiabatic oscillators. The adiabatic
oscillators can be considered as one dimensional integrable subsystems
corresponding to the Hamiltonians
\begin{equation}\label{ad1d}
h_\pm^{{\rm (ad)}} = {1\over 2} P^2 + U_\pm^{{\rm (ad)}}(Q) \, ,
\end{equation}
where $U_\pm^{{\rm (ad)}}(Q)$ is given by (\ref{adpot}).
The connection between the Bloch variables in the fixed and the adiabatic
basis is given by\\[3mm]
\begin{tabular}{ll}
$ x = - c(Q) x^{{\rm (ad)}} - \sqrt{1-c(Q)^2}\,z^{{\rm (ad)}} \hspace*{1cm}$&
$ x^{{\rm (ad)}} = - c(Q) x - \sqrt{1-c(Q)^2}\,z $\\
$ y = -y^{{\rm (ad)}} $&
$ y^{{\rm (ad)}} = -y $\\
$ z = c(Q) z^{{\rm (ad)}} - \sqrt{1 - c(Q)^2} x^{{\rm (ad)}} $&
$ z^{{\rm (ad)}} = c(Q) z - \sqrt{1 - c(Q)^2} x $
\end{tabular}
\begin{equation}\label{trbloch}
\sqrt{1-z^2}\,\sin\phi=-\sqrt{1-(z^{{\rm (ad)}})^2}\,\sin\phi^{{\rm (ad)}}
\end{equation}
Using these transformation formulas one can show that the equations of
motion (\ref{eommix}) derived in the fixed basis are actually
equivalent to those in the adiabatic basis (\ref{eomad}).
\subsection{Fixed points and bifurcation}
Essential information about the phase space of the excitonic-vibronic
coupled dimer is contained in the location and the stability
properties of the fixed points of the mixed quantum-classical
dynamics. Setting in the equations of motion for the fixed basis
(\ref{eommix}) all time derivatives to zero, we find for any
stationary state
$$
Q_s = -{1\over r}\sqrt{p\over2}z_s \,,\hspace*{1cm} P_s = 0 \,,\hspace*{1cm} y_s = 0
$$
\begin{equation}\label{statcon}
z_s - p x_s z_s = 0 \,.
\end{equation}
The stability properties of a fixed point are determined by a
linearization of the equations of motion using canonical variables
\cite{ES}.
It is appropriate to subdivide all stationary points according to
whether they are located in the bonding region $x_s > 0$ or in the
antibonding region $x_s < 0$. There is no transition between these two
groups when the parameters of the system are varied since $x_s = 0$ is
excluded by (\ref{statcon}). This terminology is in accordance with
molecular physics where it is common to refer to the state $x=1$ with
symmetric site occupation amplitudes $c_1 = c_2$ as bonding and to the
state $x=-1$ with antisymmetric amplitudes $c_1=-c_2$ as antibonding.
\subsubsection{Bonding region ($x_s > 0$)}
We consider the bonding region first. The location of the fixed points
is obtained from (\ref{statcon}) using the additional restriction
\begin{equation}
x_s^2 + z_s^2 = 1\,.
\end{equation}
One finds the following solutions in dependence on the value of the
dimensionless coupling strength $p$:
\begin{description}
\item{(A) $0 \le p \le 1$:}
In this case (\ref{statcon}) allows for a single solution only.
\begin{equation} {\bf g:}\hspace*{1cm}
x_s = 1 \,,\hspace*{1cm} z_s = 0 \,,\hspace*{1cm} Q_s = 0 \,,\hspace*{1cm} E_s =-{1\over 2}
\end{equation}
This point is the bonding ground state corresponding to a symmetric
combination of the excitonic amplitudes $c_1 = c_2 = 1/\sqrt{2}$.
$\bf g$ is stable elliptic.
\item{(B) $p \ge 1$:}
A bifurcation has occurred and we obtain three stationary points.
\begin{equation}
\begin{array}{ccc}
{\bf g_{\pm}:} & x_s = {1\over p} & z_s = \pm{\sqrt{p^2-1}\over p} \\
& Q_s=\pm{\sqrt{p^2-1}\over\sqrt{2p\,}r} & E_s= -{p^2+1\over4p}
\end{array}
\end{equation}
These two points are stable elliptic.
\begin{equation}
{\bf h:}\hspace*{1cm}
x_s = 1\,,\hspace*{1cm} z_s =0 \,,\hspace*{1cm} Q_s = 0 \,,\hspace*{1cm} E_s = -{1\over 2}
\end{equation}
The point $\bf h$ is at the position of the former ground state, but in
contrast to $\bf g$ it is unstable hyperbolic.
\end{description}
\noindent
The parameter $p$ governs a pitchfork bifurcation: The ground state g
below the bifurcation ($p < 1$) splits into two degenerate ground
states $\bf g_\pm$ above bifurcation ($p > 1$). At the former ground state
a hyperbolic point $\bf h$ appears. This situation is also obvious from
fig.~\ref{adpotplot}(a).
\subsubsection{Antibonding region ($x_s < 0$)}
Independent on the coupling strength $p$ we have in this region only one
solution of (\ref{statcon}) (see fig.\ \ref{adpotplot}(b)):
\begin{equation}
{\bf e:}
z_s = 0 \,,\hspace*{1cm} x_s= -1 \,,\hspace*{1cm} Q_s = 0 \,,\hspace*{1cm} E_s = +{1\over 2}
\end{equation}
This stationary state corresponds to an
antisymmetric combination of the excitonic amplitudes $c_1 = -c_2 =
1/\sqrt{2}$. $\bf e$ is stable for
\begin{equation}\label{stababond}
\frac{|r^2 - 1|}{r} > 2\sqrt{p}\,,
\end{equation}
which holds when the system is not in resonance and in particular for
the adiabatic case $r \ll 1$.
\paragraph*{}
Since the equations of motion in the fixed and in the adiabatic basis
are equivalent, it is clear that the fixed same points (\ref{statcon})
can also be obtained from (\ref{eomad}). Setting in (\ref{eomad}) the
time derivatives of $x$, $y$ and $Q$ equal to zero, one finds for the
stationary states of the adiabatic case
\begin{eqnarray}
\label{ans}
x_s^{{\rm (ad)}} = 0 \,,\hspace*{1cm}
y_s^{{\rm (ad)}} = 0 \,,\hspace*{1cm}
P_s^{{\rm (ad)}} = 0\,,
\end{eqnarray}
leaving for the stationary values of $z_s^{{\rm (ad)}}$ the poles
\begin{equation}\label{adz}
z_s^{{\rm (ad)}} = \pm 1
\end{equation}
It is worth noting that in the adiabatic basis the stationary states
are always located at $z^{{\rm (ad)}}_s=\pm 1$ and this will be the case for any
system treated in mixed quantum-classical description and restricted
to the two lowest adiabatic levels. A specific feature of using the
adiabatic basis is the independence of the location of the fixed
points on the explicit form of the nonadiabatic coupling function
since this function enters the equations of motion in form of the
products $\varphi_{12}(Q)P$ and $\varphi_{12}(Q)x^{{\rm (ad)}}$ which
drop out at a fixed point because of (\ref{ans}) and (\ref{adz}).
From (\ref{trbloch}) it is moreover easy to see that the fixed points
in the bonding region are located within the lower adiabatic potential
while the antibonding fixed points belong to the upper one.
The eq.\ $\dot {P}=0$ reduces to
\begin{equation}\label{adQ}
(w(Q) + pz_s^{{\rm (ad)}})Q=0\,.
\end{equation}
For $z^{{\rm (ad)}}_s=+ 1$ the only solution of (\ref{adQ}) is $Q_s=0$, whereas
for $z^{{\rm (ad)}}_s=-1$ one obtains additional solutions for $p>1$. These
solutions are easily seen to correspond to the bifurcation discussed
above.
\subsection{Integrable Approximations}
Before we investigate the dynamics of the complete coupled equations
of motion (\ref{eommix}) or (\ref{eomad}) we would like to mention two
integrable approximations to the model. The first and trivial
integrable approximation is to set in the equations of motion in the
fixed basis (\ref{eommix}) $p=0$ which results in a decoupling of the
excitonic and vibronic motions. The second and more interesting
integrable approximation is obtained by neglecting the nonadiabatic
coupling function $\varphi_{12}=0$ in the equations of motion
(\ref{eomad}) of the adiabatic basis which defines the adiabatic
approximation from a dynamical point of view. In this approximation
some of the nonlinear features of the model are still contained in the
integrable adiabatic reference oscillators (\ref{ad1d}). In particular
the lower adiabatic potential (\ref{adpot}) displays the bifurcation
from a single minimum structure to the characteristic double well
structure when the parameter $p$ (\ref{par}) passes through the
bifurcation value $p=1$. It is also important to note that the fixed
point structure is not changed when the nonadiabatic couplings are
switched off: Neglecting $\varphi_{12}$ in (\ref{eomad}) results in
the same fixed points equations as in the case including the
nonadiabatic couplings. From the formal side the neglect of
$\varphi_{12}$ not necessarily leads to $z_s^{{\rm (ad)}} = \pm 1$: According to
the equations of motion(\ref{eomad}) $\varphi_{12}=0$ implies $z_s^{{\rm (ad)}}
= const$. Then in the dynamics of the adiabatic approximation both
adiabatic modes can be occupied and only the transitions between them
are switched off. The oscillator equations become autonomous
describing regular motions according to the classical Hamilton
function (\ref{ad1dz}) with $z_s^{{\rm (ad)}}$ as a parameter. The oscillator
coordinate $Q(t)$ enters the Bloch equations for $x(t)$ and $y(t)$.
The equations for the latter describe the regular motion on a circle
generated by an intersection of the Bloch sphere with the plane
$z_s^{{\rm (ad)}} = const$ on which the phase oscillations between the modes are
realized.
In the following we demonstrate that the regular structures associated
with the adiabatic approximation are present in both the mixed and
fully quantized descriptions. At the same time we show that the
complete coupled system of Bloch-oscillator equations, i.~e.\
including the nonadiabatic couplings, displays dynamical chaos. This
identifies the nonadiabatic couplings as a source of nonintegrability
and chaos in the mixed description of the system and rises the
question about the signatures of this chaos after full quantization
is performed. The latter problem will be addressed in the last
section.
\subsection{Dynamical Properties}
\label{sec:dynmix}
The dynamical properties of the coupled Bloch-oscillator equations
(\ref{eommix}) were analyzed by a direct numerical integration. Some of
our results, such as the presence of chaos in the mixed description of
the excitonic-vibronic coupled dimer, were reported in \cite{ES}.
Therefore the aim of this section is twofold: On the one side we
reconsider the findings in \cite{ES} relating the dynamical structures
to the adiabatic approximation, in which the integrable reference
systems (\ref{ad1d}) can be defined. This clarifies the role of the
nonadiabatic couplings in the formation of the dynamics of the model,
which was not done before. On the other side we provide the necessary
characterization of the phase space structure, such as the location of
the separatrix dividing the phase space into trapped and detrapped
solutions, and the identification of the regions and associated
parameters belonging to the regular and chaotic parts of the dynamics,
respectively. The latter points will provide the basis to perform the
comparison of the mixed description with the full quantum evolution in
the next section.
The numerical integration of the equations of motion in the mixed
description can be performed both in the fixed (\ref{eommix}) and the
adiabatic basis (\ref{eomad}). The integration in the fixed basis,
however, is numerically simpler and the fixed Bloch variables provide
a more convenient frame for the excitonic motion. Hence we used the
fixed basis for a numerical integration. It must be stressed, however,
that the representation of the excitonic system in the fixed and the
adiabatic basis are equivalent, if the nonadiabatic couplings are
included. The connection between both representations is given by the
eqs. (\ref{trbloch}).
In view of the existence of two integrals of the motion three
variables from the total of five variables of the system are
independent. Therefore a standard two dimensional Poincar\'e surface
of section is defined by fixing one variable. According to the choice
of variables sections can be defined for both the oscillatory and
excitonic subsystems. The dynamics of the system was found to depend
crucially on the choice of the total energy with respect to the
characteristic energies of the system such as the minima of the
adiabatic potentials and above the bifurcation the energy
corresponding to the hyperbolic fixed point $E_h$. Above the
bifurcation the separatrix structure, which divides the phase space into
characteristic parts, is present. The regular phase space structure
following from the integrable adiabatic reference oscillators
(\ref{ad1d}) above the bifurcation is shown in fig.\ \ref{adpotplot}.
In fig.\ \ref{pq-0.8}(a) a Poincar\'e section in oscillator variables
is presented for the value $p=0.8$ which is below the bifurcation
value $p=1$. In this case the adiabatic potential $U_-(Q)$ has a
single minimum. The total energy is chosen at $E=0$, i.\~e.\ well
below the minimum of the upper adiabatic potential. Therefore the
influence of this potential is small and the oscillator dynamics can
be expected to be close to the regular dynamics of the lower reference
oscillator associated with $U_-(Q)$. This is indeed confirmed by fig.\
\ref{pq-0.8}(a). There is, however, a chain of small resonance islands
in the outer part of the section, which is due to resonance between
the oscillator motion and the occupation oscillations between the
adiabatic modes corresponding to the finite $\dot z^{{\rm (ad)}}$ in the case of
the presence of the nonadiabatic couplings. The interaction between
the occupation oscillations due to the finite $\dot z^{{\rm (ad)}} $ and the
oscillator motion becomes much more pronounced for higher energies. A
corresponding Poincar\'e section is displayed in fig.\ \ref{pq-0.8}(b)
for the same value $p=0.8$ of the coupling constant, but with the
energy now chosen above the minimum of the upper adiabatic potential.
This choice of the energy allows according to (\ref{ad1dz}) for a much
broader range for the variation of the variable $z^{{\rm (ad)}}$ and
consequently the nonadiabatic couplings are more effective.
Correspondingly we observe now several resonance chains.
Increasing the coupling above the bifurcation value $p=1$, but fixing
the total energy below $E_h$ one expects regular oscillations around
the displaced minima of the double well structure in $U_-(Q)$. A
Poincar\'e section in the oscillator variables corresponding to this
behavior is shown in fig.\ \ref{pq-3.4}(a), where $p=3.4$ and the
total energy is below the saddle point of the potential $U_-(Q)$.
Increasing the energy to a value slightly above $E_h$ one finds
sections displaying oscillations resembling the separatrix structure
as shown in fig.\ \ref{pq-3.4}(b).
For energies well above $E_h$ chaotic trajectories do exist.
Characteristic examples are provided by the Poincar\'e sections in
oscillator variables displayed in fig.\ \ref{pq-3.4}(c) and (d), where
the regions of regular and chaotic behavior of the oscillator
subsystem are shown for two cases of total energy above $E_h$. In the
case (c) the total energy is below the case (d). It is seen how with
increasing energy the regular part of the oscillator phase space
becomes smaller and the chaotic part increases. The corresponding
regular and chaotic components of the excitonic subsystem are located
in the antibonding and bonding regions of the Bloch sphere,
respectively (see also fig.\ \ref{03} below). Relating the location
of the dynamics on the Bloch sphere to the energy of the excitonic
subsystem we find that for chaotic trajectories the excitonic
subsystem is in an energetically low state within the bonding region
of the Bloch sphere whereas for regular trajectories the excitonic
subsystem is in its energetically high state within the antibonding
region. Correspondingly, the energy of the vibronic subsystem is high
for chaotic dynamics and low for regular dynamics, because the total
energy is the same for all the trajectories displayed in each of the
figures \ref{pq-3.4}(c) and (d). Hence the destruction of the regular
dynamics is connected with the energy of the vibronic subsystem:
Regular dynamics is realized for oscillator states with low energy,
small amplitude oscillations and consequently small effective
coupling, whereas high oscillator energy destroys the regular
structures and results in global chaos.
The dynamics of the oscillator subsystem is complemented by the
Poincar\'e sections on the surface of the Bloch sphere showing the
behavior of the excitonic subsystem. In the figs.\ \ref{03}(a)-(d)
such a typical set of Poincar\'e sections is presented for different
energies and above the bifurcation ($p = 2.0$). The sections
correspond to the left turning point of the oscillator. For low energy
one finds regular dynamics in the region of the bifurcated ground
states. These regular trajectories represent the self trapped
solutions of the system in which the exciton is preferentially at one
of the sites of the dimer and correspond to the one sided oscillations
of the vibronic subsystem of fig.\ \ref{pq-3.4}(a). Increasing the
energy local chaos starts in the vicinity of the hyperbolic point
$E_h$. The local chaos can be considered as a perturbation of the
dynamics near the saddle of the lower potential $U_-(Q)$ due
the nonadiabatic couplings of the adiabatic oscillators. With
increasing energy chaos spreads over the Bloch sphere leaving only
regular islands in the region of antibonding states associated with
the upper adiabatic potential and in accordance with the dynamics of
the vibronic subsystem discussed above. For high enough energy the
coupling between the adiabatic reference oscillators almost completely
destroys regular structures and results in global chaos.
\section{Quantum Evolution}\label{qevol}
We now turn to the dynamics in the full quantum description of the
model considering in the Hamiltonian (\ref{hsym}) the coordinate $Q$
and the momentum $P$ as non-commuting quantum variables. We focus on
the features of the evolution in the adiabatic parameter region for
$r\ll 1$. The evolution of the full quantum state vector of the
system satisfying some fixed initial condition is computed from the
eigenstate representation of the Hamiltonian. For a realistic
description of the system in the adiabatic parameter region a large
number of eigenstates had to be used in the expansion.
Correspondingly, the diagonalization of the Hamiltonian (\ref{hsym})
with $Q$ and $P$ being quantum operators was performed using a large
set of oscillator eigenfunctions for the undisplaced oscillator as a
basis, i.~e.\ the basis was constructed from the product states
$|n,\nu\rangle:=|n\rangle \otimes |\nu\rangle$, where the index $n=1,2$ labels the
two sites of the dimer and $\nu=0,1,\dots$ stands for the oscillator
quantum number. In this basis the quantized version of the
Hamiltonian (\ref{hsym}) is represented by the matrix
\begin{equation}
\langle n,\nu| H | n',\nu '\rangle =
-{[1-(-1)^{\nu+\nu'}]\over 4}\delta_{\nu ,\nu'} \,+\,
r \left( \nu + {1 \over 2}\right)\delta_{n, n'}\delta_{\nu,\nu '} \,+
\end{equation}
\begin{equation}\label{matel}
+\,
{\sqrt{p\,r}\over 2}(-1)^n\,(\sqrt{\nu'}\delta_{\nu,\nu ' -1} +
\sqrt{\nu}\delta_{\nu ,\nu ' + 1})\,\delta_{n, n'}\,.
\end{equation}
The typical number of oscillator eigenfunctions used was $750$
yielding a total of $1500$ basis states.
The properties of the stationary eigenstates, the fine structure of
the spectrum and in particular the influence of the adiabatic
reference oscillators and the role of the nonadiabatic couplings in
the formation of the spectrum were reported in \cite{inc}. Here we
consider the nonstationary properties of the full quantum system based
on this eigenstate expansion and demonstrate how the nonlinear
features of the dynamics in the mixed quantum-classical description
are reflected in the time dependence of the full quantum state vector
$|\Psi(t)\rangle$.
We investigated the evolution of wave packets initially prepared in
the product state
\begin{equation}\label{wp}
|\Phi,\alpha\rangle = |\Phi_{z_0,\phi_0}\rangle \otimes |\alpha_{Q_0,P_0}\rangle
\end{equation}
where $\Phi$ is an excitonic two component wave function which is
specified up to an irrelevant global phase by the expectation values
of the Bloch variables $z$ and $\phi$ (see (\ref{defphi})). $\alpha$
represents a standard coherent state in the oscillator variables,
which is specified by the complex parameter
\begin{equation}\label{aqp}
\alpha(Q,P) = \sqrt{r\over2}\langle\alpha|\hat Q|\alpha\rangle +
{i\over\sqrt{2r}}\langle\alpha|\hat P|\alpha\rangle
\end{equation}
with $Q$ and $P$ being the corresponding expectation values of
position and momentum.
In order to map the motion of the full state vector $|\Psi(t)\rangle$
constructed from the eigenstate expansion according to the initial
condition onto an analogue of the phase space of the mixed
description, in which the oscillator is treated classically,
we used for the oscillator subsystem the Husimi distribution,
which is an appropriate quantum analogue to the classical
phase space distribution (see e.\ g.\ \cite{Tak}). It is defined
by projecting $|\Psi(t)\rangle$ on the manifold of coherent states
\begin{equation}\label{husdist}
h_{z,\phi}(Q,P):=|\langle\Phi_{z,\phi},\alpha_{Q,P}|\Psi(t)\rangle|^2,
\end{equation}
where now $Q$ and $P$ are varied in the oscillator plane while
$z$ and $\phi$ are fixed parameters.
Without the interaction between the subsystems a wave packet prepared
in a coherent oscillator state would travel undistorted along the
classical trajectory started at $(Q_0,P_0)$. A weak coupling below the
bifurcation ($p<1$) results in a similar picture (not displayed) with
the wave packet after some initial period almost uniformly covering
the classical trajectories such as those displayed in figs.\
\ref{pq-0.8}(a) and (b).
Of particular interest is the effect of the separatrix structure
characterising the mixed description above the bifurcation ($p>1$) on
the propagation of the oscillator wave packet. For a system with a
proper classical limit and a separatrix in the classical phase space
the correspondence to the quantum evolution was studied e.~g.\ in
\cite{Rei}. Similar to this we found that the presence of the
separatrix is clearly reflected in the wave packet dynamics when the
energy is fixed at $E_h$. In the set of figs.\ \ref{wp-ini} the
evolution of a quantum state prepared initially right at the
hyperbolic fixed point $\bf h$ is presented. The relevant system
parameters are $p=2$ and $r=0.01$ and the Husimi distribution
(\ref{husdist}) for the projection onto the excitonic state $z=0$,
$\phi=0$ is displayed. It is seen how the oscillator wave packet
spreads along the unstable direction of the separatrix structure. The
asymmetric distortion of the wave packet in the beginning of the
propagation, when the support of the Husimi distribution is given by
the unstable direction of the separatrix, is remarkable. For long
times the wave packet covers the separatrix structure more uniformly
(see fig.\ \ref{wp-lt}(a)).
In the set of figs. \ref{hus-hp} contour plots for an analogous wave
packet propagation started at the hyperbolic point but for a larger
adiabatic parameter $r=0.1$ are presented. The propagation along the
separatrix structure, which is now indicated by a full line, is again
evident. This indicates that the well known classical-quantum
correspondence in the case of regular dynamics, namely that the
quantum distribution corresponds to the orbit of the corresponding
classical system, can be extended to systems treated in a mixed
quantum-classical description. A more detailed comparison of the
results for $r=0.01$ (fig.\ \ref{wp-ini}) and $r=0.1$ (fig.\
\ref{hus-hp}) reveals as expected that the width of the wave packet
transversal to the underlying classical structure is reduced as the
system is closer to the adiabatic limit. We conclude that in the
adiabatic regime regular structures such as a separatrix in the
formally classical phase space of the mixed description can serve to
forecast qualitatively the evolution of a wave packet in the fully
quantized system.
In fig.\ \ref{wp-lt} we compare the Husimi distributions for one and the
same wave packet projected onto two different excitonic states in
order to reveal the quantum correlations between the excitonic and the
vibronic subsystems. In fig.\ \ref{wp-lt}(a) we chose $z=0$ and $\phi=0$
corresponding to equal site occupation probabilities whereas in fig.\
\ref{wp-lt}(b) the wave packet is projected onto $z=1$, i.~e.\ an
excitonic state completely localized at one of the dimer sites. It is
seen that for the case of an equal site occupation the oscillator
evolution proceeds along both branches of the separatrix structure
whereas for the one sided projection the oscillator is preferentially
located on the branch of separatrix corresponding adiabatically to the
occupied site. This behavior reflects the property of the quantum
system to include coherently all the variants of motion of the mixed
quantum-classical system weighted with the corresponding probability
in analogy to the semiclassical propagator of a system with proper
classical limit, which is given as a sum over classical trajectories.
Finally we address the problem of how the qualitative differences
between the regular and the chaotic dynamics of the system in the
mixed description are reflected in the evolution of the fully
quantized system, i.~e.\ whether there are signatures of the dynamic
chaos in the mixed description in the time dependent state vector of
the fully quantized system. For simple systems quantized in one step
and chaotic in the classical limit the differences in the quantum
evolution between initial conditions selected in the classical regular
and chaotic parts of the phase space of the system are well known: If
e.~g.\ the initial conditions of the quantum system are selected in
the regular part of the classical phase space the time dependence of
the appropriately chosen quantum expectation values follow ("shadow")
the corresponding classical values over a substantial amount of time,
whereas for initial conditions chosen in the chaotic part of the
classical phase space these dependences start to deviate from each
other almost immediately (see e.\ g.\ \cite{BBH}). In order to
investigate this connection in our case we have selected different
initial conditions in the regular and chaotic parts of the Bloch
sphere of the system and compared the evolution in the mixed
description with that of expectation values obtained from the fully
quantized system. In fig.\ \ref{icond} the location of three different
initial conditions on the Bloch sphere of the excitonic subsystem
belonging to the main regular (A) and chaotic (B) regions of the
dynamics in the mixed description, as well as a small regular island
(C) embedded in a large chaotic surrounding are shown. For a
comparison of the dynamics in the mixed and fully quantized
descriptions for these cases we selected the variables $Q(t)$ and
$z(t)$ displayed in the upper parts in the set of figs.\
\ref{tdepA}-\ref{tdepC}.
We first compare the dynamics for initial conditions located in the
main regular (antibonding) and main chaotic (bonding) regions. Since
the initial state of the fully quantized system is chosen as a product
state with factorizing expectation values for which the decoupling
implicit in the derivation of (\ref{eommix}) is justified, there is
always an interval at the beginning of the time evolution where the
mixed description follows closely the quantum data. Then, however,
there is indeed a striking difference between initial conditions
selected in the regular and the chaotic parts of the phase space of
the mixed system: For initial conditions in the regular part ({\bf A},
fig.\ \ref{tdepA}) the quantum expectation value $Q(t)$ follows
closely the classical trajectory of the mixed description over several
periods and then, apart from a slowly growing phase shift, both
dependences keep a similar oscillatory form, whereas for an initial
condition in the chaotic part ({\bf B}, fig.\ \ref{tdepB}) the
corresponding curves are completely different and the deviation
between both starts already after a fourth of the oscillator period.
This confirms for our case the general behavior of classically chaotic
systems to produce a fast breakdown of the validity of quasiclassical
approximations when quantum effects become important. The comparison
for the occupation difference $z(t)$ of the excitonic sites is not so
direct, because the exciton constitutes the fast subsystem resulting
in rapid oscillations of $z(t)$ in the mixed description. However, for
the regular case we observe that the slowly changing mean value of
$z(t)$ obtained from the mixed description is related to the quantum
data, though amplitude and phase of the superimposed rapid
oscillations are different after a few periods of the excitonic
subsystem. In the chaotic case the breakdown of the mixed description
for shorter times is evident and there is no correspondence for the
mean values.
The gradual development of quantum correlations between both
subsystems, which are absent in the initially factorized state, can be
quantified by calculating the effective Bloch radius
\begin{equation}\label{qbr}
R(t)=\sqrt{x(t)^2+y(t)^2+z(t)^2}
\end{equation}
of the excitonic subsystem using the time dependent expectation values
of $\sigma_x$, $\sigma_y$ and $\sigma_z$. Note that the reduced density
matrix $\rho(t)$ for the excitonic subsystem, obtained from the full
density matrix by taking the trace over the oscillator states, is
related to $R(t)$ via ${\rm Tr}\, \rho(t)^2=(1/2)(1 + R(t)^2)$. For the
factorized and correspondingly uncorrelated initial quantum state the
value of the Bloch radius is $R=1$ and $R(t)$ will decrease in the
course of time according to the degree to which quantum correlations
lead to an entanglement between both subsystems. In the lower parts
of the figs.\ \ref{tdepA}-\ref{tdepC} the dependence of the Bloch
radius on the time is displayed for a long time interval. The
difference between the behavior for initial conditions chosen in the
regular and chaotic parts of the phase space of the mixed description
is remarkable: For initial conditions in the regular part of the phase
space after an initial drop $R(t)$ stabilizes at a value close to $1$,
whereas for the initial conditions in the chaotic part the descent is
much more pronounced and the long time value of $R(t)$ is much lower,
thus indicating stronger quantum correlations in the chaotic case. The
correlations between the subsystems are the reason for the breakdown
of the mixed description which implicitly contains the factorization
of expectation values. Therefore the smaller value of $R(t)$ observed
for the state prepared in the chaotic region confirms the faster
breakdown of the mixed description as compared to a regular initial
state. However, it is important to note that the striking difference
between the values of $R(t)$ is not restricted to this initial period
but extends to much longer times (which are on the other hand small
compared to the time for quantum recurrences). In this respect our
results indicate time dependent quantum signatures of chaos of the
mixed description which are beyond the well known different time
scales for the breakdown of quasiclassical approximations.
Finally we present the example for a quantum state prepared on a
regular island embedded into chaotic regions of the mixed
quantum-classical phase space ({\bf C}, fig.\ \ref{tdepC}). The
structure of the selected island is shown in the lower part of fig.\
\ref{icond}. For such a state the situation is specific due to the
spreading of the quantum state out of the regular island. After some
initial time in which the quantum dynamics probes the regular region
of the the mixed dynamics the wave packet enters the region in which
the mixed dynamics is chaotic. Correspondingly we find for an initial
time interval that the agreement between the mixed and the full
quantum description is as good as expected for regular dynamics
whereas for long times the quantum system shows the typical behavior
of a chaotic state. This is evident from the time dependence of the
Bloch radius which is displayed in the lower part of fig.\ \ref{tdepC}
on a sufficiently large time scale.
\pagebreak
\section{Conclusions}
1. We considered the nonlinear dynamical properties of a coupled
quasiparticle-oscillator system and demonstrated that the regular
structures of the mixed quantum-classical description such as the
fixed points and the presence of a separatrix are associated with the
corresponding adiabatic approximation, in which the nonadiabatic
couplings are switched off and integrable reference systems can be
defined. Comparing the evolution of quantum wave packets to the mixed
quantum-classical description we found that regular structures of the
mixed description can serve as a support for wave packet propagation
in the fully quantized system in the adiabatic regime. This should be
of interest for other systems to which a stepwise quantization must be
applied due to their more complex structure, e.~g.\ for the purpose of
forecasting the qualitative properties of propagating wave packets
using the mixed description as a reference system.
2. The nonadiabatic couplings, the inclusion of which is beyond the
adiabatic approximation, are identified as the source of dynamical
chaos observed in the mixed quantum-classical description. This
suggests that nonadiabatic couplings can be a general source of
nonintegrability and chaos also in other systems treated along a
stepwise quantization. Signatures of this type of chaos can then be
expected on the fully quantized level of description similar to what
we found for the coupled quasiparticle-oscillator system. In
particular, the breakdown of the mixed description is enhanced for
states prepared in a chaotic region of the phase space and the long
time evolution of these states is characterized by much stronger
quantum correlations between the subsystems.
3. Our results are related to the general question of how the idea of
the Born-Oppenheimer approach to analyze complex systems by a stepwise
quantization can be extended to a dynamical description. A more
systematic investigation of this question using other model systems is
certainly of interest in view of the widespread use of this approach.
\section{Acknowledgement}
Financial support from the Deutsche Forschungsgemeinschaft (DFG)
is gratefully acknowledged.
\pagebreak
| proofpile-arXiv_065-400 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
In the last few years numerous theoretical and experimental
studies have focused on the electron spectrum in semileptonic inclusive $B\to
X_c\ell\bar{\nu_{\ell}}$ decays.
The electron spectrum from free quark decays receives both
perturbative and nonperturbative corrections.
Knowledge of the shape of the spectrum can provide insights into
nonperturbative effects in $B$ meson decays, and thereby also
give some information on the weak mixing angle $|V_{cb}|$.
In the framework of Heavy Quark Effective Theory (HQET) it is possible to show
that the quark level decay rate is the first term in a power series expansion
in the small parameter $\Lambda_{QCD}/m_b$ \cite{CGG}.
For infinitely heavy quarks the free quark model is an exact description of
heavy meson physics. At finite quark masses the first few terms in the heavy quark
expansion have to be taken into account. Expressions for these nonperturbative
corrections to the lepton spectrum are known to
order $(\Lambda_{QCD}/m_b)^3$ \cite{mark,mannel,bd,gk} and the ${\cal O}(\alpha_s)$
perturbative
corrections to the free quark decay were given in \cite{jk}.
The dominant remaining uncertainties are the two-loop corrections to the quark
level decay rate and the perturbative corrections to the coefficients of
the HQET matrix elements in the operator product expansion.
Here we examine the former. While a full two-loop calculation of the electron
spectrum is a rather daunting task, it is possible to calculate the piece of the
two-loop correction that is proportional to $\beta_0=11-2/3 n_f$ with relative
ease by performing the one-loop QCD corrections with a massive gluon. The
$\alpha_s^2\beta_0$ parts of the two-loop correction may then be obtained from a
dispersion integral over the gluon mass\cite{sv}.
If there are no gluons in the tree level graph, the $\alpha_s^2\beta_0$ part of the
two-loop contribution is believed to dominate the full $\alpha_s^2$
result because $\beta_0$ is rather large. Several examples supporting this
belief are listed in \cite{asmark}.
A recent calculation \cite{asmark} of the $\alpha_s^2\beta_0$ correction to
the total inclusive rate for $B\to X_c\ell\bar{\nu_{\ell}}$ decays showed that the $\alpha_s^2\beta_0$
parts of the two-loop correction are approximately half as big as the
one-loop contribution, resulting in a rather low BLM scale \cite{BLM} of
$\mu_{BLM}=0.13m_b$. For the electron spectrum we find that this part
of the second order
correction also amounts to about 50\% of the order $\alpha_s$ contribution,
at all electron energies except those close to the endpoint.
Close to the endpoint the corrections are roughly equal in magnitude.
The HQET matrix elements $\lambda_1$ and $\bar\Lambda$ can be extracted from the
electron
spectrum in $B\to X_c\ell\bar{\nu_{\ell}}$ decays \cite{gklw}. Even though this method of
obtaining HQET matrix elements was found to be rather insensitive to the first
order perturbative corrections, it is useful to extract $\lambda_1, \bar\Lambda$
including the $\alpha_s^2\beta_0$ corrections. Then these matrix elements can be used to
relate the pole quark mass
to the $\overline{\rm MS}$ masses at order $\alpha_s^2\beta_0$. Similarly, one can include the
$\alpha_s^2\beta_0$ parts of the two-loop contribution in the theoretical prediction for the total
rate, which is needed for the determination of $|V_{cb}|$. Since the $\alpha_s^2\beta_0$
corrections are rather large the resulting changes in the quark masses and
$|V_{cb}|$ are not negligible.
In Sect.~II and III we give analytic
expressions for the contributions from virtual and real gluon radiation.
The last phase space integral in the virtual correction and
the last two integrals in the bremsstrahlung are done numerically. Readers not
interested in calculational details are advised to skip these sections.
In Sect.~IV we combine the results from the
previous two sections to obtain the $\alpha_s^2\beta_0$ corrections to the electron
spectrum, and discuss
the implications for the extraction of $\bar\Lambda,\lambda_1$, the $\overline{\rm MS}$
quark masses, and $|V_{cb}|$. In Appendix \ref{ap2} we give an interpolating
polynomial which reproduces the two-loop correction calculated here.
\section{Virtual Corrections}
The corrections from massive virtual gluons can be calculated in complete analogy
to the usual one-loop QCD corrections. The ultraviolet divergence in the vertex
correction cancels when combined with the quark wave function renormalizations.
There is no infrared divergence since we do the calculation with a massive gluon.
The virtual one-loop correction to the differential rate can be written as
\begin{eqnarray}
\frac{ {\rm d} \Gamma^{(1)}_{virt}(\hat{\mu}) }{ {\rm d} y}&=&\alpha_s^{(V)}
\frac{|V_{cb}|^2G_F^2 m_b^5}{48 \pi^4}\int {\rm d} \hat{q}^2
\Big[ 2(y- \hat{q}^2 )( \hat{q}^2 +1-r^2-y) (a_1+a_{wr}) -2 r \hat{q}^2 a_2 \nonumber \\
&&+( \hat{q}^2 (y-1)+y(1-r^2)-y^2)a_3\Big] \label{v1loop}
\end{eqnarray}
where $\hat{\mu}=\mu/m_b$ is the rescaled gluon mass, and $y=2E_e/m_b$, $r=m_c/m_b$, and
$ \hat{q}^2 =q^2/m_b^2$ are the rescaled electron energy, charm mass, and momentum
transfer, respectively. The limits for the integration over $ \hat{q}^2 $ are
\begin{equation}
0\le \hat{q}^2 \le\frac{y(1-y-r^2)}{1-y}, \qquad 0\le y\le 1-r^2.
\end{equation}
The functions $a_{wr}( \hat{q}^2 )$ and $a_i( \hat{q}^2 ),\ i=1,2,3$ are the contributions from
the wave function renormalization and the vertex correction respectively.
They can be expressed in terms of
the scalar two- and three-point functions $B_0$ and $C_0$ \cite{TV}, and the
derivative $B^\prime_0=\partial B_0(a,b,c)/\partial a$. Explicit expressions for
these functions are given in Appendix \ref{ap1}.
Using the standard decomposition for the vector and tensor loop integrals \cite{TV}
we obtain
\begin{eqnarray}
a_1 &=& -2+4C_{00}+2(C_{11}+C_1+r^2C_{22}+r^2C_2)+2(1- \hat{q}^2 +r^2)(C_{12}+C_0+C_1+C_2)\nonumber,\\
a_2 &=& 2r(C_1+C_2), \qquad
a_3 = -4(C_{11}+C_{12}+C_1)-4r^2( C_{12}+C_{22}+C_2),\\
a_{wr}&=&\frac{1}{2}\Bigg[ 2-B_0(1,\hat{\mu}^2,1)-B_0(r^2,\hat{\mu}^2,r^2)
+(1-\hat{\mu}^2)\Big(B_0(1,\hat{\mu}^2,1)-B_0(0,\hat{\mu}^2,1)\Big)\\
&&+\frac{(r^2-\hat{\mu}^2)}{ r^2} \Big(B_0(r^2,\hat{\mu}^2,r^2)-B_0(0,\hat{\mu}^2,r^2)\Big)
+2(2+\hat{\mu}^2)B^\prime_0(1,\hat{\mu}^2,1)\nonumber \\
&&+2(2r^2+\hat{\mu}^2)B^\prime_0(r^2,\hat{\mu}^2,r^2)\Bigg] \nonumber .
\end{eqnarray}
Defining $f_1=1+r^2- \hat{q}^2 $ and $f_2=(f_1^2-4r^2)$ the coefficient functions take
the form
\begin{eqnarray}
C_{00}&=&\frac{1}{4f_2}\Bigg[ f_2
+\hat{\mu}^2(f_1-2)B_0(1,1,\hat{\mu}^2)+\hat{\mu}^2(f_1-2r^2)B_0(r^2,r^2,\hat{\mu}^2)\nonumber\\
&&+(f_2+2 \hat{q}^2 \hat{\mu}^2) B_0( \hat{q}^2 ,1,r^2)+2\hat{\mu}^2(f_2+ \hat{q}^2 \hat{\mu}^2)C_0(1, \hat{q}^2 ,r^2,\hat{\mu}^2,1,r^2)\Bigg],\\
C_{11} &=& \frac{r^2}{f_2}+\frac{(f_1-2r^2)(1-r^2)}{2 \hat{q}^2 f_2}B_0(0,1,r^2)
+\frac{f_1(\hat{\mu}^2-1)}{2f_2}B_0(0,1,\hat{\mu}^2)\\
&&+ \frac{3r^2\hat{\mu}^2(f_1-2r^2)}{f_2^2}B_0(r^2,r^2,\hat{\mu}^2)+
\frac{2 \hat{q}^2 \hat{\mu}^2(f_2+6 \hat{q}^2 r^2)-f_2(f_2+2 \hat{q}^2 r^2)}{2 \hat{q}^2 f_2^2}B_0( \hat{q}^2 ,1,r^2)\nonumber\\
&&+ \frac{\hat{\mu}^2 \Big( 6r^2(f_1-2)-f_2(f_1+2)\Big) } {2f_2^2}B_0(1,1,\hat{\mu}^2)\nonumber\\
&&+ \frac{2 \hat{\mu}^2 r^2 f_2+\hat{\mu}^4(f_2+6 \hat{q}^2 r^2) }{f_2^2}C_0(1, \hat{q}^2 ,r^2,\hat{\mu}^2,1,r^2)\nonumber,\\
C_{22}&=&\frac{1}{f_2}+\frac{(1-r^2)(2-f_1)}{2 \hat{q}^2 f_2}B_0(0,1,r^2)
+\frac{2 \hat{q}^2 \hat{\mu}^2(f_2+6 \hat{q}^2 )-f_2(f_2+2 \hat{q}^2 )}{2 \hat{q}^2 f_2^2}B_0( \hat{q}^2 ,1,r^2)\nonumber\\
&&+\frac{(\hat{\mu}^2-r^2)f_1}{2r^2f_2}B_0(0,r^2,\hat{\mu}^2)
+\frac{3\hat{\mu}^2(f_1-2)}{f_2^2}B_0(1,1,\hat{\mu}^2)\\
&&+\frac{\hat{\mu}^2\Big(6r^2(f_1-2r^2)-f_2(f_1+2r^2)\Big)}{2r^2f_2^2}B_0(r^2,r^2,\hat{\mu}^2)\nonumber\\
&&+\frac{\hat{\mu}^2\Big( 2f_2+\hat{\mu}^2(f_2+6 \hat{q}^2 )\Big)}{f_2^2}C_0(1, \hat{q}^2 ,r^2,\hat{\mu}^2,1,r^2) \nonumber ,\\
C_{12}&=&\frac{-f_1}{2f_2}+\frac{(1-r^2)(2r^2-f_1)}{2 \hat{q}^2 f_2}B_0(0,1,r^2)
+\frac{r^2-\hat{\mu}^2}{f_2}B_0(0,r^2,\hat{\mu}^2)\nonumber\\
&&+\frac{\hat{\mu}^2\Big(6(f_1-2r^2)-f_2\Big)}{2f_2^2}B_0(1,1,\hat{\mu}^2)
+\frac{\hat{\mu}^2\Big(6r^2(f_1-2)-f_2\Big)}{2f_2^2}B_0(r^2,r^2,\hat{\mu}^2)\\
&&+\frac{f_2(f_2+ \hat{q}^2 f_1)-2 \hat{\mu}^2 \hat{q}^2 (3 \hat{q}^2 f_1+f_2)}{2 \hat{q}^2 f_2^2}B_0( \hat{q}^2 ,1,r^2)\nonumber\\
&&+\frac{\hat{\mu}^2\Big(-f_1 f_2-\hat{\mu}^2(3 \hat{q}^2 f_1 +f_2) \Big)}{f_2^2}C_0(1, \hat{q}^2 ,r^2,\hat{\mu}^2,1,r^2)\nonumber,\\
C_1&=&\frac{1}{f_2}\Bigg[f_1 B_0(1,1,\hat{\mu}^2)+(2r^2-f_1)B_0( \hat{q}^2 ,1,r^2)
-2r^2B_0(r^2,r^2,\hat{\mu}^2)\nonumber \\
&&+\hat{\mu}^2(2r^2-f_1)C_0(1, \hat{q}^2 ,r^2,\hat{\mu}^2,1,r^2)\Bigg],\\
C_{2}&=&\frac{1}{f_2}\Bigg[ -2 B_0(1,1,\hat{\mu}^2)+(2-f_1)B_0( \hat{q}^2 ,1,r^2)+f_1B_0(r^2,r^2,\hat{\mu}^2)\nonumber \\
&&+\hat{\mu}^2(2-f_1)C_0(1, \hat{q}^2 ,r^2,\hat{\mu}^2,1,r^2)\Bigg].
\end{eqnarray}
The infinite parts of the regularized two-point functions can be shown to cancel in
eq. (\ref{v1loop}). In the limit $\hat{\mu}\to 0$ the vertex correction diverges
logarithmically. This divergence will be canceled by corresponding divergences in the
bremsstrahlung contributions discussed in the next section.
\section{Bremsstrahlung}
The bremsstrahlung correction is found in the usual manner, by inserting a real
gluon on the $c$ and $b$ quark lines. The calculation here is complicated by the
four-body phase space with two massive final states. We follow the standard
procedure of decomposing the four-body phase space into a two- and a three-body
phase space by introducing the four-momentum $P=p_c+p_g$. In the rest frame of the
$b$ quark this decomposition reads
\begin{equation}
dR_4 = dP^2\ dR_3(m_b; p_e, p_{\bar{\nu}}, P) dR_2(P; p_c, p_g).
\end{equation}
The ${\mathcal{O}}(\alpha_s)$ bremsstrahlung correction to the
differential rate is given in terms of dimensionless variables ($\hat{P^0}=P^0/m_b$,
$\hat{P}^2=P^2/m_b^2$) by
\begin{eqnarray}
{d \Gamma_{brems}^{(1)}(\hat{\mu}) \over {\rm d} y } &=& \alpha_s^{(V)}{G_F^2\, |V_{cb}|^2\, m_b^5 \over 192\ \pi^4 }\
\int d\hat{P}^2\ d\hat{P^0}\ (\hat{P^0}^2-\hat{P}^2)^{-5/2} \biggl[ 2 b_1 (1-2\hat{P^0}+\hat{P}^2) \nonumber \\
& & \qquad + b_2 (2-2\hat{P^0}-y) y + b_3 (1-y-\hat{P}^2) (2\hat{P^0}+y-\hat{P}^2-1) \nonumber \\
& & \qquad + b_4 (1-y-\hat{P}^2) y + b_5 (2\hat{P^0} +y-2) (1-2\hat{P^0}-y+\hat{P}^2) \biggl] .
\label{r1loop}
\end{eqnarray}
For convenience the above rate has been written in terms of the
coefficients $b_i$
\begin{eqnarray}
b_1 &=& (\hat{P^0}^2-\hat{P}^2) \left( \hat{P}^2(c_2-c_1)+\hat{P^0}^2 c_1+c_3-\hat{P^0}(c_4+c_5) \right),\\
b_2 &=& (\hat{P^0}^2-\hat{P}^2)\hat{P}^2 c_1 +3\hat{P}^4 c_2+(\hat{P}^2+2\hat{P^0}^2) c_3 -3\hat{P^0}\hat{P}^2 (c_4+c_5) , \\
b_3 &=& (\hat{P^0}^2-\hat{P}^2) c_1 + (\hat{P}^2+2\hat{P^0}) c_2+3 c_3 - 3 \hat{P^0} (c_4+c_5) , \\
b_4 &=& -\hat{P^0}(\hat{P^0}^2-\hat{P}^2)c_1-3\hat{P^0}\hat{P}^2 c_2-3\hat{P^0} c_3+(\hat{P^0}^2+2\hat{P}^2) c_4+3\hat{P^0}^2 c_5,\\
b_5 &=& -\hat{P^0}(\hat{P^0}^2-\hat{P}^2) c_1-3\hat{P^0}\hat{P}^2 c_2-3\hat{P^0} c_3 + 3\hat{P^0}^2c_4 + (\hat{P^0}^2+2\hat{P}^2) c_5 ,
\end{eqnarray}
which are linear combinations of
\begin{eqnarray}
c_1 &=& {4 (v_+^2 - v_-^2) \over h}
+ { 2 [ (h+\hat{\mu}^2-2\hat{P^0})^2+(\hat{\mu}^2-2\hat{P^0})^2+2 \hat{\mu}^2 (1+r^2) ] \over
h} \ln( {2 v_+ -\hat{\mu}^2 \over 2 v_- - \hat{\mu}^2}) \nonumber \\
\quad &+& {4 (2+\hat{\mu}^2)(h-2\hat{P^0})(v_+-v_-) \over (2 v_+ -\hat{\mu}^2) (2 v_- -\hat{\mu}^2) }
- {8 [(z+\hat{\mu}^2)(\hat{P^0}-h)+z\hat{P^0}] (v_+-v_-) \over h^2 } , \\
c_2 &=& {2 (v_+^2 - v_-^2) (2-h) \over h}
- { [h \hat{\mu}^2(3\hat{\mu}^2-4\hat{P^0})+4\hat{\mu}^2(2\hat{P^0}-1)-16\hat{P^0}^2] \over h}
\ln( {2 v_+ -\hat{\mu}^2 \over 2 v_- - \hat{\mu}^2}) \nonumber \\
\quad &+& {2 (\hat{\mu}^4-4)(2\hat{P^0}-\hat{\mu}^2)(v_+-v_-) \over (2 v_+ -\hat{\mu}^2) (2 v_- -\hat{\mu}^2) }
- { 4 [h^2 (\hat{\mu}^2-\hat{P^0})+2\hat{P^0} (\hat{\mu}^2+2\hat{P}^2)] (v_+-v_-) \over h^2 } , \\
c_3 &=& {[(\hat{P}^2+r^2) (h^2+2(\hat{\mu}^2-2\hat{P^0})(h-2\hat{P^0})) -h\hat{\mu}^4 +4 r^2 \hat{P}^2 \hat{\mu}^2] \over h}
\ln( {2 v_+ -\hat{\mu}^2 \over 2 v_- - \hat{\mu}^2}) \nonumber \\
\quad &+& {2 (2+\hat{\mu}^2) (\hat{\mu}^2-r^2-\hat{P}^2)(2\hat{P^0}-\hat{\mu}^2-h) (v_+-v_-)
\over (2 v_+ -\hat{\mu}^2) (2 v_- -\hat{\mu}^2) } \nonumber \\
\quad &-& {4 [(\hat{P}^2+r^2)(h\hat{P^0}-h z+\hat{\mu}^2\hat{P^0})+4r^2\hat{P}^2\hat{P^0}] (v_+-v_-) \over h^2 } , \\
c_4 &=& {4 (\hat{P^0}-\hat{P}^2) (v_+^2 - v_-^2) \over h} + {2 (2+\hat{\mu}^2)(2\hat{P^0}-\hat{\mu}^2)(\hat{\mu}^2-2\hat{P^0}+h)
(v_+-v_-)\over(2 v_+ -\hat{\mu}^2)(2 v_- -\hat{\mu}^2) } \nonumber \\
\quad &-& {2 [4\hat{P^0}^2(2\hat{P}^2+\hat{\mu}^2)+(\hat{\mu}^2-r^2) h (2\hat{P}^2+h-4\hat{P^0})+h\hat{P}^2(\hat{P}^2+r^2-8\hat{P^0})]
(v_+-v_-) \over h^2 } \nonumber \\
\quad &-& {2 \over h} [2(\hat{\mu}^2-\hat{P^0})(h \hat{\mu}^2-2\hat{P^0} h+4\hat{P^0}^2)+h^2(1-\hat{P^0}+\hat{\mu}^2)-
2\hat{\mu}^2(r^2+r^2\hat{P^0}+\hat{P^0}) \nonumber \\
&+& \hat{\mu}^4(1+r^2-2\hat{P^0})] \ln( {2 v_+ -\hat{\mu}^2 \over 2 v_- - \hat{\mu}^2}) , \\
c_5 &=& {2 (\hat{\mu}^4-4) (\hat{P}^2+r^2-\hat{\mu}^2)(v_+-v_-)\over(2 v_+ -\hat{\mu}^2)(2 v_- -\hat{\mu}^2)} \nonumber \\
&-& {2 [\hat{\mu}^2h^2+(\hat{P}^2+r^2)(2\hat{\mu}^2+2h-h^2)+8r^2\hat{P}^2](v_+-v_-)\over h^2} \nonumber \\
&-& {2 [\hat{\mu}^4h-(\hat{P}^2+r^2)(\hat{\mu}^2h+4\hat{P^0})+2\hat{P}^2 \hat{\mu}^2] \over h}
\ln( {2 v_+ -\hat{\mu}^2 \over 2 v_- - \hat{\mu}^2}) .
\end{eqnarray}
In the expressions for the $c_i$ we have put $h=\hat{P}^2-r^2$ and
\begin{equation}
v_\pm = { (\hat{P}^2+\hat{\mu}^2-r^2)\hat{P^0} \pm \sqrt{\hat{P^0}^2-\hat{P}^2} \sqrt{(\hat{P}^2+\hat{\mu}^2-r^2)^2-4 \hat{\mu}^2\hat{P}^2}
\over 2\hat{P}^2 }.
\end{equation}
The integrals in eq. (\ref{r1loop}) are done numerically between the kinematic
limits
\begin{eqnarray}
{(1-y)^2+\hat{P}^2 \over 2 (1-y)} \le &\hat{P^0}& \le {1+\hat{P}^2 \over 2} ,\\
(\hat{\mu} + r)^2 \le &\hat{P}^2& \le 1-y.
\end{eqnarray}
To improve the numerical stability for small $\hat{\mu}^2$ we found it useful to do
the $\hat{P^0}$ integral with the variable $\ln(\hat{P^0}-r^2)$. The remaining limits for
this four-body decay are
\begin{equation}
0 \le \hat{\mu} \le \sqrt{1-y} - r, \qquad 0 \le y \le 1-r^2.
\end{equation}
\section{The $\alpha_s^2\beta_0$ correction}
Combining the corrections from virtual and real gluon radiation,
eqs. (\ref{v1loop},\ref{r1loop}), we obtain
\begin{equation}\label{glue}
\frac{ {\rm d} \Gamma^{(1)}(\hat{\mu})}{ {\rm d} y} = \frac{ {\rm d} \Gamma^{(1)}_{virt}(\hat{\mu})}{ {\rm d} y}
+\frac{ {\rm d} \Gamma^{(1)}_{brems}(\hat{\mu})}{ {\rm d} y}\Theta(\sqrt{1-y}-r-\hat{\mu}).
\end{equation}
In the $\hat{\mu}\to 0$ limit eq. (\ref{glue}) yields the one-loop correction to the
electron spectrum.
We have checked that our expression reproduces the result in \cite{jk} in this
limit. The $\alpha_s^2\beta_0$ part of the two-loop correction is related to the one-loop
expression calculated with a massive gluon by \cite{sv}
\begin{equation}
\frac{ {\rm d} \Gamma^{(2)}}{ {\rm d} y}=-\frac{{\alpha_s^{(V)}}\beta_0}{4\pi}
\int\limits_{0}^{\infty}\frac{ {\rm d} \hat{\mu}^2}{\hat{\mu}^2}
\left( \frac{ {\rm d} \Gamma^{(1)}(\hat{\mu})}{ {\rm d} y}-
\frac{1}{1+\hat{\mu}^2}\frac{ {\rm d} \Gamma^{(1)}(0)}{ {\rm d} y}\right).
\end{equation}
Note that $\alpha_s^{(V)}$, defined in the V-scheme of BLM \cite{BLM}, is related
to the more familiar $\bar\alpha_s$ defined in the $\overline{\rm MS}$ scheme by
\begin{equation}
\alpha_s^{(V)}=\bar\alpha_s+\frac{5}{3}\frac{\bar\alpha_s^2}{4\pi}\beta_0+\cdots.
\end{equation}
$\alpha_s$ is evaluated at $m_b$ unless stated otherwise.
In the $\overline{\rm MS}$ scheme the $\bar\alpha_s^2\beta_0$ part of the two-loop
correction reads
\begin{equation}
\frac{ {\rm d} \Gamma^{(2)}}{ {\rm d} y}=\frac{5}{3}\frac{\bar\alpha_s\beta_0}{4\pi}
\frac{ {\rm d} \Gamma^{(1)}(0)}{ {\rm d} y}
-\frac{\bar\alpha_s\beta_0}{4\pi}\int\limits_{0}^{\infty}\frac{ {\rm d} \hat{\mu}^2}{\hat{\mu}^2}
\left( \frac{ {\rm d} \Gamma^{(1)}(\hat{\mu})}{ {\rm d} y}-
\frac{1}{1+\hat{\mu}^2}\frac{ {\rm d} \Gamma^{(1)}(0)}{ {\rm d} y}\right).
\end{equation}
The dispersion integral has to be done with some care.
We found that using $\ln(\hat{\mu}^2)$ instead of $\hat{\mu}^2$ as the
integration variable simplifies the numerical evaluation considerably.
In Fig. 1 we plot the $\bar\alpha_s^2\beta_0$ part of the two-loop correction\footnote{
In Appendix \ref{ap2} we give a polynomial fit to our results that
reproduces the numerical results to better than 1\% for mass ratios in the range
$0.29\le r\le 0.37$.}
and for comparison the one-loop correction to the electron spectrum,
using $r=0.29$, $\bar\alpha_s=0.2$, $n_f=3$, and dividing by
$\Gamma_0=G_F^2|V_{cb}|^2m_b^5/192\pi^3$.
Except for electron energies close to the endpoint, the $\alpha_s^2\beta_0$ corrections are
about half as big as the first order corrections. The perturbation series
appears to be controlled but the higher order corrections clearly are not
negligible.
Integrating over the electron energy we reproduce the result for the
correction to the total rate given in Ref. \cite{asmark}.
In \cite{gklw} the HQET matrix elements $\bar\Lambda,\lambda_1$ were extracted from
the lepton spectrum using the experimentally accessible observables
\begin{equation}
R_1=\frac{\int_{1.5{\rm GeV}} {\rm d} E_\ell \, E_\ell {\rm d} \Gamma/ {\rm d} E_\ell}
{\int_{1.5{\rm GeV}} {\rm d} E_\ell \, {\rm d} \Gamma/ {\rm d} E_\ell}, \qquad
R_2=\frac{\int_{1.7{\rm GeV}} {\rm d} E_\ell \, {\rm d} \Gamma/ {\rm d} E_\ell}
{\int_{1.5{\rm GeV}} {\rm d} E_\ell \, {\rm d} \Gamma/ {\rm d} E_\ell}.
\end{equation}
It is straightforward to calculate the $\bar\alpha_s^2\beta_0$ corrections to these
quantities. In the spirit of HQET, we use the spin averaged meson masses
$\overline{m}_B=5.314{\rm GeV}$, and $\overline{m}_D=1.975{\rm GeV}$
instead of quark masses. Keeping only two-loop corrections that are proportional
to $\beta_0$, and neglecting terms of order $\bar\alpha_s^2\beta_0\Lambda_{QCD}/\overline{m}_B$ we find
\begin{eqnarray}\label{r1r2}
R_1&=&1.8059-0.035\frac{\bar\alpha_s}{\pi}-0.082\frac{\bar\alpha_s^2\beta_0}{\pi^2}+\cdots,\\
R_2&=&0.6581-0.039\frac{\bar\alpha_s}{\pi}-0.098\frac{\bar\alpha_s^2\beta_0}{\pi^2}+\cdots\nonumber,
\end{eqnarray}
where the ellipsis denote the other contributions including
nonperturbative corrections discussed in \cite{gk,gklw}.
The BLM scales for these
quantities are $\mu_{BLM}(R_1) = 0.01 \overline{m}_B$, and $\mu_{BLM}(R_2) = 0.007\overline{m}_B,$
reflecting the fact that the second order corrections are larger than the first
order. This is a result of the almost complete cancellation of the first order
perturbative corrections from the denominators and numerators in $R_{1,2}$.
In eq.(\ref{r1r2}) the BLM scales for the numerators and denominators are
separately comparable to the BLM scale for the total rate
$\mu_{BLM}\approx 0.1\overline{m}_B$. Therefore the very low BLM scales of $R_{1,2}$
do not necessarily indicate badly behaved perturbative series.
In order to demonstrate the impact of the $\bar\alpha_s^2\beta_0$ corrections on the
extraction of $\bar\Lambda,\lambda_1$, we repeat the analysis of \cite{gklw} neglecting
nonperturbative corrections of order $(\Lambda_{QCD}/m_b)^3$ which may be
substantial \cite{gk}. Because of the higher order nonperturbative
corrections, large theoretical uncertainties have to be assigned to the extracted
values of $\bar\Lambda,\lambda_1$. We find that the central values are moved from
$\bar\Lambda=0.39\pm0.11{\rm GeV},\,\lambda_1=-0.19\pm 0.10 {\rm GeV}^2$ to
$\bar\Lambda=0.33{\rm GeV},\,\lambda_1=-0.17$.
The shift in the values of the HQET matrix elements lies well within the
$1\sigma$ statistical error of the previously extracted values.
However, using the values of the HQET matrix elements extracted at a given order
in $\alpha_s$ to predict physical observables at the same order in $\alpha_s$,
guarantees that the renormalon ambiguity in $\bar\Lambda$ and $\lambda_1$ will
cancel\cite{renorm,renorm2} if the
expansion is continued to sufficiently high orders in $\alpha_s$. Thus
including the $\bar\alpha_s^2\beta_0$ parts in the determination of $\bar\Lambda,\lambda_1$
allows one to calculate the $\overline{\rm MS}$ quark masses consistently
at order $\bar\alpha_s^2\beta_0$.
To second order in $\Lambda_{QCD}/m_q$ and to order $\bar\alpha_s^2\beta_0$ we have
\begin{equation}
\overline{m}_q(m_q)=\left(\overline{m}_{Meson}-
\bar\Lambda+\frac{\lambda_1}{2m_q}+\cdots\right)
\left(1-\frac{4\bar\alpha_s(m_q)}{3\pi} -1.56\frac{\bar\alpha_s^2(m_q)\beta_0}{\pi^2}+\cdots\right),
\end{equation}
where $m_q$ is the $b$ or $c$ quark pole mass and $\overline{m}_{Meson}$ is the
corresponding spin averaged meson mass. With $\bar\alpha_s(m_b)=0.22,\,\bar\alpha_s(m_c)=0.39$ this yields
$\overline{m}_b(m_b)=4.16{\rm GeV}, \overline{m}_c(m_c)=0.99{\rm GeV}$ for the
$\overline{\rm MS}$ quark masses, albeit
with large theoretical uncertainties due to the effect of the higher order
nonperturbative corrections on the extraction of $\bar\Lambda,\lambda_1$\cite{gk}.
The value of $\overline{m}_b(m_b)$ is in good agreement with lattice
calculations $\overline{m}_b(m_b)=4.17\pm 0.06{\rm GeV}$ and $\overline{m}_b(m_b)=
4.0\pm 0.01{\rm GeV}$\cite{lat}.
The weak mixing angle $|V_{cb}|$ can be determined by comparing the theoretical
prediction for the total rate with experimental measurements.
Including all corrections discussed in \cite{gklw} we find at order $\alpha_s^2\beta_0$
\begin{equation}
|V_{cb}|=0.043\left(\frac{ Br(B\to X_c\ell\bar{\nu_{\ell}})}{0.105} \frac{ 1.55{\rm ps}}{\tau_B}\right)^{1/2}.
\end{equation}
\section{Conclusions}
We have calculated the ${\cal O}(\alpha_s^2\beta_0)$ corrections to the electron
spectrum in $b\to c\ell\bar{\nu_{\ell}}$ decays which turn out to be rather large,
about 50\% of the one-loop corrections. These corrections can be included in the
extraction of the HQET matrix elements $\bar\Lambda,\lambda_1$. We obtain
$\bar\Lambda=0.33{\rm GeV}$ and $\lambda_1=-0.17{\rm GeV}^2$, both somewhat lower than the
values extracted at ${\cal O}(\alpha_s)$. Using these values and including
${\cal O}(\alpha_s^2\beta_0)$ corrections we obtain $\overline{m}_b(m_b)=4.16{\rm GeV},\,
\overline{m}_c(m_c)=0.99{\rm GeV}$ for the $\overline{\rm MS}$ quark masses and
$|V_{cb}|=0.043( Br(B\to X_c\ell\bar{\nu_{\ell}})/0.105\times 1.55{\rm ps}/\tau_B)^{1/2}$.
These results have large theoretical uncertainties due to the effect of
nonperturbative corrections of order $(\Lambda_{QCD}/m_b)^3$ on the extraction of
$\bar\Lambda,\lambda_1$.
\acknowledgments
We would like to thank Anton Kapustin, Zoltan Ligeti and Mark Wise for helpful
discussions. This work was supported in part by the Department of Energy
under grant DE-FG03-92-ER 40701.
| proofpile-arXiv_065-401 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
The evolution of living organisms is a fascinating phenomenon that has
intrigued the imagination of the scientific and non-scientific community.
However, the formulation of mathematical models falls necessarily to drastic
simplifications. For example, evolution has often been considered as a
``walk'' in a rugged landscape. Following this line, Bak and Sneppen (BS)
have proposed a model of biological evolution \cite{bs} that has become
quite interesting to the physics community due to its simplicity and the new
insight it provides to the problem. It has been shown that this model
evolves to a self-organized critical state (SOC), and is kept there by the
means of avalanches of evolutionary activity. This is appealing for a model
of biological evolution, since it has been observed that life on Earth could
be in a SOC state \cite{kauf,newm}. Nevertheless, models based in fitness
landscapes, or in a concept of fitness different from the biological one,
have been criticized from a biological point of view \cite{newm,jong}.
Since one of the characterizing aspects of life, and perhaps the most
fundamental one, is that of self-replication, it is our belief that more
realistic models should involve a dynamic population for each species.
The starting point of combining population dynamics with evolution is the
association of the rates of birth and death and the carrying capacity with
phenotypes (observable features that arise from the genotype and are, then,
subject to mutation) \cite{roug}. The fitness, namely the expected number of
offsprings produced by an individual, arise from them. In this way, the
process of natural selection is directed by the ecological interactions
instead of by a non-biological notion of relative fitness.
Extinction is an essential component of evolution. The great majority of
species that have ever lived on Earth are now extinct\cite{raup}.
There exist competing
hypothesis that account extinction as originating from within the biosystem,
or from external causes --what has been called ``bad genes or bad luck''. In
any case, the pattern of extinctions and of surviving species or groups of
species is certainly an interesting problem to model, to understand, and
eventually to check with the fossil record.
We show in this contribution a simple model of a large ecological
system in evolution. This produces features of extinction similar to those
claimed for the biosystem on Earth. We have chosen to study an ecological
model in which each species consists of a population interacting with the
others, that reproduces and evolves in time. The system is supposed to be a
food chain, and the interactions to be predator-prey. Mutations that change
the interactions are supposed to occur randomly at a low rate. Extinctions
of populations result from the predator-prey dynamics. This approach can be
thought as middle way between the microscopic simulation of ``artificial
life'' by Ray and others \cite{ray}, and the coarse-grain description of
models like BS's.
\section{The model}
Our model ecosystem consists of a number of species that interact and evolve
in time. In the course of its time evolution
the populations grow and shrink following a set of equations.
Eventually, some of the species become
extinct as a result of their interaction with the others. Every now and then
we change one of the phenotypic features of one of the species,
mimicking a random mutation of its genome. This modification produces a
perturbation in the dynamics of the ecosystem, and eventually leads to the
extinctions.
To be more precise, let's consider a simple example of a food web, namely a
one-dimensional food chain. $N$ species interact in such a way that the
species $i$ feeds on the species $i-1$, and is eaten by the species $i+1$.
The species $1$ is an autotroph: it feeds at a constant rate on an
``environment''. The species $N$, the top of the chain, is not eaten by any
species, but dies giving its mass to the environment. Each species has a
population that evolves in time and interacts with its neighbors in the
chain. Furthermore, we consider this evolution in discrete time, which
is often more realistic than a continuous one \cite{roug} and simpler to
simulate in a computer.
As has been said above, each species acts as a predator with respect to the
one preceding it in the chain, and as a prey with respect to the one
following it. As a further simplification, we suppose that there are
no intrinsic birth and death rates, apart from those arising from the
predation and prey contributions. Let's propose the equations governing this
behaviors \cite{murray}. As a predator, the ``population'' (a continuous
density) of the species $i$, $n^i$, changes from time $t$ to time $t+1$
according to:
\begin{equation}
\Delta n_t^i=k_in_t^{i-1}n_t^i\left( 1-n_t^i/c_i\right) , \label{predator}
\end{equation}
where $k_i$ is a rate of growth of the predator population and $c_i$ is a
carrying capacity that accounts for a limitation imposed by the environment.
Note that (\ref{predator}) includes this carrying capacity in a logistic
factor to avoid an unbounded growth of the population. Also observe that the
growth is proportional to the population of preys, without a ``satiation''
factor. Similarly, as a prey, the population of the species $i$ will
diminish according to:
\begin{equation}
\Delta n_t^i=-g_in_t^in_t^{i+1}.
\end{equation}
The parameters $k_i$, $g_i$ and $c_i$ are the phenotypic features of our
species. In the course of the evolution we allow them to change,
mimicking random mutations. Moreover, they are the same for all the
individuals of each single population. We do not model races, traits,
polymorphisms or any phenotypic variation within a species, and when a
mutation occurs it is assumed that the whole population ``moves'' instantly
to the new state. In this sense, we are modelling the co-evolution of
the species and disregarding the evolution of a single one as well as
other important phenomena like the formation of new species \cite{vand}.
Combining the two roles of predator and prey that each species performs, and
the special status of the ends of the chain, we can write the following set
of equations for the evolution of the system:
\begin{equation}
\left\{
\begin{array}{lcl}
\Delta n_t^i & = & k_in_t^{i-1}n_t^i\left( 1-n_t^i/c_i\right)
-g_in_t^{i+1}n_t^i \mbox{\ \ \ \ for $i=1 \dots N$} \\
n^0 & = & n^{N+1} = 1,
\end{array}
\right. \label{model}
\end{equation}
where we have introduced two fictitious species, $0$ and $N+1$, to take
account for the border condition.
We make two simplifications to the system (\ref{model}): 1) we
assume that all the carrying capacities are equal, and equal to $1$;
2) we assume that $g_i=k_{i+1}$. In this way we reduce the number
of parameters that define the phenotypic features of the ecosystem.
The dynamics of the system is as follows. At time $t=0$, all the populations
and interactions are chosen at random with uniform distribution in the
interval $\left( 0,1\right) $. Then the populations begin to evolve
according to the system (\ref{model}). In the course of the evolution driven
by eq.(\ref{model}) a population can go to zero. As this can happen
asymptotically, we consider a species extinct if its population drops
below a given threshold. This is reasonable since actual biological populations
are discrete. In order to keep constant the number of species
we replace an extinct one with a
new one, which can be thought as a species coming to occupy the niche left
by the extinct one \cite{footnote}. The new population, and the (two) new interactions with
its neighbors in the chain, are also drawn at random from a uniform
distribution in the interval $\left( 0,1\right) $. On top of this dynamics
of predation and extinctions, we introduce random mutations. In each time
step a mutation is produced with probability $p$; the species to mutate is
chosen at random and the mutation itself consists of the replacement of the
species with a new one, with a new population and new interactions with its
predators and preys (all random in $(0,1)$).
Observe that we do not introduce the fitness of a species as a dynamical
variable. We do not even need to compute it from the ``phenotypes'' $k_i$.
The fitness, the degree of adaptation of a species to the ecosystem, arises
from the phenotypes, the populations, and the dynamics, and it determines
whether a species will thrive or become extinct. Chance is introduced by
the random mutations (and the random replacement of extinct species). It
provides the material the natural selection works on. This, in turn,
determines the survival of the fittest by simply eliminating from the system
those species that cannot cope with the competing environment. We believe in
this way we avoid a fundamental problem in the models of evolution as a walk
in a fitness landscape, namely that the concepts of fitness is not the
biological one \cite{jong}.
\section{Results of the numerical simulation}
We have run our model for several chain sizes, ranging from $50$ to $1000$
sites, and for times of about $10^7$ steps. In the results reported below
we let the system evolve, during a transient period, from the initial
random state to an organized one.
In fig.\ \ref{pob} we show a typical evolution of the whole population, $%
\sum_1^N n^i$.
Although each population greatly changes in the course of time (what
is not shown in the picture), we observe that the whole population
remains relatively stable. This is due to the saturation factor in the
predation term of the evolution equations.
This whole population shows a short time oscillatory dynamics
governed by the competition between species through the set of equations (%
\ref{model}), and a long time evolution characterized by periods of relative
stasis and periods of fast change. This feature is the effect of mutations
and extinction of some species. Without the extinctions and mutations, the
dynamics of the system should probably be chaotic. But it is not this
feature that we want to analyze here. Instead, we shall focus on the pattern
of extinctions.
As the set of $k_i$ represents the phenotypes of the whole ecosystem, its
distribution, $P(k)$, can be used to characterize its state. Let's observe what
happens in the course of time, including the transient mentioned above.
Initially the $k_i$ are chosen at random, and thus its distribution is flat
in $(0,1)$, with mean $0.5$. This is shown in fig.\ \ref{k} as a full line. As
time passes, and as a result of the dynamics, this distribution shifts to a
non-uniform one, as shown in fig.\ \ref{k} with dashed lines.
The whole distribution shifts towards lower values of the interaction,
showing a tendency of the system to reduce the coupling between the species.
In the course of
the evolution this distribution fluctuates following the pattern of mutations
and extinctions, but preserves its form.
Fig.\ \ref{kmed} shows the above mentioned fluctuations in $P(k)$
as the evolution of the mean value of $k_i$ in the
system, after the transient.
It corresponds to the same run as fig.\ \ref{pob}, and the same time
window is shown. Similarly to that, it displays a pattern of periods of stasis
interrupted by periods of fast change, but without the short time
oscillations displayed by the population.
There are periods of stasis of all lengths, to a degree that the unique
scale of the figure cannot display. This feature of a lack of a typical
length will be analyzed immediately.
Observe in this figure that the mean value
oscillates around $0.24$, corresponding to a distribution like that
shown in fig.\ \ref{k} with a dotted line.
The extinction events also display this characteristic pattern of periods of
stasis and periods of change, without a typical size.
In order to characterize this, we have chosen the time between
two consecutive extinction events, $\tau$, which distribution is shown
in fig.\ \ref{ext} for several system sizes an
probabilities of mutation. Observe that
they follow a power law for several decades of large values of $\tau$,
before a region where the effects of
the finite size of the system start to appear. This is a sign that the
system has self-organized into a critical state.
In other words, the extinction events are
distributed in the time axis in such a way that the time between extinctions
does not have a characteristic duration --as should have if the
distribution were exponential. Extinctions appear to come in
bursts, or avalanches, of any size.
In fig.\ \ref{aval} the pattern of extinction events of the system
is seen in the course of time.
The graphic displays time in the abscissa and the index in the food
chain in the ordinate.
Each dot marks the moment in which a species has become
extinct. Each cross, a species that suffers a chance mutation. It can be
seen that some mutations trigger avalanches of extinction, and that these
propagate in the ``prey'' direction. (Bear in mind that an extinct species
is replaced by a random new one, most probably with a larger population than
its predecessor, and observe that this has a negative impact in the
corresponding {\em prey}.) It is also apparent that this avalanches have
a complex shape in space-time. It is not easy to measure their size since,
as can be seen in fig.\ \ref{aval}, they overlap.
See, for example, a mutation that is {\em not} followed by any avalanche
(lower left), another that triggers a very small one (lower right),
and several that start events of varying size.
In any case, let's define a time step, $\Delta t$, divide the
time axis with it, and count the number of extinctions in each interval.
Now, let's call the fraction of species that have become extinct in each
interval the {\em size}, $S$, of the extinction. $S$ will obviously depend
on the time step and on the size of the system: $S=S(\Delta t,N)$. If the
system is in a critical state this function will obey some scaling law on
the variable $N$. In fig.\ \ref{scaling} we have scaled the distribution of
the system size $S(N)$, $P(S,N)$, obtained for different system sizes according
to the ansatz:
\begin{equation}
P(S,N)=N^\beta f(S\cdot N^\nu ).
\end{equation}
We can observe that the four curves collapse to a single one, showing the
scaling behaviour that is typical of a critical state.
\section{Conclusions}
We have introduced a simple model of co-evolution and extinction
in a food chain. This consists of a finite chain of species
of predators and preys. Their populations evolve in time following
Lotka-Volterra-like equations. Evolution is mimicked by randomly
changing a phenotype. Natural selection is provided by the deterministic
behaviour of the dynamical system, that produces the extinction
of any species that cannot cope with its interactions. No relative
fitness or fitness landscape had to be invoked. Nevertheless,
the pattern of extinctions displayed by this toy ecosystem appears to
be similar to that proposed for the biosystem on Earth.
Namely, the system seems to be in a critical state, in which
extinctions occur in avalanches. The time between extinctions, and the
lifetime of any species follow distributions that behave
like power-laws of time, implying that there is no typical size
for the time that a species remains in the system.
I should be of interest, in a future work, to study the precise
instability that produces the shift of the distribution of the
interactions towards low values. The analytical treatment of this
problem is currently under study.
The author greatly acknowledges Ruben Weht for invaluable discussions,
and thank Hilda Cerdeira for a careful reading of the manuscript.
| proofpile-arXiv_065-402 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
Since the previous HERA workshop in 1991
significant progress has been made on the theoretical side
in understanding the production of heavy quarks
in electron proton collisions. Improvements in available
experimental techniques and particularly
the expected increase in luminosity amply justify this effort. In general
the progress consists of the calculation of all $O(\alpha_s)$ corrections
to the processes of interest, thus improving the accuracy of the theoretical
predictions both in shape and normalization. At the time of the previous
workshop the only NLO calculations available were for the case of
inclusive photoproduction \cite{photoincl}. In the meantime
NLO calculations have also been performed for inclusive electroproduction
\cite{LRSNF,LRSNH,LRSND}, and both have been
extended to the fully differential cases \cite{FMNR,HS,H}.
Therefore, meaningful and extensive comparisons between theory and
data can now be made.
In what follows we review how the deeply inelastic
electroproduction process allows us to explore, in detail,
three areas of perturbative QCD in particular.
We first discuss the inclusive case, via the structure function
$F_2(x,Q^2,m^2)$. We show that this structure function for the case of
charm suffers from only very modest theoretical uncertainty,
that its NLO corrections are not too large, and that it is sensitive to
the shape of the small-$x$ gluon density.
Next we treat single particle differential distributions
in the charm kinematical
variables, and also charm-anticharm correlations. Because many distributions
can be studied, many QCD tests can be performed. Examples are tests
of the production mechanism (boson-gluon fusion), studies of gluon
radiation patterns, and dependence on scales
such as deep-inelastic momentum transfer $Q$, the heavy quark
mass $m$ (with enough
luminosity one can detect a sizable sample of bottom quarks), the transverse
momentum of the charm quark, etc. Finally, in the last section, we review the
theoretical status of the boson-gluon fusion description of charm production
at small and very large $Q$. In
essence, it involves answering the question: when is charm a parton?
\section{Structure Functions and Gluon Density}
This section has some overlap with the more detailed review on
heavy flavour structure functions in the structure function section. Here
we only present the most salient features.
The reaction under study is
\begin{equation}
e^-(p_e) + P(p) \rightarrow e^-(p_e') +
Q(p_1)(\bar{Q}(p_1))+X\,,\label{one}
\end{equation}
where $P(p)$ is a proton with momentum $p$, $Q(p_1)(\bar{Q}(p_1))$
is a heavy (anti)-quark with momentum $p_1$ ($p_1^2 = m^2$) and
$X$ is any hadronic state allowed. Its cross section
may be expressed as
\begin{equation}
\label{three}
\frac{d^2\sigma}{dxdQ^2} =
\frac{2\pi\alpha^2}{x\,Q^4}
\left[ ( 1 + (1-y)^2 )
F_2(x,Q^2,m^2) -y^2
F_L(x,Q^2,m^2) \right]\,,
\end{equation}
where
\begin{equation}
q = p_e - p_e'\,, \qquad Q^2 = -q^2\,,\qquad x = \frac{Q^2}{2p\cdot q}
\,, \qquad y = \frac{p\cdot q}{p \cdot p_e}\,. \label{four}
\end{equation}
The inclusive structure functions $F_2$ and $F_L$ were calculated
to next-to-leading order (NLO)
in Ref.~\cite{LRSNF}. The results can be written as
\begin{eqnarray}
F_{k}(x,Q^2,m^2) &=&
\frac{Q^2 \alpha_s}{4\pi^2 m^2}
\int_x^{z_{\rm max}} \frac{dz}{z} \Big[ \,e_H^2 f_g(\frac{x}{z},\mu^2)
c^{(0)}_{k,g} \,\Big] \nonumber \\&&
+\frac{Q^2 \alpha_s^2}{\pi m^2}
\int_x^{z_{\rm max}} \frac{dz}{z} \Big[ \,e_H^2 f_g(\frac{x}{z},\mu^2)
(c^{(1)}_{k,g} + \bar c^{(1)}_{k,g} \ln \frac{\mu^2}{m^2}) \nonumber \\ &&
+\sum_{i=q,\bar q} \Big[ e_H^2\,f_i(\frac{x}{z},\mu^2)
(c^{(1)}_{k,i} + \bar c^{(1)}_{k,i} \ln \frac{\mu^2}{m^2})
+ e^2_{L,i}\, f_i(\frac{x}{z},\mu^2) d^{(1)}_{k,i} \, \Big] \,\Big] \,,
\label{strfns}
\end{eqnarray}
where $k = 2,L$ and the upper boundary on the integration is given by
$z_{\rm max} = Q^2/(Q^2+4m^2)$. The functions $f_i(x,\mu^2)\,, (i=g,q,\bar q)$
denote the parton densities in the proton and $\mu$ stands for the
mass factorization scale,
which has been put equal to the renormalization scale. The
$c^{(l)}_{k,i}(\eta, \xi)\,,\bar c^{(l)}_{k,i}
(\eta, \xi)\,,
(i=g\,,q\,,\bar q\,;l=0,1)$
and $d^{(l)}_{k,i}(\eta, \xi)$,
$(i=q\,,\bar q\,;l=0,1)$
are coefficient functions and are represented in the
$\overline{\rm MS}$ scheme.
They depend on the scaling variables $\eta$ and $\xi$ defined by
\begin{equation}
\eta = \frac{s}{4m^2} - 1\quad \qquad \xi = \frac{Q^2}{m^2}\,.
\end{equation}
where $s$ is the square of the c.m. energy of the
virtual photon-parton subprocess
which implies that in (\ref{strfns}) $z=Q^2/(Q^2+s)$. In eq.~(\ref{strfns}) we
distinguished between the coefficient functions with respect to their origin.
The coefficient functions indicated by
$c^{(l)}_{k,i}(\eta, \xi),\bar c^{(l)}_{k,i}(\eta, \xi)$
originate from the partonic
subprocesses where the virtual photon is coupled to the heavy quark, whereas
the quantity $d^{(l)}_{k,i}(\eta, \xi)$
comes from the subprocess where the virtual
photon interacts with the light quark.
Hence the former are multiplied by the charge squared
of the heavy quark $e_H^2$, and the latter
by the charge squared of the light quark $e_L^2$ respectively
(both in units of $e$).
Terms proportional to $e_H e_L$ integrate
to zero for the inclusive structure functions.
Furthermore we have isolated the factorization scale dependent logarithm
$\ln(\mu^2/m^2)$. A fast program using fits to the coefficient functions
\cite{RSN} is available.
The first thing to note about eq.~(\ref{strfns}) is that the lowest
order term contains only the gluon density. Light quark densities only
come in at next order, and this is the reason $F_2(x,Q^2,m^2)$ is promising
as a gluon probe.
To judge its use as such,
we must examine some of the characteristics of this observable. These
are: the size of the $O(\alpha_s)$ corrections,
the scale dependence, the mass dependence, its sensitivity
to different gluon densities, and the relative size of the
light quark contribution. These are the issues we investigate in
this section. We take the charm mass 1.5 GeV,
the bottom mass 5 GeV, the factorization scale equal to $\sqrt{Q^2+m^2}$
and choose at NLO the CTEQ4M \cite{cteq4} set of parton densities,
with a two-loop running coupling constant for five flavors and
$\Lambda = 202 $ MeV, and at LO the corresponding CTEQ4L set, with
a one-loop running coupling with five flavors and $\Lambda = 181 $ MeV.
\begin{figure}[htbp]
\hspace*{1.5cm}
\epsfig{file=fig1.ps,bbllx=0pt,bblly=20pt,bburx=575pt,bbury=880pt,%
width=10cm,angle=-90}
\caption[junk]{{\it
$F_2(x,Q^2,m^2)$ vs. $x$ at LO and NLO
for two values of $Q^2$. The shaded areas indicate the uncertainty
due to varying the charm mass from 1.3 to 1.7 GeV.
}}
\label{FIGF2C1}
\end{figure}
In Fig.~\ref{FIGF2C1} we display $F_2(x,Q^2,m^2)$ vs. $x$ for
two values of $Q^2$ at LO and NLO.
The scale dependence is much reduced by including
the NLO corrections (when varying $\mu$ from $2$ to $1/2$ times
the default choice, the structure function varies from,
at LO, at most 20\% and 13\% at
$Q^2 = 10$ and $50$ GeV$^2$ respectively, to at most 5\% and 3\%
at NLO), but the dominant uncertainty is due to the charm
mass and stays roughly constant, amounting at NLO maximally to about 16\%
for $Q^2=10$ GeV$^2$
and 10\% for $Q^2=50$ GeV$^2$. The feature that the LO result is
mostly larger than the NLO ones is due to the use
of LO parton densities and one-loop $\alpha_s$,
and scale choice.
Had we used NLO densities and a two-loop $\alpha_s$, or
chosen the scale $\mu$ equal to $m$,
the LO result would have been below the NLO result. In the
first case the size
of the corrections is then about 40\% at the central
values at $Q^2 = 10$ GeV$^2$, and 25\% at $Q^2 = 50$ GeV$^2$,
and in the second case, at small $x$, about 20\% and 30\% respectively.
\begin{figure}[htbp]
\hspace*{1.5cm}
\epsfig{file=fig2.ps,bbllx=0pt,bblly=20pt,bburx=575pt,bbury=880pt,%
width=10cm,angle=-90}
\caption[junk]{{\it
$F_2(x,Q^2,m^2)$ vs. $x$ at NLO for two choices of parton densities.
The shaded areas again indicate the uncertainty
due to varying the charm mass from 1.3 to 1.7 GeV.
}}
\label{FIGF2C2}
\end{figure}
In the next figure, Fig.~\ref{FIGF2C2}, we show for the same values of $Q^2$
an important property, namely the
sensitivity of the NLO $F_2$ to different parton density parametrizations.
In this case we compare the CTEQ2MF set \cite{cteq2}, whose gluon
density stays quite
flat when $x$ becomes small, and the GRV94 set \cite{GRV94}, which
has a steeply rising gluon density. One sees
that the difference is visible in the structure function.
Finally we remark that the contribution of light quarks to the
charm structure function is typically less than 5\%.
The bottom quark structure function is suppressed by electric charge
and phase space effects and amounts to less than 2\% (5\%)
at $Q^2=10\,(50)$ GeV$^2$ of the charm structure function. Previous
investigations of the scale and parton density dependence of $F_2$
using the same NLO computer codes are available in \cite{vogt}
and \cite{grs}.
We conclude that $F_2(x,Q^2,m^2)$ for charm production is an
excellent probe to infer the gluon density in the proton at
small $x$.
The NLO theoretical prediction suffers from fairly little
uncertainty, and the QCD corrections are not too large.
See the section on structure
functions in these proceedings for many more details, where also a comparison
with (preliminary) data is shown. Therefore in view of a large
integrated luminosity, a theoretically well-behaved
observable, and promising initial experimental studies \cite{H1paper,zeus}
a precise measurement at HERA of the gluon density should be possible.
\section{Single Particle Distributions and Heavy Quark Correlations}
In this section we leave the fully inclusive case and examine in more
detail the structure of the final state of the reaction
\begin{equation}
e^-(p_e) + P(p) \rightarrow e^-(p_e') +
Q(p_1)+\bar{Q}(p_2)+X\,.
\label{react2}
\end{equation}
By studying various differential distributions of the heavy quarks
we can learn more about the dynamics of the
production process than from the structure
function alone.
Single particle distributions $dF_2(x,Q^2,m^2,v)/dv$, where
$v$ is the transverse momentum $p_T$ or rapidity $y$ of the
charm quark, were presented in NLO in
\cite{LRSND} for various choices of $x$ and $Q^2$. The LO distributions
differed significantly from the NLO ones, so that
the effect of $O(\alpha_s)$ corrections on such distributions
cannot be described by a simple K-factor.
The $O(\alpha_s)$ corrections to $F_k(x,Q^2,m^2)$ in a fully
differential form were calculated in Ref.~\cite{HS} using
the subtraction method. Recently \cite{H}, these fully differential
structure functions were incorporated in a Monte-Carlo style program
resulting in the $O(\alpha_s)$ corrections for reaction (\ref{react2}).
The program for the full cross section, generated
according to Eq.\ (\ref{three}), allows one to study correlations
in the lab frame. The phase space integration is done numerically.
Therefore, it is possible to implement experimental cuts.
It furthermore allows the use of a Peterson type fragmentation function.
For details about the calculational techniques we refer to
Ref.~\cite{HS,H}. Here we show mainly results.
\begin{figure}[htbp]
\hspace*{.5cm}
\epsfig{file=fig3.ps,bbllx=0pt,bblly=200pt,bburx=575pt,bbury=600pt,%
width=13cm,angle=0}
\caption[junk]{{\it
Differential cross sections and ZEUS data.
}}
\label{FIGF2C3}
\end{figure}
Shown in Fig.~\ref{FIGF2C3} are various distributions $d\sigma/dv$
for the reaction (\ref{react2}), where the heavy (anti)quark
has fragmented into a $D^*$ meson, with $v$ representing
(a) the $D^*$ transverse momentum
$p_T^{{D^*}}$ (b) its pseudorapidity $\eta^{{D^*}}$ (c) the hadronic
final state invariant mass $W$ (d) $Q^2$
for the kinematic range 5 GeV$^2 < Q^2 < $ 100 GeV$^2$,
$0<y<0.7$, $1.3\,{\rm GeV} < p_T^{D^*} < 9 {\rm GeV}$
and $|\eta^{{D^*}}| < 1.5$. The data are from a recent
ZEUS analysis \cite{zeus}.
The NLO theory curves have been produced by
using the GRV \cite{GRV94} parton density set, with Peterson
fragmentation \cite{peterson}.
The dashed line is for $\mu=2m$, $m=1.35$ GeV and
$\epsilon = 0.035$, whereas
the solid line is for $\mu=2\sqrt{Q^2+4m^2}$, $m=1.65$ GeV
and $\epsilon = 0.06$. From Fig.~\ref{FIGF2C3}
and studies in Ref.~\cite{H1paper}
it is clear that a wide range of
studies can be and are being performed already at the single
particle inclusive level. Preliminary conclusions \cite{H1paper,zeus}
are that the data follow the shape of the NLO predictions quite well,
but lie above the theory curves. The H1 collaboration
\cite{H1paper} has recently shown clearly
from the $d\ln\sigma/dx_D$
distribution that the charm production mechanism is
indeed boson-gluon fusion,
(after earlier indications from the EMC collaboration \cite{EMC})
as opposed to one where the charm quark is taken from the sea.
Here $x_D=2|\vec{p}_{D^*}|/W$ in the $\gamma^* P$ c.m. frame
Next we examine a few charm-anticharm correlations.
\begin{figure}[htbp]
\hspace*{1.5cm}
\epsfig{file=fig4.ps,bbllx=0pt,bblly=70pt,bburx=575pt,bbury=800pt,%
width=10cm,angle=-90}
\caption[junk]{{\it
Differential distributions $dF_2(x,Q^2,m^2,p_{cc})/dp_T^{cc}$
and $dF_2(x,Q^2,m^2,p_{cc})/d\Delta\phi^{cc}$
at $x=0.001$ and $Q^2 = 10$ GeV$^2$ (solid) and 100 GeV$^2$ (dashed).
}}
\label{FIGF2C4}
\end{figure}
At the
experimental level such correlations are more difficult to
measure since it requires the identification of both heavy
quarks in the final state. However, with the expected large
luminosity that both ZEUS and H1 will collect, such studies
are likely to be done.
As an example we show in Fig.~\ref{FIGF2C4} the $p_T$ distribution
of the pair, $p_T^{cc}$, and the
distribution in their azimuthal angle difference, $\Delta\phi^{cc}$
in the $\gamma^* P$ c.m. frame for a particular choice of $x$ and $Q^2$.
For these figures we used the MRSA$'$ densities \cite{mrsap}.
Both distributions are a measure of the recoiling gluon jet.
In summary, differential distributions of deep-inelastic heavy
quark production offer a rich variety of studies of the
QCD production mechanism. Fruitful experimental
studies, even with low statistics, have been done \cite{H1paper,zeus},
and with a large integrated luminosity we therefore fully expect many more.
We finally point out that besides a LO shower Monte Carlo program
\cite{aroma}, now also a NLO program is available for producing
differential distributions.
\section{When is Charm a Parton?}
We return to the inclusive case to ask the fundamental question in
the title. The question can be more accurately phrased as follows:
intuitively one expects that at truly large $Q^2$ the charm quark
should be described as a light quark, i.e. as a constituent parton
of the proton, whereas at small $Q^2$ (of order $m^2$) the boson-gluon
fusion mechanism, in which the charm quark can only be
excited by a hard scattering, is the correct description. This has
been demonstrated recently by H1 \cite{H1paper} and ZEUS in \cite{zeus}.
In this section we examine where the transition between the two
pictures occurs.
At LO this issue was investigated in \cite{riol}. A picture that
consistently combines both descriptions, the so-called variable
flavor number scheme, is presented and worked out to
LO in \cite{acot}. Here we exhibit where the transition occurs at NLO
\cite{BMMSN}.
In other words we will locate the onset of the
large $Q^2$ asymptotic region,
where the exact partonic coefficient functions of \cite{LRSNF}
are dominated by large logarithms $\ln(Q^2/m^2)$. These logarithms
are controlled by the renormalization group, and, when resummed,
effectively constitute the charm parton density.
Here we however restrict ourselves to the onset of the
asymptotics. Let us be somewhat more precise. In (\ref{strfns})
we can rewrite e.g. all terms proportional to $e_H^2$ as
\begin{equation}
x \int_x^{zmax}\frac{dz}{z}\Big\{
\Sigma(\frac{x}{z},\mu^2) H_{i,q}(z,\frac{Q^2}{m^2},\frac{m^2}{\mu^2})+
G(\frac{x}{z},\mu^2) H_{i,g}(z,\frac{Q^2}{m^2},\frac{m^2}{\mu^2})\Big\}
\end{equation}
where $G(x,\mu^2)$ is the gluon density and
$\Sigma(x,\mu^2) = \sum_{i=q,\bar{q}}f_{i}(x,\mu^2)$ is the singlet
combination of quark densities.
In the asymptotic regime one may write
\begin{equation}
H^{(k)}_{i,j}(z,\frac{Q^2}{m^2},\frac{m^2}{\mu^2})
= \sum_{l=0}^{k} a_{i,j}^{(k)}(z,\frac{m^2}{\mu^2})
\ln^l\frac{Q^2}{m^2}\,.
\end{equation}
The effort lies in determining the coefficients $a_{i,j}^{(k)}$.
Similar expressions hold for the other coefficients in (\ref{strfns}).
Taking the limit of the coefficients in \cite{LRSNF} is extremely
complicated. Rather, a trick \cite{BMMSN} was used, exploiting
the close relationship of the $\ln(Q^2/m^2)$ logarithms with
collinear (mass) singularities. The ingredients are the massless two-loop
coefficient functions of \cite{zn} and certain two-loop operator matrix
elements.
\begin{figure}[htbp]
\hspace*{1.5cm}
\epsfig{file=fig5.ps,bbllx=0pt,bblly=250pt,bburx=575pt,bbury=700pt,%
width=12cm,angle=0}
\caption[junk]{{\it
Ratio of the asymptotic to
exact expressions for $F_2(x,Q^2,m^2)$ for the case of charm.
}}
\label{FIGF2C5}
\end{figure}
The trick, dubbed ``inverse mass factorization'', essentially amounts
to reinserting into the IR safe massless coefficient functions the
collinear singularities represented by the logarithms $\ln(Q^2/m^2)$.
See \cite{BMMSN} for details.
There is another advantage to obtaining the asymptotic expresssions.
The terms in eq.~(\ref{strfns}) proportional to $e_L^2$ have been integrated
and full analytical expressions for them exist \cite{BMMSN}, but
in the other terms in eq.~(\ref{strfns}) two integrals still need
to be done numerically. Therefore in the large $Q^2$ region the asymptotic
formula is able to give the same results much faster, as the latter
formula needs no numerical integrations.
In Fig.~\ref{FIGF2C5} we show the ratio of the asymptotic to
exact expressions for $F_2(x,Q^2,m^2)$ for the case of charm
as a function of $Q^2$ for four different $x$ values.
Here the GRV \cite{GRV94} parton density set was used, for
three light flavors.
We see that, surprisingly, already at $Q^2$ of order 20-30 GeV$^2$
the asymptotic formula is practically identical to the exact result, indicating
that at these not so large $Q^2$ values, and for the inclusive structure
function, the charm quark behaves
already very much like a parton.
This is in apparent contradiction with the findings \cite{H1paper},
mentioned in the previous section, that the production
mechanism is boson-gluon fusion, and illustrates that, interestingly,
the question in the title can have a different answer for inclusive
quantities than for differential distributions having multiple scales.
We finally note that with the results shown in this section
also the first important step is made for extending
the variable flavour number scheme to NLO.
\section{Conclusions}
In the above we have reviewed the many interesting facets of
deep-inelastic production of heavy quarks. The possibility
of selecting the heavy quarks among the final state particles
affords a window into the heart of the scattering process,
and allows tests and measurements of some of the most
fundamental aspects of perturbative QCD: the direct determination
of the gluon density, many and varied studies of the heavy
quark production dynamics, and insight into how and when
a heavy quark becomes a parton.
| proofpile-arXiv_065-403 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
The electric polarisability of the charged pion $\alpha_E$ can be inferred
from the amplitude for low energy Compton scattering
$ \gamma + \pi^+ \rightarrow \gamma + \pi^+ $.
This amplitude cannot be measured
at low energies directly, but can be determined from measurements on
related processes like $\pi N \rightarrow \pi N \gamma$, $ \gamma N
\rightarrow \gamma N \pi$ and $ \gamma \gamma \rightarrow \pi \pi$.
The measured values for $\alpha_E$,
in units of $10^{-4}\; {{\rm fm}}^3$,
are $6.8 \pm 1.4 \pm 1.2$ \cite{Antipov},
$ 20 \pm 12$ \cite{Aibergenov} and $2.2 \pm 1.6$ \cite{Babusci}, respectively.
Alternatively, the polarisability can be predicted theoretically by relating
it to other
quantities which are better known experimentally. In chiral perturbation
theory, it can be shown \cite{Holstein} that the pion polarisability
is given by
\begin{equation}
\alpha_E = \frac{ \alpha \, F_A}{m_{\pi} F_{\pi}^2},
\label{alpha}
\end{equation}
where $F_{\pi}$ is the pion decay
constant and $F_A$ is the axial structure-dependent form factor for radiative
charged pion decay $\pi \rightarrow e \nu \gamma$ \cite{Bryman}.
The latter is often re-expressed in the form
\begin{equation}
F_A \equiv \gamma\,{F_V}\, ,
\end{equation}
because the ratio $\gamma$ can be measured
in radiative pion decay experiments more accurately than $F_A$ itself,
while the corresponding vector form factor $F_V$ is determined to be
\cite{PDG}\footnote{Our definitions of $F_V$ and $F_A$ differ from
those used by the Particle Data Group \cite{PDG} by a factor of two.}
$F_V = 0.0131 \pm 0.0003$
by using the conserved
vector current (CVC) hypothesis to relate $\pi \rightarrow e \nu \gamma$
and $\pi^0 \rightarrow \gamma\gamma$ decays.
$\gamma$ has been measured in three $\pi \rightarrow e \nu \gamma$
experiments, giving the values: $0.25 \pm 0.12$ \cite{lampf};
$0.52 \pm 0.06$ \cite{sin}; and $0.41 \pm 0.23$ \cite{istra}. The weighted
average $ \gamma = 0.46 \pm 0.06$
can be combined with the above equations to give
\begin{equation}
\alpha_E = (2.80 \pm 0.36) \, 10^{-4} \; {{\rm fm}}^3 \; .
\label{chiral}
\end{equation}
This result is often referred to as the chiral theory prediction for
the pion polarisability \cite{Holstein}. However $\alpha_E$, or
equivalently $F_A$, can be
determined in
other ways. In particular, the latter occurs in the Das-Mathur-Okubo
(DMO) sum rule \cite{Das}
\begin{equation}
I = F_{\pi}^2 \frac{\langle r_{\pi}^2 \rangle }{3} - {F_A}\,,
\label{DMO}
\end{equation}
where
\begin{equation}
I\equiv\int \frac{{\rm d}s}{s} \rho_{V-A}(s) \,,
\label{i}
\end{equation}
with $\rho_{V-A}(s) = \rho_V(s) - \rho_A(s)$ being the difference
in the spectral functions of the vector and axial-vector
isovector current correlators, while
$\langle r_{\pi}^2\rangle $
is the pion mean-square charge radius. Using its
standard value $\langle r_{\pi}^2\rangle = 0.439 \pm 0.008 \; {{\rm fm}}^2$
\cite{Amendolia} and eqs. (\ref{alpha}), (\ref{chiral}) one gets:
\begin{equation}\label{iexp}
I_{DMO}=(26.6 \pm 1.0)\cdot 10^{-3}
\end{equation}
Alternatively, if the integral $I$ is known, eq. (\ref{DMO}) can be rewritten
in the form of a prediction for the polarisability:
\begin{equation}
\alpha_E = \frac{\alpha}{m_{\pi}} \biggl(
\frac{\langle r_{\pi}^2 \rangle }{3}
- \frac{I}{F_{\pi}^2} \biggr).
\label{DMOpredict}
\end{equation}
Recent attempts to analyse this relation have resulted in some contradiction
with the chiral prediction.
Lavelle et al. \cite{Lavelle}
use related QCD sum rules to estimate the integral $I$ and obtain
$\alpha_E = (5.60 \pm 0.50) \, 10^{-4} \; {{\rm fm}}^3$.
Benmerrouche et. al. \cite{Bennerrouche} apply certain sum rule
inequalities to obtain a lower bound on the polarisability
(\ref{DMOpredict}) as a function of
${\langle r_{\pi}^2 \rangle }$. Their analysis also
tends to prefer larger $\alpha_E$ and/or smaller
${\langle r_{\pi}^2\rangle }$ values.
In the following we use available experimental data to reconstruct the
hadronic spectral function $\rho_{V-A}(s)$, in order to calculate
the integral
\begin{equation}\label{i0}
I_0(s_0) \equiv \int_{4m_{\pi}^2}^{s_0}
\frac{{\rm d}s}{s} \rho_{V-A}(s) \; .
\end{equation}
for $s_0\simeq M_{\tau}^2$, test the saturation of the DMO sum rule
(\ref{DMO}) and its compatibility with the chiral prediction (\ref{chiral}).
We also test the saturation of the first Weinberg sum rule \cite{Weinberg}:
\begin{eqnarray}
W_1(s_0) \equiv \int_{4m_{\pi}^2}^{s_0}
{\rm d}s \, \rho_{V-A}(s)\;;\;\;\;\;\;\;\;\;\;\;
W_1(s_0)\;\bigl|_{s_0\to\infty} = F_{\pi}^2
\label{w1}
\end{eqnarray}
and use the latter to improve convergence and
obtain a more accurate estimate for the integral $I$:
\begin{eqnarray}\label{i1}
I_1(s_0) & = & \int_{4m_{\pi}^2}^{s_0} \frac{{\rm d}s}{s}
\rho_{V-A}(s) \nonumber \\
& + & {{\beta}\over{s_0}}
\left[ F_{\pi}^{2} - \int_{4m_{\pi}^2}^{s_0} {\rm d}s \, \rho_{V-A}(s)
\right]
\end{eqnarray}
Here the parameter $\beta$ is arbitrary and can be chosen to minimize the
estimated error in $I_1$ \cite{markar}.
Yet another way of reducing the uncertainty in our estimate of $I$ is to
use the Laplace-transformed version of the DMO sum rule \cite{marg}:
\begin{equation}
I_2(M^2) = F_{\pi}^2 \frac{\langle r_{\pi}^2 \rangle}{3} - {F_A}
\end{equation}
with $M^2$ being the Borel parameter in the integral
\begin{eqnarray}
I_2(M^2) &\equiv& \int \frac{{\rm d}s}{s} \exp{\left( \frac{-s}{M^2} \right)} \,
\rho_{V-A}(s) + \frac{F_{\pi}^2}{M^2} \nonumber \\
& - & \frac{C_6 \langle O^6\rangle}{6 M^6} -
\frac{C_8 \langle O^8\rangle }{24 M^8} + \ldots \;.
\label{i2}
\end{eqnarray}
Here $ C_6 \langle O^6\rangle $ and $ C_8 \langle O^8\rangle $
are the four-quark vacuum condensates of dimension 6 and 8, whose values
could be estimated theoretically or taken from previous analyses
\cite{markar,kar}.
All the three integrals (\ref{i0}), (\ref{i1}) and (\ref{i2}) obviously reduce
to (\ref{i}) as $s_0, M^2 \to \infty$.
\section{Evaluation of the spectral densities}
Recently ALEPH published a comprehensive and consistent
set of $\tau$ branching fractions \cite{aleph}, where in many cases
the errors are
smaller than previous world averages. We have used these values to
normalize the contributions of specific hadronic final states, while
various available experimental data have been used to determine the
shapes of these contributions. Unless stated otherwise,
each shape was fitted with a single
relativistic Breit-Wigner distribution with appropriately chosen threshold
behaviour.
\subsection{ Vector current contributions.}
A recent comparative study {\cite{eidel}}
of corresponding final states in $\tau$ decays and
$e^+e^-$ annihilation
has found no significant violation of CVC or isospin symmetry.
In order to determine the shapes of the hadronic spectra, we have used
mostly $\tau$ decay data, complemented by $e^+e^-$ data in some
cases.
{{ $\pi^-\pi^0$ :}} ${{\rm BR}}=25.30\pm0.20\%$ \cite{aleph},
and the $s$-dependence was
described by the three interfering resonances $\rho(770)$, $\rho(1450)$
and $\rho(1700)$, with the parameters taken from
\cite{PDG} and \cite{pi2}.
{{ $3\pi^{\pm}\pi^0$ :}} ${{\rm BR}}=4.50\pm0.12\%$, including $\pi^-\omega$
final state \cite{aleph}. The shape was determined by
fitting the spectrum measured by ARGUS \cite{pi4a}.
{{ $\pi^{-}3\pi^0$ :}} ${{\rm BR}}=1.17\pm0.14\%$ \cite{aleph}.
The $s$-dependence
is related to that of the reaction $e^+e^-\to 2\pi^+2\pi^-$. We have fitted
the latter measured by OLYA and DM2 \cite{pi4b}.
{{ $6\pi$ :}} various charge contributions give the overall
${{\rm BR}}=0.13\pm0.06\%$ \cite{aleph}, fairly close to CVC expectations
\cite{eidel}.
{{ $\pi^-\pi^0\eta$ :}} ${{\rm BR}}=0.17\pm0.03\%$ \cite{pi0}.
The $s$-dependence
was determined by fitting the distribution measured by CLEO
\cite{pi0}.
{{ $K^-K^0$ :}} ${{\rm BR}}=0.26\pm0.09\%$ \cite{aleph}.
Again, the fit of the CLEO measurement \cite{cleok} was performed.
\subsection{ Axial current contributions.}
The final states with odd number of pions contribute to the axial-vector
current.
Here, $\tau$ decay is the only source of precise information.
{{ $\pi^-$ :}} ${{\rm BR}}=11.06\pm0.18\%$ \cite{aleph}. The single pion
contribution has a trivial $s$-dependence and hence is
explicitly taken into account in theoretical formulae. The quoted
branching ratio corresponds to $F_{\pi}=93.2$ MeV.
{{ $3\pi^{\pm}$ and $\pi^-2\pi^0$ :}}
${{\rm BR}}=8.90\pm0.20\%$ and ${{\rm BR}}=9.21\pm0.17\%$,
respectively \cite{aleph}. Theoretical models \cite{pi3th}
assume that these two modes are identical in both shape and normalization.
The $s$-dependence has been analyzed in \cite{opal},
where the parameters of two theoretical models describing this decay
have been determined. We have used the average of these two distributions,
with their difference taken as an estimate of the shape uncertainty.
{{ $3\pi^{\pm}2\pi^0$ :}} ${{\rm BR}}=0.50\pm0.09\%$,
including $\pi^-\pi^0\omega$
final state \cite{aleph}. The shape was fitted using the
CLEO measurement \cite{pi5a}.
{{ $5\pi^{\pm}$ and $\pi^-4\pi^0$ :}}
${{\rm BR}}=0.08\pm0.02\%$ and $BR=0.11\pm0.10\%$,
respectively \cite{aleph}. We have assumed that these two terms have the same
$s$-dependence measured in \cite{pi5b}.
\subsection{ $K{\overline K} \pi$ modes.}
$K{\overline K} \pi$ modes can
contribute to both vector and axial-vector currents, and various theoretical
models cover the widest possible range of predictions \cite{kkpith}.
According to \cite{aleph}, all three $K{\overline K} \pi$ modes
(${\overline K}^0K^0\pi^-, K^-K^0\pi^0$ and
$K^-K^+\pi^-$) add up to BR$({\overline K} K \pi)=0.56\pm0.18\%$, in
agreement with other measurements (see \cite{cleok}).
The measured $s$-dependence suggests that these final
states are dominated by $K^*K$ decays \cite{cleok}.
We have fitted the latter, taking into
account the fact that due to parity constraints, vector and axial-vector
$K^*K$ terms have different threshold behaviour. A parameter
$\xi$ was defined as the portion of $K{\overline K} \pi$ final state with
axial-vector quantum numbers, so that
\begin{eqnarray}
{\rm BR}({\overline K} K\pi)_{V}&=&(1-\xi)\;{\rm BR}({\overline K}K\pi)
\nonumber\\
{\rm BR}({\overline K} K\pi)_{A}&=&\xi\;{\rm BR}({\overline K} K\pi)\,.
\end{eqnarray}
\section{Results and conclusions}
\begin{figure}[htb]
\begin{center}
\epsfig{file=fig1.eps,width=10cm,clip=}
\end{center}
\vspace{1cm}
\caption{\tenrm
Difference of vector and axial-vector hadronic spectral densities.
In figs.1-5: the three curves correspond
to $\xi=0$, $0.5$ and 1 from top to bottom;
the errors originating from the shape variation and those coming
from the errors in the branching fractions are roughly equal and have
been added in quadrature to form the error bars, shown only for $\xi=0.5$.}
\label{fig1}
\end{figure}
\begin{figure}[p]
\begin{center}
\epsfig{file=fig2.eps,width=10cm,clip=}
\end{center}
\vspace{1cm}
\caption{\tenrm
Saturation of the DMO sum rule integral (8).
The thick dashed line is the chiral prediction for the asymptotic value
(6) and the dotted lines show its errors.}
\label{fig2}
\begin{center}
\epsfig{file=fig3.eps,width=10cm,clip=}
\end{center}
\vspace{1cm}
\caption{\tenrm
Saturation of the first Weinberg sum rule (9).
The dashed line shows the expected asymptotic value $F_{\pi}^2$.}
\label{fig3}
\end{figure}
The resulting spectral function is shown in Fig.1.
The results of its integration according to (\ref{i0})
are presented in fig.2 as a function of the upper bound $s_0$.
One can see that as $s_0$
increases, $I_0$ converges towards an asymptotic value which we
estimate to be{\footnote{In the following, the first error
corresponds to the quadratic sum of the errors in the branching ratios
and the assumed shapes, while the second one arises from to the variation
of $\xi$ in the interval $0.5\pm 0.5$.}
\begin{equation}\label{i0m}
I_0 \equiv I_0(\infty) = ( 27.5 \pm 1.4 \pm 1.2 ) \cdot 10^{-3},
\end{equation}
in good agreement with the chiral value (\ref{iexp}).
The saturation of the Weinberg sum rule (\ref{w1}) is shown in fig.3.
One sees that the expected value $F_{\pi}^2$ is well within the errors,
and $\xi \simeq 0.25\div0.3$ seems to be preferred. No
significant deviation from this sum rule is expected theoretically
\cite{Floratos}, so we use (\ref{i1}) to calculate our second
estimate of the integral $I$. The results of this
integration are presented in fig.4, with the asymptotic value
\begin{equation}\label{i1m}
I_1 \equiv I_1(\infty) = ( 27.0 \pm 0.5 \pm 0.1 ) \cdot 10^{-3},
\end{equation}
corresponding to $\beta\approx 1.18$.
One sees that the convergence has improved, the errors
are indeed much smaller, and the $\xi$-dependence is very weak.
\begin{figure}[p]
\begin{center}
\epsfig{file=fig4.eps,width=10cm,clip=}
\end{center}
\vspace{1cm}
\caption{\tenrm
Saturation of the modified DMO sum rule integral (10).
The chiral prediction is also shown as in fig.2.}
\label{fig4}
\begin{center}
\epsfig{file=fig5.eps,width=10cm,clip=}
\end{center}
\vspace{1cm}
\caption{\tenrm
The Laplace-transformed sum rule (12) as a function of
the Borel parameter $M^2$, compared
to the chiral prediction.}
\label{fig5}
\end{figure}
Now we use (\ref{i}) to obtain our third estimate of the spectral
integral. The integration results are plotted against the
Borel parameter $M^2$ in fig.5, assuming standard values for dimension
6 and 8 condensates.
The results are independent
of $M^2$ for $M^2 > 1 \, GeV^2$, indicating that higher order terms
are negligible in this region, and giving
\begin{equation}\label{i2m}
I_2\equiv I_2(\infty) = ( 27.2 \pm 0.4 \pm 0.2 \pm 0.3) \, 10^{-3} \; ,
\end{equation}
where the last error reflects the sensitivity of (\ref{i}) to the variation
of the condensate values.
One sees that these three numbers (\ref{i0m}) -- (\ref{i2m}) are in
good agreement with each other and with the chiral prediction (\ref{iexp}).
Substitution of our most precise result (\ref{i1m}) into (\ref{DMOpredict})
yields for the standard value of the pion charge radius quoted above:
\begin{equation}\label{alem}
\alpha_E = ( 2.64 \pm 0.36 ) \, 10^{-4} \; {{\rm fm}}^3,
\end{equation}
in good agreement
with (\ref{chiral}) and the smallest of the measured values, \cite{Babusci}.
Note that by substituting a larger value
$\langle r_{\pi}^2\rangle = 0.463 \pm 0.006 \; {{\rm fm}}^2$ \cite{gesh},
one obtains
$\alpha_E = (3.44 \pm 0.30) \, 10^{-4} \; {{\rm fm}}^3$,
about two standard deviations higher than
(\ref{chiral}).
In conclusion, we have used recent precise data
to reconstruct the difference in vector and axial-vector hadronic
spectral densities and to study the saturation of Das-Mathur-Okubo
and the first Weinberg sum rules. Two methods of improving convergence
and decreasing the errors have been used.
Within the present level of accuracy, we have found perfect consistence
between $\tau$ decay data, chiral and QCD sum rules, the standard value
of $\langle r_{\pi}^2\rangle$, the average value of $\gamma$ and the chiral
prediction for $\alpha_E$.
Helpful discussions and correspondence with R. Alemany, R. Barlow, M.Lavelle
and P. Poffenberger are gratefully acknowledged.
\newpage
| proofpile-arXiv_065-404 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Dynamics and time orientation}
We introduce a treatment of dynamics which may be readily
generalized for a subsequent application to cosmology.
\subsection{Predynamical and physical time}
Dynamics in general is time dependence of a state of a
physical system. A definition of the state may involve the
direction of time, which is not given a priori. Therefore we
introduce a predynamical time, or pretime, for short, $\tau$
as a point of the oriented real axis $T$. For the direction of
physical time, $t$, there are two possibilities: $t=\tau$
and $t=-\tau$. Definitions of the state and dynamics should
be given in terms of the pretime, and the problem of
physical time orientation is to be solved on the basis of
dynamics.
\subsection{Dynamical process}
We start with the notion of a dynamical process, which
plays a central role in dynamics. Let $\Omega$ be a set of
pure states $\omega$, $\Delta$ be a connected subset of
$T$, i.e., an interval:
\begin{equation}
\Delta=(\tau_{1},\tau_{2}),[\tau_{1},\tau_{2}),
(\tau_{1},\tau_{2}],[\tau_{1},\tau_{2}],\quad
-\infty\leq \tau_{1}<\tau_{2}\leq\infty.
\label{1.2.1}
\end{equation}
A dynamical process ${\cal P}_{\Delta}$ on ${\Delta}$
is a function
from $\Delta$ to $\Omega$:\begin{equation}
{\cal P}_{\Delta}:\Delta\rightarrow \Omega,\quad \Delta\ni\tau
\mapsto {\cal P}_{\Delta}(\tau)=\omega_{\tau}\in\Omega.
\label{1.2.2}
\end{equation}
In fact, it would suffice for ${\cal P}_{\Delta}(\tau)$
to be defined almost everywhere on $\Delta$.
A restriction and extension of a process are defined as
those of a function with regard to the fact that the
domain of the process is connected.
A left (right) prolongation of a process ${\cal P}_{\Delta}$ to
$\Delta'$, $\Delta'\ni\tau'<\tau\in\Delta$ ($\Delta\ni\tau
<\tau'\in\Delta'$), is a process ${\cal P}_{\Delta'}$, such that
there exists a process ${\cal P}_{\Delta\cup\Delta'}$ with the
restrictions ${\cal P}_{\Delta}$ and ${\cal P}_{\Delta'}$.
\subsection{Dynamics}
Dynamics on $\Delta$, ${\cal D}_{\Delta}$, is a family of
processes
on $\Delta$:
\begin{equation}
{\cal D}_{\Delta}=\left\{ {\cal P}_{\Delta} \right\}.
\label{1.3.1}
\end{equation}
A restriction and extension of a dynamics boil down to those
of corresponding processes.
\subsection{Deterministic process and deterministic
dynamics}
A deterministic process ${\cal P}_{\Delta}$ is defined as follows:
For every restriction of ${\cal P}_{\Delta}$ the only extension to
$\Delta$ is ${\cal P}_{\Delta}$ itself.
A deterministic dynamics is a family of deterministic
processes.
\subsection{Indeterministic point, process, and dynamics}
An interior isolated indeterministic point
$\tau\in{\rm int}\:\Delta$
of a process ${\cal P}_{\Delta}$ is defined as follows:
There
exists $\theta>0$, such that
(i) $(\tau-\theta,\tau+\theta)\subset\Delta$;
(ii) left prolongations of ${\cal P}_{\Delta}|_{[\tau,
\tau+\theta) }$
to ($\tau-\theta,\tau$) and right ones of ${\cal P}_{\Delta}|_
{(\tau-\theta,\tau]}$ to ($\tau,\tau+\theta$) are
deterministic processes;
(iii) cardinal numbers ${\rm card^{left}}$ and
${\rm card^{right}}$ of sets of those prolongations meet
the condition ${\rm card^{left}+card^{right}}>2$.
We assume that there are only isolated indeterministic
points.
An indeterministic process is that with indeterministic
points. An indeterministic dynamics is one with
indeterministic processes.
\subsection{Orientable dynamics and time orientation:
The future and the past}
Let $\tau\in{\rm int}\:\Delta$ be an indeterministic point
of a process ${\cal P}_{\Delta}$. We introduce five dynamics
related to the point as follows:
(i) ${\cal D}_{(\tau-\theta,\tau)}$, ${\cal D}_{(\tau,
\tau+\theta)}$ are deterministic,
${\cal D}_{(\tau-\theta,\tau)}\ni {\cal P}_{\Delta}|_{
(\tau-\theta,\tau)}$, ${\cal D}_{(\tau,\tau+\theta)}\ni
{\cal P}_{\Delta}|_{(\tau,\tau+\theta)}$;
(ii) ${\cal D}_{(\tau-\theta,\tau+\theta)}=\{{\cal P}
_{(\tau-\theta,\tau+\theta)}^{\alpha},\alpha\in {\cal A}\}$,
${\cal D}_{(\tau-\theta,\tau+\theta)}|_{(\tau-\theta,\tau)}=
{\cal D}_{(\tau-\theta,\tau)}$, ${\cal D}_{(\tau-\theta,\tau+
\theta)}|_{(\tau,\tau+\theta)}={\cal D}_{(\tau,\tau+\theta)}$;
(iii) a graph where points are elements of ${\cal D}_{(\tau-
\theta,\tau)}$ and ${\cal D}_{(\tau,\tau+\theta)}$ and lines
connecting related points are elements of ${\cal D}
_{(\tau-\theta,\tau+\theta)}$ is connected and complete, i.e.,
involves all processes associated with the indeterministic
point;
(iv) ${\cal D}_{(\tau-\theta,\tau+\theta)}^{{\rm right}\;
\alpha}=
\{{\cal P}^{\alpha'}\in{\cal D}_{(\tau-\theta,\tau+\theta)}:
{\cal P}^{\alpha'}|_{(\tau-\theta,\tau)}={\cal P}^{\alpha}|
_{(\tau-\theta,\tau)}\}$,
${\cal D}_{(\tau-\theta,\tau+\theta)}^{{\rm left}\;\alpha}=
\{{\cal P}^{\alpha'}\in{\cal D}_{(\tau-\theta,\tau+\theta)}:
{\cal P}^{\alpha'}|_{(\tau,\tau+\theta)}={\cal P}^{\alpha}|
_{(\tau,\tau+\theta)}\}$,
${\rm card^{right\;\alpha},\; card^{left\;\alpha}}$
being corresponding cardinal numbers.
An indeterministic point is symmetric (asymmetric) if
${\rm card^{right\;\alpha}=(\ne)card^{left\;\alpha}}$.
A symmetric dynamics is
that with symmetric indeterministic points only.
An indeterministic dynamics is orientable if for all
indeterministic points either
\begin{equation}
{\rm card^{right\;\alpha}}>
{\rm card^{left\;\alpha}}\quad{\rm and}\quad
{\rm card}\:D_{(\tau,\tau+\theta)}>
{\rm card}\:D_{(\tau-\theta,\tau)}
\label{1.6.1}
\end{equation}
or
\begin{equation}
{\rm card^{right\;\alpha}<card^{left\;\alpha}}\quad
{\rm and}\quad
{\rm card}\:D_{(\tau,\tau+\theta)}<
{\rm card}\:D_{(\tau-\theta,\tau)}.
\label{1.6.2}
\end{equation}
An orientable dynamics is oriented as follows: The future
corresponds to the greater of
${\rm card^{right\;\alpha}}$,
${\rm card}\:D_{(\tau,\tau+\theta)}$
and
${\rm card^{left\;\alpha}}$,
${\rm card}\:D_{(\tau-\theta,\tau)}$,
i.e.,
\begin{equation}
{\rm card^{future\;\alpha}>card^{past\;\alpha}},\quad
{\rm card}\:D_{\rm future}>{\rm card}\:D_{\rm past};
\label{1.6.3}
\end{equation}
so that the physical time is
\begin{equation}
t=+(-)\tau\quad {\rm for}\;{\rm card^{right\;\alpha}>(<)
\;card^{left\;\alpha}},\;
{\rm card}\:D_{(\tau,\tau+\theta)}>(<)\;
{\rm card}\:D_{(\tau-\theta,\tau)}.
\label{1.6.4}
\end{equation}
This defines time orientation, or the arrow of time.
\subsection{Nonpredeterminability and the question of
reconstructibility}
For an oriented dynamics we have
\begin{equation}
{\rm card^{future\;\alpha}+card^{past\;\alpha}}>2,
\label{1.7.1}
\end{equation}
\begin{equation}
{\rm card^{future\;\alpha}>card^{past\;\alpha}}\geq 1,
\label{1.7.2}
\end{equation}
so that
\begin{equation}
{\rm card^{future\;\alpha}}>1.
\label{1.7.3}
\end{equation}
This implies that the future is not predeterminate.
If
\begin{equation}
{\rm card^{past\;\alpha}}=1,
\label{1.7.4}
\end{equation}
the past is reconstructible.
In any case, the inequality (\ref{1.6.1}) implies that the
reconstructibility of the past is greater than the
predictability of the future. This feature is inherent in
an oriented dynamics.
The phenomenon of memory should be related to dynamics
orientation.
\subsection{Probabilistic dynamics}
Let $\tau$ be an indeterministic point of a process
${\cal P}_{\Delta}$.
Time evolution implies transitions from one of the sets
${\cal D}_{(\tau-\theta,\tau)}$, ${\cal D}_{(\tau,
\tau+\theta)}$
to the other: from ${\cal D}^{\rm initial}$ to ${\cal D}
^{\rm final}$.
We assume that for their cardinal numbers
\begin{equation}
{\rm card^{final}\geq card^{initial}}
\label{1.8.1}
\end{equation}
holds.
Let there exist $i\to f$ transition probabilities, or
conditional probabilities $w(f/i)$, where $i$ and $f$ are
indexes of elements of ${\cal D}^{\rm initial}$ and
${\cal D}^{\rm final}$
respectively. The probabilities meet the equation
\begin{equation}
\sum_{f}w(f/i)=1.
\label{1.8.2}
\end{equation}
Taking into account the relations
\begin{equation}
{\rm card^{initial}}=\sum_{i}1=\sum_{i}\sum_{f}w(f/i)=
\sum_{f}\sum_{i}w(f/i)\leq \sum_{f}1={\rm card^{final}},
\label{1.8.3}
\end{equation}
we put
\begin{equation}
\sum_{i}w(f/i)\leq 1.
\label{1.8.4}
\end{equation}
By Bayes formula, the a posteriori probability is
\begin{equation}
w(i/f)=\frac{w(i)w(f/i)}{\sum_{i'}w(i')w(f/i')}.
\label{1.8.5}
\end{equation}
We put for the a priori probability
\begin{equation}
w(i)={\rm const},
\label{1.8.6}
\end{equation}
then
\begin{equation}
w(i/f)=\frac{w(f/i)}{\sum_{i'}w(f/i')}\geq w(f/i).
\label{1.8.7}
\end{equation}
Thus
\begin{equation}
w(f/i)\leq w(i/f)\leq 1.
\label{1.8.8}
\end{equation}
For a symmetric dynamics,
\begin{equation}
{\rm card^{final}=card^{initial}},\quad \sum_{i}w(f/i)=1,
\label{1.8.9}
\end{equation}
so that
\begin{equation}
w(i/f)=w(f/i).
\label{1.8.10}
\end{equation}
Specifically, the increase of entropy is, on the average, the
same for the future and for the past:
\begin{equation}
-\sum_{f}w(f/i)\ln w(f/i)\approx-\sum_{i}w(i/f)\ln w(i/f).
\label{1.8.11}
\end{equation}
\subsection{Irreversibility and orientation}
It should be particularly emphasized that irreversibility
does not imply dynamics orientability and, by the same
token, time orientation.
Indeed, a reversible dynamics is defined as follows. Let
${\cal P}_{\Delta}$ be a process with a symmetric domain, i.e.,
$\Delta=(\tau_{1},\tau_{2})$ or $[\tau_{1},\tau_{2}]$.
The inverse process, ${\cal P}_{\Delta}^{\rm inv}$, is
defined by
\begin{equation}
{\cal P}_{\Delta}^{\rm inv}(\tau)={\cal P}_{\Delta}(\tau_{1}+
\tau_{2}-\tau),
\quad \tau\in\Delta.
\label{1.9.1}
\end{equation}
Let $S$ be a transformation of $\Omega$, $S:\Omega\to\Omega$.
The transformed process, $S{\cal P}_{\Delta}$, is defined by
\begin{equation}
S{\cal P}_{\Delta}(\tau)=S({\cal P}_{\Delta}(\tau)),\quad
\tau\in\Delta.
\label{1.9.2}
\end{equation}
A dynamics ${\cal D}_{\Delta'}$ is reversible if there exists
a bijection $S:\Omega\to\Omega$, such that
\begin{equation}
S\;{\rm is\;an\;involution}\;(S^{2}\;{\rm is\; identity)
\; and}\; {\cal P}
_{\Delta}\in {\cal D}_{\Delta}
\Rightarrow {\cal P}_{\Delta}^{\rm rev}\equiv S{\cal P}_
{\Delta}^{\rm inv}
\in {\cal D}_{\Delta}\; {\rm for\; all}\; \Delta\subset
\Delta'
\label{1.9.3}
\end{equation}
(rev stands for reverse).
The nonexistence of $S$ does not imply the orientability of
${\cal D}_{\Delta'}$.
Here is an example. Let a dynamical equation be of the form
\begin{equation}
\frac{d^{2}x}{d\tau^{2}}=-\alpha\frac{dx}{d\tau}.
\label{1.9.4}
\end{equation}
All dynamical processes ${\cal P}_{(-\infty,\infty)}$ are
given by
\begin{equation}
{\cal P}_{(-\infty,\infty)}(\tau)=\omega_{\tau}=\left( x(\tau),
\frac{dx(\tau)}{d\tau} \right),
\label{1.9.5}
\end{equation}
\begin{equation}
x(\tau)=c_{1}+c_{2}e^{-\alpha\tau};
\label{1.9.6}
\end{equation}
they are deterministic.
The dynamics ${\cal D}_{(-\infty,\infty)}$
is irreversible but deterministic and, therefore, not
orientable.
On the other hand, a dynamics with asymmetric indeterministic
points is irreversible---in view of inequality ${\rm card
^{right\:\alpha}\ne card^{left\:\alpha}}$. Specifically, an
orientable
dynamics is irreversible.
\section{Dynamics of standard indeterministic quantum
theory}
Let us consider the dynamics of standard, or orthodox
indeterministic quantum theory from the standpoint
developed in the previous section.
\subsection{Standard dynamical process}
In standard quantum theory, indeterminism originates from
quantum jumps. A standard indeterministic dynamical process
${\cal P}_{(-\infty,\infty)}$ may be described as follows. Let
$\tau_{k},\;k\in K=\{0,\pm 1,\pm 2,...\}$, be indeterministic
points, i.e., points of jumps. The process is denoted by
\begin{equation}
{\cal P}_{(-\infty,\infty)}^{\{j_{k},k\in K\}},\;j_{k}\in
J=\{1,2,...,
j_{\rm max}\},\;j_{\rm max}\leq \infty.
\label{2.1.1}
\end{equation}
The definition of this process reduces to that of its
restrictions to the intervals
\begin{equation}
\Delta_{k}=(\tau_{k},\tau_{k+1}),\quad k\in K,
\label{2.1.2}
\end{equation}
\begin{equation}
{\cal P}_{\Delta_{k}}^{j_{k}}\equiv {\cal P}_{(-\infty,
\infty)}^{\{j
_{k},k\in K\}}|_{\Delta_{k}}.
\label{2.1.3}
\end{equation}
The process (\ref{2.1.3}) is defined as follows:
\begin{equation}
{\cal P}_{\Delta_{k}}^{j_{k}}(\tau)=\omega^{j_{k}}_{\tau}=
(\Psi^{j_{k}}
(\tau),\cdot\Psi^{j_{k}}(\tau)),\quad \tau\in\Delta_{k},
\label{2.1.4}
\end{equation}
where $\Psi^{j_{k}}$ is a state vector,
\begin{equation}
\Psi^{j_{k}}(\tau)=U(\tau,\tau_{k})\Psi_{j_{k}},
\label{2.1.5}
\end{equation}
\begin{equation}
A_{k}\Psi_{j_{k}}=a_{j_{k}}\Psi_{j_{k}},
\label{2.1.6}
\end{equation}
where the $A_{k}$ is an observable,
and $U$ is a unitary operator
of time evolution.
This description seemingly fixes the time orientation, namely,
in view of eq.(\ref{2.1.5}),
\begin{equation}
t=\tau.
\label{2.1.7}
\end{equation}
But there is another possibility for describing the process
considered.
\subsection{Reverse description}
In place of eqs.(\ref{2.1.5}),(\ref{2.1.6}), we may put
\begin{equation}
\Psi^{j_{k}}(\tau)=U(\tau,\tau_{k+1})\Psi_{j_{k+1}}^{\rm rev},
\label{2.2.1}
\end{equation}
\begin{equation}
A_{k+1}^{\rm rev}\Psi_{j_{k+1}}^{\rm rev}=a_{j_{k+1}}
^{\rm rev}\Psi_{j_{k+1}}^{\rm rev},
\label{2.2.2}
\end{equation}
where
\begin{equation}
\Psi_{j_{k+1}}^{\rm rev}=U(\tau_{k+1},\tau_{k})\Psi_{j_{k}},
\label{2.2.3}
\end{equation}
\begin{equation}
A_{k+1}^{\rm rev}=U(\tau_{k+1},\tau_{k})A_{k}U(\tau_{k},
\tau_{k+1}),
\label{2.2.4}
\end{equation}
\begin{equation}
a_{j_{k+1}}^{\rm rev}=a_{j_{k}}.
\label{2.2.5}
\end{equation}
This description implies, in view of eq.(\ref{2.2.1}), the time
orientation
\begin{equation}
t=-\tau.
\label{2.2.6}
\end{equation}
The two descriptions are completely equivalent physically.
\subsection{Standard quantum dynamics}
We have for an indeterministic point $\tau_{k}$
\begin{equation}
{\cal D}_{k}^{\rm left}\equiv{\cal D}
_{(\tau_{k}-\theta,\tau_{k})}=
\{{\cal P}_{\Delta_{k-1}}^{j_{k-1}}|_{(\tau_{k}-\theta,
\tau_{k})}
,j_{k-1}\in J\},
{\cal D}_{k}^{\rm right}\equiv{\cal D}
_{(\tau_{k},\tau_{k}+\theta)}=
\{{\cal P}_{\Delta_{k}}^{j_{k}}|_{(\tau_{k},\tau_{k}+\theta)},
j_{k}\in J\},
\label{2.3.1}
\end{equation}
\begin{equation}
{\rm card}\:{\cal D}_{k}^{\rm left}={\rm card}\:{\cal D}_{k}
^{\rm right}=
{\rm card}\:J.
\label{2.3.2}
\end{equation}
Thus standard quantum dynamics is not orientable.
\subsection{Standard probabilistic quantum dynamics}
We have for the time orientation $t=\tau$
\begin{equation}
w(j_{k+1}/j_{k})=w_{j_{k+1}\gets j_{k}}=|(\Psi^{j_{k+1}}
(\tau_{k+1}+0),\Psi^{j_{k}}(\tau_{k+1}-0))|^{2}=
|(\Psi_{j_{k+1}},U(\tau_{k+1},\tau_{k})\Psi_{j_{k}})|^{2},
\label{2.4.1}
\end{equation}
\begin{equation}
w_{j_{k+m}\gets j_{k+m-1}\gets\ldots\gets j_{k+1}\gets j_{k}}
=w(j_{k+m}/j_{k+m-1})\cdots w(j_{k+1}/j_{k});
\label{2.4.2}
\end{equation}
for the time orientation $t=-\tau$
\begin{equation}
w(j_{k}/j_{k+1})=w_{j_{k}\gets j_{k+1}}=
|(\Psi^{j_{k}}(\tau_{k+1}-0),\Psi^{j_{k+1}}(\tau_{k+1}+0))|
^{2}=w(j_{k+1}/j_{k}),
\label{2.4.3}
\end{equation}
\begin{equation}
w_{j_{k}\gets\ldots\gets j_{k+m}}=w_{j_{k+m}\gets\ldots\gets j
_{k}}.
\label{2.4.4}
\end{equation}
The probabilities satisfy the equations
\begin{equation}
\sum_{j_{k+1}}w(j_{k+1}/j_{k})=\sum_{j_{k}}
w(j_{k+1}/j_{k})=1.
\label{2.4.5}
\end{equation}
We obtain for $t=\tau$ by Bayes formula, under the condition
$w(j_{k})={\rm const}$,
\begin{equation}
w(j_{k}/j_{k+1})=w(j_{k+1}/j_{k}),
\label{2.4.6}
\end{equation}
which coincides with eq.(\ref{2.4.3}).
\subsection{Nonorientability of standard indeterministic
quantum dynamics}
Summing up the results of this section, we conclude that
the dynamics of standard indeterministic quantum theory is
nonorientable and, by the same token, does not fix the
orientation of physical time.
\section{Dynamics of indeterministic quantum gravity}
As in standard quantum theory, in indeterministic quantum
gravity indeterminism originates from quantum jumps. But
the origin of the jumps in the latter theory differs radically
from that in the former one.
A quantum jump is the reduction of a state vector to one of
its components. In standard quantum theory, the cause of the
jump is coherence breaking between the components. In
indeterministic quantum gravity, the cause is energy difference
between the components, the difference occurring at a crossing of
energy levels.
According to the paper [2], a jump occurs at the tangency of
two levels. But level tangency imposes too severe constraints
on the occurrence of the jump. Here we introduce a scheme in
which the jump occurs at a simple crossing of two levels.
\subsection{Level crossing}
Let $\tau=0$ be the point of a crossing of levels $l=1,2$;
$P_{1\tau},P_
{2\tau}$ be the projectors for the corresponding
states in a neighborhood of the point:
\begin{equation}
P_{l\tau}\leftrightarrow \omega_{ml\tau}=
(\Psi_{l\tau},\cdot\Psi_{l\tau})
\label{3.1.1}
\end{equation}
($m$ stands for matter), and
\begin{equation}
P_{\tau}=P_{1\tau}+P_{2\tau}.
\label{3.1.2}
\end{equation}
The part of the Hamiltonian $H_{\tau}$ related to the
two levels is a projected Hamiltonian
\begin{equation}
H_{\tau}^{\rm proj}=P_{\tau}H_{\tau}P_{\tau}=
\epsilon_{1\tau}P_{1\tau}+\epsilon_{2\tau}P_
{2\tau}
\label{3.1.3}
\end{equation}
$(H_{\tau}^{\rm proj}\;{\rm is}\;\tilde H_{t}\;
{\rm in}\;[2])$. The metric tensor is
\begin{equation}
g=d\tau\otimes d\tau-h_{\tau}
\label{3.1.4}
\end{equation}
$(h_{\tau}\;{\rm is}\;\tilde g_{t}\;{\rm in}\;[2])$.
We have
\begin{equation}
H_{\tau}^{\rm proj}=H^{\rm proj}[h_{\tau},\dot h_{\tau}],
\label{3.1.5}
\end{equation}
where dot denotes the derivative with respect to the pretime
$\tau$,
\begin{equation}
H_{0}^{\rm proj}=\epsilon_{0}P_{0}=H^{\rm proj}
[h_{0},\dot h_{0}],\quad \epsilon_{0}=\epsilon_{10}=
\epsilon_{20}.
\label{3.1.6}
\end{equation}
\subsection{Creation projector and creation state}
We have in the first order in $\tau$
\begin{equation}
H_{\tau}^{\rm proj}=H_{0}^{\rm proj}+\dot H_{0}^{\rm proj}\tau
=H_{0}^{\rm proj}+\dot H^{\rm proj}[h_{0},\dot h_{0},\ddot h
_{0}]\tau.
\label{3.2.1}
\end{equation}
Furthermore,
\begin{equation}
\ddot h_{0}=\ddot h[h_{0},\dot h_{0},P^{\rm creat}],
\label{3.2.2}
\end{equation}
where $P^{\rm creat}$ is a one-dimensional projector which
creates $\ddot h_{0}$ and, by the same token, the Hamiltonian
$H_{\tau}^{\rm proj}$ eq.(\ref{3.2.1}). This creation projector
satisfies
\begin{equation}
P^{\rm creat}P_{0}=P^{\rm creat}
\label{3.2.3}
\end{equation}
and corresponds to a creation state $\omega_{m}^{\rm creat}$
belonging to a state subspace determined by $P_{0}$. For
the sake of brevity, we write
\begin{equation}
H_{\tau}^{\rm proj}=H_{0}^{\rm proj}+v\tau,
\label{3.2.4}
\end{equation}
\begin{equation}
v=\dot H^{\rm proj}[h_{0},\dot h_{0},\ddot h[h_{0},
\dot h_{0},P^{\rm creat}]].
\label{3.2.5}
\end{equation}
\subsection{Diagonal Hamiltonian}
The diagonalization of the Hamiltonian $H_{\tau}^{\rm proj}$
eq.(\ref{3.2.4}) gives
\begin{equation}
H_{\tau}^{\rm proj}=(\epsilon_{0}+\epsilon_{\tau}^{+})
P^{+}
+(\epsilon_{0}+\epsilon_{\tau}^{-})P^{-},
\label{3.3.1}
\end{equation}
where
\begin{equation}
\epsilon_{\tau}^{\pm}=\tau\frac{v_{11}+
v_{22}}{2}\pm |\tau|\sqrt{\frac{(v_{11}-v_{22})^{2}}{4}
+|v_{12}|^{2}},
\label{3.3.2}
\end{equation}
\begin{equation}
P^{\pm}\leftrightarrow\omega_{m}^{\pm}=(\Psi^{\pm},\cdot
\Psi^{\pm}),
\label{3.3.3}
\end{equation}
\begin{equation}
\Psi^{+}=e^{{\rm i}\beta}\cos\vartheta\:
\Psi_{1}+\sin\vartheta
\:\Psi_{2},\quad\Psi^{-}=-e^{{\rm i}\beta}\sin\vartheta\:
\Psi_{1}
+\cos\vartheta\:\Psi_{2},
\label{3.3.4}
\end{equation}
\begin{equation}
e^{{\rm i}\beta}=\frac{\tau}{|\tau|}\frac{v_{12}}{|v_{12}|},
\label{3.3.5}
\end{equation}
\begin{equation}
\tan\vartheta=\frac{|v_{12}|}{(\tau/|\tau|)(v_{11}-v_{22})/2
+\sqrt{(v_{11}-v_{22})^{2}/4+|v_{12}|^{2}}},
\label{3.3.6}
\end{equation}
\begin{equation}
\cot\vartheta=\frac{|v_{12}|}{(-\tau/|\tau|)(v_{11}-v_{22})
/2
+\sqrt{(v_{11}-v_{22})^{2}/4+|v_{12}|^{2}}},
\label{3.3.7}
\end{equation}
\begin{equation}
v_{ll'}=(\Psi_{l},v\Psi_{l'}),
\label{3.3.8}
\end{equation}
and $\{\Psi_{1},\Psi_{2}\}$ is a basis in the
two-dimensional Hilbert subspace ${\cal H}_{0}^{(2)}$
determined by $P_{0}$.
We have
\begin{equation}
\epsilon_{-\tau}^{\pm}=-\epsilon_{\tau}^{\mp},
\label{3.3.9}
\end{equation}
\begin{equation}
\tau\to -\tau\Rightarrow e^{{\rm i}\beta}\to-e^{{\rm i}\beta},
\tan\vartheta\leftrightarrow \cot\vartheta,\sin\vartheta
\leftrightarrow\cos\vartheta,\Psi^{+}\leftrightarrow\Psi^{-},
P^{+}\leftrightarrow P^{-}.
\label{3.3.10}
\end{equation}
\subsection{Germ projector, germ state, and germ process}
A germ projector $P^{\rm germ}$ is one of the two
projectors
$P^{\pm}$ eq.(\ref{3.3.3}); it gives rise to a germ
process ${\cal P}^{\rm germ}$---a process in a proximity
of the point $\tau=0$; ${\cal P}_{(0,\theta)}^{\rm germ\;right}$
and
${\cal P}_{(-\theta,0)}^{\rm germ\;left}$ are defined by
(i) ${\cal P}^{\rm germ}$ is deterministic;
(ii) $\lim_{\tau\to+0}{\cal P}_{(0,\theta)}^{\rm germ\;right}
(\tau)=\omega_{m}^{\rm germ\;right}\leftrightarrow P
^{\rm germ\;right}$;
(iii) $\lim_{\tau\to-0}{\cal P}_{(-\theta,0)}^{\rm germ\; left}
(\tau)=\omega_{m}^{\rm germ\;left}\leftrightarrow P
^{\rm germ\;left}$.
For $\Psi\in{\cal H}_{0}^{(2)}$ we put
\begin{equation}
\Psi=e^{{\rm i}\alpha}\cos\varphi\:\Psi_{1}+\sin
\varphi\:\Psi_{2},
\label{3.4.1}
\end{equation}
so that
\begin{equation}
P^{\rm creat}\leftrightarrow(\alpha,\varphi).
\label{3.4.2}
\end{equation}
The operator $v$ eq.(\ref{3.2.5}) is a function of
$(\alpha,\varphi)$, so that $\beta,\theta$
eqs.(\ref{3.3.5}),(\ref{3.3.6}),(\ref{3.3.7})
are such functions as well,
\begin{equation}
(\alpha,\varphi)\to(\beta,\theta).
\label{3.4.3}
\end{equation}
We assume that there exist the inverse functions,
\begin{equation}
(\beta,\theta)\to(\alpha,\varphi),
\label{3.4.4}
\end{equation}
so that there exists a bijection
\begin{equation}
(\alpha,\varphi)\leftrightarrow(\beta,\theta).
\label{3.4.5}
\end{equation}
As
\begin{equation}
P^{+}+P^{-}=P_{0},
\label{3.4.6}
\end{equation}
so that
\begin{equation}
P^{+}\leftrightarrow P^{-}
\label{3.4.7}
\end{equation}
and
\begin{equation}
P^{\rm germ}\to\{P^{+},P^{-}\}\leftrightarrow
(\beta,\theta),
\label{3.4.8}
\end{equation}
we have
\begin{equation}
{\cal P}^{\rm germ}\leftrightarrow P^{\rm germ}\to(\beta,
\theta)
\leftrightarrow (\alpha,\varphi)\leftrightarrow P^{\rm
creat}.
\label{3.4.9}
\end{equation}
Thus
\begin{equation}
{\cal P}^{\rm germ}\to P^{\rm creat}.
\label{3.4.10}
\end{equation}
\subsection{Regular crossing}
Let for $\tau<0$
\begin{equation}
\omega_{m}^{\rm creat\:left}=\omega_{m}^{\rm germ\:left}
\label{3.5.1}
\end{equation}
hold. Then it is natural to put for $\tau>0$
\begin{equation}
\omega_{m}^{\rm creat\:right}=\omega_{m}^{\rm creat\:left}
\equiv\omega_{m}^{\rm creat}
\label{3.5.2}
\end{equation}
and, in view of eqs.(\ref{3.3.9}),(\ref{3.3.10}),
\begin{equation}
\omega_{m}^{\rm germ\:right}=\omega_{m}^{\rm germ\:left}.
\label{3.5.3}
\end{equation}
Thus, there exists a germ process ${\cal P}_{(-\theta,\theta)}
^{\rm germ}$, such that ${\cal P}_{(-\theta,0)}^{\rm germ\:
left}$ and
${\cal P}_{(0,\theta)}^{\rm germ\:right}$ are its restrictions,
\begin{equation}
\lim_{\tau\to-0}{\cal P}^{\rm germ\:left}(\tau)=\lim_{\tau\to+0}
{\cal P}^{\rm germ\:right}(\tau)={\cal P}_{(-\theta,\theta)}
^{\rm germ}(0)=
\omega_{m}^{\rm creat},
\label{3.5.4}
\end{equation}
and there is no jump. The point $\tau=0$ and the process
${\cal P}_{(-\theta,\theta)}^{\rm germ}$ are deterministic.
\subsection{Singular crossing and quantum jump}
Now let
\begin{equation}
\omega_{m}^{\rm creat\:left}\ne\omega_{m}^{\rm germ\:left}.
\label{3.6.1}
\end{equation}
There is no possibility for a continuous process
${\cal P}_{(-\theta,\theta)}^{\rm germ}$. Since $\omega_{m0}$
is not
determined by the process ${\cal P}_{(-\theta,0)}^{\rm germ\:
left}$,
we put
\begin{equation}
\omega_{m0}=\lim_{\tau\to-0}\omega_{m\tau}=\omega_{m}^{\rm
germ\:left}.
\label{3.6.2}
\end{equation}
Furthermore, it is natural to put
\begin{equation}
\omega_{m}^{\rm creat\:right}=\omega_{m0},
\label{3.6.3}
\end{equation}
so that
\begin{equation}
\omega_{m}^{\rm creat\:right}=\omega_{m}^{\rm germ\:left}=
\lim_{\tau\to-0}\omega_{m\tau}=\omega_{m-0}.
\label{3.6.4}
\end{equation}
We have, by
eqs.(\ref{3.4.2}),(\ref{3.4.3}),(\ref{3.4.8}),(\ref{3.6.4}),
a quantum jump
\begin{equation}
P_{-0}^{\rm left}\leftrightarrow \omega_{m-0}
\stackrel{\rm jump}{\longrightarrow}\omega_{m+0}^{l}
\leftrightarrow
P_{+0}^{{\rm right}\:l},\quad l=\pm,
\label{3.6.5}
\end{equation}
with a transition probabilities related to it
\begin{equation}
w({\cal P}_{(0,\theta)}^{{\rm germ\:right}\:l}
/{\cal P}_{(-\theta,0)}
^{\rm germ\:left})={\rm Tr}\{P_{+0}^{{\rm right}\:l}
P_{-0}^{\rm left}\},\quad l=\pm.
\label{3.6.6}
\end{equation}
In the case of a regular crossing, eq.(\ref{3.6.6}) gives
$w=1$ or 0; this case is an idealized limiting one.
Thus, a singular crossing gives rise to a quantum jump.
\subsection{Orientability of dynamics and arrow of time}
A point which corresponds to a singular crossing is
indeterministic. We have for the cardinal numbers related to
it
\begin{equation}
{\rm card^{future\:\alpha}=card^{right\:\alpha}=2>1=
card^{left\:\alpha}=card^{past\:\alpha}},\quad \alpha=l=\pm.
\label{3.7.1}
\end{equation}
Thus the dynamics of indeterministic quantum
gravity is orientable;
it determines the arrow of time given by
\begin{equation}
t=\tau.
\label{3.7.2}
\end{equation}
We find for the probabilities of subsection 1.8
\begin{equation}
w(f/i)={\rm Tr}\{P_{+0}^{{\rm right}\:l}
P_{-0}^{\rm left}\},\quad i=1,\quad f=l=\pm,
\label{3.7.3}
\end{equation}
\begin{equation}
\sum_{f}w(f/i)={\rm Tr}\{P_{0}P
_{-0}^{\rm left}\}=
{\rm Tr}\{P_{-0}^{\rm left}\}=1,
\label{3.7.4}
\end{equation}
\begin{equation}
\sum_{i}w(f/i)=w(f/1)\leq 1,
\label{3.7.5}
\end{equation}
\begin{equation}
w(i/f)=1,
\label{3.7.6}
\end{equation}
so that
\begin{equation}
w(f/i)\leq w(i/f)=1.
\label{3.7.7}
\end{equation}
\subsection{Nonpredeterminability of the future and
reconstructibility of the past}
The dynamics developed is indeterministic, therefore the
future
is not predeterminate and may be forecasted only on a
probabilistic level. On the other hand, in view of
eqs.(\ref{3.4.10}),(\ref{3.6.4}), we have
\begin{equation}
{\cal P}_{(0,\theta)}^{\rm germ\:future}\to \omega_{m}
^{\rm create\:right}=\omega_{m}^{\rm germ\:left}
\leftrightarrow
{\cal P}_{(-\theta,0)}^{\rm germ\:past},
\label{3.8.1}
\end{equation}
so that
\begin{equation}
{\cal P}_{(0,\theta)}^{\rm germ\:future}\to {\cal P}
_{(-\theta,0)}
^{\rm germ\:past}.
\label{3.8.2}
\end{equation}
Thus, the past is reconstructible uniquely.
| proofpile-arXiv_065-405 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
proofpile-arXiv_065-406 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
|
\section{Introduction}
Mixing phenomena in neutral $B$ meson systems provide us with an
important probe of standard model flavordynamics and its
interplay with the strong interaction. As is well-known, non-zero
off-diagonal elements of the mixing matrix in the flavor basis
$\{|B_s\rangle,\, |\bar B_s\rangle\}$ are generated in second order
in the weak interaction through `box diagrams'. In the
$B_s$ system\footnote{For $B_d$ mesons there is further CKM suppression
and their lifetime difference will not be considered here.}
the off-diagonal elements obey the pattern
\begin{equation}
\left|\frac{\Gamma_{12}}{M_{12}}\right|\sim {\cal O}\!\left(\frac{m_b^2}
{m_t^2}\right).
\end{equation}
The mass and lifetime
difference between eigenstates are given by (`H' for `heavy',
`L' for `light')
\begin{equation}\label{delmex}
\Delta M_{B_s} \equiv M_H-M_L=2 \,|M_{12}|,\\[0.1cm]
\end{equation}
\begin{equation}\label{delgex}
\Delta \Gamma_{B_s} \equiv \Gamma_L-\Gamma_H = -\frac{2\,\mbox{Re}\,(
M_{12}^*\Gamma_{12})}{|M_{12}|}\approx -2 \Gamma_{12},
\end{equation}
up to very small corrections (assuming standard model CP violation).
Anticipating the magnitudes of the eigenvalues,
we have defined both $\Delta M_{B_s}$ and $\Delta\Gamma_{B_s}$
to be positive. Note that the lighter state is CP even and decays
more rapidly than the heavier state.
The lifetime difference is an interesting quantity in several
respects. Contrary to the neutral kaon system, it is calculable
by short-distance methods and directly probes the spectator quark
dynamics which generates lifetime differences among all $b$ hadrons.
If the mass difference $\Delta M_{B_s}$ turns out to be large,
the lifetime difference also tends to be large and may well be the
first direct observation of mixing for $B_s$ mesons. If
$\Delta\Gamma_{B_s}$ is sizeable, CP violation in the
$B_s$ system can be observed without
flavor-tagging \cite{DUN1}.
The following sections summarize the calculation of Ref.~\cite{BBD}
and discuss some of the implications of a non-zero $\Delta\Gamma_{B_s}$.
\section{Heavy quark expansion of $\Delta\Gamma_{B_s}$}
The mass difference is dominated by the top-quark box diagram, which
reduces to a local $\Delta B=2$ vertex on a momentum scale smaller than
$M_W$. The lifetime difference, on the other hand, is generated by
real intermediate states and is not yet local on this scale. But
the $b$ quark mass $m_b$ provides an additional short-distance
scale that leads to a large energy release
(compared to $\Lambda_{QCD}$) into the intermediate states. Thus,
at typical hadronic scales the decay is again a local process. The
lifetime difference can then be treated by the same operator product
expansion that applies to the average $B_s$ lifetime and
other $b$ hadrons \cite{BIG}.
Summing over all intermediate states, the off-diagonal element
$\Gamma_{21}$ of the decay width matrix is given by
\begin{equation}\label{g21t}
\Gamma_{21}=\frac{1}{2M_{B_s}}
\langle\bar B_s|\,\mbox{Im}\,i\!\int\!\!d^4xT\,{\cal H}_{eff}(x)
{\cal H}_{eff}(0)|B_s\rangle
\end{equation}
with
\begin{eqnarray}\label{heff}
{\cal H}_{eff}&=&\frac{G_F}{\sqrt{2}}V^*_{cb}V_{cs}
\big(C_1(\mu) (\bar b_ic_j)_{V-A}(\bar c_js_i)_{V-A}
\nonumber\\
&& +\,C_2(\mu) (\bar b_ic_i)_{V-A}(\bar c_js_j)_{V-A}\big).
\end{eqnarray}
Cabibbo suppressed and penguin operators in $
{\cal H}_{eff}$ have not been written
explicitly. In leading logarithmic approximation, the Wilson coefficients
are given by
$C_{2,1}=(C_+\pm C_-)/2$, where
\begin{equation}
C_+(\mu)\!=\!\left[\frac{\alpha_s(M_W)}{\alpha_s(\mu)}\right]^{6/23}\!\!\!\!
C_-(\mu)\!=\!\left[\frac{\alpha_s(M_W)}{\alpha_s(\mu)}\right]^{-12/23}
\end{equation}
\noindent and $\mu$ is of order $m_b$.
The heavy quark expansion expresses $\Delta\Gamma_{B_s}$ as a series in local
$\Delta B=2$-operators. In the following we keep $1/m_b$-corrections to
the leading term in the expansion. Keeping these terms fixes various
ambiguities of the leading order calculation, such as whether the
quark mass $m_b$ or meson mass $M_{B_s}$ should be used, and establishes
the reliability of the leading order expression obtained in
Ref.~\cite{LO,VOL}. Compared to the `exclusive approach'
pursued in Ref.~\cite{ALE93} that
adds the contributions to $\Delta\Gamma_{B_s}$ from individual intermediate states, the
inclusive approach is model-independent. The operator product expansion
provides a systematic approximation in $\Lambda_{QCD}/m_b$, but it
relies on the assumption of `local duality'. The accuracy to which
one should expect duality to hold is difficult to quantify, except for
models \cite{CHI96} and eventually by comparison with data. We shall
assume that duality violations will be less than 10\% for $\Delta\Gamma_{B_s}$.
To leading order in the heavy quark expansion, the long distance
contributions to $\Delta\Gamma_{B_s}$ are parameterized by the matrix elements of
two dimension six operators
\begin{eqnarray}\label{qqs}
Q &=& (\bar b_is_i)_{V-A}(\bar b_js_j)_{V-A},\\[0.1cm]
Q_S\!\!&=& (\bar b_is_i)_{S-P}(\bar b_js_j)_{S-P}
\end{eqnarray}
between a $\bar{B}_s$ and $B_s$ state. We write these matrix elements
as
\begin{eqnarray}\label{meqs}
\langle Q\rangle &=& \frac{8}{3}\, f^2_{B_s}M^2_{B_s}\,B,
\\[0.1cm]
\langle Q_S\rangle\!\! &=& - \frac{5}{6} \,f^2_{B_s}M^2_{B_s}
\frac{M^2_{B_s}}{(m_b+m_s)^2}\,B_S,
\end{eqnarray}
where $f_{B_s}$ is the $B_s$ decay constant.
The `bag' parameters $B$ and $B_S$ are defined such that
$B=B_S=1$ corresponds to factorization. $B$ also appears in the
mass difference, while $B_S$ is specific to $\Delta\Gamma_{B_s}$.
The matrix elements of these operators are not independent of $m_b$.
Their $m_b$-dependence could be extracted with the help of
heavy quark effective theory. There seems to be no gain
in doing so, since the number of independent
nonperturbative parameters is not reduced even at leading order
in $1/m_b$ and since we work to subleading order in $1/m_b$ even
more parameters would appear. The
matrix elements of the local $\Delta B=2$-operators should therefore be
computed in `full' QCD, for instance on the lattice.
Including $1/m_b$-corrections, the width difference is found to be
\begin{eqnarray}\label{tres}
\Delta\Gamma_{B_s} &=& \frac{G^2_F m^2_b}{12\pi M_{B_s}}(V^*_{cb}V_{cs})^2
\sqrt{1-4z}
\nonumber\\
&&\hspace*{-1.7cm}\,\cdot\bigg[\left((1-z)K_1+
\frac{1}{2}(1-4z)K_2\right)\langle Q
\rangle\\
&&\hspace*{-1.7cm}\,+\,(1+2z)\left(K_1-K_2\right)\langle Q_S\rangle +
\hat{\delta}_{1/m} + \hat{\delta}_{rem} \bigg],
\nonumber
\end{eqnarray}
where $z=m^2_c/m^2_b$ and
\begin{equation}\label{k1k2}
K_1=N_c C^2_1+2C_1 C_2\qquad K_2=C^2_2 .
\end{equation}
The $1/m_b$-corrections are summarized in
\begin{eqnarray}\label{oneoverm}
\hat{\delta}_{1/m} &=& (1+2 z)\Big[K_1\,(-2\langle R_1\rangle
-2\langle R_2\rangle)
+\,K_2\,(\langle R_0\rangle -2\langle \tilde{R}_1
\rangle -2\langle \tilde{R}_2\rangle)\Big]
\nonumber\\
&&-\,\frac{12 z^2}{1-4 z}\Big[K_1\,(\langle R_2\rangle
+2\langle R_3\rangle)
+\,K_2\,(\langle \tilde{R}_2\rangle
+2\langle \tilde{R}_3\rangle)\Big] .
\end{eqnarray}
The operators $R_i$ and $\tilde{R_i}$ involve derivatives on quark
fields or are proportional to the strange quark mass $m_s$, which we
count as $\Lambda_{QCD}$. For instance,
\begin{eqnarray}
\label{r0qt}
R_1\!&=&\!\frac{m_s}{m_b}(\bar b_is_i)_{S-P}(\bar b_js_j)_{S+P},\\
\label{rrt2}
R_2\!&=&\!\frac{1}{m^2_b}(\bar b_i {\overleftarrow D}_{\!\rho}
D^\rho s_i)_{V-A}( \bar b_j s_j)_{V-A}.
\label{rrt3}
\end{eqnarray}
The complete set can be found in Ref.~\cite{BBD}. Operators with gluon
fields contribute only at order $(\Lambda_{QCD}/m_b)^2$. Since the
matrix elements of the $R_i$, $\tilde{R}_i$ are $1/m_b$-suppressed
compared to those of $Q$ and $Q_S$, we estimate them in the factorization
approximation, assuming factorization at a scale of order $m_b$ (A smaller
scale would be preferable, but would require us to calculate the
anomalous dimension matrix.). Then all matrix elements can be
expressed in terms of quark masses and the $B_s$ mass and decay constant.
No new nonperturbative parameters enter at order $1/m_b$ in this
approximation.
The term $\hat{\delta}_{rem}$ denotes the contributions from
Cabibbo-suppressed decay modes and pengiun operators. They can be
estimated \cite{BBD} to be below $\pm 3\%$ and about $-5\%$,
respectively, relative to the leading order contribution. We neglect
this term in the following numerical analysis.
\section{Numerical estimate}
\begin{table}[t]
\addtolength{\arraycolsep}{-0.01cm}
\renewcommand{\arraystretch}{1.3}
\caption{\label{table1}
Dependence of $a$, $b$ and $c$ on the $b$-quark mass (in GeV)
and renormalization
scale for fixed values of all other short-distance parameters. The last
column gives $(\Delta\Gamma/\Gamma)_{B_s}$ for $B=B_S=1$ (at
given $\mu$), $f_{B_s}=210\,$MeV.}
$$
\begin{array}{|c|c||c|c|c|c|}
\hline
m_b & \mu & a & b & c &
(\Delta\Gamma/\Gamma)_{B_s} \\
\hline\hline
4.8 & m_b & 0.009 & 0.211 & -0.065 & 0.155 \\ \hline
4.6 & m_b & 0.015 & 0.239 & -0.096 & 0.158 \\ \hline
5.0 & m_b & 0.004 & 0.187 & -0.039 & 0.151 \\ \hline
4.8 & 2 m_b & 0.017 & 0.181 & -0.058 & 0.140 \\ \hline
4.8 & m_b/2 & 0.006 & 0.251 & -0.076 & 0.181 \\ \hline
\end{array}
$$
\end{table}
It is useful to separate the dependence on the long-distance parameters
$f_{B_s}$, $B$ and $B_S$ and write $(\Delta\Gamma/\Gamma)_{B_s}$ as
\begin{equation}
\left(\frac{\Delta\Gamma}{\Gamma}\right)_{B_s} =
\Big[a B + b B_S + c\Big]\left(\frac{f_{B_s}}{210\,\mbox{MeV}}\right)^2,
\end{equation}
where $c$ incorporates the explicit $1/m_b$-corrections. In the
numerical analysis, we express $\Gamma_{B_s}$ as the theoretical value of
the semileptonic width divided by the semileptonic branching ratio.
The following parameters are kept fixed: $m_b-m_c=3.4\,$GeV,
$m_s=200\,$MeV, $\Lambda^{(5)}_{LO}=200\,
$MeV, $M_{B_s}=5.37\,$GeV, $B(B_s\to X e\nu)=10.4\%$. Then $a$, $b$ and
$c$ depend only
on $m_b$ and the renormalization scale $\mu$. For some values of
$m_b$ and $\mu$, the coefficients $a$, $b$, $c$ are listed in
Tab.~\ref{table1}. For a central choice of parameters, which we take
as $m_b=4.8\,$GeV, $\mu=m_b$, $B=B_S=1$ and $f_{B_s}=210\,$MeV, we
obtain $(\Delta\Gamma/\Gamma)_{B_s} = 0.220 - 0.065 = 0.155$, where the
leading term and the $1/m_b$-correction are
separately quoted. We note that the $V-A$ `bag' parameter $B$ has a very
small coefficient and is practically negligible. The $1/m_b$-corrections
are not small and decrease the prediction for $\Delta\Gamma_{B_s}$ by about 30\%.
The largest theoretical uncertainties arise from the decay constant
$f_{B_s}$ and the second `bag' parameter $B_S$. In the large-$N_c$
limit, one has $B_S=6/5$, while estimating $B_S$ by keeping the
logarithmic dependence on $m_b$ (but not $1/m_b$-corrections as required
here for consistency) and assuming factorization at the scale $1\,$GeV
gives \cite{VOL} $B_S=0.88$. $B_S$ has never been studied by either
QCD sum rules or lattice methods. In order to estimate the range of
allowed $\Delta\Gamma_{B_s}$ conservatively, we vary $B_S=1\pm0.3$,
$f_{B_s}=(210\pm30)\,$MeV and obtain
\begin{equation}\label{dgnum1}
\left(\frac{\Delta\Gamma}{\Gamma}\right)_{B_s} = 0.16^{+0.11}_{-0.09}.
\end{equation}
This estimate could be drastically improved with improved knowledge of
$B_S$ and $f_{B_s}$.
\section{Measuring $\Delta\Gamma_{B_s}$}
In principle, both $\Gamma_L$ and $\Gamma_H$ can be measured by
following the time-dependence of flavor-specific modes \cite{DUN1},
such as $\bar{B}_s\to D_s l\nu$, given by
\begin{equation}
e^{-\Gamma_H t}+e^{-\Gamma_L t} .
\end{equation}
In practice, this is a tough measurement. Alternatively, since the
average $B_s$ lifetime is predicted~\cite{BBD} to be equal to the
$B_d$ lifetime within $1\%$, it is sufficient to measure either
$\Gamma_L$ or $\Gamma_H$.
The two-body decay $B_s\to D_s^+ D_s^-$ has a pure CP even final state
and measures $\Gamma_L$. Since $D^0$ and $D^{\pm}$ do not decay into
$\phi$ as often as $D_s$, the $\phi\phi X$ final state tags a
$B_s$-enriched $B$ meson sample, whose decay distribution informs us
about $\Gamma_L$.
A cleaner channel is $B_s\to J/\psi\phi$, which has both CP even and
CP odd contributions. These could be disentangled by studying the
angular correlations \cite{DIG96}. In practice, this might not be
necessary, as the CP even contribution is expected \cite{ALE93}
to be dominant by
more than an order of magnitude. In any case, the
inequality
\begin{equation} \Gamma_L \geq 1/\tau(B_s\to J/\psi\phi)
\end{equation}
holds. CDF \cite{MES} has fully reconstructed 58 $B_s\to J/\psi\phi$
decays from
run Ia+Ib and determined $\tau(B_s\to J/\psi\phi)=
1.34^{+0.23}_{-0.19}\pm 0.05\,$ps. Together with
$\tau(B_d)=1.54\pm 0.04\,$ps, assuming equal average $B_d$ and $B_s$
lifetimes, this yields
\begin{equation}
\left(\frac{\Delta\Gamma}{\Gamma}\right)_{B_s}\geq 0.3\pm 0.4,
\end{equation}
which still fails to be significant.
In the Tevatron run II, as well as at HERA-B, one expects $10^3-10^4$
reconstructed $J/\psi\phi$, which will give a precise measurement of
$\Delta\Gamma_{B_s}$.
\section{Implications of non-zero $\Delta\Gamma_{B_s}$}
\subsection{CKM elements}
Once $\Delta\Gamma_{B_s}$ is measured (possibly before $\Delta M_{B_s}$ is measured!),
an alternative route to obtain the mass difference could use this
measurement combined with the theoretical prediction for $(\Delta M/
\Delta\Gamma)_{B_s}$ \cite{DUN1,BP}. The decay constant $f_{B_s}$ drops
out in this ratio, as well as the dependence on CKM elements, since
$|(V_{cb}V_{cs})/(V_{ts}V_{tb})|^2= 1\pm 0.03$ by CKM unitarity. However,
the dependence on long-distance matrix elements does not
cancel even at leading order in $1/m_b$ and the prediction depends on
the ratio of `bag' parameters $B_S/B$, which is not very well-known
presently. We obtain $\Delta\Gamma/\Delta M=(5.6\pm 2.6)\cdot 10^{-3}$,
where the largest error ($\pm 2.3$) arises from varying $B_S/B$ between
0.7 and 1.3.
When lattice measurements yield an accurate value of $B_S/B$ as well
as control over the $SU(3)$ flavor-symmetry breaking in $B f_B^2$, the
above indirect determination of $\Delta M_{B_s}$ in conjunction with
the measured mass difference in the $B_d$ system provides an
alternative way of determining the CKM ratio $|V_{ts}/V_{td}|$,
especially if the latter is around its largest currently
allowed value. In contrast, the ratio
$\Gamma(B\to K^*\gamma)/\Gamma(B\to\{\varrho,\omega\}\gamma)$
is best suited for extracting small $|V_{ts}/V_{td}|$ ratios,
provided the long distance effects can be sufficiently well
understood.
\subsection{CP violation}
The existence of a non-zero $\Delta\Gamma_{B_s}$ allows the observation of
mixing-induced CP asymmetries without tagging the initial $B_s$ or
$\bar{B}_s$ \cite{DUN1,DF}. These measurements are difficult, but the
gain in statistics, when tagging is obviated, makes them worthwhile
to be considered. The mass difference drops out in the time
dependence of untagged
samples, which is given by
\begin{equation}
A_+ (e^{-\Gamma_L t}+e^{-\Gamma_H t})
+ A_-(e^{-\Gamma_L t}-e^{-\Gamma_H t}).
\end{equation}
$A_-$ carries CKM phase information even in the absence of direct
CP violation.
In combination with an analysis of angular distributions, a measurement
of the CKM angle $\gamma$ from exclusive $B_s$ decays governed by the
$\bar{b}\to c\bar{c}\bar{s}$ or $\bar{b}\to\bar{c}u\bar{s}$ transition
can be considered \cite{DF}.
\section*{Acknowledgements}
I am grateful to my collaborators G.~Buchalla and I.~Dunietz for
sharing their insights into problems related to this work with me.
\section*{References}
| proofpile-arXiv_065-407 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
Since the early days of quantum field theory (QFT) 1+1 dimensional models
have attracted much attention. They have been extremely valuable to
develop general ideas and intuition about the structure of QFT.
The eldest and perhaps most popular of these 1+1 D models bear the names of
Thirring \cite{Thirring} -- Dirac fermions interacting with a local
current-current interaction -- and Schwinger \cite{Schwinger} -- quantum
electrodynamics with fermions. The models originated
in particle physics and
therefore, in order to have Lorentz invariance, were considered mainly on
infinite space ${\bf R}$ (see e.g.\ \cite{CHu,CW,CRW} and references therein).
Then one has to deal with infrared (infinite space volume) divergences in
addition to singularities coming from the ultraviolet (short distances).
In case the fermions are massless, both
models are soluble \cite{Schwinger,Klaiber,swieca} and a very detailed
picture of their properties can be obtained. Another related model
originated from solid state physics and is due to Luttinger
\cite{Luttinger} -- massless Dirac fermions on spacetime $S^1\times{\bf R}$
interacting with a non-local current-current interaction
(Lorentz invariance is nothing natural to ask for in solid
state physics).
The Luttinger
model shows that an interacting fermion system in one space dimension
need not behave qualitatively similar to free fermions
but rather has properties similar to a boson system. Such behaviour
is generic for 1+1 D interacting fermion models and is denoted as Luttinger
liquid in solid state physics, in contrast to Landau liquids common in 3+1 D.
To consider the Luttinger model on compact space has the enormous technical
advantage that infra red
(IR) problems are absent, and one can concentrate on the
short distance (UV) properties which are rather simple due
to the non-locality of the interaction. In fact, this allows
a construction of the interacting model on the Fock space of {\em free}
fermions \cite{MattisLieb,HSU,CH}
and one directly can make
use of mathematical results from the representation theory of the affine
Kac-Moody algebras. Such an approach was recently given for QCD with
massless fermions \cite{LS3}.
As shown by Manton \cite{Manton}, the Schwinger model on compact space $S^1$
allows a
complete understanding of the UV divergences and anomalies and their
intriguing interplay with gauge invariance and vacuum structure.
In the present paper we study the extension of the Luttinger model
obtained by coupling it to a dynamical electromagnetic field. For
vanishing Luttinger (4-point) interaction our model therefore reduces to
the Schwinger model as studied by Manton \cite{Manton}, and for vanishing
electric charge to the Luttinger model \cite{MattisLieb}.
Since our approach is in Minkowski space and provides a
direct construction of the field-- and observable algebras of the model on
a physical Hilbert space, it is conceptually quite different from the path
integral approach, and we believe it adds to the physical understanding of
these models.
The plan of the paper is as follows. In Section 2 the construction of
the model is given. To fix notation, we first summarize the classical
Hamiltonian formalism. We then construct the physical Hilbert
space and discuss the
non-trivial implications of anomalies (Schwinger terms) and gauge
invariance. In Section 3 the model is solved by bosonization, and a
method for calculating all Green functions is explained. As an example the
equal time 2-point functions are given. In Section 4 we comment on
regularization and renormalization in our setting. We discuss the limit to
the Thirring-Schwinger model where the 4-point interaction becomes local
and space infinite. We end with a short summary in Section 5. A
summary of the mathematical results needed and some details of calculations
are deferred to the appendix.
\section{Constructing the model}
\subsection{Notation}
Spacetime is the cylinder with $x=x^1\in \Lambda\equiv
[-L/2,L/2]$ the spatial coordinate and $t=x^0\in{\bf R}$ time. We have one
Dirac Fermion field $\psi_{\sigma}(\vec{x})$ and one Photon field
$A_\nu(\vec{x})$ (here and in the following,
$\sigma,\sigma'\in\{+,-\}$ are spin indices,
$\mu,\nu\in\{0,1\}$ are spacetime indices, and
$\vec{x}=(t,x),\vec{y}=(t',y)$ are spacetime
arguments).
The action defining the Luttinger-Schwinger model
is\footnote{unless otherwise stated,
repeated indices are summed over throughout the paper}$^,$
\footnote{$\partial_\nu\equiv\partial/\partial x^\nu$;
our metric tensor is $g_{\mu\nu}=diag(1,-1)$}
\begin{eqnarray}
\label{1}
{\cal S} = \int d^2\vec{x}\left(-\frac{1}{4}F_{\mu\nu}(\vec x)F^{\mu\nu}(\vec x) +
\bar\psi(\vec x) \gamma^\nu\left(-{\rm i}\partial_\nu +
e A_\nu(\vec x)\right)\psi(\vec x)\right) \nonumber \\ \nopagebreak
- \int d^2\vec{x}\int d^2\vec{y}\, j_\mu (\vec x)v(\vec x
-\vec y) j^\mu(\vec y)
\end{eqnarray}
where $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu
A_\mu$
and $\gamma^\nu\equiv (\gamma^\nu)_{\sigma\sigma'}$ are Dirac matrices
which we take as $\gamma^0=\sigma_1$ and
$\gamma^1={\rm i}\sigma_2$, and $\gamma_5=-\gamma^0\gamma^1=\sigma_3$
($\sigma_i$ are Pauli spin matrices).
As usual, the fermion currents are
$
j_\nu = \bar\psi\gamma_\nu\psi,
$
and we assume the 4--point interaction to be instantaneous (local in time)
\begin{equation}
v(\vec x-\vec y) = \delta(t-t')V(x-y)
\end{equation}
where the interaction potential is parity
invariant, $V(x)=V(-x)$.
As in case of the Luttinger model \cite{HSU} we will also have to assume
that this potential is not `too strong', or more precisely that the Fourier
coefficients
\alpheqn
\begin{equation}
\label{condition}
W_k = \frac{1}{8\pi}\int_{\Lam}{\rm d}{x}\, V(x)\ee{-{\rm i} kx} = W_{-k}= W_k^*,\quad
k\frac{L}{2\pi} \in{\bf Z}
\end{equation}
of the potential obey the conditions
\begin{equation}
\label{condition1}
-1-\frac{e^2}{\pi k^2} < W_k < 1\quad\forall k\quad
\mbox{ and }\quad \sum_k |kW_k^2|<\infty .
\end{equation}
\reseteqn
{}From the action \Ref{1} we obtain the canonical momenta $\Pi_{A_0(x)}
\simeq 0$, $\Pi_{A_1(x)} = F_{01}(x) = E(x)$ etc.\
(here and in the following, we set $t=0$ and make
explicit the dependence on the spatial coordinate only) resulting in the
Hamiltonian ($\psi^*\equiv \bar\psi\gamma^0$)
\begin{eqnarray}
\label{4}
H=\int_{\Lam} {\rm d}{x} \left( \frac{1}{2} E(x)^2 +
\psi^*(x)\gamma_5\left(-{\rm i}\partial_1 + e A_1(x)\right)
\psi(x)\right) + 4\int_{\Lam} {\rm d}{x} {\rm d}{y} \, \rho^+(x)V(x-y)\rho^-(y),
\end{eqnarray}
and the Gauss' law
\begin{equation}
\label{5}
G(x) = -\partial_1 E(x) + e \rho(x) \simeq 0 \, .
\end{equation}
We introduced chiral fermion currents
\begin{equation}
\rho^\pm(x) = \psi^*(x)\frac{1}{2}(1\pm\gamma_5)\psi(x) \label{chiralcu}
\end{equation}
so that fermion charge-- and momentum density $\rho=j^0$ and $j=j^1$
can be written as
\begin{eqnarray}
\rho(x) &=& \rho^+(x) + \rho^-(x)\nonumber \\ \nopagebreak
j(x) &=& \rho^+(x) - \rho^-(x).
\end{eqnarray}
\subsection{Observables}
\label{obs}
The observables of the model are all gauge invariant operators.
They leave invariant physical states. The ground state
expectation values of these operators are the Green functions we are interested
in. For later reference we write down the action of static gauge
transformations i.e.\ differentiable maps $\Lambda\to{\rm U}(1), x\mapsto \ee{{\rm i}
\alpha(x)}$,
\begin{eqnarray}
\label{gaugetrafo}
\psi_\sigma(x)&\to& \ee{{\rm i} \alpha(x)}\psi_\sigma(x) \nonumber \\ \nopagebreak
A_1(x) &\to& A_1(x) - \frac{1}{e}\frac{\partial \alpha(x)}{\partial x} \\
E(x) &\to& E(x)\, . \nonumber
\end{eqnarray}
These obviously leave our Hamiltonian and Gauss' law invariant. We note
that every gauge transformation can be decomposed into a {\em small} and a
{\em large} gauge transformation,
$
\alpha(x)=\alpha_{small}(x) + \alpha_{large}(x),
$
where
\begin{equation}
\alpha_{large}(x) = n\frac{2\pi x}{L} \quad (n\in{\bf Z}), \quad
\alpha_{small}\left(-\frac{L}{2}\right)=
\alpha_{small}\left(\frac{L}{2}\right)
\end{equation}
with $n=\frac{\alpha(L/2)-\alpha(-L/2)}{2\pi L}$. The large gauge
transformations correspond to $\Pi_1(S^1)={\bf Z}$ and play an important role in
the following, as expected from general arguments \cite{Jackiw}.
It is important to note that Gauss' law \Ref{5} requires physical states
only to be invariant under small (but {\em not} under large) gauge
transformations.
All gauge invariant objects which one can construct
from $A_1(x)$ (at fixed time) are functions of
\begin{equation}
Y= \frac{1}{2\pi} \int_{\Lambda}{\rm d} y A_1(y) \, .
\end{equation}
In fact, $Y$ above is only invariant with respect to small gauge
transformations and changes by multiples of $1/ e$ under the large ones.
Thus the quantity which is invariant under all gauge transformations is
$\ee{{\rm i} 2\pi e Y}$ which is equal to the Wilson line (holonomy)
\begin{equation}
\label{Wilson}
W[A_1] = \ee{{\rm i} e\int_{\Lambda}{\rm d} y A_1(y)} \: .
\end{equation}
The fermion fields are not gauge invariant, but by attaching
parallel transporters to them one obtains field operators
\begin{equation}
\label{chi}
\chi_\sigma(x) = \ee{{\rm i} e\int_r^x{\rm d} y A_1(y) }\psi_\sigma(x)\, ,
\quad r\in\Lambda
\end{equation}
which obviously are invariant under all (small and large) gauge
transformations \Ref{gaugetrafo} with $\alpha(r)=0$; $r$ is a spatial
point which we can choose arbitrarily. Note that these fields also obey
CAR but are {\em not} antiperiodic: they obey $\chi_\sigma(L/2)=
-W[A_1]\chi_\sigma(-L/2)$ where $W[A_1]$ is the Wilson line above.
Bilinears of these operators are the meson operators
$$
M_{\sigma\sigma'}(x,y) = \chi^*_\sigma(x)\chi_{\sigma'}(y) \, .
$$
These are invariant under all static gauge
transformations and thus can be used as building blocks of the Green
functions we are interested in.
\subsection{The quantum model}
In the following we find it convenient to work in Fourier space.
We introduce the following useful notation. Fourier space for even
(periodic) functions is
\alpheqn
\begin{equation}
\label{a}
\Lambda^*\equiv \left\{\left. k=\frac{2\pi}{L} n \right| n\in{\bf Z}\right\}\quad \, .
\end{equation}
As we use fermions with odd (anti--periodic) boundary conditions we
also need
\begin{equation}
\Lambda^*_{odd}\equiv \left\{\left. k=\frac{2\pi}{L} \left(n+\frac{1}{2}\right)
\right| n\in{\bf Z}\right\}.
\end{equation}
{}For functions $\hat f$ on Fourier space we write
\begin{equation}
\hat{\int}_{\dLam} \hat{\rm d} k \hat f(k) \equiv \sum_{k\in\Lambda^*} \frac{2\pi}{L} \hat f(k)
\end{equation}
\reseteqn
and similarly for $\Lambda^*_{odd}$ (we will use the same symbols $\delta$ and
$\hat\delta$ also in the latter case). Then the appropriate
$\delta$-function satisfying $\hat{\int}_{\dLam}\hat{\rm d}
q\, \hat{\delta}(k-q)\hat f(q)=\hat f(k)$ is
$\hat{\delta}(k-q) \equiv \frac{L}{2\pi} \delta_{k,q}$.
{}For the Fourier transformed operators we use the following conventions,
\alpheqn
\begin{equation}
\label{10a}
\hat\psi^{}_{\sigma}(q) =
\int_{\Lam} \frac{{\rm d}{x}}{\sqrt{2\pi}} \psi^{}_{\sigma}(x)
\ee{-{\rm i} qx},\quad
\hat\psi^{*}_{\sigma}(q)=\hat\psi^{}_{\sigma}(q)^*
\quad (q\in\Lambda^*_{odd})
\end{equation}
(as mentioned, we use anti--periodic boundary conditions for the fermions),
\begin{equation}
\label{A1}
\hat A_1(k) = \int_{\Lam} \frac{{\rm d}{x}}{2\pi}
A_1(x) \ee{-{\rm i} kx} \quad (k\in\Lambda^*)
\end{equation}
and in the other cases
\begin{equation}
\label{other}
\hat Y(k) = \int_{\Lam} {\rm d}{x}\,
Y(x) \ee{-{\rm i} kx}\quad (k\in\Lambda^*)\quad \mbox{ for $Y=E,\rho^{\pm},\rho,j,V$}
\end{equation}
\reseteqn
Following \cite{HSU} we also find it convenient to introduce $W_k=\hat
V(k)/8\pi$ (cf.\ \Ref{condition}). With that the non-trivial C(A)CR in
Fourier space are
\begin{eqnarray}
\label{fcacr}
\ccr{\hat A_1(p)}{\hat E(k)} &=& {\rm i}\hat{\delta}(k+p) \nonumber \\ \nopagebreak
\car{\hat\psi_{\sigma}(q)}{\hat\psi^*_{\sigma'}(q')}
&=& \delta_{\sigma\sigma'}\hat{\delta}(q-q') .
\end{eqnarray}
The essential physical requirement determining the construction of the
model and implying a non-trivial quantum structure is positivity of
the Hamiltonian on the physical Hilbert space.
It is well-known that it forces one to use a
non-trivial representation of the field operators of the model. The
essential simplification in (1+1) (and not possible in higher)
dimensions is that one can use a quasi-free representation for the
fermion field operators corresponding to ``filling up the Dirac sea''
associated with the {\em free} fermion Hamiltonian, and for the
photon operators one can use a naive boson representation.
This will be verified for our model
for the class of potentials $V$ obeying
(\ref{condition},b).
So the full Hilbert space of the model is ${\cal H} = {\cal H}_{\rm
Photon}\otimes {\cal H}_{\rm Fermion}$.
For ${\cal H}_{\rm Photon}$ we take the boson Fock space
generated by boson field operators $b^{*}(k)$ obeying CCR
\alpheqn
\begin{equation}
\ccr{b(k)}{b^*(p)} = \hat{\delta}(k-q)\quad\mbox{etc.}
\end{equation}
and a vacuum\footnote{Note that the term ``vacuum'' here and in the
following does {\em not} mean that
this state has anything to do with the ground state of the model; it is
just one convenient state from which all other states in the Hilbert space
can be generated by applying the field operators.} $\Omega_{\rm P}$ such that
\begin{equation}
b(k)\Omega_{\rm P} = 0 \quad \forall k\in\Lambda^* .
\end{equation}
\reseteqn
We then set
\alpheqn
\begin{equation}
\label{photon}
\hat A_1(k) = \frac{1}{s}\left(b(k) + b^*(k)\right)
\quad \hat E(k) = -\frac{{\rm i} s}{2}\left(b(k)-b^*(k)\right)
\end{equation}\reseteqn
where
$
s^4 = \pi e^2
$
(the reason for choosing this factor $s$ will become clear later). We
will use below normal ordering $\xx\cdots\xx$ of bilinears in the Photon
field operators with respect to the vacuum $\Omega_{\rm P}$, for example
$\xx b(k)b^*(p)\xx\, = b^*(p)b(k)$.
{}For ${\cal H}_{\rm Fermion}$ we take the Fermion Fock space with vacuum
$\Omega_{\rm F}$ such that
\begin{eqnarray}
\label{11}
\frac{1}{2}(1 \pm \gamma_5) \hat\psi(\pm q) \,\Omega_{\rm F} &=&
\frac{1}{2}(1 \mp \gamma_5) \hat\psi^*(\mp q) \,\Omega_{\rm F} =0
\quad \forall q > 0\, .
\end{eqnarray}
The presence of the Dirac sea requires
normal-ordering $\normal{\cdots}$ of the Fermion bilinears such as
$\hat H_0=\hat{\int}_{\dLamF}\hat{\rm d} q \normal{q\,\hat\psi^*(q)\gamma_5\hat
\psi(q)}$ and $\hat \rho_\pm$ (\ref{chiralcu}).
This modifies their naive commutator relations following
from the CAR as Schwinger terms show up \cite{GLrev,CR,A}.
In our case, the relevant commutators are:
\begin{eqnarray}
\ccr{\hat\rho^\pm(k)}{\hat\rho^{\pm}(p)}&=&
\pm k\hat{\delta}(k+p)\,, \nonumber\\
\ccr{\hat\rho^\pm(k)}{\hat\rho^{\mp}(p)} &=& 0 \label{12} \, , \\
\ccr{\hat H_0}{\hat\rho^\pm(k)} &=& \pm k
\hat\rho^\pm(k) \nonumber \, .
\end{eqnarray}
We note that
\begin{equation}
\label{vacF}
\hat\rho^+(k)\Omega_{{\rm F}} = \hat\rho^-(-k)\Omega_{{\rm F}} =
0\quad\forall k>0
\end{equation}
which together with \Ref{12} shows that the $\hat\rho^+(k)$ (resp.
$\hat\rho^-(k)$) give a highest (resp. lowest) weight representation of
the Heisenberg algebra.
We can now write the Gauss' law operators in Fourier space as
\begin{equation}
\label{14}
\hat G(k) = -{\rm i} k\hat E(k) + e\hat\rho(k),
\end{equation}
so eqs. \Ref{12} imply
\[
\ccr{\hat G(k)}{\hat\rho^\pm(p)} =
\pm k e\hat{\delta}(k+p).
\]
Due to the presence of the Schwinger terms, these Fermion
currents no longer commute with the
Gauss' law generators, hence they are not gauge invariant
and no observables of the model.
To obtain Fermion currents obeying the classical relations (without
Schwinger terms), we note that $
\ccr{\hat G(k)}{\hat A_1(p)} =
k\hat{\delta}(k+p),$
hence the operators
\begin{equation}
\label{16}
\tilde\rho^\pm(k) \equiv \hat\rho^\pm(k)
\pm e\hat A_1(k)
\end{equation}
commute with the Gauss law generators and are thus the observables of the
model corresponding to the chiral Fermion currents on the quantum
level.
Recalling the normalization is only unique up to finite terms, it is
natural to regard the $\tilde \rho^\pm(k)$ as the fermion
currents obtained by a {\em gauge covariant normal ordering}
preserving the classical transformation properties under gauge
transformations. Indeed, these currents can be shown to be
identical to those obtained by the gauge invariant point splitting method.
Similarly, the naive Hamiltonian $\hat H=\hat H_1+\hat H_2$,
\begin{eqnarray*}
\hat H_1 &=& \hat H_0 +\hat{\int}_{\dLam}\hat{\rm d} k \xx \left(
\frac{1}{4\pi}\hat E(k)\hat E(-k) + e\hat A_1(k)\hat j(-k)\right)\xx \\
\hat H_2 &=& \hat{\int}_{\dLam}\hat{\rm d} k \, \hat\rho^+(k)W_k \hat\rho^-(-k)
\end{eqnarray*}
is not gauge invariant: $\hat H_1$ -- which is the naive Hamiltonian of the
Schwinger model -- obeys
\[
\ccr{\hat G(k)}{\hat H_1} = 2ke^2 \hat A_1(k)
\]
and therefore becomes gauge invariant only after adding a photon mass
term \cite{Manton}
\[
\hat{\int}_{\dLam}\hat{\rm d} k \, e^2 \hat A_1(k)\hat A_1(-k)
\]
(note that in position space this mass term has the usual form
$\frac{e^2}{2\pi}\int_{\Lam}{\rm d}{x}A_1(x)^2$, i.e.\ the photon mass--squared is
$e^2/\pi$). Also the Luttinger--interaction term $\hat H_2$
becomes gauge invariant only if one replaces the non--gauge
invariant currents $\hat\rho^\pm$ by the gauge invariant $\tilde\rho^\pm$
ones.
Thus we obtain the gauge invariant Hamiltonian of the
Luttinger--Schwinger model as follows,
\begin{eqnarray}
H= \hat H_0 + \hat{\int}_{\dLam}\hat{\rm d} k \xx\left( \frac{1}{4\pi}
\hat E(k) \hat E(-k) + e\hat A_1(k)
\hat j(-k) + e^2 \hat A_1(k)\hat A_1(-k)
\right.\nonumber \\ \nopagebreak\left.\frac{}{}\!\!\!+
\left[\hat\rho^+(k)+e\hat A_1(k)\right]
W_k\left[\hat\rho^-(-k)-e\hat A_1(-k)\right]\right)\xx.
\end{eqnarray}
We can now explain the choice (\ref{photon},b) for the representation of the
Photon field: the factor $s$ is determined such that the free Photon
Hamiltonian
is equal to $\hat{\int}_{\dLam}\hat{\rm d} k \sqrt{\frac{e^2}{\pi}}\, b^*(k)b(k)$.
\subsection{Bosonization}
Kronig's identity\footnote{in the modern literature this is often referred
to as (special case of the) Sugawara construction} allows us to rewrite the
free Hamiltonian as
$
\hat H_0 = \frac{1}{2}\hat{\int}_{\dLam}\hat{\rm d} k \, \xx \left(
\hat\rho^+(k)\hat\rho^+(-k) \right.$ $\left.+ \hat\rho^-(k)
\hat\rho^-(-k) \right) \xx
$
(cf.\ Appendix A for the precise definition of normal ordering;
for simplicity of notation we do not distinguish the normal
ordering symbol for the photon fields and the fermion currents). With
that, it follows from eq.\ \Ref{16} that
\begin{equation}
\label{20}
H = \hat{\int}_{\dLam}\hat{\rm d} k\, \xx \left(\frac{1}{2}\left(
\tilde\rho^+(k)\tilde\rho^+(-k) + \tilde\rho^-(k)
\tilde\rho^-(-k) \right) + \frac{1}{4\pi}\hat E(k) \hat E(-k) +
\tilde\rho^+(k) W_k \tilde\rho^-(-k)\right)\xx
\end{equation}
which is now explicitly gauge invariant.
\section{Solution of the model}
\subsection{Gauge Fixing}
The only gauge invariant degree of freedom of the Photon field at fixed
time is the holonomy $\int_\Lambda{\rm d}{x} A_1(x)$ and one can gauge away
all Fourier modes $\hat A_1(k)$ of the gauge field except the one for
$k=0$. Thus we can impose the gauge condition
\begin{equation}
\label{22}
\hat A_1(k) = \delta_{k,0} Y \makebox{, \qquad } A_1(x)=\frac{2\pi}{L}Y
\end{equation}
and solve the Gauss' law $\hat G(k)\simeq 0$ (cf.\ eq.\ \Ref{14}) as
\begin{equation}
\hat E(k) \simeq
\frac{e\hat\rho(k)}{{\rm i} k} \quad {\mbox{for $k\neq 0$}}.
\end{equation}
This determines all components of $E$ except those conjugate to
$Y$: $\hat E(0) = \frac{L}{2\pi} \frac{\partial}{{\rm i} \partial Y}$. After
that we are left with the ($k=0$)--component of Gauss' law, {\em viz.}
\begin{equation}
eQ_0\simeq 0,\quad Q_0 = \hat\rho(0) = \hat\rho^+ (0) + \hat\rho^- (0)\, .
\end{equation}
Inserting this into \Ref{20}, gives the Hamiltonian of the model on the physical
Hilbert space ${\cal H}_{\rm phys}={\cal L}^2({\bf R},{\rm d}Y)\otimes {\cal H}'_{\rm
Fermion}$
(where ${\cal H}'_{\rm Fermion}$ is the zero charge sector of the
fermionic Fock space):
\begin{eqnarray}
\label{24}
H = -\frac{L}{8\pi^2} \frac{\partial^2}{\partial Y^2} +
\frac{\pi}{L}\left(\left(\hat\rho^+(0) + eY\right)^2 +
\left(\hat\rho^-(0) - eY\right)^2 + \left(\hat\rho^+(0) +
eY\right) 2 W_0 \left(\hat\rho^-(0) - eY\right) \right) +
\nonumber \\ \nopagebreak
\hat{\int}_{\dLam\backslash \{0\}}\hat{\rm d} k\, \xx \left( \frac{e^2}{4\pi k^2} \hat\rho(k)\hat\rho(-k)
+ \frac{1}{2}\left( \hat\rho^+(-k)\hat\rho^+(k) +
\hat\rho^-(k)\hat\rho^-(-k) \right) +
\hat\rho^+(k)W_k \hat\rho^-(-k)\right)\xx.
\end{eqnarray}
\subsection{Diagonalization of the Hamiltonian}
\label{zeromode}
{}Following \cite{HSU} we now write
\begin{equation}
H = \frac{2\pi}{L}\sum_{k\geq 0} h_k.
\end{equation}
Introducing boson creation-- and annihilation operators
\alpheqn
\begin{equation}
\label{crho}
c(k) = \left\{\bma{cc} \frac{1}{\sqrt{|k|}}\hat\rho^+(k) & \mbox{ for
$k>0$}\\
\frac{1}{\sqrt{|k|}}\hat\rho^-(k) & \mbox{ for $k<0$} \end{array}\right.
\end{equation}
obeying usual CCR
\begin{equation}
\ccr{c(k)}{c^*(p)} = \hat{\delta}(k-p)\quad \mbox{etc.}.
\end{equation}
We then get for $h_{k>0}$
\begin{equation}
h_k =\left(k + \frac{e^2}{2\pi k}\right)\left(c^*(k)c(k)+c^*(-k)c(-k)
\right) + \left( kW_k +
\frac{e^2}{2\pi k}\right)\left(c^*(k)c^*(-k)+c(k)c(-k) \right). \end{equation}
\reseteqn
{}For $k=0$ we introduce the quantum mechanical variables
\alpheqn
\begin{eqnarray}
\label{qmv}
P &=& \left( \hat\rho^+(0) -\hat\rho^-(0) + 2eY \right)\, , \nonumber \\ \nopagebreak
X &=& {\rm i}\frac{L}{2\pi}\frac{1}{2e} \frac{\partial}{\partial Y}
\end{eqnarray}
obeying Heisenberg relations, $\ccr{P}{X}=-{\rm i} L/2\pi$,
which allow us
to write $h_0$ as Hamiltonian of a harmonic oscillator,
\begin{equation}
\label{zeromom0}
h_0= \frac{e^2}{\pi} X^2 + \frac{1}{4}(1-W_0)\, P^2 + \frac{1}{4} (1+W_0)\, Q_0^2
-\frac{1}{2}\sqrt{\frac{e^2}{\pi}}\frac{L}{2\pi}
\end{equation}
\reseteqn
(the last term stems from normal ordering $\,\xx\cdots\xx\,$
and is irrelevant for the following).
We can now solve the model by diagonalizing its decoupled
Fourier modes $h_k$ separately, with the help of
a boson Bogoliubov transformation preserving the CCR,
\begin{equation}
\label{BT}
C(k) = \cosh(\lambda_k) c(k) + \sinh(\lambda_k) c^*(-k)
\end{equation}
where $\lambda_k=\lambda_{-k}$. This leads to
\alpheqn
\begin{equation}
h_k = \omega_k\left(C^*(k)C(k) + C^*(-k)C(-k)\right) -2\eta_k
\frac{L}{2\pi}
\end{equation}
if we choose
\begin{equation}
\tanh(2\lambda_k) = \frac{2\pi k^2 W_k + e^2}{2\pi k^2 + e^2} \label{th}\, .
\end{equation}
Then
\begin{equation}
\omega_k^2 = k^2(1-W_k^2) + \frac{e^2}{\pi}(1-W_k)
\end{equation}
and
\begin{equation}
\eta_k = \frac{1}{2}\left(|k| + \frac{e^2}{2\pi|k|} - \omega_k \right)
\quad (k\neq 0) .
\end{equation}
\reseteqn
The zero--momentum piece $h_0$ is just
a harmonic oscillator and can be written as
\alpheqn
\begin{equation}
\label{zeromom}
h_0 = \omega_0 C^*(0) C(0) + \frac{1}{4} (1+W_0)\, Q_0^2 - \eta_0\frac{L}{2\pi}
\end{equation}
with
\begin{equation}
\label{Cnull}
C(0) = \frac{1}{\sqrt 2}\left(rX + \frac{1}{r}{\rm i} P\right),\quad
r^4=\frac{e^2}{\pi}\frac{4}{1-W_0},
\end{equation}
energy--squared
\begin{equation}
\omega_0^2 = \frac{e^2}{\pi}(1- W_0)
\end{equation}
and zero point energy
\begin{equation}
\eta_0 =\frac{1}{2}\left(\sqrt{\frac{e^2}{\pi}} -
\sqrt{\frac{e^2}{\pi}(1-W_0)}\, \right).
\end{equation}
\reseteqn
Thus we get the Hamiltonian in the following form
\alpheqn
\begin{equation}
H=\hat{\int}_{\dLam}\hat{\rm d} k \omega_k C^*(k)C(k) -L E_0
\end{equation}
with the ground state energy density given by
\begin{equation}
E_0=\frac{1}{2\pi}\hat{\int}_{\dLam}\hat{\rm d} k\, \eta_k.
\end{equation}
\reseteqn
(Note that for large $|k|$,
$
\eta_k = \frac{1}{2}\left( \frac{1}{2}|k W_k^2| + \frac{e^2}{2\pi
|k|}W_k\right)\left(1+{\cal O}\left(\frac{1}{|k|}\right)\right),
$
hence $E_0$ is finite due to our assumptions \Ref{condition1} on
the potential.)
We now construct the unitary operator ${\cal U}$ implementing the
Bogoliubov transformation \Ref{BT}, i.e.\
\begin{equation}
C(k) = {\cal U} c(k){\cal U}^* \quad \forall k\in\Lambda^* .
\end{equation} It is easy to see that operators ${\cal U}_k$ satisfying $C(\pm k)
= {\cal U}_k
c(\pm k){\cal U}_k^*$ for all $k>0$ are given by
\alpheqn
\begin{equation}
\label{cU}
{\cal U}_k = \ee{S_k},\quad S_k = \lambda_k \left( c(k)c(-k) - c^*(k)c^*(-k)
\right) \end{equation} which are unitary since the operators $S_k$ are
screw-hermitian.\footnote{i.e.\ ${\rm i} S_k$ is selfadjoint}
Thus,
\begin{equation}
{\cal U} = \ee{S},\quad S=\sum_{k>0}S_k .
\end{equation} This operator $S$ can be shown to exist and defines an
anti-selfadjoint operator if and only if
\begin{equation}
\sum_{k>0}|k||\lambda_k|^2<\infty \label{24c},
\end{equation}\reseteqn
and therefore (\ref{24c}) is necessary and sufficient for the unitary
operator ${\cal U}$ to exist. This latter condition is equivalent to the
second one in \Ref{condition1} and thus fulfilled by assumption. Note
that
\begin{equation}
\label{hdiag}
{\cal U}^* H {\cal U} = \frac{2\pi}{L}h_0 + \frac{2\pi}{L}\sum_{k>0} \left(
\omega_k\left( c^*(k)c(k) + c^*(-k)c(-k)\right) -2\eta_k\frac{L}{2\pi}\right)
\equiv H_D
\end{equation}
and therefore ${\cal U}$ is the unitary operator diagonalizing the
non--zero modes of our Hamiltonian.
\subsection{Gauge invariant states}
\label{GND}
By the gauge fixing above we reduced the Hilbert space from ${\cal H}$ to
${\cal H}'_{{\rm phys}}$ containing all states invariant under {\em small} gauge
transformations, i.e. of the form $\ee{{\rm i}\alpha(x)}$ with
$\alpha(L/2)=\alpha(-L/2)$. There are, however, still large gauge
transformations present which are generated by $\ee{\ii2\pi x/L}$. It is
important to note that physical states need not be invariant under these
latter transformations, but it is useful to construct states with simple
transformation properties. This is the origin of the $\theta$--vacuum.
The large gauge transformation $\ee{\ii2\pi x/L}$ acts on the fields
as follows
\begin{eqnarray}
\label{large}
\psi(x)&\stackrel{R}{\to}& \ee{{\rm i} 2\pi x/L}\psi(x) = (R_+R_-)^{-1}
\psi(x)(R_+R_-)\, ,\nonumber \\ \nopagebreak
eY&\stackrel{R}{\to}& eY - 1
\end{eqnarray}
where $R_\pm$ are the implementers of
$\ee{{\rm i} 2\pi x/L}$ in the chiral sectors of the
fermions and are discussed in detail
in Appendix A. The large gauge transformation $R$ obviously generates
a group ${\bf Z}$, $n\to R^n$, and we denote this group as ${\bf Z}_R$.
Our aim is to construct the states in
${\cal H}_{{\rm phys}}$ which carry an irreducible representation of
${\bf Z}_R$ and especially the ground states of our model.
We start with recalling that the Fermion Fock space can be decomposed in
sectors of different chiral charges $\hat\rho^\pm(0)$,
$${\cal H}_{{\rm Fermion}} =\bigoplus_{n_+,n_-\in{\bf Z}}{\cal H}^{(n_+,n_-)}$$
where
$$
{\cal H}^{(n_+,n_-)}=\left\{\left. \Psi\in {\cal H}_{{\rm Fermion}}\right|
\hat\rho^\pm(0)\Psi = n_\pm \Psi \right\} =
R_+^{n_+}R_-^{-n_-}{\cal H}^{(0,0)}
$$
(for a more detailed discussion see Appendix A).
Thus,
\begin{equation}
{\cal H}_{{\rm phys}}= {\cal L}^2({\bf R},{\rm d} Y)\otimes {\cal H}_{{\rm Fermion}}'
\end{equation}
where
\begin{equation}
{\cal H}_{{\rm Fermion}}' = \bigoplus_{n\in{\bf Z}}{\cal H}^{(n,-n)},
\quad {\cal H}^{(n,-n)} = (R_+ R_-)^n{\cal H}^{(0,0)}
\end{equation}
is the zero charge subspace of the Fermion Fock space and we
use the Schr\"odinger representation for the physical
degree of freedom $Y = \int_{\Lam} {\rm d}{x} A_1(x)/2\pi$ of the photon
field as discussed in the last subsection.
${\cal H}_{{\rm phys}}$ can therefore be spanned by
states
\alpheqn
\begin{equation}
\label{tetn}
\Psi(n) = \phi\mbox{$(Y+\frac{n}{e}) $} (R_+R_-)^n
\Psi,\quad \phi \in {\cal L}^2({\bf R},{\rm d} Y),\, \Psi\in{\cal H}^{(0,0)}
\end{equation}
which, under a large gauge transformation \Ref{large}, transform as
\begin{equation}
\Psi(n)\stackrel{R}{\to} \Psi(n-1).
\end{equation}
\reseteqn
Thus the states transforming under an irreducible representation of ${\bf Z}_R$
are given by
\begin{equation}
\label{tet}
\Psi^\theta= \sum_{n\in{\bf Z}}\ee{{\rm i}\theta n}\Psi(n)
\stackrel{R}{\to} \ee{{\rm i}\theta}\Psi^\theta
\end{equation}
It is easy to calculate the inner products of these states,
\begin{equation}
<\Psi_1^\theta,\Psi_2^{\theta'}>=
2\pi\delta_{2\pi}(\theta-\theta')
<\Psi_1,\Psi_2>_{{\rm F}}(\phi_1,\phi_2)_{{\cal L}^2}
\end{equation}
($2\pi\delta_{2\pi}(\theta)=\sum_{n\in{\bf Z}}\ee{{\rm i} n\theta}$,
since $<(R_+R_-)^n\Psi_1,(R_+R_-)^m\Psi_2> =
\delta_{n,m}<\Psi_1,\Psi_2>_{{\rm F}}$; $<\cdot,\cdot>_{{\rm F}}$ and
$<\cdot,\cdot>_{{\cal L}^2}$ are the inner products in ${\cal H}_{{\rm Fermion}}$
and ${\cal L}^2({\bf R},{\rm d} Y)$, respectively).
Thus the states $\Psi^\theta$ actually are not elements in
${\cal H}_{{\rm phys}}$ (they do not have a finite norm).
In our calculation of Green functions below we find it useful to use the
notation
\begin{equation}
\label{reg}
<\Psi_1^\theta,\Psi_2^{\theta}>_\theta \equiv
<\Psi_1,\Psi_2>_{{\rm F}}(\phi_1,\phi_2)_{{\cal L}^2}
\end{equation}
which can be regarded as redefinition of the inner product using a simple
multiplicative regularization (dropping the infinite term
$2\pi\delta_{2\pi}(0)$).
We now construct the ground states of our model. As expected, the
quantum mechanical variables $P,X$ \Ref{qmv} describing the zero mode $h_0$
of the Hamiltonian have a simple representation on the
$\theta$-states
\Ref{tet},
\begin{eqnarray}
P\Psi^\theta = \sum_{n\in{\bf Z}}\ee{{\rm i}\theta n}
2e\mbox{$(Y+\frac{n}{e})$}
\phi\mbox{$(Y+\frac{n}{e}) $}(R_+R_-)^n\Psi \, ,\nonumber \\ \nopagebreak
X\Psi^\theta = \sum_{n\in{\bf Z}}\ee{{\rm i}\theta n}
\frac{{\rm i}}{2e}\frac{L}{2\pi}\frac{\partial}{\partial Y}
\phi\mbox{$(Y+\frac{n}{e}) $} (R_+R_-)^n\Psi.
\end{eqnarray}
Thus the ground states of $h_0$ annihilated by $C(0)$ are of the form
\Ref{tetn} with
\begin{equation}
\label{gst}
\phi_0(Y)
= \left(\frac{\pi}{4e^2\alpha}\right)^\frac{1}{4} \exp\left(-\alpha(2eY)^2\right)
\end{equation}
where
$
\label{alpha}
\alpha=\frac{1}{L}\sqrt{\frac{\pi^3}{2e^2}(1-W_0)},
$
and the other eigenstates are the harmonic oscillator eigenfunctions
$\phi_n\propto C^*(0)^n\phi_0$. From $C(k)={\cal U} c(k){\cal U}^*$ and
$c(k)\Omega_{{\rm F}}=0$ it is clear that the ground state of all $h_{k>0}$
is ${\cal U} \Omega_{{\rm F}}$. We conclude that the ground states of our
model obeying $H \Psi_0^\theta = L E_0\Psi_0^\theta$ are given by
\begin{equation}
\label{vac}
\Psi_0^\theta= \sum_{n\in{\bf Z}}\ee{{\rm i}\theta n}
\phi_0\mbox{$(Y+\frac{n}{e}) $}(R_+R_-)^n {\cal U}\Omega_{{\rm F}}.
\end{equation}
\subsection{Gauge invariant Green functions}
\label{green}
The observables of our model now are operators on ${\cal H}_{\rm phys}$ where
$\int_\Lambda {\rm d} x A_1(x)$ is represented by $2\pi Y$.
We recall that the fully gauge invariant field operators are the
$\chi$, \Ref{chi}, which are
represented in the present gauge fixed setting by
$$\chi_\sigma(x)=\ee{{\rm i} 2\pi e Y(x-r)/L}\psi_\sigma(x).$$
These operators depend on the $r\in\Lambda$ chosen.
Bilinears such as meson operators are, however, independent of $r$ and give
rise to translational invariant equal time Green functions.
Moreover, on the quantum level not only the
Wilson line $W[A_1]$ \Ref{Wilson} but actually even
\begin{equation}
e \int_\Lambda{\rm d} x A_1(x) + \mbox{$\frac{1}{2}$} Q_5 \equiv w[A_1]
\end{equation}
is gauge invariant (note that $W[A_1] = \ee{{\rm i} w[A_1]}$).
This operator is represented by $e Y+\mbox{$\frac{1}{2}$} Q_5 = P/2$ (cf. \Ref{qmv}).
The gauge invariant equal time Green functions of the model are the ground
state expectation values of products $(\cdots)$ of meson operators and
functionals $F[P,X]$ of the zero mode operators $P$, $X$. Since we only
consider $(\cdots)$ which are also invariant under large gauge
transformations, the transition amplitudes
$\left<\Psi_1^\theta,(\cdots)\Psi_2^{\theta'} \right>$ are always proportional
to $2\pi\delta(\theta-\theta')$. Thus the Green functions we consider can be
defined as
\begin{equation}
\label{greenf}
\left< \Psi_0^\theta, F[P,X] \,
\chi^*_{\sigma_1}(x_1)\chi_{\tau_1}(y_1) \cdots
\chi^*_{\sigma_N}(x_N) \chi_{\tau_N}(y_N)\Psi_0^\theta \right>_\theta
\end{equation}
(note that $\left< \Psi_0^\theta, \Psi_0^\theta \right>_\theta =1$,
cf. \Ref{reg}).
{}Following \cite{HSU} it is useful to define {\em interacting
fermion fields}
\alpheqn
\begin{equation}
\label{intf}
\Psi_\sigma(x) = {\cal U}^* \psi_\sigma(x)\, {\cal U}
\end{equation}
such that (\ref{greenf}) becomes
\begin{equation}
\mbox{Eq.\ \Ref{greenf}} =
\left<\Omega^\theta, F[P,X]\, \Psi^*_{\sigma_1}(x_1)
\Psi_{\tau_1}(y_1)\cdots \Psi^*_{\sigma_N}(x_N)
\Psi_{\tau_N}(y_N) \Omega^{\theta'}\right>
\end{equation}
where
\begin{equation}
\label{freevac}
\Omega^\theta= \sum_{n\in{\bf Z}}\ee{{\rm i}\theta n}
\phi_0\mbox{$(Y+\frac{n}{e}) $}(R_+R_-)^n \Omega_{{\rm F}}
\end{equation}
\reseteqn
is the $\theta$--state constructed from the free fermion vacuum.
The strategy to calculate Green functions of the model using bosonization
techniques is the following: the relation \Ref{k2} of appendix A
can be used to move
the operators $R_\pm$ and
combine them to some power of $(R_+R_-)$. The operators $Q_\pm$
when applied to physical states become simple ${\bf C}$--numbers:
$Q_\pm (R_+R_-)^n = (R_+R_-)^n (\pm n+Q_\pm)$ for all integers
$n$, and $Q_\pm\Omega_{\rm F} = 0$. For the exponentials of boson operators
we use the decomposition into creation and annihilation parts
outlined in A.4. The normal ordering procedure gives a
product of exponentials of commutators which are (${\bf C}$-number) functions.
For the correlation functions of meson operators
$\chi_\sigma^*(x)\chi_{\sigma'}(y)$ we obtain:
\alpheqn
\begin{eqnarray}
\left<\Psi_0^\theta,\chi^*_{\pm}(x)
\chi_{\pm}(y)\Psi_0^{\theta} \right>_\theta &=&
\ee{-\frac{\pi}{4L}m(x-y)^2} \ee{\Delta(x-y)}g_0^\pm(x-y) \label{2p1}\, , \\
\left<\Psi_0^\theta,\chi_{\pm}^*(x)
\chi_{\mp}(y)\Psi_0^{\theta} \right>_\theta &=& \ee{\mp {\rm i}\theta}
\ee{-i\frac{2\pi}{L}(x-y)} \ee{-\frac{\pi m}{4L}((x-y)+\frac{2}{m})^2} C(L) \ee{D(x-y)}
\label{2p2}
\, .
\end{eqnarray}
\reseteqn
with
\begin{eqnarray}
\Delta &=&
\sum_{k>0}\frac{2\pi}{Lk}\sinh^2(\lambda_k)[\ee{{\rm i} kx }+
\ee{-{\rm i} kx }-2] \nonumber\, , \\
D(x) &=& -\sum_{k>0} \frac{\pi}{Lk} \sinh(2\lambda_k)
[ \ee{{\rm i} kx }+ \ee{-{\rm i} kx }-2] \label{DeDC}\, , \\
C(L) &=& \frac{1}{L}\exp[\sum_{k>0} \frac{2\pi}{kL}(\sinh(2\lambda_k)-2\sinh^2(\lambda_k))]\,.
\nonumber
\end{eqnarray}
where $g_0^\pm(x)=\frac{1}{L}\frac{e^{\mp i\frac{\pi}{L}x}}{1-e^{\pm i\frac{2\pi}{L}
(x\pm i\varepsilon)}}$
is the 2-point function of free fermions, and the Schwinger mass is
renormalized to $m^2=\frac{e^2}{\pi(1-W_0)}$.
Note that the Green function \Ref{2p2}
depends on $\theta$ and is non--zero due to chiral symmetry breaking as in
the Schwinger model. As expected, for vanishing electromagnetic coupling,
$e=0$, this Green function vanishes (due to the factor $\ee{-\pi/mL}$ appearing
in (\ref{2p2})).
{}From (\ref{2p2}) we can calculate the chiral condensate by
setting $x=y$, and in the limit $L\to\infty$ we obtain
\begin{equation}
\lim_{L\to\infty}
\left<\Psi_0^\theta,\chi_{\pm}^*(x) \chi_{\mp}(x)\Psi_0^{\theta} \right>_\theta
= \lim_{L\to\infty} \ee{\mp {\rm i}\theta} \ee{-\frac{\pi}{mL}}
C(L) \nonumber
= \ee{\mp {\rm i}\theta} C
\end{equation}
with a constant $C$ which can be calculated in principle
from eq.\ (\ref{DeDC}). In
the special case of the Schwinger model ($W_k=0$), $C$ can be computed
and we recover the well--known result $C_{W_k=0}=\frac{m}{4\pi}\ee{\gamma}$ where
$\gamma=0.577\ldots$ is Eulers constant
(see e.g.\ \cite{SaWi,Hoso}).
\section{Multiplicative regularization and the Thirring-Schwinger model}
We recall that the Thirring model is formally obtained
from the Luttinger model in the limits
\begin{equation}
\label{LT}
L\to \infty, \quad V(x)\to g \delta(x)
\end{equation}
i.e.\ when the interaction becomes local and space becomes infinite.
The first
limit amounts to remove the IR cut--off of our
model. By inspection it can be easily done in all Green functions.
The second limit in \Ref{LT} is non--trivial: we recall, that condition
\Ref{condition1} on the Luttinger potential requires
sufficient decay of the Fourier modes $W_k$ of the interaction, and this is
violated in the Thirring model where $W_k=W_0$ is independent of $k$. This
latter condition was necessary for the interacting model to be
well--defined on the Hilbert space of the non--interacting model.
A better understanding can be obtained by explicitly
performing the limit \Ref{LT} in the present setting.
The idea is to find a family of Luttinger
potentials $\{ V_\ell(x)\}_{\ell>0}$ becoming local for $\ell\downarrow 0$,
i.e.\ for all $\ell>0$ the condition \Ref{condition1} is fulfilled and
$\lim_{\ell\downarrow 0} V_\ell(x)=g\delta(x)$. Then for all $\ell>0$
everything is well-defined on the free Hilbert space and one can work out in
detail how to regularize such that the correlation functions make sense
for $\ell\downarrow 0$.
We note that a direct construction of the Thirring model in a
framework similar to the one here has been completed in \cite{CRW}. This
construction seems to be, however, different from the one outlined below.
{}For the case of Luttinger-Schwinger model we split the function $\Delta(x)$
into a part corresponding to the pure Luttinger model and a part which
describes the additional Schwinger coupling,
i.~e.~$\Delta(x)=(\Delta(x)-\Delta^{e=0}(x))+\Delta^{e=0}(x)$.
The limit $W_k=W_0={\rm const.}$ exists for $\Delta-\Delta^{e=0}$. As $L\to\infty$, the sum
in (\ref{DeDC}) turns into an integral and we obtain
\begin{eqnarray}
\Delta(x)-\Delta^{e=0}(x)=&&\!\!\!\!\!\!\! \frac{1}{\sqrt{1-W_0^2}} \int_0^\infty {\rm d}k\, \left(
\frac{1}{\sqrt{k^2+\mu^2}}-1\right)(\cos (kx) -1)+ \nonumber\\
&& \!\!\!\!\!\!\!
\sqrt{\frac{1+W_0}{1-W_0}}\, \int_0^\infty {\rm d}k\, \frac{\mu^2}{k^2}\frac{1}{\sqrt{k^2+\mu^2}}
(\cos (kx) -1) \, .
\end{eqnarray}
The first integral becomes $K_0(|\mu x|)+\ln\frac{|\mu x|}{2}+\gamma$ and the
expression in the last line is a second integral $(n=2)$
of $K_0$ defined iteratively
by Ki$_n(x)=\int_x^\infty $Ki$_{n-1}(t) \, dt$, Ki$_0=K_0$ \cite{Abram}.
Moreover we
introduced a new mass by $\mu^2=e^2/(\pi(1+W_0))$. Note that the
singularities at the origin of the Bessel function are removed by the
additional terms, consistent with $\Delta(0)=0$.
No regularization has been necessary so far. Renormalization comes along with
$\Delta^{e=0}$.
We choose a Luttinger-interaction such that
$(1-W_k^2)^{-1/2}-1=2a^2 e^{-\ell k}$ where $\ell$
defines the range of the interaction.
For this choice we find
\begin{eqnarray}
\Delta^{e=0}(x)=2a^2\ln\left|\frac{\ell}{x+i\ell}\right|
\end{eqnarray}
and obviously the Thirring limit makes sense only if one removes the singular
part $\ln \ell$ which can be done by a wave function renormalization of the form
\begin{eqnarray}
\chi_\pm(x) \to \tilde\chi_\pm(x)=Z^{1/2}(a,\ell)\chi_\pm(x)
\quad \makebox{with}
\quad Z^{1/2}(a,\ell)=\ell^{-a^2} \, . \label{Th3}
\end{eqnarray}
A similar discussion holds for the chirality mixing correlation function.
The 2-point function of the Thirring-Schwinger model therefore
become
\alpheqn
\begin{eqnarray}
\langle\Psi_0^\theta,\tilde\chi_\pm^*(x)\tilde\chi_\pm(0)\Psi_0^\theta\rangle_\theta
&=&e^{\Delta_{\rm reg}(x)}g_0^\pm(x)\, ,\\
\left<\Psi_0^\theta,\tilde\chi_{\pm}^*(x)
\tilde\chi_{\mp}(0)\Psi_0^{\theta} \right>_\theta &=& \ee{\mp {\rm i}\theta}
C_{\rm reg}\, \ee{D_{\rm reg}(x)}
\, .
\end{eqnarray}
\reseteqn
If we define $\tau_0$ by $\tanh(2\tau_0)=W_0$ we can write
\begin{eqnarray}
\Delta_{\rm reg}(x)&\!=&\!\cosh (2\tau_0)\left[ K_0(|\mu x|)+\ln\frac{|\mu x|}{2}+\gamma\right]
+\nonumber\\
&&\!\!\frac{1}{2}e^{2\tau_0}\left[1-\frac{\pi}{2}|\mu x|-{\rm Ki}_2(|\mu x|)\right]+
(\cosh (2\tau_0)-1)\ln |x| \, , \nonumber\\
D_{\rm reg}(x) &\!=&\! -\sinh (2\tau_0)\left[ K_0(|\mu x|)+\ln\frac{|\mu x|}{2}+\gamma\right]
- \\
&&\!\! \frac{1}{2}e^{2\tau_0}\left[1-\frac{\pi}{2}|\mu x|-{\rm Ki}_2(|\mu x|)
\right] \, ,\nonumber\\
\ln C_{\rm reg} &\!=&\! \gamma+\ln\frac{1}{2\pi}+e^{-2\tau_0}\ln \frac{\mu}{2}
\nonumber \, .
\end{eqnarray}
We checked that all Green functions of the Thirring-Schwinger model have
a well-defined limit after the wave function renormalization.
We would like to stress that this procedure can be naturally interpreted as
low--energy limit of the Luttinger--Schwinger model:
if one is interested only in
Green functions describing correlations of far--apart fermions, the precise
form of the Luttinger interaction $V(x)$ should be irrelevant and only the
total interaction strength $g = \int{\rm d} x\, V(x)$ should matter. Thus as
far
as these correlators are concerned, they should be equal to the ones of the
Thirring model corresponding to this coupling $g$.
\section{Conclusion}
We formulated and solved the Luttinger-Schwinger model in the Hamiltonian
formalism. Structural issues like gauge invariance, the role of anomalies
and the structure of the physical states were discussed in detail. The
necessary tools for computing all equal time correlation functions were
prepared and illustrated by calculating the 2--point Green functions. From
this the chiral condensate and critical exponents were computed. We could
also clarify how the non trivial short distance behavior of the
Thirring-Schwinger
model arises in a limit from the Luttinger-Schwinger model.
\app
\section*{Appendix A: Bosons from fermions and vice versa}
In this appendix we summarize the basics for the bosonization used in the
main text to solve the Luttinger--Schwinger model. Bosonization is known
in the physics literature since quite some time
(\cite{CR,PressSegal,Kac,Mickelsson}), for a discussion of the
older history see \cite{HSU}).
We consider the fermion Fock space ${\cal H}_{\rm Fermion}$ generated
by the fermion field operators
from the vacuum $\Omega_{\rm F}$ as described in
the main text. We note that ${\cal H}_{\rm Fermion}={\cal H}_{+}\otimes{\cal H}_{-}$
where ${\cal H}_{\pm}$ are generated by the left-- and right--handed chiral
components $\hat\psi_+$ and $\hat\psi_-$ of our Dirac fermions.
Bosonization can be
formulated for the chiral components $\hat\psi_\pm$ separately
as it leaves
the two chiral sectors ${\cal H}_{\pm}$ completely decoupled. For our purpose
it is more convenient to treat both chiral sectors together.
\subsection*{A.1 Structure of fermion Fock space}
We start by introducing two unitary operators $R_\pm$
which are defined up to an irrelevant phase factor (which we will leave
unspecified) by the following equations,
\begin{equation}
\hat\psi_\pm(k)R_\pm = R_\pm \hat\psi_\pm (k - \frac{2\pi}{L})
\end{equation}
and $R_\pm$ commutes with $\hat\psi_\mp$. A proof of existence and an
explicit construction of these operators can be found in \cite{Ruij}.
Here we just summarize their physical meaning and special properties.
It is easy to see that $R_\pm$ are just the implementors of Bogoliubov
transformations given by the {\em large gauge transformations}
$\psi_\pm(x)\mapsto \ee{{\rm i} 2\pi x/L}\psi_\pm(x)$
and $\psi_\mp(x)\mapsto \psi_\mp(x)$, hence $R_+R_-$ and
$R_+R_-^{-1}$ implement
the vector-- and the axial large gauge transformations
$\ee{{\rm i} 2\pi x/L}$ and $\ee{{\rm i}\gamma_5 2\pi x/L }$, respectively.
These have non--trivial winding number\footnote{
the w.n. of a smooth gauge transformation $\Lambda\to{\rm U}(1)$,
$x\mapsto \ee{{\rm i}\alpha(x)}$ is the integer
$\frac{1}{2\pi}\left(\alpha(L)-\alpha(0)\right)$.
}
and change the vacuum $\Omega_{\rm F}$ to states containing (anti-)
particles.
The latter follows from the commutator relations with the chiral
fermion currents
\begin{eqnarray}
(R_\pm)^{-1} \hat\rho^\pm (k) R_\pm = \hat\rho^\pm (k) \pm
\delta_{k,0} \label{k2}\, .
\end{eqnarray}
The essential point of bosonization is that the total Hilbert space
${\cal H}_{\rm Fermion}$ can be generated from $\Omega_{\rm F}$ by the chiral
fermion currents $\hat\rho^{\pm}(k)$ and $R_\pm$. More
precisely, for all pairs of integers $n_+,n_-\in{\bf Z}$ we introduce the
subspaces ${\cal D}^{(n_+,n_-)}$ of ${\cal H}_{{\rm Fermion}}$ containing all linear
combinations of vectors
\begin{equation}
\label{lin}
\hat\rho^+(k_1) \cdots \hat\rho^+(k_{m_+}) \hat\rho^-(q_1)\cdots
\hat\rho^-(k_{m_-} ) R_+^{n_+} R_-^{-n_-} \Omega_F
\end{equation}
where $m_\pm\in{\bf N}_0$ and $k_i,q_i\in\Lambda^*$.
The basic result of the boson--fermion correspondence is the
following
{\bf Lemma:} The space
\begin{equation}
{\cal D} \equiv \bigoplus_{n_+,n_-\in{\bf Z}} D^{(n_+,n_-)} .
\end{equation}
is dense in ${\cal H}_{\rm Fermion}$ (for a proof see e.g.\ \cite{CR}).
{\em Remark:} This Lemma gives the following picture of the
structure of the Fock space ${\cal H}_{\rm Fermion}$: It splits into
{\em superselection sectors} ${\cal H}^{(n_+,n_-)}$ (which are the closure of
${\cal D}^{(n_+,n_-)}$) containing the eigenstates of the chiral charges
$Q_\pm$ with eigenvalues $n_\pm$. The fermion currents $\hat\rho^\pm(k)$
leave all these sectors invariant, and the operators $R_\pm$ intertwine
different sectors, $R_+: {\cal H}^{(n_+,n_-)}\to {\cal H}^{(n_+ +1,n_-)}$ and
$R_-: {\cal H}^{(n_+,n_-)}\to {\cal H}^{(n_+,n_- - 1)}$.
\subsection*{A.2 Kronig's identity}
The basic formula underlying the solution of our model is
\begin{equation}
\label{kronig}
\hat H_0 = \frac{\pi}{L}\left( Q_+^2 + Q_-^2 \right) +
\frac{2\pi}{L} \sum_{k>0}\left( \hat\rho^+(-k)\hat\rho^+(k) +
\hat\rho^-(k) \hat\rho^-(-k) \right) .
\end{equation}
It expresses the free Dirac Hamiltonian in terms of bilinears of the
chiral fermion currents.
\subsection*{A.3 Boson--fermion correspondence}
The boson--fermion correspondence provides explicit formulas of the
fermion operators $\psi_\pm(x)$ in terms of operators
$\hat\rho^\pm(k)$ and $R_\pm$,
\aalpheqn
\begin{equation}
\label{limit}
\psi_\pm(x) = \lim_{\varepsilon\searrow 0} \psi_\pm(x;\varepsilon)
\end{equation}
(this limit can e.g. be understood in the weak sense for states in
${\cal D}$), where
\begin{eqnarray}
\psi_\pm(x;\varepsilon) = \frac{1}{\sqrt{L}}
S_\pm(x) \normal{\exp(K_\pm(x;\varepsilon))} \label{bfc}
\end{eqnarray}
with
\begin{equation}
\label{S}
S_\pm(x)=
\ee{\pm {\rm i} \pi x Q_\pm/L } (R_\pm)^{\mp 1}\ee{\pm {\rm i} \pi x Q_\pm/L } =
\ee{\mp \pi{\rm i} x/L} (R_\pm)^{\mp 1}\ee{\pm {\rm i} 2\pi x Q_\pm/L }
\end{equation}
and
\begin{equation}
\label{Kpm}
K_\pm(x;\varepsilon) =\mp \frac{2\pi}{L} \sum_{k\in\Lambda^*\backslash\{0\}}
\frac{\hat\rho^\pm(-k)}{k}\ee{-{\rm i} kx}\ee{-\varepsilon|k|} = -
K_\pm(x;\varepsilon)^*. \end{equation}
\areseteqn
More explicitly, the normal ordering $\normal{\cdots}$ is with respect
to the fermion vacuum $\Omega_{{\rm F}}$ (cf.\ \Ref{vacF}),
\aalpheqn
\begin{equation}
\normal{\exp(K_\pm(x;\varepsilon))} = \exp(K^{(-)}_\pm(x;\varepsilon))
\exp(K^{(+)}_\pm(x;\varepsilon))
\end{equation}
where
\begin{equation}
K^{(\sigma)}_\pm(x;\varepsilon) = \sigma\frac{2\pi}{L}\sum_{k>0}
\frac{\hat\rho^\pm(\pm \sigma k)}{k}\ee{\pm\sigma {\rm i} kx}\ee{-\varepsilon|k|},
\quad \sigma=+,-
\end{equation}
\areseteqn
is such that $K_\pm = K_\pm^{(-)}+K_\pm^{(+)}$ and
$K_\pm^{(+)}\Omega_{{\rm F}} = 0$ (cf.\ \Ref{vacF}).
\subsection*{A.4 Interacting fermions}
{}From the definition of the interacting fermion fields $\Psi(x)$ (\ref{intf})
and the
representation of free fermions in terms of bosons, we are led to investigate
the interacting kernel $\tilde K_\pm(x)={\cal U}^* K_\pm(x) {\cal U}$:
$$
\tilde{K}_\pm(x) =\mp \frac{2\pi}{L} \sum_{k\in\Lambda^*\backslash\{0\}}
\frac{1}{k} \left( \cosh(\lambda_k)\, \hat\rho^\pm(-k) -
\sinh(\lambda_k)\, \hat\rho^\mp(-k) \right) \ee{-{\rm i} kx}\ee{-\varepsilon|k|}\, .
$$
It is convenient to write
\newcommand{{K\! s}}{{K\! s}}
\newcommand{{K\! c}}{{K\! c}}
\newcommand{{K\! s/c}}{{K\! s/c}}
\aalpheqn
\begin{equation}
\tilde{K}_\pm = \tilde{K}_\pm^{(+)} + \tilde K_\pm^{(-)},\quad
\tilde{K}_\pm^{(\sigma)} = Kc_\pm^{(\sigma)} -Ks_\mp^{(\sigma)}
\end{equation}
where the upper index refers to the creation-- ($\sigma=-$) and
annihilation-- ($\sigma=+$) parts of operators and
\begin{eqnarray}
{K\! c}^{(\sigma)}_\pm(x) &=& \sigma\sum_{k>0}\frac{2\pi}{Lk}\cosh(\lambda_k)
\hat\rho^\pm(\pm\sigma k)\ee{\mp\sigma {\rm i} kx }\ee{-\varepsilon k} \, , \nonumber \\ \nopagebreak
{K\! s}^{(\sigma)}_\pm(x) &=& \sigma\sum_{k>0}\frac{2\pi}{Lk}\sinh(\lambda_k)
\hat\rho^\pm(\pm\sigma k)\ee{\mp\sigma {\rm i} kx }\ee{-\varepsilon k} \: .
\end{eqnarray}
\areseteqn
The nonzero commutators of these operators are
\begin{eqnarray}
\ccr{{K\! c}^{(+)}_\pm(x)}{{K\! c}^{(-)}_\pm(y)} &=&
- \sum_{k>0}\frac{2\pi}{Lk}\cosh^2(\lambda_k)
\ee{ \mp {\rm i} k(x-y) }\ee{-2\varepsilon k} \, , \nonumber \\ \nopagebreak
\ccr{{K\! c}^{(+)}_\pm(x)}{{K\! s}^{(-)}_\pm(y)} &=&
- \sum_{k>0}\frac{\pi}{Lk}\sinh(2\lambda_k)
\ee{ \mp{\rm i} k(x-y) }\ee{-2\varepsilon k}\, , \\
\ccr{{K\! s}^{(+)}_\pm(x)}{{K\! s}^{(-)}_\pm(y)} &=&
- \sum_{k>0}\frac{2\pi}{Lk}\sinh^2(\lambda_k)
\ee{ \mp{\rm i} k(x-y) }\ee{-2\varepsilon k} \: .
\nonumber
\end{eqnarray}
We find the following normal ordered expression for the interacting fermions
\begin{equation}
\Psi_\pm(x) = \frac{1}{\sqrt{L}} z S_\pm(x)
\normal{\ee{\tilde K_\pm(x) }}
\end{equation}
where $z = \ee{- \sum_{k>0}
\frac{2\pi}{Lk}\sinh^2(\lambda_k) }$.
\appende
\begin{center}{\bf Acknowledgments}\end{center}
E.L. would like to thank the Erwin Schr\"odinger International Institute
in Vienna for hospitality where part of this work was done, and the
``\"Osterreichische Forschungsgemeinschaft'' for partial financial support
in May/June 1994 when this work was begun. He would also like to thank
S.G. Rajeev and Mats Wallin for usefull discussions. The authors thank the
referee for valuable suggestions concerning the presentation of their results.
| proofpile-arXiv_065-408 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
This paper is focused on the actual problem of heavy flavor physics,
exclusive s.l. decays of low-lying bottom and charm baryons. Recently,
activity in this field has obtained a great interest due to the
experiments worked out by the CLEO Collaboration \cite{CLEO} on the
observation of the heavy-to-light s.l. decay $\Lambda_c^+\to\Lambda e^+\nu_e$.
Also the ALEPH~\cite{ALEPH} and OPAL~\cite{OPAL} Collaborations expect in
the near future to observe the exclusive mode $\Lambda_b\to\Lambda_c\ell\nu$.
In ref. \cite{Aniv} a model for QCD bound states composed from
light and heavy quarks was proposed. Actually, this model is
the Lagrangian formulation of the NJL model with separable interaction
\cite{Goldman} but its advantage consists in the possibility of
studying of baryons as the relativistic systems of three quarks.
The framework was developed for light mesons~\cite{Aniv} and
baryons~\cite{Aniv,PSI}, and also for heavy-light hadrons~\cite{Manchester}.
The purpose of present work is to give a description of properties of baryons
containing a single heavy quark within framework proposed in
ref. \cite{Aniv} and developed in ref. \cite{PSI,Manchester}.
Namely, we report the calculation of observables of semileptonic decays of
bottom and charm baryons: Isgur-Wise functions, asymmetry parameters,
decay rates and distributions.
\section{Model}
Our approach \cite{Aniv} is based on the interaction Lagragians describing
the transition of hadrons into constituent quarks and {\it vice versa}:
\begin{eqnarray}\label{strong}
{\cal L}_B^{\rm int}(x)=g_B\bar B(x)\hspace*{-0.2cm}\int \hspace*{-0.2cm}
dy_1...\hspace*{-0.2cm}\int \hspace*{-0.2cm}dy_3
\delta\left(x-\frac{\sum\limits_i m_iy_i}{\sum\limits_i m_i}\right)
F\left(\sum\limits_{i<j}\frac{(y_i-y_j)^2}{18}\right)
J_B(y_1,y_2,y_3)+h.c.\nonumber
\end{eqnarray}
with $J_B(y_1,y_2,y_3)$ being the 3-quark current with quantum numbers
of a baryon $B$:
\begin{eqnarray}\label{current}
J_B(y_1,y_2,y_3)=\Gamma_1 q^{a_1}(y_1)q^{a_2}(y_2)C\Gamma_2 q^{a_3}(y_3)
\varepsilon^{a_1a_2a_3}\nonumber
\end{eqnarray}
Here $\Gamma_{1(2)}$ are the Dirac matrices, $C=\gamma^0\gamma^2$ is
the charge conjugation matrix, and $a_i$ are the color indices.
We assume that the momentum distribution of the constituents inside
a baryon is modeled by an effective relativistic vertex function
which depends on the sum of relative coordinates only
$F\left(\frac{1}{18}\sum\limits_{i<j}(y_i-y_j)^2\right)$ in the configuration
space where $y_i$ (i=1,2,3) are the spatial 4-coordinates of quarks with
masses $m_i$, respectively. They are expressed through the center of mass
coordinate $(x)$ and relative Jacobi coordinates $(\xi_1,\xi_2)$. The shape
of vertex function is chosen to guarantee ultraviolet convergence of matrix
elements. At the same time the vertex function is a phenomenological
description of the long distance QCD interactions between quarks and gluons.
In the case of light baryons we shall work in the limit of isospin
invariance by assuming the masses of $u$ and $d$ quarks are equal each other,
$m_u=m_d=m$. Breaking of the unitary SU(3) symmetry is taken into account
via a difference of strange and nonstrange quark masses $m_s-m\neq 0$.
In the case of heavy-light baryonic currents we suppose that heavy quark
is much larger than light quark $(m_Q\gg m_{q_1},m_{q_2})$,
i.e. a heavy quark is in the c.m. of heavy-light baryon.
Now we discuss the model parameters. First, there are the baryon-quark
coupling constants and the vertex function in the Lagrangian
${\cal L}_B^{\rm int}(x)$.
The coupling constant $g_B$ is calculated from {\it the compositeness
condition} that means that the renormalization
constant of the baryon wave function is equal to zero,
$Z_B=1-g_B^2\Sigma^\prime_B(M_B)=0$, with $\Sigma_B$ being the baryon mass
operator.
The vertex function is an arbitrary function except that it should make the
Feynman diagrams ultraviolet finite, as we have mentioned above.
We choose in this paper a Gaussian vertex function for simplicity.
In Minkowski space we write $F(k^2_1+k^2_2)=\exp[(k^2_1+k^2_2)/\Lambda_B^2]$
where $\Lambda_B$ is the Gaussian range parameter which is
related to the size of a baryon. It was found \cite{PSI} that for nucleons
$(B=N)$ the value $\Lambda_N=1.25$ GeV gives a good description of the
nucleon static characteristics (magnetic moments, charge radii)
and also form factors in space-like region of $Q^2$ transfer up to 1 GeV$^2$.
In this work we will use the value $\Lambda_{B_q}\equiv\Lambda_N=1.25$ GeV
for light baryons and consider the value $\Lambda_{B_Q}$ for the heavy-light
baryons as an adjustable parameter.
As far as the quark propagators are concerned we shall use the standard form
of light quark propagator with a mass $m_q$
\begin{eqnarray}\label{Slight}
<0|{\rm T}(q(x)\bar q(y))|0>=
\int{d^4k\over (2\pi)^4i}e^{-ik(x-y)}S_q(k), \,\,\,\,\,\,
S_q(k)={1\over m_q-\not\! k}\nonumber
\end{eqnarray}
and the form
\begin{eqnarray}\label{Sheavy}
S(k+v\bar\Lambda_{\{q_1q_2\}})=
\frac{(1+\not\! v)}{2(v\cdot k+\bar\Lambda_{\{q_1q_2\}}+i\epsilon)}
\nonumber
\end{eqnarray}
for heavy quark propagator obtained in the heavy quark limit (HQL)
$m_Q\to\infty$. The notation are the following:
$\bar\Lambda_{\{q_1q_2\}}=M_{\{Qq_1q_2\}}-m_Q$ is the difference between
masses of heavy baryon $M_{\{Qq_1q_2\}}\equiv M_{B_Q}$
and heavy quark $m_Q$ in the HQL, $v$ is the four-velocity of heavy baryon.
It is seen that the value $\bar\Lambda_{\{q_1q_2\}}$
depends on a flavor of light quarks $q_1$ and $q_2$. Neglecting
the SU(2)-isotopic breaking gives three independent parameters:
$\bar\Lambda\equiv\bar\Lambda_{uu}=\bar\Lambda_{dd}=\bar\Lambda_{du}$,
$\bar\Lambda_{s}\equiv\bar\Lambda_{us}=\bar\Lambda_{ds}$, and
$\bar\Lambda_{ss}$.
Of course, the deficiency of such a choice of light
quark propagator is lack of confinement. This
could be corrected by changing the analytic properties of the propagator.
We leave that to a future study. For the time being we shall
avoid the appearance of unphysical imaginary parts in the Feynman diagrams
by restricting the calculations to the following condition:
the baryon mass must be less than the sum of constituent quark masses
$M_B<\sum\limits_i m_{q_i}$.
In the case of heavy-light baryons the restriction
$M_B<\sum\limits_i m_{q_i}$ trivially gives that the parameter
$\bar\Lambda_{\{q_1q_2\}}$ must be less than the
sum of light quark masses $\bar\Lambda_{\{q_1q_2\}} < m_{q_1}+m_{q_2}$. The
last constraint serves as the upper limit for a choice of parameter
$\bar\Lambda_{\{q_1q_2\}}$.
Parameters $\Lambda_{B_Q}$, $m_s$, $\bar\Lambda$ are fixed in this paper
from the description of data on $\Lambda^+_c\to\Lambda^0+e^+ +\nu_e$ decay.
It is found that $\Lambda_Q$=2.5 GeV, $m_s$=570 MeV and
$\bar\Lambda$=710 MeV.
Parameters $\bar\Lambda_s$ and $\bar\Lambda_{\{ss\}}$ cannot be adjusted
at this moment since the experimental data on the decays of heavy-light
baryons having the strange quarks (one or two) are not available. In this
paper we use $\bar\Lambda_s=$850 MeV and $\bar\Lambda_{\{ss\}}=$1000 MeV.
\section{Results}
In this section we give the numerical results for the observables of
semileptonic decays of bottom and charm baryons:
the baryonic Isgur-Wise functions, decay rates and asymmetry parameters.
We check that $\xi_1$ and $\xi_2$ functions are
satisfied to the model-independent Bjorken-Xu inequalities.
Also the description of the $\Lambda^+_c\to\Lambda^0+e^+ +\nu_e$
decay which was recently measured by
CLEO Collaboration \cite{CLEO} is given. In what follows we will use
the following values for CKM matrix elements: $|V_{bc}|$=0.04,
$|V_{cs}|$=0.975.
In our calculations of heavy-to-heavy matrix elements we are restricted
only by one variant of three-quark current for each kind of heavy-light
baryon: {\it Scalar current} for $\Lambda_Q$-type baryons and
{\it Vector current} for $\Omega_Q$-type baryons \cite{Shuryak,Manchester}.
The functions $\zeta$ and $\xi_1$ have the upper limit
$\Phi_0(\omega)=\frac{\ln(\omega+\sqrt{\omega^2-1})}{\sqrt{\omega^2-1}}$.
It is easy to show that $\zeta(\omega)=\xi_1(\omega)=\Phi_0(\omega)$
when $\bar\lambda=0$. The radii of $\zeta$ and $\xi_1$ have
have the lower bound $\zeta\geq 1/3$ and $\xi_1\geq 1/3$.
Increasing of the $\bar\lambda$ value leads to
the suppression of IW-functions in the physical
kinematical region for variable $\omega$.
The IW-functions $\xi_1$ and $\xi_2$ must satisfy two
model-independent Bjorken-Xu inequalities \cite{Xu}
derived from the Bjorken sum rule for semileptonic $\Omega_b$ decays to
ground and low-lying negative-parity excited charmed baryon states in
the HQL
\begin{eqnarray}
& &1\geq B(\omega)=\frac{2+\omega^2}{3}\xi_1^2(\omega)+
\frac{(\omega^2-1)^2}{3}\xi_2^2(\omega)
+\frac{2}{3}(\omega-\omega^3)\xi_1(\omega)\xi_2(\omega)
\label{ineq1}\\
& &\rho^2_{\xi_1}\geq \frac{1}{3}-\frac{2}{3}\xi_2(1)
\label{ineq2}
\end{eqnarray}
\noindent
The inequality (\ref{ineq2}) for the slope of the $\xi_1$-function
is fulfilled automatically because of $\rho^2_{\xi_1} \geq 1/3$ and
$\xi_2(1) > 0$.
From the inequality (\ref{ineq1})
one finds the upper limit for the function $\xi_1(\omega)$:
$\xi_1(\omega)\leq\sqrt{3/(2+\omega^2)}$
In Fig.1 we plot the $\zeta$ function in the kinematical region
$1\leq \omega \leq \omega_{max}$.
For a comparison the results of other phenomenological
approaches are drawn. There are data of QCD sum
rule~\cite{Grozin}, IMF models~\cite{Kroll,Koerner2},
MIT bag model~\cite{Zalewski}, a simple quark model (SQM)~\cite{Mark1} and
the dipole formula~\cite{Koerner2}. Our result is close to the result of
QCD sum rules~\cite{Grozin}.
In Table 1 our results for total rates are compared with
the predictions of other phenomenological approaches:
constituent quark model \cite{DESY},
spectator quark model \cite{Singleton}, nonrelativistic
quark model \cite{Cheng}.
\newpage
\begin{center}
{\bf Table 1.} Model Results for Rates of Bottom Baryons
(in $10^{10}$ sec$^{-1}$)\\
\end{center}
\begin{center}
\def1.{1.}
\begin{tabular}{|c|c|c|c|c|} \hline
Process & Ref. \cite{Singleton} & Ref. \cite{Cheng} & Ref. \cite{DESY}
& Our results\\
\hline\hline
$\Lambda_b^0\to\Lambda_c^+ e^-\bar{\nu}_e$ & 5.9 & 5.1 & 5.14 & 5.39
\\
\hline
$\Xi_b^0\to\Xi_c^+ e^-\bar{\nu}_e$ & 7.2 & 5.3 & 5.21& 5.27
\\
\hline
$\Sigma_b^+\to\Sigma_c^{++} e^-\bar{\nu}_e$
& 4.3 & & & 2.23 \\
\hline
$\Sigma_b^{+}\to\Sigma_c^{\star ++} e^-\bar{\nu}_e$
& & & &4.56 \\
\hline
$\Omega_b^-\to\Omega_c^0 e^-\bar{\nu}_e$
& 5.4 & 2.3 & 1.52 & 1.87\\
\hline
$\Omega_b^-\to\Omega_c^{\star 0} e^-\bar{\nu}_e$
& & & 3.41 & 4.01 \\
\hline\hline
\end{tabular}
\end{center}
\vspace*{0.4cm}
Now we consider the heavy-to-light semileptonic modes.
Particular the process $\Lambda^+_c\to\Lambda^0+e^++\nu_e$ which was recently
investigated by CLEO Collaboration \cite{CLEO} is studied in details.
At the HQL ($m_C\to\infty$), the weak hadronic
current of this process is defined by two form factors $f_1$ and $f_2$
\cite{DESY,Cheng}.
Supposing identical dipole forms of the form factors
(as in the model of K\"{o}rner and Kr\"{a}mer \cite{DESY}),
CLEO found that $R=f_2/f_1=$-0.25$\pm$0.14$\pm$0.08. Our form factors have
different $q^2$ dependence. In other words, the quantity $R=f_2/f_1$
has a $q^2$ dependence in our approach. In Fig.10 we plot the results
for $R$ in the kinematical region $1\leq \omega \leq \omega_{max}$ for
different magnitudes of $\bar\Lambda$ parameter.
Here $\omega$ is the scalar product of four velocities of
$\Lambda_c^+$ and $\Lambda^0$ baryons.
It is seen that growth of the $\bar\Lambda$ leads to the
increasing of ratio $R$. The best fit of experimental data is achieved
when our parameters are equal to $m_s=$570 MeV, $\Lambda_Q=$2.5 GeV
and $\bar\Lambda=$710 MeV. In this case the $\omega$-dependence of the
form factors $f_1$, $f_2$ and their ratio $R$ are drawn in Fig.11.
Particularly, we get $f_1(q^2_{max})$=0.8, $f_2(q^2_{max})$=-0.18,
$R$=-0.22 at zero recoil ($\omega$=1 or q$^2$=q$^2_{max}$) and
$f_1(0)$=0.38, $f_2(0)$=-0.06, $R$=-0.16 at maximum recoil
($\omega=\omega_{max}$ or $q^2$=0).
One has to remark that our results at $q^2_{max}$ are closed to the results
of nonrelativistic quark model \cite{Cheng}:
$f_1(q^2_{max})$=0.75, $f_2(q^2_{max})$=-0.17, $R$=-0.23.
Also our result for $R$ weakly deviate from the experimental
data \cite{CLEO} $R=-0.25 \pm 0.14 \pm 0.08$ and the result of
nonrelativistic quark model (Ref. \cite{Cheng}). Our prediction for
the decay rate
$\Gamma(\Lambda^+_c\to\Lambda^0e^+\nu_e)$=7.22$\times$ 10$^{10}$ sec$^{-1}$
and asymmetry parameter $\alpha_{\Lambda_c}$=-0.812 also coincides with the
experimental data $\Gamma_{exp}$=7.0$\pm$ 2.5 $\times$ 10$^{10}$ sec$^{-1}$
and $\alpha_{\Lambda_c}^{exp}$=-0.82$^{+0.09+0.06}_{-0.06-0.03}$ and
the data of Ref. \cite{Cheng} $\Gamma$=7.1 $\times$ 10$^{10}$ sec$^{-1}.$
One has to remark that the success in the reproducing of experimental
results is connected with the using of the $\Lambda^0$ three-quark current
in the $SU(3)$-flavor symmetric form.
By analogy, in the nonrelativistic quark model \cite{Cheng} the assuming
the $SU(3)$ flavor symmetry leads to the presence of the flavor-suppression
factor $N_{\Lambda_c\Lambda}=1/\sqrt{3}$ in matrix element of
$\Lambda_c^+\to\Lambda^0 e^+\nu_e$ decay. If the $SU(3)$ symmetric
structure of $\Lambda^0$ hyperon is not taken into account the
predicted rate for $\Lambda_c^+\to\Lambda^0 e^+\nu_e$ became too large
(see, discussion in ref. \cite{DESY,Cheng}).
Finally, in Table 2 we give our predictions for some modes of
semileptonic heavy-to-lights transitions.
Also the results of other approaches are tabulated.
\vspace*{0.4cm}
\begin{center}
{\bf Table 2.} Heavy-to-Light Decay Rates (in 10$^{10}$ s$^{-1}$).
\end{center}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline
Process & Quantity & Ref.\cite{Singleton} & Ref.\cite{Cheng} &
Ref.\cite{Datta}
& Our & Experiment \\
\hline\hline
$\Lambda_c^+\to\Lambda^0 e^+\nu_e$ & $\Gamma$ & 9.8 & 7.1 &
5.36 & 7.22 & 7.0$\pm$ 2.5 \\
\hline
$\Xi_c^0\to\Xi^- e^+\nu_e$ & $\Gamma$ & 8.5 & 7.4 & & 8.16 & \\
\hline
$\Lambda_b^0\to p e^-\bar\nu_e$ & $\Gamma/|V_{bu}|^2$ & & & 6.48$\times$ 10$^2$ &
7.47$\times$ 10$^2$ &\\
\hline
$\Lambda_c^+\to ne^+\nu_e$ & $\Gamma/|V_{cd}|^2$ & & & &
0.26$\times$ 10$^2$ & \\
\hline\hline
\end{tabular}
\end{center}
\vspace*{.5cm}
\section{Acknowledgements}
We would like to thank J\"{u}rgen K\"{o}rner and Peter Kroll for useful
discussions. This work was supported in part by the INTAS Grant 94-739,
the Heisenberg-Landau Program by the Russian Fund of
Fundamental Research (RFFR) under contract 96-02-17435-a and the
State Committee of the Russian Federation for
Education (project N 95-0-6.3-67,
Grand Center at S.-Petersburg State University).
| proofpile-arXiv_065-409 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
\no
Phase ordering kinetics, critical and low temperature dynamics of pure
and random systems are the subject of active research \cite{Brayrev}.
Of particular
interest are the approximate methods to deal with non linear
dynamical equations, which often amount to a self-consistent
resummation of perturbation theory \cite{BCKM}. A much debated case is
the `mode-coupling' approximation, used to describe liquids
approaching their
frozen (glass) phase. Interestingly, this mode-coupling
approximation for systems without disorder can alternatively be
seen as the exact equations for
an associated {\it disordered} model of the spin-glass type
\cite{Krai,frhz,BCKM}.
The simplest mode coupling approximation for the $\f^4$ theory is however
not very good. For example, it predicts for the static critical exponent
$\eta$ the value $2 - \frac{d}{2} $ independently of the number
$n$ of components of the field $\f$. Furthermore, the underlying disordered
model is not stable \cite{BCKM}.
A better behaved resummation scheme is the ``Self-Consistent Screening
Approximation" (SCSA) introduced by Bray in the context of the
static
$\f^4$ theory \cite{bray1,bray2}, and used in other contexts
\cite{MY,LD}. It amounts to resumming self
consistently all the diagrams appearing in the large-$n$
expansion, including those of order $1 \over n$.
Again, this approximation becomes exact for a particular mean-field
like spin-glass model \cite{BCKM}, which turns out to be well defined for all
temperatures and thus ensures that the approximation is well behaved.
The aim of the present paper is to generalize the SCSA
equations to describe the dynamics of the $\f^4$ theory {\it at the critical
point}, and to predict a value for the dynamical exponent $z$.
In section \ref{scsa} we shall introduce the dynamical SCSA and the dynamical
equations in their general form. From section \ref{statics} and throughout
the rest of the paper we assume that time-translation invariance (TTI) and the
fluctuation-dissipation theorem hold at least down to the critical point.
Bray's equations will be recovered as the static limit of our
dynamical equations. The reliability of the SCSA is discussed
quantitatively in the $0$-dimensional static case.
In section \ref{dynamics} we study the equations right at the critical
temperature where dynamical scaling is supposed to hold.
The full solution of these coupled equations, involving {\it scaling functions}
gives in principle
the dynamical exponent $z$ within the SCSA approximation. Unfortunately, as
is often the case \cite{KPZ}, these equations are very hard
to solve, either analytically or even numerically. In section \ref{ansatz1} and
\ref{ansatz2} we thus propose two different ans\"atze for the scaling
functions, which are however much constrained by general requirements.
The second ansatz leads to the exact $O(\eps^2)$ result in the $\eps=4-d$
RG expansion of Halperin, Hohenberg and Ma \cite{HHM}. The numerical value
of the exponent $z$ only very weakly depends on the chosen ansatz,
and turns out to be quite close to the best available Monte-Carlo estimate
for the Ising model in $d=3$ \cite{Heuer}.
\section{The Self Consistent Screening Approximation}
\label{scsa}
Let us consider the coarse-grained Hamiltonian density
\be
{\cal H}[{\bf \f} (\vec{x})] = \frac{1}{2}(\nabla \f (\vec{x}))^2 +
\frac{\mu}{2} \f^2 (\vec{x}) -
\frac{g}{8} \f^4 (\vec{x}),
\label{h1}
\ee
\no
where ${ \f(\vec{x}) }$ is an $n$ component field and
$\vec{x}$ is the $d$ dimensional space variable. With $\f^2 (\vec{x})$
and $\f^4
(\vec{x})$ we indicate respectively $|{\f(\vec{x}) }|^2$ and $(|{
\f(\vec{x}) }|^2 )^2$. The coupling constant
$g$ is negative and of order $n^{-1}$; $\mu$ is a (temperature dependent)
mass term which
vanishes at the mean-field transition point.
The partition function is
\be
Z = \int {\cal D} {\bf \f} e^{-\int d^{d}x \ \frac{{\cal H}[{\bf \f}
(\vec{x})]}{T}},
\ee
\no
In order to introduce the Self Consistent Screening Approximation one
starts from
a large-$n$ expansion formalism. We re-write $Z$ with a gaussian
transformation introducing an
auxiliary field $\s$
\be
Z = \int {\cal D} \s {\cal D} {\bf \f} e^{-\int d^{d}x \ \frac{ {\cal
H}[{\bf \f} (\vec{x}),\s(\vec{x})]}{T}} ,
\ee
\no
$H[{\bf \f} (\vec{x}),\s(\vec{x})]$ being now the Hamiltonian density of
two coupled fields
${\bf \f} (\vec{x})$ and $\s(\vec{x})$.
\be
H[\s,{\bf \f} ] = \frac{1}{2}(\nabla \f (\vec{x}))^2
+ \frac{\mu}{2}
\f^2 (\vec{x}) + \frac{1}{2}\s^2 (\vec{x}) -
\frac{\sqrt{g}}{2}
\s (\vec{x}) \f^2 (\vec{x}) .
\ee
The SCSA amounts to consider the renormalization of the order
$1/n$ diagrams in the Dyson expansion for the correlation functions of the
two fields ${\bf \f} (\vec{x})$ and $\s(\vec{x})$.
Using this resummation scheme Bray \cite{bray1} obtained interesting
results for the static
exponent $\n$ which describes the small momentum behaviour of
the correlation functions. The static SCSA equations for
$\langle {\bf \f} (\vec{x})
{\bf \f} (\vec{x}^{\prime}) \rangle$ (plain line) and
$\langle \s(\vec{x}) \s(\vec{x}^{\prime}) \rangle$ (``dashed" line) are reported
diagrammatically in figure 1.
The bare quantities are indicated respectively with a thinner plain
line and with a dashed line.
\begin{figure}
\epsfbox{diagst.ps}
\vspace{0.5cm}
\caption[]{}
\end{figure}
Our goal is to develop a dynamical generalization of this expansion for
non-conserved Langevin dynamics, starting from the SCSA Hamiltonian. We thus
obtain the following equations of motion for ${\bf \f} (\vec{x},t)$ and
$\s(\vec{x},t)$:
\begin{eqnarray}
\dot{{\bf \f} }(\vec{x},t)& = &-(\nabla^2 + \mu)
{\bf \f} (\vec{x},t) + \sqrt{g} {\bf \f}
(\vec{x},t) \s (\vec{x},t) +
\eta_{\f} (\vec{x},t)
\label{langf} \\
\dot{\s}(\vec{x},t)& = & - \s (\vec{x},t) + \frac{\sqrt{g}}{2}
\f^2(\vec{x},t) + \eta_{\s}(\vec{x},t).
\label{langs}
\end{eqnarray}
\noindent
with two independent thermal noises $\eta_{\f}, \eta_{\s}$.
Let us now consider the two-point functions
\begin{eqnarray}
G_{\f}(\vec{x},\vec{x}^{\prime},t,t^{\prime}) &=& \left<
\frac{\partial{{\bf \f}(\vec{x},t)}} {\partial{\n(\vec{x}^{\prime},t
^{\prime})}}
\right> \\ C_{\f}(\vec{x},\vec{x}^{\prime},t,t^{\prime}) &=& < {\bf \f}
(\vec{x},t)
{\bf \f}(\vec{x}^{\prime},t^{\prime}) >,
\end{eqnarray}
and the corresponding functions for the field $\s$. The SCSA dynamical
equations, which can be seen as a Mode-Coupling approximation on the set of
equations (2.5-6) (see figure 2) then read:
\begin{eqnarray} \nonumber
\Sigma_{\f}(t_1,t_2) & = &
n \frac{g}{2} \delta(t_1-t_2) \int_{0}^{t_{1}} dt_{3} C_{\f}(t_{3} ,t_{3})
G_{\s}^{0} (t_{1},t_{3}) +
\\
&+& g [G_{\f}(t_{1},t_{2})C_{\sigma}(t_{1},t_{2})
+
G_{\s}(t_{1},t_{2})C_{\f}(t_{1},t_{2})] \label{2times1}
\end{eqnarray}
\be
\Sigma_{\s}(t_1,t_2) = n g G_{\f}(t_{1},t_{2})
C_{\f}(t_{1},t_{2})
\ee
\begin{eqnarray}
D_{\f}(t_1,t_2) &=& 2T \delta (t_1-t_2) +
g C_{ \f}(t_1,t_2) C_{ \s}(t_1,t_2)
\\
D_{\s}(t_1,t_2) &=&
2T \delta (t_1-t_2) + n \frac{g}{2} C_{\f}^2 (t_1,t_2),
\label{2times}
\end{eqnarray}
where we have dropped the space coordinates $\vec x$ for clarity,
and introduced the self-energies $\Sigma$, defined as:
\be
G(t,t^{\prime}) = G^0(t,t^{\prime}) + \int_{0}^{t} dt_{1} \int_{t'}^{t_1}
dt_2 G^{0}(t,t_{1}) \Sigma(t_{1},t_{2}) G (t_{2},t^{\prime}),
\ee
(the label $^0$ refers to the bare quantity), and the `renormalized noises'
$D$, defined as:
\be
C(t,t^{\prime}) = \int_{0}^{t}dt_{1} \int_{0}^{t^{\prime}}dt_{2}
G(t,t_{1})D(t_{1},t_{2})G(t^{\prime},t_{2}).
\label{cdef}
\ee
\begin{figure}
\epsfbox{diagdy.ps}
\vspace{0cm}
\caption[]{}
\end{figure}
We shall limit ourselves to consider the above equations
in a regime of stationary dynamics.
That is to say that we will make use of the assumption of
time translational invariance (only differences of times matter),
which allows one to show that
the fluctuation dissipation theorem (FDT) is valid, i.e:
\be
\theta(t-t^{\prime}) \frac{\partial C(t-t^{\prime})}{\partial{t^{\prime}}} =
T G(t-t^{\prime}).
\ee
Extensions of these methods to non stationary low temperature
regime, where this theorem is violated \cite{CK}, will
be subject of further work. In the following, we shall set
the energy scales by choosing $T=1$, and vary the mass term $\mu$ to
reach the critical point.
\section{Static Limit}
\label{statics}
With these assumptions equations (\ref{2times}) reduce to only
two coupled independent equations which have the simplest form in Fourier
space
\begin{eqnarray} \nonumber
\Sigma_{\f}(k,\w) &=& g \int \left[ C_{\s}(k-
k^{\prime},
\w - \w^{\prime}) G_{\f}(k^{\prime},\w^{\prime}) +
C_{\f}(k-k^{\prime},\w - \w^{\prime})
G_{\s}(k^{\prime},\w^{\prime}) \right] Dk'D\w'
\\
&& + \frac{ n g}{2} G_{\s}^{0}(k=0,\w=0)
\int C_{\f}(k^{\prime},\w^{\prime}) Dk'D\w'
\label{sigf} \\
\Sigma_{\s}(k,\w) &=& {n g} \int C_{\f}(k-
k^{\prime},\w - \w^{\prime})
G_{\f}(k^{\prime},\w^{\prime})
Dk'D\w' .
\label{sigs}
\end{eqnarray}
where $Dk' \equiv \frac{d^d k'}{(2\pi)^d}$ and $D\w' \equiv \frac{d\w'}{2\pi}$.
Using the fact that $C(k,t=0) \equiv {\cal C}(k)$ is equal to $G(k,\omega=0)$
(from FDT and the Kramers-Kronig (KK) relations), and using again
the KK relations, it is easy to check that for $\w=0$ one recovers exactly
the static
SCSA equations \cite{bray1}, namely
\begin{eqnarray}
{\cal C}_{\f}(k) & = & \frac{1}{\mu + k^2 - g \int Dk'
{\cal C}_{\f}(k-k'){\cal C}_{\s}(k') -\frac{g n}{2}
\int Dk' {\cal C}_{\f}(k')} \nonumber \\ {\cal C}_{\s}(k)
& = & \frac{1}{1 - \frac{g}{2} n
\int Dk'{\cal C}_{\f}(k-k'){\cal C}_{\f}(k')},
\label{cstatiche}
\end{eqnarray}
\noindent
In order to test the validity of this approximation, it is
interesting to consider the case of zero spatial dimensions \cite{bray2}. Let us
set $n=1$ which is a bad case for the SCSA which should become more
accurate the larger $n$ is. We will compare equations (\ref{cstatiche}) with
the exact static
correlation function which in zero dimension can be calculated
analytically and is
\be
{\cal C}_{exact} = -{1\over { \mu}} + {\mu \over g} -
{{\mu \,{\rm K}_{-\frac{3}{4}}(
{{{\mu^2} }\over {4\,g}})}\over
{2\,g\,{\rm K}_{1\over 4}(
{{- \,{\mu^2} }\over {4\,g}})}} -
{{\mu \,{\rm K}_{5\over 4}(
{{{\mu^2} }\over {4\,g}})}\over
{2\,g\,{\rm K}_{1\over 4}(
{{- \,{\mu^2} }\over {4\,g}})}},
\ee
\no
where $K_{n}(a)$ is the modified Bessel function of the second kind.
Equations (\ref{cstatiche}) give for ${\cal C}_{\f}$:
\be
{\cal C}_{scsa} = \frac{1}{\left(\mu -
n \frac{g}{2} {\cal C}_{scsa} - g \frac{{\cal C}_{scsa}}
{ \left( 1 - \frac{g}{2} n
{\cal C}_{scsa}^2 \right)}\right)} .
\ee
From plotting the relative
difference of the two correlation functions versus the coupling (see
figure 3)
we can see that SCSA is quite close to the exact theory.
In particular, the asymptotic behaviour in the $|g| \rightarrow \infty$
limit of the two
functions is
\be
\lim_{|g| \rightarrow \infty} \sqrt{|g|}
{\cal C}_{scsa} = 2(\sqrt{2} - 1) {\mbox{ and }}
\lim_{|g| \rightarrow \infty} \sqrt{|g|} C_{exact} =
{{2\,{\sqrt{2}}\,{\Gamma}({3\over 4})}\over
{{\Gamma}({1\over 4})}}.
\ee
For all $g$, the relative difference is actually bounded by:
\be
\frac{\left| {\cal C}_{exact} - {\cal C}_{scsa} \right|}
{{\cal C}_{exact}}
< 1 - {{{\sqrt{-1 + {\sqrt{2}}}}\,{\rm \Gamma}({1\over 4})}\over
{2\,{\rm \Gamma}({3\over 4})}}
= 0.0479...
\ee
\no
We can also compare the small$-g$
expansions of the two theories which give
\begin{eqnarray}
{\cal C}_{exact} &=& \frac{1}{ \mu}(1 + \frac{3}{2 \mu^2} g +
\frac{21}{4 \mu^4} g^2) \\
{\cal C}_{scsa} &=& \frac{1}{ \mu}(1 + \frac{3}{2 \mu^2} g +
\frac{5}
{\mu^4} g^2)
\end{eqnarray}
showing explicitly how the two theories
differ already at order $g^{2}$.
The self consistent nature of the approximation however keeps the SCSA in good
agreement with the exact theory even for large values of the coupling constant
as remarked before.
It is instructive, in passing, to compare the SCSA with the simple
Hartree ($n = \infty$) resummation scheme, which is also the
Gaussian variational result. One defines $F_{H} = \min{\{F\} }$ where
\be
F = F_{0} + <H-H_{0}>,
\ee
with
\be
F_{0} = - \ln \int {\cal D} {\bf \f} e^{-\frac{\tilde{\mu}\f^2}{2}} =
- \ln{(\frac{2 \pi}{\tilde{\mu}})}
\ee
\be
<H_{0}> = \frac{1}{2 }
\ee
\be
<H> = \int {\cal D} {\bf \f} e^{-\frac{\tilde{\mu}\f^2}{2}} \left(
\frac{\mu}{2}\f^{2} - \frac{g}{8}\f^{4} \right) = \left(
\frac{\mu}{2\tilde{\mu}} - \frac{3g}{8 \tilde{\mu}^{2}} \right)
\ee
Minimising $F$ with respect to $\tilde{\mu}$ we find
\be
\mu_{H} = \frac{\mu + \sqrt{\mu^2 - 6g}}{2},
\ee
and consequently
\be
{\cal C}_{H} = <\f^2>_{\mu_{H}} = \frac{2}{\mu + \sqrt{\mu^2 - 6g}}.
\ee
As can be seen from figure 3, the SCSA turns out to be fairly better
than the Hartree variational approach (at least in this particular case
of $n=1$ and $d=0$).
\begin{figure}
\epsfbox{comparison.ps}
\vspace{0.5cm}
\caption[]{Relative difference between the exact result and the Hartree
(${\cal C}_H$) and the SCSA (${\cal C}_{scsa}$) approximations,
in the case $n=1,d=0$.}
\end{figure}
\section{Critical Dynamics}
\label{dynamics}
We shall now work right at the critical point $\mu_{c}$ such that
the renormalised mass vanishes (therefore eliminating the `tadpole'
contribution in Eq. \ref{2times1}). We shall search for solutions under
the general dynamic scaling form (valid in the small-$k$ and
small-$\w$ limit):
\begin{eqnarray}
G_{\f}(k,\w) &=& \frac{1}{k^{\Delta}} n_{\f}(\frac{\w}{k^{z}})
\hspace{2truecm} G_{\s}(k,\w) =
\frac{1}{k^{\Delta^{\prime}}} n_{\s}(\frac{\w}{k^{z}})
\nonumber
\\ C_{\f}(k,\w) &=& \frac{2}{ \w k^{\Delta}} Im \left[
n_{\f}(\frac{\w}{k^{z}}) \right ]
\hspace{.7truecm} C_{\s}(k,\w) = \frac{2}{ \w k^{\Delta^{\prime}}}
Im \left[
n_{\s}(\frac{\w}{k^{z}})
\right ].
\label{generalscaling}
\end{eqnarray} where we have defined $\Delta = 2- \n$, and used FDT.
\no
Setting first $\w=0$, one finds by matching the momentum dependence of the
left and right hand sides of (\ref{sigf}-\ref{sigs}) that:
\be
\Delta^{\prime} = d - 2 \Delta = d - 4 + 2 \n.
\ee
Note that in mean field, $z=2$, $\Delta =2$, $\eta=0$ and $\Delta'=0$.
Identification of the prefactors yields:
\be
n_{\s}(0) n_{\f}^2 (0) = - \frac{2}{ f(\eta,d) n g }
\label{relstat}
\ee
where
\be
f(\eta,d) = \frac{1}{(4 \pi)^{d/2}} \frac{\Gamma[\Delta - \frac{d}{2}]
\Gamma[\frac{d-\Delta}{2}]^2 }{\Gamma[d-\Delta] \Gamma[\frac{\Delta}{2}]^2} .
\label{fbray}
\ee
and an extra equation fixing $\eta$ as a function of $d$ and $n$, which
we do not write explicitely \cite{bray1}.
Now let us consider the other case where $k=0$ and $\w > 0$ (but small).
Taking the imaginary part of (\ref{sigf}-\ref{sigs}), one obtains:
\begin{eqnarray}
Im \left[ \Sigma_{\f}(0,\w) \right] &=& \frac{S \w }{n n_{\f} (0)}
\int q^{\Delta -1} dq ds
\frac{Im\left[ f_{\f} (\frac{(\w -s)}{q^{z}}) \right]
Im\left[ f_{\s} (\frac{s}{q^{z}}) \right] }{s (\w -s)}
\label{fk=0} \\
Im \left[ \Sigma_{\s}(0,\w)\right] &=& \frac{S}{n_{\s}(0)} \int
q^{\Delta ^{\prime} -1} dq d s \frac{Im\left[ f_{\f}
(\frac{(\w -s)}{q^{z}}) \right] Im\left[ f_{\f}
(\frac{(s)}{q^{z}}) \right] }{s},
\label{sk=0}
\end{eqnarray}
\no
where $f_{\f,\s}(x) = n_{\f,\s}(x)/n_{\f,\s}(0)$. We also defined
\be
S = \frac{2 n g \Omega_{d}}{(2 \pi)^{(d+1)}} n_{\f}^2(0) n_{\s}(0)
\equiv -\frac{4 \Omega_{d}}{f(\eta,d)(2 \pi)^{(d+1)}} \label{eq1}
\ee
In general the scaling functions can be written
\begin{eqnarray}
Im[f_{\f}(x)] & \doteq & A \tilde{f_{\f}} (a x) \\ \nonumber
Im[f_{\s}(x)] & \doteq & A'\tilde{f_{\s}} (a^{\prime} x),
\end{eqnarray}
\no
with by convention $\lim_{u \to \infty}u^{\Delta/z} \tilde{f_{\f}}(u) =1$ and
$\lim_{u \to \infty} u^{\Delta'/z} \tilde{f_{\s}}(u)=1$. This asymptotic behaviour
is required for the $k \to 0$ limit to be well defined, if
(\ref{generalscaling}) is correct. Furthermore, the small-$\w$
behaviour of
the imaginary part of the response function is expected to be regular
for $k$ finite, and hence $\tilde{f}(u) \propto u$ for $u \to 0$.
$A,A'$ are coefficients setting the scale of the imaginary part of
the response function while $a,a'$ are coefficients
setting the frequency scales.
Using the fact that the imaginary and real part of the response function are
power-laws at large frequencies, which imply that their ratio is
$\tan\left( \frac{\pi \Delta}{2z} \right)$
(resp. $\tan\left( \frac{\pi \Delta'}{2z} \right)$), one finds that:
\begin{eqnarray} \nonumber
\frac{a^{\Delta/z} }{A} \sin^{2}
\left( \frac{\pi \Delta}{2z} \right) &=&
\frac{ S }{nz} \int_{0}^{\infty}
\frac{dx}{x^{1 + \Delta/z}} \int_{-\infty}^{\infty} \frac{du}{u(1-u)}
Im\left[ f_{\f} (x (1-u)) \right] Im\left[ f_{\s}(x u) \right] \\
\frac{\alpha'^{\Delta^{\prime}/z} }{A'}
\sin^{2} \left( \frac{\pi \Delta^{\prime}}{2z} \right) &=& \frac{ S }{z}
\int_{0}^{\infty} \frac{dx}{x^{1 + \Delta^{\prime}/z}}
\int_{-\infty}^{\infty} \frac{du}{u} Im\left[ f_{\f} (x (1-u)) \right]
Im\left[ f_{\f} (x u) \right]
\label{scsagens}
\end{eqnarray}
It is easy to show that these equations actually only depend on the value of
the {\it ratio} of frequency scales $y=\frac{a'}{a}$. The coefficient $A$
can be fixed using the KK relation, since the involved integral converges,
which means that the small-$k$
behaviour of the real part of the correlation function is
fully determined by the imaginary part in the scaling region $\omega, k \to 0$.
Hence:
\be
1 = \frac{A}{\pi} \int_{-\infty}^{\infty} dx \frac{\tilde{f_{\f}} (x )}{x}.
\label{kk}
\ee
The corresponding integral for $\tilde{f_{\s}}$ does not converge for
large $x$, meaning that the non-scaling region is needed to saturate the
sum-rule. Hence, we must use another relation to fix $A'$, which we
choose to be the small-$\w$ expansion of Eq. (\ref{sk=0}).
Thus, {\it if} the functions $\tilde{f_{\f}},\tilde{f_{\s}}$ were
known, we would have four equations to fix four constants: $A,A',y$,
and, of course, the dynamical exponent $z$, in
terms of $d$ and $n$. $\tilde{f_{\f}},\tilde{f_{\s}}$
are in principle also fixed by the full equations for
arbitrary $\frac{\omega}{k^z}$. However,
as in other similar cases \cite{KPZ}, these equations are very
hard to solve, either analytically or numerically. We will
thus propose ans\"atze for these functions, which have to satisfy the
above general requirements. Note that once $A,A',a,a'$ have been pulled
out, the only
freedom is in the {\it shape} of these functions. We shall thus work
with two such ans\"atze, which will turn out to give very similar answers
for $z$. This
was also the case in the context of the KPZ equation \cite{KPZ}.
\section{Ansatz 1}
\label{ansatz1}
The simplest ansatz one can think of, which generalizes the mean field
shape:
\be
\tilde{f_{\f}}(x) = \frac{x}{(1+x^2)}
\label{fmf}
\ee
reads:
\begin{eqnarray}
\tilde{f_{\f}}(x) &=& \frac{x}{\left(1+x^2 \right)^{\alpha}} \\
\tilde{f_{\s}}(x) &=& \frac{x}{\left(1+x^2\right)^{\ap}} ,
\end{eqnarray}
where we have set
\begin{eqnarray}
\alpha &\doteq& \frac{\Delta + z}{2z} \\
\ap &\doteq& \frac{\Delta^{\prime} + z}{2z}.
\label{alphas}
\end{eqnarray}
(Note that $\alpha=1$ in mean field).
These functions have indeed the correct asymptotic behaviours; they go linearly
to zero for small values of the argument and behave as power laws
($\tilde{f_{\f}}(x) \simeq x^{-\frac{\Delta}{z}} $ and $
\tilde{f_{\s}}(x) \simeq x^{-\frac{\Delta^{\prime}}{z}}$) in the large-$x$ limit.
We can now use (\ref{kk}) to determine $A$
\be
A = \sqrt{\pi} \frac{\Gamma[\alpha]}{\Gamma\left[\alpha - \frac{1}{2} \right]}.
\ee
The small-$\w$ expansion of $Im \Sigma_{\s}(k,\w) $ can be matched with that of
the right hand side of Eq.(\ref{sk=0}) leading to
the following equation
\be
y = -\frac{2 A^2}{A' f(\eta,d) (2 \pi)^{d+1}}
\int_{-\infty}^{\infty} d^d q \frac{1}{|q|^{\Delta} |1-q|^{\Delta + z}}
\int_{-\infty}^{\infty} dt
\frac{1}{(1+t^2)^{\alpha} \left(1+ (\frac{|q|^{z} t}{|1-q|^{z}} )^2 \right)^{\alpha}}
\label{ggg}
\ee
After some algebraic manipulations we obtain for the last three equations:
\begin{eqnarray} \nonumber
\sin^{2} \left( \frac{\pi \Delta}{2z} \right) &=&- \frac{A^2 A'y S}{2 nz}
B\[[1 -\frac{\Delta}{2z}, \frac{d}{2z}\]] \\& &
\int_{-\infty}^{\infty} \frac{du}{ |u|^{2-\frac{\Delta}{z}}} F\[[\ap,1
-\frac{\Delta}{2z},\alpha + \ap,1- y^2 \frac{(1-u)^2}{u^2}\]] \label{scsa11}
\end{eqnarray}
\begin{eqnarray} \nonumber
\sin^{2} \left( \frac{\pi \Delta^{\prime}}{2z} \right) &=&- \frac{A^2 A^{\prime}S}{2z}
y^{-\frac{\Delta^{\prime}}{z}} B \[[1
-\frac{\Delta^{\prime}}{2z}, \frac{d}{2z}\]]\\
& & \int_{-\infty}^{\infty}
\frac{u du}{|u|^{2 -\frac{\Delta^{\prime}}{z}}} F\[[\alpha,1 -\frac{\Delta^{\prime}}{2z},2
\alpha,\frac{2u-1}{u^2}\]]
\label{scsa12}
\end{eqnarray}
\begin{eqnarray} \nonumber
y &=& \pi \frac{A^2 S}{z A' \Omega_{d}}
B\[[\frac{1}{2},2 \alpha -\frac{1}{2}\]] \\
& & \int_{0}^{\infty}
dq q^{d-2-\Delta}
\int_{|1-q|^{2z}}^{|1+q|^{2z}} \frac{dx}{x^{\frac{\Delta +3z- 2}{2z}}}
F\[[\alpha,\frac{1}{2},2\alpha,1-\frac{q^{2z}}{x}\]],
\label{scsa13}
\end{eqnarray}
where $B[a,b]$ and $F[a,b,c,x]$ are the Euler Beta and Hypergeometric
functions and where the last equation (\ref{scsa13}) was written for the
special case $d=3$ which we shall consider below.
We can solve analytically Eqs. (\ref{ggg},\ref{scsa11},\ref{scsa12})
at order $\eps^2$ to compare
with the exact RG treatment of \cite{HHM}. At lowest order we obtain:
\begin{eqnarray}
c &=& \frac{8 \ln{2}}{ \pi}
\frac{ \arctan \sqrt{\frac{1 - y^2}
{y^2}}}{\sqrt{1 - y^2}}-1 \\
A' &=& - \frac{\pi \eps}{4} \\
y &=& \frac{4 \ln 2 }{\pi}
\label{oeps2}
\end{eqnarray}
where we have defined, following \cite{HHM},
\be
z = 2 + c \eta .
\label{formz}
\ee
The order $O(\eps^2)$ RG result reads, $c =6\ln{\frac{4}{3}} -1 =0.7261$.
The form (\ref{formz}) means that to lowest order $z$ depends on $n$
only through the static exponent $\n$. On the other hand,
Eqs. (\ref{oeps2}) give
\be
c = 0.8376,
\ee
in slight disagreement with the exact result.
This comes from the fact that while our ansatz for $\tilde f_{\f}$
is exact in the limit $\epsilon \to 0$,
the corresponding ansatz for $\tilde f_{\s}$ is already wrong at lowest
order since it does not satisfy Eq. (\ref{sk=0}).
In our second ansatz, we thus keep
the same shape for $\tilde f_{\f}$, but choose
for $\tilde f_{\s}$ a form which is
exact when $\epsilon \to 0$.
\section{Ansatz 2}
\label{ansatz2}
Knowing the mean field form for $f_{\f}(x)$ we can, at lowest
order in $\epsilon$, write for $Im[f_{\s}(x)]$
\be
Im f_{\s}(x) = 2^{d-4} \frac{ f(\eta,d)}{\pi^{d/2}}
\frac{(2\pi)^d}{\Gamma[2-\frac{d}{2}]} Im \[[ \frac{1}{\xi(x)} \]]
\label{gsgen}
\ee
where
\be
\xi(x) = 1 - \frac{\eps}{2} \int_{0}^{1} dt \log\[[ 1-t^2 - 2 i x (1-t) \]].
\ee
It is then straightforward to generalize $Im[f_{\s}(x)]$ to general dimensions
as:
\be
\tilde {f_{\s}} \propto Im \[[ (2 - i x )^{1 -
\frac{\Delta^{\prime}}{z}} - (1 - i x )^{1 - \frac{\Delta^{\prime}}{z}} \]]
\label{fsfin}
\ee
with a prefactor ensuring that the coefficient of $x^{-\frac{\Delta^{\prime}}{z}}$
for large $x$ is unity. Eq. (5.8) is now
replaced by: \be
\sin^2 \((\frac{\pi \Delta}{2 z} \)) = \frac{A^2 A' S b}{n z} \int_{0}^{\infty}
\frac{{\mbox d} r}{r^{\frac{\Delta}{z}}} \int_{-\infty}^{\infty} {\mbox d} u
\frac{Im \[[ (2 - i \frac{\pi}{2
\ln{2}} (y r u) )^{1 - \frac{\Delta^{\prime}}{z}} -
(1 - i \frac{\pi}{2 \ln{2}}
(y r u) )^{1 - \frac{\Delta^{\prime}}{z}} \]]}{u \[[1 + r^2 (1-u)^2 \]]^{\alpha}}.
\label{scsa21}
\ee
where now $b$ is given by:
\be
b = \frac{2 \ln{2}}{\pi \((2^{ - \frac{\Delta^{\prime}}{z}} -1\))
\((\frac{\Delta^{\prime}}{z} -1 \)) }.
\ee
We finally obtain a set of equations for $z$ of
the same kind as (\ref{scsa11}-\ref{scsa13}) but which now exact up to
$O(\epsilon^2)$, as we have checked directly.
\section{Numerical Results}
We solved numerically both sets of equations in $d= 3$ for $n=1,...,10$.
We used the values of $\eta(d=3,n)$ that can be derived from the
formula reported in
\cite{bray1}.
The values obtained for $z$ are reported in the following table.
\medskip
\begin{equation}
\begin{array}{|c|c|c|} \hline
\par
n & z \mbox{ (ansatz 1 )} & z \mbox{ (ansatz 2 )}\\ \hline
1& 2.119 & 2.113 \\ \hline
2 & 2.071 & 2.069 \\ \hline
3 & 2.050& 2.049 \\ \hline
4 & 2.038 & 2.038 \\ \hline
5 & 2.031& 2.031 \\ \hline
6 & 2.0258& 2.0258 \\ \hline
7 & 2.0223& 2.0222 \\ \hline
8 & 2.0196& 2.0195 \\ \hline
9 & 2.0174& 2.0174 \\ \hline
10 & 2.0157& 2.0157 \\ \hline
\end{array}
\nonumber
\end{equation}
\medskip
As it was hoped, the results are fairly independent from
the ansatz used, which is more and more true for large $n$. The result
for $n=1$ is rather close to the best Monte-Carlo estimate
of ref. \cite{Heuer}, which gives $z=2.09 \pm 0.02$. Let us note
however
that the SCSA overestimates significantly $\eta$ in $d=3$.
In figure (4) we compare the two different choices for the
scaling function $f_{\s}(x)$ with their relative values of the
parameters $y$, $A'$ and $z$, and in the case
$n=1,d=3$. We notice that the constraints for small $x$ and
large $x$ restrict very much the freedom on the shape of this function.
\begin{figure}
\epsfbox{diff.ps}
\vspace{0.5cm}
\caption[]{The two ans\"atze for the functions $f_{\s}(x)$, $n=1$}
\end{figure}
Finally, a linear regression of our results for $n=1-10$
gives $z \simeq 2 + c \eta$ with $c=0.64$, which is lower
that the $O(\eps^2)$ result, but larger that
the exact result for $d=3$, $n \to \infty$, i.e. $c=\frac{1}{2}$ \cite{HHM}.
\section{Conclusions}
The aim of this paper was extend the static Self-Consistent screening
approximation to dynamics, in particular to calculate the properties of the
critical dynamics of the $\phi^4$ model. Although the resulting
equations cannot be fully solved, a much constrained ansatz leads to a value
of the exponent $z$ in rather good agreement with Monte-Carlo data.
Our work was originally inspired by glassy dynamics: the
SCSA equations actually describe in exactly the dynamics of some mean-field
spin glass like models. It would be interesting to study these equations in
the low temperature phase, where dynamics becomes non stationary (aging)
and FDT is lost. For $\phi^4$ models, this corresponds to a coarsening
regime \cite{Brayrev}. It would be
interesting to know whether the SCSA equations describe properly this
regime, and can compete with other approximation schemes
\cite{Brayrev,NB}.
\vskip 1cm
{\it Acknowledgments} It is a pleasure for us to thank A.~Barrat,
A.~Bray, L.F.~Cugliandolo, J.~Kurchan,
E.~Maglione, M.~M\'ezard, R.~Monasson, G.~Parisi and P.~Ranieri for
very instructive discussions.
\bibliographystyle{IEEE}
| proofpile-arXiv_065-410 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
Recent remarkable developments in superstring theory
lead to the discovery that five known superstring
theories in ten dimensions are related by duality
transformations and that there are also $M$-theory
in $11d$ and $F$-theory in $12d$ that are useful
in the study of the moduli space of quantum vacua \cite {FILQ}
-\cite{Vaf}.
Duality requires the presence of extremal black holes
in the superstring spectra. A derivation of the Bekenstein-Hawking
formula for the entropy of certain extremal black holes was given
using the D-brane approach \cite{SV}-\cite{pol2}.
In all these developments
the study of p-brane solutions of the supergravity equations
play an important role \cite{DGHR}-\cite{berg}.
In this paper we shall consider the following action
\begin{equation}
I=\frac{1}{2\kappa ^{2}}\int d^{D}x\sqrt{-g}
(R-\frac{1}{2}(\nabla \phi) ^{2}
-\frac{1}{2(q+1)!}e^{-\alpha \phi}F^{2}_{q+1}
-\frac{1}{2(d+1)!}e^{\beta \phi}G^{2}_{d+1}),
\label{011}
\end{equation}
It describes the interaction of the gravitation field $g_{MN}$
with the dilaton $\phi$ and with two antisymmetric fields:
$F_{q+1} $ is a closed $q+1$-differential form and $G_{d+1}$
is a closed $d+1$-differential form. Various supergravity
theories contain the terms from (\ref{011}).
The aim of this paper is to present a solution of (\ref{011})
with the metric of the form
\begin{equation}
ds^{2}=H_{1}^{-2a_{1}}H_{2}^{-2a_{2}}\eta_{\mu \nu} dy^{\mu} dy^{\nu}+
H_{1}^{-2b_{1}}H_{2}^{-2a_{2}}\delta_{nm}dz^{n}dz^{m}+
H_{1}^{-2b_{1}}H_{2}^{-2b_{2}}\delta_{\alpha\beta}dx^{\alpha}dx^{\beta},
\label{12}
\end{equation}
where $\eta_{\mu\nu}$ is a flat Minkowski metric,
$$\mu, ~\nu = 0,...,q-1;~~m,n=1,2,...,d-q,$$ and
$$\alpha,\beta =1,...,D-d.$$
For definitness we assume that $ D>d\geq q$.
The parameters $a_{i}$ and
$b_{i}$ in the solution (\ref{12}) are rational functions of
the parameters in the action (\ref{011}):
\begin{equation}
a_{1}=\frac{2\tilde q}{\alpha^{2}(D-2)+2q\tilde q},
~~~~a_{2}=\frac{\alpha^{2}(D-2)}
{\alpha^{2}d(D-2)+2\tilde d q^{2}}
\label{13}
\end{equation}
\begin{equation}
b_{1}=-\frac{2q}{\alpha^{2}(D-2)+2q\tilde q},
~~~~b_{2}=-\frac{\alpha^{2}d(D-2)}{\tilde d[
\alpha^{2}d(D-2)+2\tilde d q^{2}]}
\label{14}
\end{equation}
where
\begin{equation}
\tilde d=D-d-2,~~~~\tilde q=D-q-2.
\label{14a}
\end{equation}
Our solution (\ref{12}) is valid only if the following
relation between parameters in the action is satisfied
\begin{equation}
\alpha\beta=\frac{2q\tilde d}{D-2}
\label{15}
\end{equation}
There are two arbitrary harmonic functions
$H_{1}$ and $H_{2}$ of variables $x^{\alpha}$ in (\ref{12}),
\begin{equation}
\Delta H_{1}=0,~~~~~\Delta H_{2}=0.
\label{16}
\end{equation}
Non-vanishing components of the differential form are given by
\begin{equation}
{\cal A}_{\mu_{1}...\mu_{q}}=h
{\epsilon_{\mu_{1}...\mu_{q}}}H_{1}^{-1},~~~~ F=d{\cal A},
\label{18}
\end{equation}
\begin{equation}
{\cal B}_{I_{1}...I_{d}}=k
\epsilon_{I_{1}...I_{d}}H_{2}^{-1}, ~~~G=d{\cal B},~~~I=0,...d-1.
\label{18a}
\end{equation}
Here $\epsilon _{123..,q}=1$, $\epsilon _{123...d}=1$ and
$h$ and $k$ are given by the formulae
\begin{equation}
h^{2}=\frac{4(D-2)}
{\alpha ^{2}(D-2)+2q\tilde q},
\label{194}
\end{equation}
\begin{equation}
k^{2}=\frac{2\alpha^{2} (D-2)^{2}}
{{\tilde d}[\alpha ^{2}d(D-2)+2q^{2}{\tilde d}]},
\label{195n}
\end{equation}
The dilaton field is
\begin{equation}
\phi=\frac{1}{2}\beta k^{2}\ln H_{2}-\frac{1}{2}
\alpha h^{2}\ln H_{1}.
\label{17}
\end{equation}
We obtain the solution (\ref{12}) by reducing the Einstein
equations to the system of algebraic equations. To this end
we introduce a linear dependence between functions
in the Ansatz (see below).
The solution (\ref{12}) consists of three blocks, the first block
consists of variables $y$, another of variables $z$
and the other of variables $x$ and all functions depend only on $x$.
We shall call it
the three-block p-brane solution.
We shall consider also the following "dual" action
\begin{equation}
\tilde I=\frac{1}{2\kappa ^{2}}\int d^{D}x\sqrt{-g}
(R-\frac{1}{2}(\nabla \phi) ^{2}
-\frac{1}{2(q+1)!}e^{-\alpha \phi}F^{2}_{q+1}
-\frac{1}{2({\tilde d}+1)!}e^{\tilde\beta \phi}G^{2}_{{\tilde d}+1}),
\label{11}
\end{equation}
where $G_{\tilde d+1}$ is a closed $\tilde d+1$-differential form.
If $\tilde{d}$ is related to $d$ by (\ref {14a}) and
\begin{equation}
\tilde\beta =-\beta
\label{11a}
\end{equation}
then the solution for the metric (\ref{12}) with the differential
form $F$ (\ref{18}) and the dilaton (\ref{17}) is
valid also for the action (\ref{11}). An expression for
the antisymmetric
field $G$ will be different, namely
\begin{equation}
G^{\alpha_{1}...\alpha _{\tilde d+1}}=k
H_{1}^{\sigma_{1}}H_{2}^{\sigma_{2}}
\epsilon ^{\alpha _{1}...\alpha_{\tilde d+1}\beta}
\partial _{\beta} H_{2}^{-1}.
\label{1999}
\end{equation}
here $\epsilon ^{123..\tilde d+2}=1$ and
\begin{equation}
\sigma_{1}=\frac{\alpha\beta h^{2}}{2}(1-\frac{1}{\tilde d }),
~~~~\sigma_{2}=\frac{\beta k^{2}}{2}(\frac{1}{\tilde d }-1)
\label{917}
\end{equation}
The three-block p-branes solution
for the Lagrangian with one differential form for various dimensions
of the space-time was found in \cite{AV}. It
contains previously known D=10 case \cite{Tseytlin,CM}.
Equations of motion for the case of one form corresponds to equation of
motion for ansatz (\ref{12}), (\ref{18}) and (\ref{1999}) for the
dual action (\ref{11}) when $\alpha$=
$\beta$ and $q=\tilde{d}$.
Note that the metric (\ref{12}) describes also the solution for the
action with
the form $F_{q+1}$ replaced by its dual $F_{\tilde{q}+1}$ with
$\tilde{q}+q+2=D$ and $\alpha \to \tilde{\alpha}=-\alpha$.
One can also change two forms $F$ and $G$ to their dual version
without changing the metric (\ref{12}).
To illustrate our method
on a simple example we first consider in Sect. 3 the simple case when
in (\ref{011})
$d =q$ and one has only two blocks in the metric.
Then in Sect. 4 we derive the solution (\ref{12}). In Sect. 5
we consider particular cases of the solution (\ref{12})
and obtain different known solutions. In Appendix the solution
of the system
of algebraic equations is given.
\section{Two block solution}
To illustrate the method of solution in this section
we consider the simple case when the system of
algebraic equations can be easily solved. Let us consider the action
\begin{equation}
I=\frac{1}{2\kappa ^{2}}\int d^{D}z\sqrt{-g}
[R-\frac{1}{2}(\partial \phi) ^{2}-
\frac{e^{-\alpha \phi}}{2(d+1)!}F^{2}_{d+1}
-\frac{e^{\beta \phi}}{2(d+1)!}G^{2}_{d+1})
\label{31}
\end{equation}
The Einstein equations for the action (\ref{11}) read
\begin{equation}
R_{MN}-\frac{1}{2}g_{MN}R=T_{MN},
\label{19a}
\end{equation}
where the energy-momentum tensor has the form
$$T_{MN}= \frac{1}{2}
(\partial _{M}\phi \partial _{N}\phi -
\frac{1}{2}g_{MN} (\partial \phi)^{2})
$$
$$
+ \frac{1}{2d!}e^{-\alpha \phi}
( F_{MM_{1}...M_{d}}F_{N}^{M_{1}...M_{d}}-
\frac{1}{2(d+1)}g_{MN}F^{2})+
$$
\begin{equation}
\frac{1}{2d!}e^{\beta \phi}
(G_{MM_{1}...M_{d}}G_{N}^{M_{1}...M_{d}}-
\frac{1}{2(d+1)}g_{MN}G^{2})
\label{32}
\end{equation}
and one has also equations of motion
\begin{equation}
\partial _{M}(\sqrt{-g}e^{-\alpha \phi}F^{MM_{1}...M_{d}})=0
\label{33}
\end{equation}
\begin{equation}
\partial _{M}(\sqrt{-g}e^{\beta \phi}G^{MM_{1}...M_{d}})=0
\label{34}
\end{equation}
\begin{equation}
\partial _{M}(\sqrt{-g}g^{MN}\partial _{N }\phi )
+
\frac{\alpha }{2(d+1)!}\sqrt{-g}e^{-\alpha \phi}F^{2}-
\frac{\beta }{2(d+1)!}\sqrt{-g}e^{\beta \phi}G^{2}=0
\label{35}
\end{equation}
We use the following Ansatz
\begin{equation}
F=d{\cal A} , ~~~~~{\cal A}_{01...d-1}=\gamma _{1}e^{C_{1}(x)}
\label{36}
\end{equation}
\begin{equation}
G=d{\cal B} ,~~~~~ {\cal B}_{01...d-1}=\gamma _{2}e^{C_{2}(x)}
\label{37}
\end{equation}
\begin{equation}
ds^{2}=e^{2A(x)}\eta_{\alpha \beta} dy^{\alpha} dy^{\beta}
+e^{2B(x)}dx^{i}dx^{i},
\label{38}
\end{equation}
$\alpha$, $\beta$ =0,1,...,d-1, $\eta_{\alpha \beta}$
is a flat Minkowski metric, $i,j$ =d,...,D.
With the above Ansatz equations (\ref{19a}) are reduced to
the following system of equations:
\begin{eqnarray}
(d-1)\Delta A+
(\tilde{d}+1)\Delta B +
\frac{d(d-1)}{2}(\partial A)^{2} +
\frac {\tilde{d}(\tilde{d}+1)}{2}(\partial B)^{2}+
(d-1)\tilde{d}(\partial A\partial B) = \nonumber \\
-\frac{\gamma _{1}^{2}}{4}e^{\alpha \phi -2dA +2c_{1}}(\partial
c_{1})^{2}-
\frac{\gamma _{2}^{2}}{4}e^{\alpha _{2}\phi -2dA +2c_{2}}(\partial
c_{1})^{2}-
\frac{1}{4}(\partial \phi)^{2}
\label{39}
\end{eqnarray}
and
$$
-d(\partial _{m}\partial _{n} A +\partial_{m}A \partial _{n}A )
-{\tilde d}(\partial _{m}\partial _{n}B
-\partial _{m}B\partial _{n}B)
+d(\partial _{m}A\partial _{n} B+\partial _{m}B\partial _{n} A)$$
$$
+\delta _{mn}[d\Delta A + \frac{d(d+1)}{2}(\partial A)^{2}
+{\tilde d}\Delta B+ \frac{{\tilde d}({\tilde d}+1)}{2}(\partial B)^{2}
+d({\tilde d}-1)(\partial A \partial B)]=$$
$$\frac{1}{2}[\partial _{m}\phi\partial_{n} \phi -\frac{1}{2}\delta_{mn}
(\partial \phi)^{2}]
-\frac{\gamma _{1}^{2}}{4}e^{\alpha \phi -2dA +2C_{1}}
[\partial _{m}C_{1}\partial _{n}C_{1} -\frac{1}{2}\delta _{mn}(\partial
C_{1})^{2}]
$$
\begin{equation}
-\frac{\gamma _{2}^{2}}{4}e^{-\beta \phi -2dA +2C_{2}}
[-\partial _{m}C_{2}\partial _{n}C_{2} -\frac{1}{2}\delta _{mn}(\partial
C_{2})^{2}]
\label{310}
\end{equation}
Here $\tilde d=D-d-2$.
Equations for antisymmetric fields are
\begin{equation}
\partial _{m}(e^{-\alpha \phi -dA +{\tilde d}B +c_{1}}\partial
_{m}c_{1})=0
\label{311}
\end{equation}
\begin{equation}
\partial _{m}(e^{\beta \phi -dA +{\tilde d}B +c_{1}}\partial
_{m}c_{2})=0
\label{312}
\end{equation}
Equation of motion for dilaton
$$\partial _{m}(e^{-dA +{\tilde d}B}\partial _{m}\phi )
-\frac{\alpha\gamma _{1}^{2}}{2}e^{-\alpha \phi -2dA +2C_{1}}
(\partial _{m}C_{1})^{2}$$
\begin{equation}
+\frac{\beta \gamma _{1}^{2}}{2}e^{\beta \phi -2dA +2C_{2}}
(\partial _{m}C_{2})^{2} =0
\label{31m}
\end{equation}
We impose the following relations:
\begin{equation}
dA +{\tilde d}B =0,
\label{314}
\end{equation}
\begin{equation}
-\alpha \phi -2dA +2C_{1}=0,
\label{315}
\end{equation}
\begin{equation}
\beta \phi -2dA +2C_{2}=0.
\label{316}
\end{equation}
Under these conditions equations (\ref{311}), (\ref{312}) and (\ref{31m})
take the following forms, respectively,
\begin{equation}
\partial _{m}(e^{-C_{1}}\partial _{m}C_{1})=0,
\label{317}
\end{equation}
\begin{equation}
\partial _{m}(e^{-C_{2}}\partial _{m}C_{2})=0,
\label{318}
\end{equation}
\begin{equation}
\Delta \phi
-\frac{\alpha \gamma _{1}^{2}}{2}(\partial _{m} C _{1})^{2}
+\frac{\beta \gamma _{2}^{2}}{2}(\partial _{m} C _{2})^{2}=0
\label{319}
\end{equation}
>From equations (\ref{317}) , (\ref{318}) and (\ref{319}) we get
\begin{equation}
\Delta C _{1}
=(\partial C_{1})^{2},
\label{320}
\end{equation}
\begin{equation}
\Delta C _{2}
=(\partial C_{2})^{2}
\label{321}
\end{equation}
and
\begin{equation}
\phi=\varphi _{1}C_{1}+\varphi _{2}C_{2}
\label{322}
\end{equation}
with
\begin{equation}
\varphi _{1}=\frac{\alpha \gamma ^{2}}{2}
\label{323}
\end{equation}
\begin{equation}
\varphi _{2}=-\frac{\beta\gamma ^{2}}{2}
\label{324}
\end{equation}
Equations (\ref{315}) and (\ref{316}) give
\begin{equation}
A=a_{1}C_{1}+a_{2}C_{2},
\label{325}
\end{equation}
where
\begin{equation}
a_{1}=\frac{\beta }{d(\alpha +\beta )},~~~
a_{2}=\frac{\alpha }{d(\alpha +\beta )},
\label{326}
\end{equation}
and
\begin{equation}
\phi=\varphi _{1}C_{1}+\varphi _{2}C_{2}
\label{532}
\end{equation}
where
\begin{equation}
\varphi _{1}=\frac{2}{\alpha +\beta },~~~
\label{327}
\end{equation}
\begin{equation}
\varphi _{2}=-\frac{2}{\alpha +\beta }.
\label{328}
\end{equation}
Comparing (\ref{323}) with (\ref{327}) and (\ref{324}) with (\ref{328})
we conclude that
\begin{equation}
\gamma _{1}^{2}=\frac{4}{\alpha (\alpha +\beta )},~~~
\gamma _{2}^{2}=\frac{4}{\beta (\alpha +\beta )}.
\label{329}
\end{equation}
Let us now consider equation (\ref{310}). Since $C_{1}$ and $C_{2}$
are two independent functions we have to have that the terms with
$\partial _{m}C_{1}\partial _{n}C_{2}$ vanish, i.e. we have to
impose the condition
\begin{equation}
\alpha \beta=\frac{2d{\tilde d}}{d+{\tilde d}}.
\label{330}
\end{equation}
Therefore, the Ansatz (\ref{37}) is consistent with the
metric (\ref{38}) only under condition (\ref{330}). Straitforward
calculations show that for $A$, and $\phi$ given by
(\ref{325})-(\ref{326}) and (\ref{532}) in terms of two independent
functions $C_{1}$ and $C_{2}$, satisfying equations
(\ref{320}) and (\ref{321}), equations (\ref{39}) (\ref{310})
are satisfied if we impose the condition (\ref{330}).
Let us write the metric in terms of two harmonic functions:
\begin{equation}
H_{1}=-\ln C_{1},~~~ H_{2}=-\ln C_{2}
\label{331}
\end{equation}
We finally get
$$ds^{2}=H_{1}^{-\frac{2\beta }{d(\alpha +\beta )}}
H_{2}^{-\frac{2\alpha }{d(\alpha +\beta )}}
\eta_{\alpha \beta} dy^{\alpha} dy^{\beta}
$$
\begin{equation}
+H_{1}^{\frac{2\beta }{{\tilde d}(\alpha +\beta )}}
H_{2}^{\frac{2\alpha }{{\tilde d}
(\alpha +\beta )}}dx^{i}dx^{i}
\label{332}
\end{equation}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\section{Three block solution}
\setcounter{equation}{0}
In this secction we consider equations of motion
for the action (\ref{11}).
The Einstein equations for the action (\ref{11}) read
\begin{equation}
R_{MN}-\frac{1}{2}g_{MN}R=T_{MN},
\label{19}
\end{equation}
where the energy-momentum tensor is
$$
T_{MN}= \frac{1}{2}
(\partial _{M}\phi \partial _{N}\phi -
\frac{1}{2}g_{MN} (\partial \phi)^{2})
$$
$$
+ \frac{1}{2q!}e^{-\alpha \phi}(F_{MM_{1}...M_{q}}F_{N}^{M_{1}...M_{q}}-
\frac{1}{2(q+1)}g_{MN}F^{2})
$$
\begin{equation}
+ \frac{1}{2{\tilde d}!}e^{-\alpha \phi}(G_{MM_{1}...M_{{\tilde d}}}
G_{N}^{M_{1}...M_{{\tilde d}}}-
\frac{1}{2({\tilde d}+1)}g_{MN}G^{2})
\label{110}
\end{equation}
The equation of motion for the antisymmetric fields are
\begin{equation}
\partial _{M}(\sqrt{-g}e^{-\alpha \phi}F^{MM_{1}...M_{q}})=0,
\label{111a}
\end{equation}
\begin{equation}
\partial _{M}(\sqrt{-g}e^{-\beta \phi}G^{MM_{1}...M_{{\tilde d}}})=0,
\label{111}
\end{equation}
and one has the Bianchi identity
\begin{equation}
\epsilon ^{M_{1}...M_{q+2}}\partial_{M_{1}}F_{M_{2}...M_{q+2}}=0.
\label{112}
\end{equation}
\begin{equation}
\epsilon ^{M_{1}...M_{{\tilde d}+2}}
\partial_{M_{1}}G_{M_{2}...M_{{\tilde d}+2}}=0.
\label{1129}
\end{equation}
The equation of motion for the dilaton is
\begin{equation}
\partial _{M}(\sqrt{-g}g^{MN}\partial _{N }\phi )
+
\frac{\alpha}{2(q+1)!}\sqrt{-g}e^{-\alpha \phi}F^{2}
+
\frac{\beta }{2({\tilde d}+1)!}\sqrt{-g}e^{-\beta \phi}G^{2}
=0.
\label{113}
\end{equation}
We shall solve equations (\ref{19})-(\ref{113}) by using
the following Ansatz for the metric
\begin{equation}
ds^{2}=e^{2A(x)}\eta_{\mu \nu} dy^{\mu} dy^{\nu}+
e^{2F(x)}\delta_{nm}dz^{n}dz^{m}+
e^{2B(x)}\delta_{\alpha\beta}dx^{\alpha}dx^{\beta},
\label{114}
\end{equation}
where $\mu$, $\nu$ = 0,...,q-1, $\eta_{\mu\nu}$
is a flat Minkowski metric, $m,n$=$1,2,...,r$ and
$\alpha,\beta$ =$1,...,{\tilde d}+2$. Here $A$, $B$
and $C$ are functions on $x$; $\delta_{nm}$ and $\delta_{\alpha\beta}$
are Kronecker symbols.
Non-vanishing components of the differential forms are
\begin{equation}
{\cal A}_{\mu_{1}...\mu_{q}}=h{\epsilon_{\mu_{1}...\mu_{q}}}e^{C(x)},
~F=d{\cal A}
\label{115}
\end{equation}
\begin{equation}
G^{\alpha_{1}...\alpha _{{\tilde d}+1}}=\frac{1}{\sqrt{-g}}
k e^{\beta
\phi}\epsilon ^{\alpha_{1}...\alpha_{{\tilde d}+1}\gamma}
\partial _{\gamma} e^{\chi} ,
\label{116}
\end{equation}
where $h$ and $k$ are constants.
The left hand side of the Einstein equations for the metric
(\ref{114}) read
$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\eta_{\mu\nu}e^{2(A-B)}[(q-1)\Delta A
+
({\tilde d}+1)\Delta B + r\Delta F $$
$$+\frac{q(q-1)}{2}(\partial A)^{2}+
\frac{r(r+1)}{2}(\partial F)^{2}+\frac{{\tilde d}({\tilde
d}+1)}{2}(\partial B)^{2}$$
\begin{equation}
+{\tilde d}(q-1)(\partial A\partial
B)+r(q-1)(\partial A\partial F) +r{\tilde d}(\partial F\partial B)],
\label{175}
\end{equation}
$$R_{mn}-\frac{1}{2}g_{mn}R=\delta_{mn}e^{2(F-B)}[q\Delta A+
({\tilde d}+1)\Delta B + (r-1)\Delta F $$
$$ +\frac{q(q+1)}{2}(\partial A)^{2}+
\frac{r(r-1)}{2}(\partial F)^{2}+\frac{{\tilde d}({\tilde
d}+1)}{2}(\partial B)^{2}$$
\begin{equation}
+{\tilde d}q(\partial A\partial
B)+q(r-1)(\partial A\partial F) +{\tilde d}(r-1)(\partial F\partial B)],
\label{176}
\end{equation}
$$R_{\alpha \beta} -\frac{1}{2} g_{\alpha \beta} R
=-q\partial_{\alpha}\partial_{\beta} A-
{\tilde d}\partial_{\alpha}\partial_{\beta} B-
r\partial_{\alpha}\partial_{\beta} F$$
$$-q\partial_{\alpha} A\partial_{\beta}A+
{\tilde d}\partial_{\alpha} B\partial_{\beta}B-
r\partial_{\alpha} F\partial_{\beta} F$$
$$+q(\partial_{\alpha} A\partial_{\beta} B+
\partial_{\alpha} B\partial_{\beta} A)+
r(\partial_{\alpha} B\partial_{\beta} F+
\partial_{\alpha} F\partial_{\beta} B)$$
$$+\delta_{\alpha\beta}[q\Delta A+
{\tilde d}\Delta B + r\Delta F + \frac{q(q+1)}{2}(\partial A)^{2}+
\frac{r(r+1)}{2}(\partial F)^{2}+
\frac{{\tilde d}({\tilde d}-1)}{2}(\partial B)^{2}$$
\begin{equation}
q({\tilde d}-1)(\partial A\partial B)+
qr(\partial A\partial F) +r({\tilde d}-1)(\partial F\partial B)].
\label{177}
\end{equation}
For more details see \cite{AV}. Now one reduces
the $(\mu\nu)$-components of
(\ref{19}) to the equation
\begin{eqnarray}
(q-1)\Delta A+
({\tilde d}+1)\Delta B +r\Delta F \nonumber \\
+\frac{q(q-1)}{2}(\partial A)^{2} + \frac{r(r+1)}{2}(\partial F)^{2}+
\frac {{\tilde d}({\tilde d}+1)}{2}(\partial B)^{2} \nonumber \\
+{\tilde d}(q-1)(\partial A\partial B) + r(q-1)(\partial A\partial F) +
r{\tilde d}(\partial B\partial F) = \nonumber \\
-\frac{1}{4}(\partial
\phi)^{2}
-\frac{{h}^{2}}{4}(\partial C)^{2}e^{-\alpha\phi-2qA+2C}
-\frac{k^{2}}{4}(\partial \chi)^{2}
e^{2{\tilde d}B+\beta\phi+2\chi} , \label{117}
\end{eqnarray}
$(nm)$-components of (\ref{19}) to the following equation:
$$q\Delta A+({\tilde d}+1)\Delta B+(r-1)\Delta F$$
$$+\frac{q(q+1)}{2}(\partial A)^{2}+
\frac{{\tilde d}({\tilde d}+1)}{2}(\partial B)^{2}+
\frac{r(r-1)}{2}(\partial F)^{2}$$
$$+q{\tilde d}(\partial A\partial B)+q(r-1)(\partial A\partial F)+
{\tilde d}(r-1)(\partial
B\partial F)=$$
\begin{equation}
-\frac{1}{4}(\partial \phi)^{2}+\frac{h^{2}}{4}(\partial
C)^{2}e^{-\alpha\phi-2qA+2C}-
\frac{k^{2}}{4}(\partial \chi)^{2}e^{2{\tilde d}B+\beta\phi+2\chi},
\label{118}
\end{equation}
and $(\alpha\beta)$-components to the equation:
$$ -q\partial_{\alpha}\partial_{\beta} A-
{\tilde d}\partial_{\alpha}\partial_{\beta} B-
r\partial_{\alpha}\partial_{\beta} F $$
$$-q\partial_{\alpha} A\partial_{\beta} A + {\tilde d}\partial_{\alpha}
B\partial_{\beta} B - r\partial_{\alpha} F\partial_{\beta} F +
q(\partial_{\alpha} A\partial_{\beta} B +\partial_{\alpha}
B\partial_{\beta} A)$$
$$+r(\partial_{\alpha} B\partial_{\beta} F + \partial_{\alpha}
F\partial_{\beta} B) + \delta_{\alpha\beta}[q\Delta A+{\tilde d}\Delta B+
r\Delta F$$
$$+\frac{{\tilde d}({\tilde d}+1)}{2}(\partial A)^{2} +
\frac{r(r+1)}{2}(\partial F)^{2} $$
$$ +\frac{{\tilde d}({\tilde d}-1)}{2}(\partial B)^{2} +
q({\tilde d}-1)(\partial A\partial B) +
r({\tilde d}-1)(\partial F\partial B )+ qr(\partial A\partial F)] $$
$$ = \frac{1}{2}[\partial_{\alpha} \phi\partial_{\beta} \phi -
\frac{1}{2}\delta_{\alpha\beta}(\partial\phi)^{2}] -
\frac{h^{2}}{2}e^{-\alpha\phi-2qA+2C}[\partial_{\alpha} C\partial_{\beta} C
-
\frac{\delta_{\alpha\beta}}{2}(\partial C)^{2}]$$
\begin{equation}
-\frac{k^{2}}{2} e^{2{\tilde d}B+\beta\phi+
2\chi}[\partial_{\alpha} \chi\partial_{\beta} \chi
-\frac{\delta_{\alpha\beta}}{2}(\partial \chi)^{2}],
\label{119}
\end{equation}
where we use notations
$( \partial A\partial B)=\partial_{\alpha} A\partial_{\alpha} B$
and $D=q+r+{\tilde d}+2=d+{\tilde d}+2$.
The equations of motion (\ref{111}) for a part of components of the
antisymmetric field are identically satisfied and for the other part
they are reduced to a simple equation:
\begin{equation}
\partial _{\alpha}(e^{-\alpha \phi -2qA +C}\partial _{\alpha}C)=0.
\label{120}
\end{equation}
For $\alpha$-components of the antisymmetric field we
also have the Bianchi identity:
\begin{equation}
\partial_{\alpha}(e^{\alpha\phi + 2Bq + \chi }\partial_{\alpha}\chi)=0.=
\label{121}
\end{equation}
The equation of motion for the dilaton has the form
$$\partial _{\alpha}(e^{qA +{\tilde d} B + Fr}\partial _{\alpha}\phi )
-\frac{\alpha h^{2}}{2}e^{-\alpha \phi -qA + qB +rF
+2C}(\partial _{\alpha} C)^{2}
$$
\begin{equation}
+\frac{\beta k ^{2}}{2}e^{\beta \phi +2{\tilde d}B+2\chi}
(\partial _{\alpha}\chi)^{2} =0.
\label{122}
\end{equation}
We have to solve the system of equations (\ref{117})-(\ref{122})
for unknown functions$~~~~$
$ A,B,F,C,\chi,\phi$. We shall express $ A,B,F$ and $\phi $
in terms of two functions $C$ and $\chi$.
In order to get rid of exponents in (\ref{117})-(\ref{122})
we impose the following relations:
\begin{equation}
qA + rF +{\tilde d}B =0,
\label{123}
\end{equation}
\begin{equation}
2\chi +2{\tilde d}B +\beta \phi =0,
\label{124}
\end{equation}
\begin{equation}
2C - 2qA -\alpha\phi = 0.
\label{125}
\end{equation}
Under these conditions equations (\ref{120}),(\ref{121}) and (\ref{122})
will have the following forms, respectively,
\begin{equation}
\partial _{\alpha}(e^{-C}\partial _{\alpha}C)=0,~~~~~
\partial _{\alpha}(e^{-\chi}\partial _{\alpha}\chi)=0,
\label{126}
\end{equation}
\begin{equation}
\Delta \phi +\frac{\beta k ^{2}}{2}(\partial _{\alpha} \chi )^{2}-
\frac{\alpha h^{2}}{2}(\partial _{\alpha} C )^{2}=0.
\label{127}
\end{equation}
One rewrites (\ref{126}) as
\begin{equation}
\Delta C =(\partial C)^{2},~~~~~
\Delta \chi =(\partial \chi)^{2}. \label{128}
\end{equation}
Therefore (\ref{127}) will have the form
\begin{equation}
\Delta \phi +\frac{\beta k^{2}}{2}\Delta \chi -
\frac{\alpha h^{2}}{2}\Delta C =0.
\label{129}
\end{equation}
>From (\ref{129}) it is natural to set
\begin{equation}
\phi =\phi_{1}C + \phi_{2}\chi,
\label{130}
\end{equation}
where
\begin{equation}
\phi_{1}=\frac{\alpha h^{2}}{2},~~~
\phi_{2}=-\frac{\beta k^{2}}{2}.
\label{131}
\end{equation}
>From equations (\ref{123}), (\ref{124}) and (\ref{125}) it follows that
$A$, $B$ and $F$ can
be presented as
linear combinations of functions $C$ and $\chi$:
\begin{equation}
A=a_{1}C + a_{2}\chi ,
\label{132}
\end{equation}
\begin{equation}
B=b_{1}C + b_{2}\chi ,
\label{133}
\end{equation}
\begin{equation}
F=f_{1}C + f_{2}\chi ,
\label{134}
\end{equation}
where
\begin{equation}
a_{1}=\frac{4-\alpha^{2}h^{2}}{4q}, ~~~
a_{2} = \frac{\alpha\beta k^{2}}{4q},
\label{135}
\end{equation}
\begin{equation}
b_{1} = -\frac{\alpha\beta h^{2}}{4{\tilde d}},~~~
b_{2} =\frac{\beta ^{2}k^{2}-4}{4{\tilde d}},
\label{136}
\end{equation}
\begin{equation}
f_{1} = \frac{\alpha^{2}h^{2} +\alpha \beta h^{2}-4}{4r},~~~
f_{2} = \frac{4-\alpha \beta k^{2}-\beta^{2} k^{2}}{4r}.
\label{137}
\end{equation}
Let us substitute expressions (\ref{130}),(\ref{132})-(\ref{134}) for
$\phi,A,B,F$ into (\ref{117})-(\ref{119}). We get relations containing
bilinear forms over derivatives on $ C$ and $\chi$ .
We assume that the
coefficients in front of these bilinear
forms vanish. Then we get the system of twelve quartic equations
which is presented and solved in Appendix.
The system has a solution only if
$\alpha$ and $\beta$ satisfy the relation
\begin{equation}
\alpha \beta=\frac{2q{\tilde d}}{q+r+{\tilde d}}
\label{81m}
\end{equation}
In this case $h$ and $k$ are given by the formulae
\begin{equation}
h=\pm\sqrt{\frac{4(q+r+{\tilde d})}
{\alpha ^{2}(q+r+{\tilde d})+2q({\tilde d}+r)}},
\label{819}
\end{equation}
\begin{equation}
k=\pm\frac{2\alpha (q+r+{\tilde d})}{\sqrt{{\tilde d}
[2\alpha ^{2}(q+r)(q+r+{\tilde d})+4q^{2}{\tilde d}]}}
\label{195}
\end{equation}
To summarize, the action (\ref{11}) has the solution of the form
(\ref{12}) expressed in terms
of two harmonic functions $H_{1}$ and $H_{2}$
if the parameters in the action are related by (\ref{193})
and the parameters in the Ansatz $h$ and $k$ are given
by (\ref{819}),(\ref{195}).
\section {Discussion and Conclusion}
Let us discuss different particular cases of the solution (\ref{12}).
There is the relation (\ref{15}) between parameters $\alpha$
and $\beta$
in the action (\ref{011}). As a result the action
corresponds to the bosonic part of a supergravity theory
only in some dimensions.
If $D=4$ and $ q=d=1$ then
one can take $\alpha=\beta=1$ and the action
corresponds to the $SO(4)$ version of $N=4$ supergravity.
The solution (\ref{12}) takes the form
\begin{equation}
ds^{2}=-H_{1}^{-1}H_{2}^{-1}dt^{2}+H_{1}H_{2}dx^{\alpha}dx^{\alpha}
\label{195a}
\end{equation}
This supersymmetric solution has been obtained in \cite{kallosh}.
If $\alpha=\beta$ and $q=\tilde d$ then one has the solution
\begin{equation}
ds^{2}=H_{1}^{\frac{2}{D-2}}H_{2}^{\frac{2(D-q-2)}{q(D-2)}}
[(H_{1}H_{2})^{-\frac{2}{q}} \eta_{\mu \nu} dy^{\mu} dy^{\nu}+
H_{2}^{-\frac{2}{q}}dz^{m}dz^{m}+
dx^{\alpha}dx^{\alpha}],
\label{713}
\end{equation}
This solution was obtained in \cite{AV}. It contains
as a particular case for $d=10,~q=2$ the known solution
\cite{TS,CM,CH}
$$ ds^{2}=H_{1}^{-\frac{3}{4}}H_{2}^{-\frac{1}{4}}
(-dy_{0}^{2}+dy_{1}^{2})
$$
\begin{equation}
+ H_{1}^{\frac{1}{4}}H_{2}^{-\frac{1}{4}}
(dz_{2}^{2}+dz_{3}^{2}+dz_{4}^{2}+dz_{5}^{2})+
H_{1}^{\frac{1}{4}}H_{2}^{\frac{3}{4}}
(dx_{6}^{2}+dx_{7}^{2}+dx_{8}^{2}+dx_{9}^{2}).
\label{66m}
\end{equation}
This solution has been used in the D-brane derivation
of the black hole entropy \cite{SV,CM}. Note however
that the solution (\ref{66m}) corresponds to the action
(\ref{011}) with the $3$-form $F_{3}$ and the $7$-form
$G_{7}$.
To conclude, a rather general three-block solution
of the action (\ref{011}) has been constructed. It contains
as particular cases many known solutions. However the solution
is not general enough to include some known multi-block
solutions. Further work is needed to understand better
the structure hidden behind the multi-block p-brane solutions.
\section*{Acknowlegments}
This work is supported by an operating grant from
the Natural Sciences and Engineering Research Council of Canada.
I.A. and I.V. thank the Department of Physics for the kind
hospitality during their stay at Simon Fraser University.
I.V. is partially supported by the RFFI grant 9600312.
\section {Appendix}
We obtain the following system of algebraic equations
$$-a_{1}+b_{1}+\frac{q(q-1)}{2}a_{1}^{2}+\frac{r(r+1)}{2}f_{1}^{2}+
\frac{{\tilde d}({\tilde d}+1)}{2}b_{1}^{2}$$
\begin{equation}
+{\tilde d}(q-1)a_{1}b_{1}+r(q-1)a_{1}f_{1}+r{\tilde d}f_{1}b_{1}+
\frac{\phi_{1}^{2}}{4}+
\frac{h^{2}}{4}=0;
\label{178}
\end{equation}
$$-a_{2}+b_{2}+\frac{q(q-1)}{2}a_{2}^{2} +
\frac{r(r+1)}{2}f_{2}^{2} + \frac{{\tilde d}({\tilde d}+1)}{2}b_{2}^{2}$$
\begin{equation}
+{\tilde d}(q-1)a_{2}b_{2}+r(q-1)a_{2}f_{2}+r{\tilde d}b_{2}f_{2}
+\frac{\phi_{2}^{2}}{4}+\frac{k^{2}}{4}=0;
\label{179}
\end{equation}
$$q(q-1)a_{1}a_{2} +r(r+1)f_{1}f_{2}+{\tilde d}({\tilde d}+1)b_{1}b_{2}$$
\begin{equation}
+{\tilde d}(q-1)(a_{1}b_{2}+a_{2}b_{1})+r(q-1)(a_{2}f_{1}+a_{1}f_{2})+
r{\tilde d}(f_{1}b_{2}
+f_{2}b_{1})+\frac{\phi_{1}\phi_{2}}{2}=0;
\label{180}
\end{equation}
$$-f_{1}+b_{1}+\frac{q(q+1)}{2}a_{1}^{2}+\frac{r(r-1)}{2}f_{1}^{2}+
\frac{{\tilde d}({\tilde d}+1)}{2}b_{1}^{2}$$
\begin{equation}
+q {\tilde d} a_{1}b_{1}+q(r-1)a_{1}f_{1}+
{\tilde d}(r-1)f_{1}b_{1}+\frac{\phi_{1}^{2}}{4}
-\frac{h^{2}}{4}=0;
\label{181}
\end{equation}
$$-f_{2}+b_{2}+\frac{q(q+1)}{2}a_{2}^{2} +
\frac{r(r-1)}{2}f_{2}^{2} + \frac{{\tilde d}({\tilde d}+1)}{2}b_{2}^{2}+$$
\begin{equation}
+q {\tilde d} a_{2}b_{2}+q(r-1)a_{2}f_{2}+{\tilde d}(r-1)b_{2}f_{2}
+\frac{\phi_{2}^{2}}{4}+\frac{k^{2}}{4}=0;
\label{182}
\end{equation}
$$q(q+1)a_{1}a_{2} +r(r-1)f_{1}f_{2}+{\tilde d}({\tilde d}+1)b_{1}b_{2}$$
\begin{equation}
+q{\tilde d}(a_{1}b_{2}+a_{2}b_{1})+q(r-1)(a_{2}f_{1}+a_{1}f_{2})
+{\tilde d}(r-1)(f_{1}b_{2}
+f_{2}b_{1})+\frac{\phi_{1}\phi_{2}}{2}=0;
\label{183}
\end{equation}
$$-qa_{1}^{2}-rf_{1}^{2}+
{\tilde d} b_{1}^{2}+$$
\begin{equation}
+2qa_{1}b_{1}+2rf_{1}b_{1}-\frac{\phi_{1}^{2}}{2}+
\frac{h^{2}}{2}=0;
\label{184}
\end{equation}
$$-qa_{2}^{2} -rf_{2}^{2} + {\tilde d}b_{2}^{2}$$
\begin{equation}
+2qa_{2}b_{2}+2rb_{2}f_{2}
-\frac{\phi_{2}^{2}}{2}+\frac{k^{2}}{2}=0;
\label{185}
\end{equation}
$$-qa_{1}a_{2} -rf_{1}f_{2}+{\tilde d}b_{1}b_{2}$$
\begin{equation}
+q(a_{1}b_{2}+a_{2}b_{1})+r(f_{1}b_{2}
+f_{2}b_{1})-\frac{\phi_{1}\phi_{2}}{2}=0;
\label{186}
\end{equation}
$$\frac{q(q+1)}{2}a_{1}^{2}+\frac{r(r+1)}{2}f_{1}^{2}+
\frac{{\tilde d}({\tilde d}-1)}{2}b_{1}^{2}$$
\begin{equation}
+q({\tilde d}-1)a_{1}b_{1}+r({\tilde d}-1)b_{1}f_{1}+
rqf_{1}a_{1}+\frac{\phi_{1}^{2}}{4}-
\frac{h^{2}}{4}=0;
\label{187}
\end{equation}
$$\frac{q(q+1)}{2}a_{2}^{2} +
\frac{r(r+1)}{2}f_{2}^{2} + \frac{{\tilde d}({\tilde d}-1)}{2}b_{2}^{2}$$
\begin{equation}
+q({\tilde d}-1)a_{2}b_{2}+r({\tilde d}-1)a_{2}f_{2}+rqb_{2}f_{2}
+\frac{\phi_{2}^{2}}{4}-\frac{k^{2}}{4}=0;
\label{188}
\end{equation}
$$q(q+1)a_{1}a_{2} +r(r+1)f_{1}f_{2}+{\tilde d}({\tilde d}-1)b_{1}b_{2}+
q({\tilde d}-1)(a_{1}b_{2}+a_{2}b_{1})
$$
\begin{equation}
+r({\tilde d}-1)(b_{2}f_{1}+b_{1}f_{2})+rq(f_{1}a_{2}
+f_{2}a_{1})+\frac{\phi_{1}\phi_{2}}{2}=0.
\label{188a}
\end{equation}
Now let us discuss the system of eq. (\ref{178})-(\ref{188a}).
The action depends on the parameters $D,\alpha, \beta ,q,\tilde{d}$.
We have used the parameter $r$ instead of $D$ which is defined from
$$q+r+\tilde{d} +2=D$$
Therefore our action (\ref{11}) depends on five parameters
$r,\alpha, \beta ,q,\tilde{d}$
We have also two parameters $h$ and $k$ in our Ansatz
(\ref{115}),(\ref{116}).
We substitute expressions (\ref{130}),(\ref{131}),
(\ref{135})-(\ref{137})
into (\ref{178})-(\ref{188a}). Then we get
the system of twelve
quartic equations for
seven unknown variables $r,\alpha, \beta ,q,\tilde{d},h$ and $k$.
Using Maple V we found the solution of the system
of equations (\ref{178})-(\ref{188a}). The solution has the form
\begin{equation}
k=\frac{\sqrt{2q(r+{\tilde d})h^{2}-4(q+r+{\tilde d})}}
{\sqrt{{\tilde d}(qrh^{2}-2(q+r))}},
\label{190}
\end{equation}
\begin{equation}
\alpha=\frac{\sqrt{4(q+r+{\tilde d})-2q(r+{\tilde d})h^{2}}}
{h\sqrt{q+r+{\tilde d}}}
\label{191}
\end{equation}
\begin{equation}
\beta=\frac{2q{\tilde d}h}{\sqrt{(q+r+{\tilde d})
[4(q+r+{\tilde d})-2q(r+{\tilde d})h^{2}]}}
\label{192}
\end{equation}
where $r,q,\tilde{d},h$ are arbitrary. Let us rewrite
the solution in terms of parameters in the action.
>From equations (\ref{191}) and (\ref{192}) follows that
$h$ can be found from equations (\ref{178})-(\ref{188a}) only
if $\alpha$ and $\beta$ are subjected to the relation
\begin{equation}
\alpha \beta=\frac{2q{\tilde d}}{q+r+{\tilde d}}
\label{193}
\end{equation}
In this case $h^{2}$ and $k^{2}$ are given by the formulae
\begin{equation}
h^{2}=\frac{4(q+r+{\tilde d})}
{\alpha ^{2}(q+r+{\tilde d})+2q({\tilde d}+r)},
\label{}
\end{equation}
\begin{equation}
k^{2}=\frac{2\alpha^{2} (q+r+{\tilde d})^{2}}
{{\tilde d}[\alpha ^{2}(q+r)(q+r+{\tilde d})+2q^{2}{\tilde d}]}
\label{}
\end{equation}
\newpage
| proofpile-arXiv_065-411 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
In recent years, new semiconductor heterostructures have attracted
considerable interest. Multiple quantum well structures and superlattices
of II--VI compounds are the subject of intensive study because of their
interesting optical properties \cite{tersoff,flores,brasil,ichino}. With
these structures, energy gaps ranging from the UV to IR are accessible
\cite{brasil,ichino,pelhos}. In these systems the binary interfaces are
usually lattice mismatched. This lattice mismatch modifies the band
alignments, and hence modifies the device optical properties.
In searching for the desired material parameters such as
band gap, lattice matching to substrates, dielectric contact, carrier
mobility, etc., a large number of materials, have been investigated.
Recently, high--quality cubic--structured ternary and quaternary alloys
have been proposed as appropriate materials for heterostructures \cite{brasil,%
ichino}. Ternary alloys
allow a certain control of the induced strain at the interface.
The deep understanding of the physics of
the interface is important for the detailed study
of thermal, optical, and other properties of quantum--wells
and superlattices.
The electronic properties at solid--solid interfaces
depend sometimes even on details of the interaction between the two
atomic layers from the different materials in contact. Our work can
be used as a starting point to analyze those details. These are
responsible for the characteristics of interface reconstruction,
thermodynamic properties, degree of intermixing, stress, compound
formation, etc.
In previous work, we have studied the electronic structure of
the valence
band for the (001)--surface of several II--VI wide band gap
semiconductors \cite{prb50,prb51},
and different binary heterostructures \cite{infcs}.
We have obtained the (001)--projected electronic structure for both,
surfaces and binary interfaces using the known
Surface Green's Function Matching (SGFM) method \cite{g-m}.
In the (001)--surfaces in addition to the well known bulk bands and surface
resonances, we have described three
different structures in the valence band region, the so--called
surface induced bulk states ($B_h,\ B_l$, and
$B_s$). We have shown that these states
owe their origin to the creation of the surface, that is,
they depend on the surface through the boundary condition
(the wave function has to be zero at the surface),
but they are not surface resonances.
They are {\it surface--induced bulk states} \cite{prb50,prb51}.
Later we found that this kind of
induced states appear at the interface domain as well.
Therefore, more generally, we found that any frontier can induce these
states. For that reason we have redefined them
as frontier--induced semi-infinite medium (FISIM) states
since they are not, strictly speaking, {\it bulk} (infinite medium) states.
These FISIM states do not show dispersion as a function of the wave
vector {\bf k} for the surfaces studied. This is theoretically and
experimentally shown for the (001)--oriented
CdTe surface \cite{prb50,prb51,niles,gawlik}.
For the binary/binary (001)--oriented II--VI compound
interfaces, in contrast, they show some
clear dispersion \cite{infcs}.
The interest of the present
work is twofold. Firstly, we want to make practical use of our recently
set found tight--binding Hamiltonians for the ternary alloys.
They reproduce the known experimental change
with the composition of the band gap and they can be further used in
detailed studies of different physical problems as, for example,
the dependence of the transport properties on composition in
quantum well structures that avoid stress.
The second interest of
this work is the study of the evolution of the FISIM states
from a non--dispersive character to a dispersive one as
stress and different crystal composition enters into play.
We show that, if we select a ternary alloy to produce
little stress and change only slightly the composition, the
FISIM states do exists on both sides of the interface
but do not show as much dispersion. So the existence of
the FISIM states is due to the existence of a frontier alone
and the amount of dispersion is related to the existing stress
at the interface and on the chemical character of the
interface partner.
We will present in this work the valence band of some II--VI
(001)--binary/ternary alloy interfaces and we will concentrate
in particular in the
FISIM states. The method used is discussed in our previous
work. Here we only summarize the relevant features of it in
Section II for completeness; Section III is devoted to discuss
our results. Finally, we give our conclusions in section IV.
\section{The Method}
To describe the interface between two semiconductor compounds, we make
use of tight--binding Hamiltonians. The Green's function matching
method takes into account the perturbation caused by the surface or
interface exactly,
at least in principle, and we can use the bulk
tight--binding parameters (TBP) \cite{rafa,noguera,quintanar}.
This does not mean that we are
using the same TBP for the surface, or for the interface and the bulk.
Their difference is taken into account through the matching of the
Green's functions. We use the method in the form cast by
Garc\'\i a--Moliner and Velasco \cite{g-m}. They make use of the transfer
matrix approach first introduced by Falicov and Yndurain \cite{falicov}.
This approach became very useful due to the quickly converging
algorithms of L\'opez--Sancho {\it et al.} \cite{sancho} Following the
suggestions
of these authors, the algorithms for all transfer matrices needed to
deal with these systems can be found in a straightforward way
\cite{trieste}.
The Green's function for the interface, $G_I$, is given by \cite{g-m},
\begin{equation}
\label{infcs}
G_I^{-1}=G_{s(A)}^{-1} + G_{s(B)}^{-1} - I_BH^iI_A - I_AH^iI_B,
\end{equation}
where
$G_{s(A)}$ and $G_{s(B)}$ are
the surface Green's function of medium A and B, respectively.
$-{\cal I}_AH^i{\cal I}_B$
and $-{\cal I}_BH^i{\cal I}_A$ are the Hamiltonian matrices
that describe the interaction between the two media. In our model
these are $20\times20$ matrices, the input TBP for these matrices
are the average for those of the two media.
This is a reasonable approximation when both sides of the interface
have the same crystallographic structure and we take the same basis of
wave functions.
The tight--binding Hamiltonians for the
II--VI ternary alloys
are described in detail in Ref. [18]. Briefly speaking,
we have used the tight--binding
method and, under certain conditions, the virtual crystal
approximation to study the ternary alloys.
We have included an empirical bowing parameter\ in the $s-$on site TBP of
the substituted ion. This procedure gave us the correct behaviour of the
band gap value with composition \cite{ternario}.
More exactly for the
TBP of the ternary alloy, we take
\begin{equation}
\overline E_{\alpha,\alpha'}(x)= x E_{\alpha,\alpha'}^{(1)}
+ (1-x)E^{(2)}_{\alpha,\alpha'}, \qquad \alpha,\ \alpha'=s,\ p^3,\ s^*
\label{vca}
\end{equation}
for all but the $s-$on site TBP of the substituted
ion. In eq. (\ref{vca}) $E^{(1,2)}_{\alpha,\alpha'}$ are the TBP
for the compound 1 (2); $\alpha,\ \alpha'$ are the atomic
orbitals used in the basis set.
For the $s-$on site TBP of the substituted ion we use the
following expression
\begin{equation}
\overline E_{s,\nu}(x,b_\nu)= \overline E_{s,\nu}(x) +
x(1-x)b_\nu, \qquad \nu=a,\ c
\end{equation}
where $\overline E_{s,\nu}(x)$ is given by eq. (\ref{vca}) and
$b_\nu$ is the empirical bowing parameter\ per each different substitution (anion--substitution $(a)$
or cation--one $(c)$). In Table \ref{ebp} we have the empirical bowing parameter s used
in this work. We do not introduce any
further parameter \cite{ternario}.
From the knowledge of the Green's function, the local
density of states can be calculated from its imaginary part integrating
over the two--dimensional first Brillouin zone,
the dispersion relations can be obtained from the poles of the
real part. We have
applied previously this formalism to surfaces \cite{prb50,prb51,rafa,noguera},
interfaces \cite{infcs,quintanar,rafa-prb} and
superlattices \cite{rafa-moliner}.
Now we present our results.
\section{Results and discussion}
This section is devoted to the discussion of the interface--valence band of the
(001)--projected electronic band structure of II--VI binary/ternary
alloy interfaces.
We will present in this paper the (001)--CdTe/CdSe$_{.15}$Te$_{.85}$,
(001)--CdTe/Zn$_{.17}$Cd$_{.83}$Te,
(001)--ZnSe/ZnSe$_{.87}$Te$_{.13}$, and (001)--ZnSe/Zn$_{.85}$Cd$_{.15}$Se
interfaces in detail. The
interfaces studied have been chosen with a composition
$(x)$ as to give a minimum stress.
For the lattice parameter value of the materials considered see
Table II. As we can see the induced stress is small,
about 1\%. This magnitude of the induced stress allow us to
ignore its effect in our
calculation.
The real bulk bands
as well as the FISIM states, should lie very closely to our
calculated ideal case.
We adopt the same convention for the interface domain as in Ref.
\cite{infcs}.
That is to say, we consider nearest neighbors interactions in our bulk
Hamiltonians and, as a consequence, four atomic layers as the interface
domain, two belonging to medium A and two to medium B. To distinguish
between the different atomic layers we will call each atomic layer
by the medium its neighbors belong to. The atomic layer AA
will be the second from the interface into medium A. AB will be
the last atomic layer belonging to medium A and facing the first
atomic layer
of medium B and so on. So the four atomic layers that
constitute the interface domain
will be labeled AA, AB, BA, and BB.
For the interfaces aligned along the (001) direction the two media
are facing each other either through its anion or cation atomic layer.
In the alloy case, we consider a pseudobinary compound
so that the concept of anion and cation atomic layers remain meaningful.
We will consider here only anion-anion interfaces but our results
can be extented without difficulty to other kind
of interfaces.
We will project the interface electronic band
structure on each atomic layer and we will see how the different states that we
found for the free surface
and for the binary/binary interface case change or disappear at the
binary/ternary one.
It is known that the common anion interfaces
have small valence band--offset and the common cation ones have small
conduction band--offset, both of the order of some meV \cite{ichino,pelhos,%
duc}. In consequence,
we will use the boundary condition that the
top of the valence bands at the interface are aligned and choose this energy
as our zero. Accordingly, the conduction band offset will be equal to the
difference in the band gaps. The actual calculation of the band
offset is still an open theoretical question that we do not
want to address in this work \cite{infcs2}.
As a general remark, the FISIM states are not Bloch states and therefore
the {\bf k}--wave number is not expected to be a good quantum
number. The existence of a frontier (surface or interface) breaks
the symmetry. This does not actually mean that when the Schr\"odinger
equation is solved for differents values of {\bf k} (the Hamiltonian
depends explicitly on it) one should get the same eigenvalue. It is
found, theoretically \cite{prb50,prb51} as well as experimentally
\cite{niles,gawlik}, that the solution
does not depend on {\bf k} for the case of a surface.
In this case the boundary condition is that the wave
function has to be zero at the surface boundary for any value
of the derivative. It is the condition for an infinite
potential barrier. For the interface it is not so. For the
binary/binary case we got a solution that depends on the wave
vector, {\bf k}, but we should not call it {\it dispersion} since it
is not the behaviour with respect to a quantum number that we
are looking at but rather with respect to a parameter. FISIM
states are neither Bloch states nor surface states. They
do exist in the semi--infinite medium space but they do not
follow the infinite--medium symmetry of the crystal. So we
have to look for a different physical reason of their {\bf k}--dependence.
The first thing to notice is that the boundary condition is different.
For an interface, the wave function has not to be zero, is has to
be continuous together with the derivative. The boundary condition
therefore will depend on {\bf k}. This is because the Hamiltonian
describing the interaction depends on it and therefore the
wave function that solves the Schr\"odinger equation does
depend on it as well. For this reason its value and its
derivative at the border will also depend on it. One does
therefore, in general, expect a {\bf k}--dependence of the FISIM
states eigenvalues for an interface. For a surface
the boundary condition is always zero and on the contrary we
do not expect a {\bf k}--dependence.
In previous work,\cite{infcs} we have explored the behaviour
of the FISIM states at binary/binary interfaces. These
represent a strong change at the interface.
In this work, we explore the existence and behaviour of the
FISIM states at interfaces that do change slowly.
Here, ternary alloys are chosen so as to minimize stress (same
lattice constant in both sides) and the corresponding binary/ternary
alloy interface FISIM states are obtained. Their
{\bf k}--dependence as expected, is minimum. So, we
can conclude that, in general, stress is responsible
for the {\bf k}--dependence of the FISIM states. This
is in agreement with the ideas developed above.
Therefore, FISIM states are a consequence of the
existence of a frontier and their {\bf k}--dependence is a
result of the stress at it.
Furthermore, we have obtained
from this calculation two interface
states in the valence band range for the CdTe--based interfaces
and one interface state for the ZnSe--ones. Now we present
the details for each interface.
In Figs. 1--4, we show the electronic band structure of the valence band for
the interfaces studied here, (001)--CdTe/CdTe$_{.85}$Se$_{.15}$,
(001)--CdTe/Zn$_{.17}$Cd$_{.83}$Te, (001)--ZnSe/ZnSe$_{.87}$Te$_{.13}$,
and (001)--ZnSe/Zn$_{.85}$Cd$_{.15}$Se.
The dispersion relations are found from the poles
(triangles in the figures) of the real part
of the interface Green's function.
The solid--lines are a guide to the eye.
These are to be compared to the dispersion
curves found for the bulk (infinite medium) case.
The calculated eigenvalues for the FISIM states are denoted by stars,
crosses and points; the dotted lines
are intended only as a guide to the eye. We label the FISIM states
as $B_{Ih},\ B_{Il}$, and $B_{Is}$.
This convention follows the previous
free (001)--surfaces study (see Refs. \cite{prb50,prb51}).
The energy eigenvalues for all the calculated states are in Tables III and IV.
\subsection{The (001)--CdTe/CdSe$_{.15}$Te$_{.85}$ interface}
Fig. 1 shows the projected electronic structure of the valence band for
this interface per atomic layer. From the figure is evident that we
have obtained the general pattern of the projected band structure
of the II--VI semiconductor surfaces \cite{prb50,prb51}.
As we have commented above we will consider
an anion--anion interface and we will aling the top of the
valence band as our zero of energy. We have obtained that the heavy hole
(hh) and light hole (lh)
bands show more dispersion in the interface domain than in the semi--infinite
medium. They are usually low in energy about 0.7 eV an 0.4 eV,
respectively, in all the atomic layers. The spin--orbit band shows almost the
same dispersion that in the semi--infinite medium, see Table III.
As is pointed previously \cite{infcs}, the FISIM states $B_{Ih}$ and
$B_{Il}$ in the interface domain do not mix with the hh and lh bands,
as is observed in the (001)--surface case \cite{prb50,prb51}, see Fig. 1 and
Table IV. The states are lower in energy than the lh band. These
upper FISIM states show a slight dependence on {\bf k}, but in most of
the cases it is less than 0.3 eV. In contrast, the $B_{Is}$
state follows the spin--orbit band as in the semi--infinite medium. In
general, from the Fig. 1 we appreciate that the FISIM states show
better behaviour than in the binary/binary interfaces \cite{infcs}.
Moreover, in this energy interval we have obtained some states that
we identify with {\it interface states} ($IS_1$ and $IS_2$, the dotted
lines in Fig. 1, are a guide to the eye). The first one, at --1.3 eV,
in $\Gamma$, shows
notable dispersion and seems to disappear for {\bf k}--values
near the $X$--point. The second state, with more noticely
dispersion, appears at --1.9 eV in $\Gamma$ and reaches the
$X-$point in --4.3 eV. However, as we do not know about experimental
results in this system we can not give a complete comparison.
We only predict the possibility of the existence of these
interface states.
\subsection{The (001)--CdTe/Zn$_{.17}$Cd$_{.83}$Te interface}
This system shows almost the same pattern describe above. The
calculated valence band electronic structure is presented in
Fig. 2. From the Table III we observe
that the hh and lh bulk bands show more dispersion that in the
semi--infinite medium. In particular the electronic structure
projected onto the Cd--atomic layer (Fig. 2a).) shows bigger
dispersion for these bands, of about 0.8 and 0.5 eV, respectively,
than the semi--infinite medium, see Table III. For the other
atomic layers the projected electronic structure shows almost
the same pattern all together: the hh and lh bands are 0.5
and 0.3 eV below in energy with respect to the bulk values,
respectively. The spin--orbit band, however, in the interface seems
to form a barrier of about 0.2 eV from the AA--atomic layer
to the BB--atomic layer, see Table III.
In general, the $B_{Ih},\
B_{Il}$ and the $B_{Is}$ FISIM states are lower in energy
than the hh, lh, and spin--orbit bulk bands, respectively. In the same
way that in the previous case, these FISIM states do not mix
with the respective bulk bands at $X$, as in observed in the
semi--infinite medium case \cite{prb50,prb51}. However,
the FISIM states shows slight dependence on {\bf k}.
As in the previous interface,
we obtain two interface states in the present system,
label $IS_1$ and $IS_2$ in Fig. 2. The $IS_1$ state, located at
--1.3 eV in $\Gamma$, shows notable dispersion and reaches
the $X-$point between the hh and lh bulk bands. The $IS_2$ state,
with bigger dispersion than the previous one, is located
in $\Gamma$ at --1.9 eV and reaches the $X-$point at the
same values that the spin--orbit bulk band.
\subsection{The (001)--ZnSe/ZnSe$_{.87}$Te$_{.13}$ interface}
Fig. 3 shows the calculated electronic structure of the
valence band for this interface. Opposite to the CdTe--based
interfaces, discused above, the bulk bands and the FISIM
calculated states for this system shows almost the same behaviour
that in the semi--infinite medium. This observation goes for
all the calculated bands but the spin--orbit band in the $\Gamma$ point,
where we obtain, as in the previous case, a discontinuity from
the AA--atomic layer to the BB--atomic layer. In this case
the spin--orbit band seems to form a potential well, in $\Gamma$, of about
0.2 eV, see Table III. On the other hand, for this interface
we obtain that the $B_{Ih},\ B_{Il}$, and $B_{Is}$ FISIM states
mix with the hh, lh, and spin--orbit bulk bands, respectively, as is
observed in the semi--infinite medium \cite{prb50,prb51}. In this
sense this interface shows better behaviour than the other ones \cite{infcs}.
Although, as previously, we have obtained an interface state for this
system, $IS_1$. This interface state appears in $\Gamma$ at --1.7 eV, shows
noticely dispersion and seems to disappear for {\bf k}--values near
the $X-$point. However, the state appears notoriously in all the
calculated atomic layers.
\subsection{The (001)--ZnSe/Zn$_{.85}$Cd$_{.15}$Se interface}
Finally, in the Fig. 4 we show our electronic structure calculated
for this system. In the same way that the previous case, we obtain
that all the calculated states, per atomic layer, for this interface
are similar with the semi--infinite medium, see Tables III and IV.
In addition to these states, we have an interface state, $IS_1$, located
in $\Gamma$ at --1.7 eV and showing notable dispersion. The state
do not appear for all the interval between $\Gamma-X$, it seems to
disappear for {\bf k}--values near the $X-$point, as we have
commented previously for the other interfaces.
\section{Conclusions}
In conclusion, we
have calculated the electronic structure of the valence band of the
II--VI binary/ternary alloy interfaces. We have used the tight--binding
method and the surface Green's function matching method to obtain
the electronic structure projected onto each atomic layer that
constitutes the
interface domain. For the ternary alloys we have used our
tight--binding Hamiltonians described in previous work that give
good account for the changes of the band gap with composition as obtained
experimentally. Our parametrization includes an empirical bowing parameter\ for the ``$s$'' on--site
tight--binding parameter of the substituted ion and we
use the known virtual crystal approximation for the rest of
them. The systems were chosen here so that stress can be
ignored for the particular value of the compositional variable.
The calculated valence band electronic structure of these interfaces
show bulk bands with similar dispersion as for the semi--infinite
medium (a system with a surface).
The FISIM states observed in the (001)--oriented
surfaces and binary/binary interfaces appear also in this case and
show an intermediately strong {\bf k}--dependence as compare to the
previous ones.
In the interface domain the calculated states,
both the bulk bands and the FISIM states, have a composition that is
a combination
of the corresponding states of the two media forming the interface.
It is interesting to note further that we have
obtained for the binary/ternary alloy case
two interface states for the CdTe--based
heterostructures and one interface state for the ZnSe--ones that do not
show for the binary/binary interfaces at least in the energy interval
that we have considered. We will consider the binary/quaternary
and the ternary/quaternary alloy interfaces in future work.
| proofpile-arXiv_065-412 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
A photographic plate consists of an emulsion which separates grains
of AgCl (or similar) molecules. Let me present a simplified
discussion of the detection of a single photon. The photon may
dissociate an AgCl molecule. The Cl
escapes, whereas the Ag radical starts, upon photographic
development, a chemical reaction which
leaves a small, visible spot. Before detection the
one-particle photon wave may pass through {\it both} parts of a
double slit, see for instance ref.\cite{Bra}.
The geometry of the device (distance between the
slits etc.) can be chosen such that the photon is spread out over
a region covering distances much larger than a single grain, which
determines the size of the finally visible spot. The physics which
causes the photon wave to collapse is not understood.
Let us first recall where the Schr\"odinger equation
leaves us. For simplicity we assume that each grain consists of
precisely one AgCl molecule. We label the AgCl molecules on
the plate by $i=1,...,n$ and assume that the relevant features
of the photographic plate are described by products of
AgCl molecule wave functions. Initially the state is
\begin{equation}
|\Psi\rangle = |\Psi_0\rangle = \prod_{i=1}^n |\psi_i^b\rangle ,
\ \langle \Psi_0 | \Psi_0 \rangle = 1, \label{Psi0}
\end{equation}
where the
$|\psi_i^b\rangle$, $(i=1,...,n)$ indicate the {\it bound}
AgCl molecules and the overlap between different molecules is
neglected, {\it i.e.} $\langle \psi_i^b | \psi_j^b \rangle =
\delta_{ij}$. Through interaction with the photon the state
is transformed into
\begin{equation} \label{causal}
|\Psi\rangle = c_0 |\Psi_0\rangle + \sum_{j=1}^n c_j
|\Psi_j\rangle ~~{\rm with}~~ |\Psi_j\rangle =
|\psi_j^d\rangle \prod_{i\ne j} |\psi_i^b\rangle ,\
\langle \Psi_j | \Psi_j \rangle = 1\, . \label{Psi}
\end{equation}
Here $|\psi_j^d\rangle$, $(j=1,...,n)$ denotes a {\it dissociated}
AgCl molecule. A quantum measurement is constituted by
the fact that only one of the $|\Psi_j\rangle$, $(j=0,1,...,n)$
states survives, each with probability $P_j=|c_j|^2$,
$\sum_{j=0}^n P_j = 1$. The probabilities $P_j$ are related to
the photon wave function $\psi_{ph}$ by means of
\begin{equation}
P_j = const\ \int_{V_j} d^3x\ |\psi_{ph}|^2\, ,\
j=1,...,n\, . \label{probability}
\end{equation}
The constant does not depend on $j$, and $V_j$ is a
cross-sectional volume corresponding to the $j$th molecule.
Picking a branch $|\Psi_j\rangle$ becomes in this way
interpreted as observing the photon at the position $V_j$
(for a perfect
detector $c_0=0$). It is remarkable that in this process
of measurement the photon becomes destroyed through
spontaneous absorption by the dissociating molecule. To
summarize, measurements perform wave function {\it reductions} by
making decisions between alternatives proposed by the continuous, causal
time evolution part of Quantum Theory (QT). In our example
the reduction decides the location of the visible spot. Given the photon
wave function $\psi_{ph}$, QT predicts probabilities for the reduction
alternatives.
Whereas time evolution from eqn.(\ref{Psi0}) to
(\ref{Psi}) is described by the Schr\"odinger equation,
this is not true (or at least controversial) for the
measurement process. Decoherence theory, for
an overview see \cite{Zurek}, tries to establish that
conventional time evolution leads, in the situation of
eqn.(\ref{Psi}), to $\langle\Psi_j |
\Psi_k\rangle = 0$ for $j\ne k$, such that it becomes
impossible to observe contradictions with interference
effects predicted by the Schr\"odinger equation. In
contrast to this, explicit collapse models predict
deviations from conventional QT, see
for instance \cite{Ch86}. So far, no such deviations
have been measured.
The Schr\"odinger equation, more precisely its applicable
relativistic generalization $|\Psi (t)\rangle = \exp (
- H\, t)\, |\Psi (0)\rangle $, describes a continuous and
causal time evolution and I shall use the notation Quantum
Object (QO) for matter $|\Psi (t)\rangle$ as long as it
exhibits this behavior. Concerning the measurement process,
it seems to be widely believed that many body
processes, involving $\gg 10^{10}$ particles \cite{Bra}, are
responsible. In the presented example the photographic plate,
possibly also the environment beyond, would be blamed. However,
the ultimate collapse into one branch $|\Psi_j\rangle$ remains
an unexplained property. No satisfactory derivation from the known
properties of microscopic matter appears possible. Consequently, a
search for hereto overlooked new, fundamental properties of matter
is legitimate.
The question arose whether hidden variables may exist which
ensure local, continuous and causal time development for the
entire system, including measurements. Bell \cite{Bell} turned
this apparently philosophical question into physics by showing
that all such local, realistic theories are measurably distinct
from QT (Bell's inequalities). Subsequently, many experiments
were performed and local, realistic theories are now
convincingly excluded. For instance, the experiments of \cite{Aspect}
found violations of Bell's inequalities for spacelike measurements
on entangled quantum states. In such experiments one performs
measurements at distinct locations, say $\vec{x}_1$ and $\vec{x}_2$,
in time intervals small enough that any mutual influence through
communication at
or below the speed of light can be excluded. Results at $\vec{x}_1$
correlate with those at $\vec{x}_2$ (and vice versa) in a way that
{\it excludes} an interpretation as a classical correlation,
see \cite{Mermin} for a pedagogical presentation. Such
effects underline the need for qualitatively new properties of matter,
because they cannot be propagated through local, relativistic wave
equations (which also govern the interaction with the environment).
Also a consistent description of the space-time evolution of the quantum
state vector $|\Psi\rangle$ under such measurements encounters
difficulties. After accepting that a measurement at $(c\, t_1,\vec{x}_1)$
or $(c\, t_2,\vec{x}_2)$ interrupts the continuous, causal time evolution
by a discontinuous jump, once faces the problem that Lorentz
transformations can change the time ordering of spacelike events. In
a recent paper \cite{Be98a} it has been shown that a spacetime picture
for a physical state vector with relativistically covariant reduction
exists. It may be summarized as follows:
\begin{description}
\item{(1)} Measurement are performed by detectors, which are part of
the state vector, at localized spacetime positions
$(c\, t_i,\vec{x}_i)$, $i=1,2,\dots\,$.
\item{(2)} Discontinuous reductions of the state vector are defined
on certain Lorentz covariant spacelike hypersurfaces, which in some
neighbourhood of a detector include its backward light cone.
\item{(3)} The thus defined measurements happen in some reduction order,
which is {\it not} a time ordering with respect to a particular inertial
frame.
\end{description}
Based on this scenario, I pursue in the present paper an approach which
builds on the strengths of QT and tries to supplement it with new laws
for reductions, such that the conventional rules
for measurements (Born's probability interpretation) follow. These laws
are supposed to act on the {\it microscopic} level, independently of
whether macroscopic measurements are
actually carried out or not. Typically, they will effect some
interference phenomena. This implies observable consequences and
makes their eventual existence a physical issue. On the other hand,
we will see that rules can be designed in a way that most interference
effects survive entirely, whereas those affected are only weakened in
the sense of a decreased signal over background ratio (visibility). That
makes such laws difficult to detect, as most experiments work with
ensembles of particles and are happy to demonstrate a small signal over
a large (subtracted) background. Fortunately, recent years have seen
considerable improvements of experimental techniques, such that invoking
experimental input may become feasible.
The central idea of my approach is to {\it propose} that the Ability
To Perform Reductions (ATPR) between alternatives proposed by QT is a
hereto unidentified {\it elementary} property of microscopic matter.
I shall use the notation Quantum Detector (QD) to denote microscopic as
well as macroscopic matter acting in its ATPR. Matter gets such
a dual character: As QO it follows the continuous, deterministic time
evolution. As QD it has the ATPR and causes jumps in the wave function.
The goal is to explain the functioning of actually existing
macroscopic detectors from the properties of microscopic QDs.
Within our hypothetical framework central questions are now:
\begin{description}
\item{(1)} Which conglomerates of matter constitute a QD?
\item{(2)} Which are precisely the alternatives of QT standing
up for reductions?
\item{(3)} What are the rules according to which QDs make their
reductions?
\end{description}
It is unlikely, that ultimate answers can be found without
additional experimental guidance. But it is instructive to introduce
a simple hypothesis which allows (a) to illustrate the possibilities and
general direction of the approach and (b) focuses on questions about
experimental input which, quite generally, may be crucial for
achieving progress in the field.
Let us return to the detection of a photon by a photographic plate.
The simplest possibility is to attribute to each {\it single} AgCl
molecule the ATPR about collapsing the photon wave function. As this
is a spontaneous absorption, we get to the question asked in
the title of this paper. We assume that each molecule acts
independently when making its reductions and, by chance,
the $j^{th}$ molecule makes its reduction first, ahead of the
others. The alternative is to decay or to stay intact. Either choice
causes a jump in the wave function $|\Psi\rangle$ of eqn.(\ref{causal}).
Subsequently rules are given which seem (a) to be minimal and (b)
consistent with observations.
The collapse results are fixed by the rules of quantum mechanics.
Namely, to be either ($f$ stands for final)
\begin{equation} \label{collapse1}
|\Psi\rangle \to |\Psi\rangle_f = {c_j\over \sqrt{P_j}}\,
|\Psi_j\rangle ~~~{\rm with\ probability}~~ P_j = |c_j|^2
\end{equation}
or
\begin{equation} \label{collapse2}
|\Psi\rangle \to |\Psi'\rangle = \sum_{k\ne j} c'_k\, |\Psi_k\rangle ,\
c'_k= {c_k \over \sqrt{1-P_j}}, ~~~{\rm with\ probability}~~ 1-P_j\, .
\end{equation}
The $k$-sum in eqn.(\ref{collapse2}) includes $k=0$, compare
(\ref{causal}). The particular choice of the phase factors,
$c_j/\sqrt{P_j}$ and $c_k/\sqrt{1-P_j}$, assumes that a decoherence
process leads into alternative branches, whereas for collapse with
incomplete decoherence the issue would have to be resolved by the
collapse rules. Each molecule thus constitutes a QD. As their mutual
distances are short and their
relative motion is negligible, compared to the speed of light, we can
ignore the relativistic complications discussed in \cite{Be98a} (the
backward light-cone becomes an excellent approximation to
``instantaneous'').
Equation (\ref{collapse1})
implies as final result a dark spot at the position of molecule $j$.
By construction this happens with the correct probability $P_j$. As soon
as the wave function (\ref{collapse1}) rules, the reduction is
completed. This is different when molecule $j$
stays intact. Then the same rules (\ref{collapse1}) and
(\ref{collapse2}), which collapse $|\Psi\rangle$ of eqn.(\ref{causal}),
have now to be applied to the wave function $|\Psi'\rangle$ of
eqn.(\ref{collapse2}). Assume, molecule $l$ (note $l\ne j$ as the
branch $|\Psi_j\rangle$ does no longer exist) makes the next reduction.
The transformation will be either
\begin{equation} \label{collapse3}
|\Psi'\rangle \to |\Psi\rangle_f = {c'_l\over \sqrt{P'_l}}\,
|\Psi_l\rangle ~~~{\rm with\ probability}~~ P'_l=|c'_l|^2
\end{equation}
or
$$ |\Psi'\rangle \to |\Psi''\rangle = \sum_{k\ne j,l} c''_k\,
|\Psi_k\rangle ,\ c''_k= {c'_k \over \sqrt{1-P'_l}},
~~{\rm with\ probability}~~ 1-P'_l\, . $$
Putting equations (\ref{collapse2}) and (\ref{collapse3}) together,
we obtain
$$ |\Psi\rangle \to |\Psi\rangle_f = {c_l\over \sqrt{P_l}}\,
|\Psi_l\rangle ~~~{\rm with\ probability}~~ P_l,$$
{\it i.e.} precisely the correct likelihood to find the dark
spot at the position of molecule $l$. Continuing the procedure,
it is easy to see that all probabilities come out right.
Once a molecule $j$, $j=1,...,n$ has collapsed $|\Psi\rangle$
into the $|\Psi\rangle_f$ state of eqn.(4), the chemical reaction$^1$
\footnotetext[1]{{During the chemical reaction similar
collapse processes may continue. Presently, they are not of
interest to us, as our aim is to discuss the collapse of the
incoming photon wave function, which has the special property of
being transversally spread out over a macroscopic region.}}
-- initiated by the corresponding branch of each molecule -- survives
only in the neighbourhood of molecule $j$, where the visible spot
will occur. Let $t=0$ be the time at which the photon hits the
photographic plate. This time is well-defined as long as we can
assume that the photon flight time over a distance of the relevant
thickness of the photographic plate (for example $0.3~mm \Rightarrow
\triangle t = 10^{-12}~s = 1~ps$) is much smaller than the typical
collapse time. Let us denote by
\begin{equation} \label{tauf}
P^f(t)=(1-P_0)\, (1-e^{-t/\tau^f(t)})
\end{equation}
the probability that, at time $t$, the (entire) system has
decided about the location of the dark spot. Here $P_0=|c_0|^2$, see
eqn.(\ref{causal}), is the probability that the system fails to
detect. The r.h.s. of (\ref{tauf}) defines the system collapse time
$\tau^f(t)$. If $\tau^f$ is constant, it is the mean time the system
needs to make its reduction (with corresponding collapse probability
density $(\tau^f)^{-1}\, \exp (-t/\tau^f)$).
Let us assume that each molecule performs reductions on its own and that
$\rho^c_j(t)$ is the likelihood per time unit that the $j^{th}$ molecule
makes its reduction. We simplify the situation further
and consider a $\rho^c_j(t)$ that does not
depend on $j$ and is a step function: $\rho_j^c(t)=\rho^c\, \theta(t)$
with $\rho^c =$ constant. The corresponding one molecule collapse
probability is
\begin{equation} \label{tauc}
p^c(t)=(1-e^{-t/\tau^c})\, \theta(t)\, ,
\end{equation}
where $\tau^c=1/\rho^c$ is the mean collapse time of a single molecule.
The molecules make their reductions in some sequential order. For our
purposes the reductions process comes to a halt as soon as one molecule
has decayed. Assume, $n^c(t)$ molecules made their reductions. Whatever
values the $P_j$ in eqn.(\ref{collapse1}) take,
$P^f(t) = (n^c(t)/ n)\, (1-P_0)$ is the probability that the collapse
process has selected a definite location. As $n^c(t)=n\, p^c(t)$, we
conclude
\begin{equation}
P^f(t)= (1-P_0)\, p^c(t)\, .
\end{equation}
With the approximations made the system collapse time $\tau^f$, defined
by (\ref{tauf}), and the single molecule collapse time $\tau^c$,
defined by (\ref{tauc}), are identical.
Soon some arguments will be given that the constant $\tau^c$
should be regarded as upper bound of the system collapse time
$\tau^f(t)$.
Let us return to the central questions. In our discussion of
detection of a spread-out photon, I assumed the following:
(1) Each, single AgCl molecule may act as QD. (2) One alternative
stands up for reduction: decaying (through absorbing the photon)
or staying intact. (3) Each AgCl makes its reduction with a certain,
constant likelihood per time unit: $\rho^c$.
Ad (1): Assuming that a single AgCl molecule can act as QD reflects
the attempt to introduce an ATPR as a fundamental property of
microscopic matter. In our simplified discussion each AgCl molecule
is separated from the others by the emulsion and causal interactions
between them can be neglected. In a real photographic film only grains
of AgCl molecules are separated. Causal interactions between an AgCl
molecule and its neighbors within one grain cannot be neglected.
Indeed, the initiated chemical process will spread out over the
entire grain. If the collapse time is sufficiently large, competing
(ultimately alternative) chemical processes would start to evolve in
several grains. Under such circumstances, the definition of the QD
should be extended to include each causally connected region of AgCl
molecules. As a general rule, I find
it attractive to conjecture that a conglomerate of matter which (in a
reasonable approximation) can be treated as isolated QO can also be
regarded as isolated QD. Neglecting the influence of most of the world
is precisely how we get solutions out of QT. The hypothesis is, whenever
this works well for a QO, this QO may also constitute a QD whose
reduction probabilities are determined by its local quantum state,
although this quantum state may participate in discontinuous, non-local
transformations.
Ad (2): The scenario, pursued now, is that the QT alternatives up
for reductions have, quite generally, to do with absorption and emission
of particles. Here I limit the discussion to the absorption and emission
of photons, the process argued to be at the heart of every real, existing
and functioning measurement device. Of course, other physical processes
(like for instance in nuclear decay) should then be governed by similar
rules. In essence: Ruled by not yet identified {\it stochastic} laws,
superpositions of Fock space sectors with distinct particle numbers are
conjectured to collapse into particle number eigenstate sectors.
Ad (3): Our ATPR introduces an explicit arrow in time. This is attractive,
because it is a matter of fact that such an arrow exists. The canonical
conjugate variable to time is energy. Therefore, a frequency law which
relates the collapse time to the difference $\triangle E$ in energy
distribution between emerging branches is suggested
\begin{equation} \label{tcguess}
\tau^c\ =\ \tau^c (\triangle E)\, ,
\end{equation}
where the energy difference is defined as the one experienced by the
QD. For example, in case of our single AgCl molecule the difference
between absorbing or not absorbing the photon is:
$\triangle E = E_{\gamma}$, where $E_{\gamma}$ is the energy
of the photon. The total energy is (in the same way as in QT) conserved
in our approach.
The introduced system has been chosen because of its popularity in QT
text books in connection with the double slit effect.
Instead of the photographic plate other measurement devices
can be considered. For instance, the emergence of a track in a
bubble chamber through ionization by an high energy particle allows a
similar discussion. Here it is instructive to consider the emergence of
such a track in monatomic dilute gas, say hydrogen. Assume that {\it one}
incoming high energy particle has been split into two distinct
transversally sharp rays$^1$,
\footnotetext[1]{{Within our approach it might, however, happen that such
a state collapses spontaneously, because the device which caused the
split (and hence correlates with it) might act as a QD.}}
each with 50\% probability content. At time $t=0$ the two rays may hit
spacelike regions of
hydrogen gas. Each ray builds up a column of half-ionized atoms. Let us
focus on one of them, consisting of $n$ participating atoms. If one of
the atoms of our column emits a photon by re-capturing an
electron the relevant reduction has been made. A transformation of
type (\ref{collapse1}) puts all atoms of
the competing column into their unperturbed branches and the atoms
of our column into their ionized branches. There is now some ambiguity
about what should be considered a QD. Should each single atom
(including the involved electrons) act as independent QD or should all
atoms of the column together form one, single QD? In favor of the first
viewpoint is that the gas is assumed to be dilute. Hence, the mutual
influence through continuous, causal time evolution between the atoms is
negligible. On the other hand, the high energy particle correlates all
the atoms within the column (and, of course, also the other column): If
one atom performs its reduction in favor of the ionized branch, all
other are put there too. Assume the atoms act independently and the
differences in energy distribution between their branches are
$\triangle E_i$, ($i=1,...,n$), implying corresponding mean collapse
times $\tau^c_i (\triangle E_i)$. The probability that none
of them makes the reduction during the time interval $[0,t]$ becomes
$$ q^c (t) = \prod_{i=1}^n \exp [-t/\tau^c_i (\triangle E_i) ]
= \exp \left[ - \sum_{i=1}^n t/\tau^c_i (\triangle E_i) \right]\, .$$
On the other hand, if they all together form one single QD, this
probability becomes
$$ q^c (t) = \exp [-t/\tau^c(\triangle E)]\, ,~~{\rm where}~~
\triangle E = \sum_{i=1}^n \triangle E_i $$
is the total difference in energy distribution between the alternative,
macroscopic branches. Remarkably, the $q^c(t)$ of the last two equations
agree, when the law for the mean collapse time is
\begin{equation} \label{berg}
\tau^c (\triangle E)\ =\ {b\, \hbar \over \triangle E}\, ,
\end{equation}
where $b$ is a dimensionless constant. (Correspondingly, $\tau^c_i
=b\, \hbar/\triangle E_i$, $i=1,..,n$, of course.) Equation (\ref{berg})
has phenomenologically attractive features. The first one is that the
indicated ambiguity is rendered irrelevant. Another is that the
collapse time becomes large for small energy differences. Especially,
superpositions of states degenerate in energy will not collapse.
In this context {\it a measurement device is now an apparatus which
speeds up the collapse by increasing the difference in energy
distribution between quantum branches.} Before the distinct branches
become macroscopically visible, the energy difference becomes so large
that collapse happens with (practical) certainty.
Are there observable consequences beyond standard QT?
Reduction by an AgCl molecule destroys the possibility of
interference of the branches (\ref{collapse1}) and (\ref{collapse2})
of the wave function (\ref{Psi}). In particular, this does still hold
for the case of a single molecule $(n=1)$. But it appears unlikely that
anyone will, in the near future, measure interference effects between
AgCl$+\gamma$ and Ag$+$Cl. Hence, there is no contradiction.
In addition, it should be noted that our mechanism leaves the most
commonly observed interference effects intact: Namely, all those which
rely on the wave character of particles in a Fock space sector with
fixed particle number. This includes photon or other particle waves
passing through double slits and so on. Neutron interferometry which
relies on hyperfine level splitting would, in principle, be
suppressed. However, the energy differences are small such that
observable effect are unlikely.
Larger energy differences are achieved in atomic beam spectroscopy.
Ramsey fringes have been observed from interference of branches which
differ by photon quanta with energy in the $eV$ range. Figure~1 depicts
the interaction geometry of Bord\'e \cite{Bo84}, for a recent review
see~\cite{St97}. An atomic beam of two level
systems $(E_0 < E_1)$ interacts with two counterpropagating sets
of a traveling laser wave. The laser frequency is tuned
to the energy difference $\triangle E=E_1-E_0$, such that
induced absorption/emission processes take place at each of the
four interaction zones. The laser intensity is adjusted such that at
each interaction zone an incoming partial wave is (further) split
into two equally strong parts, $|\psi_0,m_0\rangle$ and
$|\psi_1,m_1\rangle$. Here $|\psi_0\rangle$ denotes an atom in its
incoming state, $|\psi_1\rangle$ an excited atom
and $m_0$, $m_1$ are the numbers of photon moments transferred.
Examples are indicated in the figure. The process leaves us with $2^n$
partial waves after the $n^{th}$ interaction zone, $n=1,2,3,4$. Of the
final sixteen partial waves $4\cdot 2=8$ interfere under detuning of the
laser frequency. Positions and directions of those eight partial waves
are along the four lines, indicated after the last interaction
zone of figure~1. The interference can be made visible by monitoring
the decay luminosity $I$ of the excited states $|\psi_1\rangle$ after
the last interaction zone. The contrast or visibility is defined by
\begin{equation} \label{contrast}
K = {I_{\max} - I_{\min} \over I_{\max} + I_{min}}\, ,
\end{equation}
where $I_{\max}$ and $I_{\min}$ are maximum and minimum of the measured
luminosities. Eight of the final sixteen partial waves are in excited
states and four of them interfere in two pairs, $|\psi_1,-1\rangle$
and $|\psi_1,1\rangle$ of the figure. Hence, the optimal contrast for the
Bord\'e geometry is
\begin{equation} \label{Kopt}
K_{opt} = {(4-4) + (8-0) \over (4+4) + (8+0)} = 0.5\, .
\end{equation}
This result is found by normalizing (in arbitrary units) the average
luminosity of each excited partial wave to one. Four decoherent branches
contribute then $I_{\max}=I_{\min}=4$, whereas the other four excited
partial waves contribute $I_{\min}=0$ and $I_{\max}=8$ (for $I_{\min}$
they annihilate one another and in the other extreme they amplify).
According to our hypothesis, integer photon numbers get restored with
a collapse time $\tau^c=\tau^c(\triangle E)$. If this happens in
range~1 of figure~1, the interference effect becomes
entirely destroyed. The likelihood for it to happen is $p^c=1-q^c$,
where $q^{c}=\exp (-t_D/\tau^c)$ and $t_D$ is the time an atom stays
in range~1. In range~2 each of the two $|\psi_1,1\rangle$ partial waves
has borrowed $1/4$ of a photon from the laser beam. To get a unique
collapse description, we invoke a minimality assumption:
The splitting of the atom
has to be constructed with the minimal number of photons possible.
The assumption seems to be natural, because two photons with
the same quantum numbers cannot be distinguished. It follows that
the system can collapse either into the two $|\psi_1,1\rangle$
partial waves or into the two $|\psi_0,0\rangle$ partial waves.
Neither collapse has observable consequences,
because the interference effects of the upper part and lower
part of figure~1 are not distinguished by measuring the decay
luminosity. In range~3 two split photons (distinct momenta) get
involved: One mediates
collapse between $|\psi_1,1\rangle$ and $|\psi_0,2\rangle$, the other
between $|\psi_0,0\rangle$ and $\psi_1,1\rangle$. These two collapse
processes are supposed to act independently. Each destroys, if it happens,
half of the interference effect. The probability for both of them to
happen is $(p^c)^2$ (using that $t_D$ is identical in range~3 and~1) and
the probability that one of them (excluding both) happens is
$p'^c=1-(q^c)^2-(p^c)^2$. Putting things together, the optimal contrast
becomes
\begin{equation} \label{Kcopt}
K_{opt}^c=16^{-1}[8-8\,p^c-4\,q^c\,p'^c-8\,q^c\,(p^c)^2]=
0.5\, \exp [ -2\, t_D/\tau^c (\triangle E)]\, .
\end{equation}
Experiments performed at the Physikalisch-Technische Bundesanstalt
(PTB) Braunschweig rely on the $^3P_1$--$^1S_0$ transition of $^{40}Ca$
which has $\lambda = 657.46$~nm, {\it i.e.} $\triangle E =1.886\,
eV$. The best contrast achieved \cite{Riehle} is approximately
$K=0.2$ with $t_D=21.6\cdot 10^{-6}\,s$. The actual experiments
are performed using pulsed laser beams applied to laser cooled
atoms in a magneto-optical trap. The times $t_D$ and $t_d$,
corresponding to the distances $D$ and $d$ of figure~1, are
then the times between the laser pulses, see~\cite{St97} for
details. Relying on the PTB result we obtain the estimate
$$ \tau^c_{\min}\, (1.89\, eV) = 2\,t_D\, /\, \ln (5/2) =
47\cdot 10^{-6}\,s < \tau^c\, (1.89\, eV) $$
which translates (\ref{berg}) into
\begin{equation} \label{bmin}
b_{\min} = 1.35\cdot 10^{11} < b\, .
\end{equation}
That the constant $b$ has to be large is no surprise, as the action
$b\hbar$ marks the transition from quantum to classical physics. The
bound (\ref{bmin}) can easily be improved by estimating conventional
effects which contribute to diminishing the contrast $K$. Beyond,
a direct measurement of a non-zero $\tau^c$
requires that all other effects can convincingly be controlled and that
still a gap between the estimated and measured contrast remains. Such
an analysis goes beyond the scope of the present paper. Here, I am
content with establishing firm, but crude, bounds on $b$.
Finally, in this paper, I derive an upper bound on $b$. Avalanche
photodiodes are the up-to-date devices for achieving
measurements in short time intervals, as needed for spacelike
measurements \cite{Aspect}. Time resolutions down to $20\,ps$ FWHM
are achieved, see \cite{Co96} for a recent review. The energy
consumption is sharply peaked in these short intervals
(order of watts), but does not translate into an immediate estimate
of the collapse time. The reason is that collapse at some later time
may lead to indistinguishable results. Claiming differently includes
the task of disproving popular decoherence ideas \cite{Zurek}.
Nevertheless, there is an easy way to estimate upper bounds by
analysis of actually working measurement devices: Our approach makes
only sense when the reduction process does keep up with the
{\it sustained} performance of every real, existing measurement device.
Then, the energy dissipation of such a device yields immediately
an upper bound on $b$. Ref.\cite{Co96} gives on p.1964 the example
of a photo avalanche diode which operates at $10^5\,cps$ and has
a mean power dissipation of $4\,mW$. This translates into an energy
consumption of about $2.5\cdot 10^{11}\,eV$ per count, {\it i.e.}
$$ \tau^c\, (2.5\cdot 10^{11}\, eV) <
\tau^c_{\max}\,(2.5\cdot 10^{11}\, eV) = 10^{-5}\, s\, ,$$
which implies
\begin{equation} \label{bmax}
b < b_{\max} = 3.8\cdot 10^{21}\, .
\end{equation}
Equation (\ref{bmin}) and (\ref{bmax}) leave a wide range open. An
analysis of existing experiments should allow to narrow things
down by a least a few orders of magnitude. Here the emphasize is
on quoting save, instead of sophisticated, bounds. Even this has
caused some efforts, the reason simply being that experimentalists
do not focus on the information needed.
In conclusion, we have discussed the possibility of attributing to
microscopic matter the ability to perform wave function reductions.
It is of interest to improve the bounds $b_{\min}$ (\ref{bmin}) and
$b_{\max}$ (\ref{bmax}) for the collapse time $\tau^c (\triangle E)$ of
equation~(\ref{berg}). From this viewpoint, I would
like to argue in favor of a paradigm shift concerning QT experiments.
It is no longer of central interest to demonstrate the existence of
one or another exotic interference effect. We know, they are there.
Most interesting is to control that interference happens for every
single, participating particle. This puts the focus on experiments
with high visibility. If one could convincingly demonstrate that
particles occasionally skip participation in an interference pattern,
such a results could pave a major inroad towards understanding of the
measurement process. The aim of pushing experiments towards optimal
visibility is of interest in itself. Independent of its validity,
the introduced collapse scenario provides an interesting classification
pattern for such results: The achieved lower bounds $b_{\min}$ should
be compiled. Concerning $b_{\max}$, one is lead to minimizing the energy
dissipation of measurement devices under sustained performance. Again,
this is a goal of interest in itself.
\medskip
\noindent
{\bf Acknowledgements:} I would like to thank Dr. Wolfgang Beirl for
his interest and useful discussions.
\medskip
| proofpile-arXiv_065-413 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\subsection{Simulation Results in $2$ Dimensions}
The critical transition in the shape of the hysteresis loop is observed
in the simulation, and expected from the renormalization group
\cite{Dahmen1,Dahmen2}, in $3$, $4$, and $5$ dimensions. We also found
that the upper critical dimension, at and above which the mean field
exponents become correct, is six. Furthermore, in one dimension, we
expect that in a thermodynamic system, with an unbounded distribution of
random fields, there will be no infinite avalanche for $R > 0$. That
will be so because if there is any randomness, there will be a spin in
the linear chain that will have the ``right'' value for its random field
to stop the first avalanche. For a bounded distribution of random
fields, the scaling behavior near the transition will not be universal\
\cite{Dahmen1}; instead, it will depend on the exact shape of the tails
of the distribution of random fields. Then, the question that remains
is: what happens in two dimensions?
From the simulation and a few arguments that we are about to show, we
conjecture that the two dimensional exponents will have the values:
$\tau + \sigma\beta\delta =2$, $\tau=3/2$, $1/\nu=0$, and $\sigma\nu
=1/2$. (The other exponents (except $z$) can be found from exponent
relations\ \cite{Dahmen1,Dahmen3} using these values.) The ``arguments''
are as follows.
It is quite possible that two is the {\it lower critical dimension}
(LCD) for our system. At the lower critical dimension, the critical
exponents are often ratios of small integers, and it is often possible
to derive exact solutions. Since the geometry in $2$ dimensions allows
for at most one system spanning avalanche, the ``breakdown of
hyperscaling'' exponent $\theta$ (see section IV B) must be zero, and
the hyperscaling relation\ \cite{Dahmen1,Dahmen3} is restored:
\begin{equation}
{1 \over \sigma\nu} = d - {\beta \over \nu}
\label{hyper_2d_equ1}
\end{equation}
We know that this relation is violated in $4$ and $5$ dimensions, and is
probably violated in $3$ dimensions. In dimensions above two, the
hyperscaling relation is modified by the exponent $\theta$ which gives a
measure of the number of spanning avalanches near the critical
transition, as a function of the system size. Figure\ \ref{span_2d_fig}
shows the number of spanning avalanches in $2$ dimensions for several
system sizes. Notice that, as assumed, there is not more than one
spanning avalanche in the system.
\begin{figure}
\centerline{
\psfig{figure=Figures/Span_aval_paper_2d_THESIS.ps,width=3truein}
}
\caption[Spanning avalanches in $2$ dimensions]
{{\bf Number of spanning avalanches in $2$ dimensions}
as a function of disorder
$R$, for several system sizes. The data points are averages between
as little as $10$ to as many as $2200$ random field configurations.
Some typical error bars near the center of the curves are shown;
error bars are smaller toward the ends. Note that there is no
more than one spanning avalanche. \label{span_2d_fig}}
\end{figure}
We use two more arguments to derive the critical exponents. In $2$
dimensions, we find that the avalanches ``look'' compact (figure\
\ref{compact_2d_fig}). (The avalanches in $3$ dimensions are not compact
(figure\ \ref{notcompact_3d_fig}).) This implies that $1/\sigma\nu = d =
2$, which leads to $\beta/\nu=0$ from equation\ (\ref{hyper_2d_equ1}).
Furthermore, it is often the case that in the lower critical dimension,
the Harris criterion\cite{Dahmen1}
\begin{equation}
{\nu \over \beta\delta} \geq {2 \over d}
\label{Harris_eqn}
\end{equation}
becomes saturated (an equality); so in $2$ dimensions we expect
$\beta\delta/\nu=1$. From this and the previous result, the exponent
which gives the decay in space of the avalanche correlation function
\begin{equation}
\eta = 2 + {\beta \over \nu} - {\beta\delta \over \nu}
\label{eta_eqn}
\end{equation}
(see references\ \cite{Dahmen1,Dahmen3} for the derivation of all the
exponent relations) becomes equal to $\eta =1$.
Since at the LCD the correlation length typically diverges exponentially
as the critical point is approached, we expect $\nu \rightarrow
\infty$, and $\beta$ can be finite. Using the exponent relation\
\cite{Dahmen1,Dahmen3}:
\begin{equation}
\tau-2=\sigma\beta(1-\delta),
\label{hyper_2d_equ3a}
\end{equation}
we further find that $\tau=3/2$ and $\tau+\sigma\beta\delta=2$.
\begin{figure}
\centerline{
\psfig{figure=Figures/2D_R.8_span_grey_THESIS.ps,width=3truein}
}
\caption[Simulation of a $400^2$ spin system at $R=0.8$]
{Simulation in $2$ dimensions of a $400^2$ spin system at $R=0.8$.
The figure shows the configuration of the system after a spanning
avalanche has just occurred (grey region). The dark area corresponds
to spins
that have not yet flipped, while the white area are spins that have
flipped earlier. Notice that the spanning avalanche (grey area)
seems compact. \label{compact_2d_fig}}
\end{figure}
We must mention that our firm conjectures about the exponents in two
dimensions must be contrasted with our lack of knowledge about the
proper scaling forms. As mentioned above, at the LCD the correlation
exponent $\nu$ typically diverges, although some combinations of
critical exponents stay finite (hence $\sigma\nu = 1/2$). Those which
diverge and those which go to zero usually must be replaced by exponents
and logs, respectively. We have used three different RG-scaling {\it
ans\"atze} to model the data in two dimensions. (1)~We used the
traditional scaling form $\xi \sim |R_c-R|^{-\nu}$, deriving $\nu =
5.3\pm 1.4$ and $R_c = 0.54\pm 0.04$. These collapses worked as well as
any, but the large value for $\nu$ (and larger value still for $1/\sigma
= 10\pm 2$) makes one suspicious. (2)~We used a scaling form suggested
by Bray and Moore\cite{BrayMoore} in the context of the equilibrium
thermal random field Ising model at the LCD, where $R_c=0$: if they
assume that $R$ is a marginal direction, then by symmetry the flows must
start with $R^3$, leading to $\xi \sim e^{(\tilde{a}/|R_c-R|^2)} \equiv
e^{(\tilde{a}/R^2)}$. This form has the fewest free parameters, and
most of the collapses were about as good as the others (except notably
for the finite-size scaling of the moments of the avalanche size
distribution, which did not collapse well once spanning avalanches
became common). (3)~We developed another possible scaling form, based on
a finite $R_c$ and $R$ marginal, which generically has a quadratic flow
under coarse-graining: here $\xi \sim e^{(\tilde{b}/|R_c-R|)}$. We find
$R_c=0.42\pm0.04$. The rational behind these three forms is shown in
appendix A.
The results from data collapses in two dimensions were obtained from
measurements of the spanning avalanches, the second moments of the
avalanche size distribution, the integrated avalanche size
distributions, and the avalanche correlations. The magnetization curves
are also obtained from the simulation, but as in the higher dimensions,
the scaling region is small (around $H_c$ and $M_c$), and the collapses
do not define the exponents well.
\begin{figure}
\centerline{
\psfig{figure=Figures/Sites_1_grey_THESIS.ps,width=3truein}
}
\caption[Largest avalanche in the hysteresis loop in a $40^3$ system, near
the critical point]
{Largest avalanche occurring in the hysteresis loop
in a $40^3$ spins system near the critical point.
The avalanche is not compact. \label{notcompact_3d_fig}}
\end{figure}
Measurements that require the knowledge of the critical randomness are
the binned avalanche size distribution from which we extract the
exponents $\tau$ and $\beta\delta$, the critical magnetic field $H_c$,
and the avalanche time measurement which gives the exponent $z$. These
measurements were not obtained {\it at} the critical disorder because
$R_c$ is not well defined as was mentioned above, and because for low
disorders (less than $0.71$ for a $7000^2$ system), the system flips in
one infinite avalanche, and such measurements are therefore not
possible. We have nevertheless estimated the values of some of these
exponents and of $H_c$, from data obtained at a larger disorder (where
there is no spanning avalanche). From the avalanche size distribution
binned in $H$ at $R=0.71$ and $L=7000$, and the magnetization curves, we
find that the critical field $H_c$ is around $1.32$. A straight line fit
through the data agrees with a possible value of $\tau=3/2$ (the
conjectured value). From the time distribution of avalanche sizes for a
system of $30000^2$ spins, at $R=0.65$, we measured (from a straight
linear fit) the exponent $\sigma\nu z$ to be $0.64$. The other exponents
were obtained from scaling collapses as follows.
Figure\ \ref{s2_2d_fig}a shows the second moments of the avalanche size
distribution for several system sizes. The collapses using the three
different scaling forms are shown in figures\ \ref{s2_2d_fig}(b-d). The
first one (figure\ \ref{s2_2d_fig}b) is:
\begin{equation}
{\langle S^2 \rangle}_{int} \sim L^{-(\tau+\sigma\beta\delta-3)/\sigma\nu}\
{\check {\cal S}}_{int}^{(2)}(L\ |r|^{\nu})
\label{2d_equ01}
\end{equation}
which is the kind of scaling form used in $3$, $4$, and $5$ dimensions.
This form assumes $\xi \sim |r|^{-\nu}$. The exponents are $(\tau +
\sigma\beta\delta-3)/\sigma\nu = -1.9$ and $\nu=5.25$, and $r=(R_c -
R)/R$ with $R_c=0.54$. The second scaling form (figure\
\ref{s2_2d_fig}c) is:
\begin{equation}
{\langle S^2 \rangle}_{int} \sim L^{-(\tau+\sigma\beta\delta-3)/\sigma\nu}\
{\bar {\cal S}}_{int}^{(2)}(L\ e^{-\tilde{a}/|R_c-R|^2})
\label{2d_equ02}
\end{equation}
which is obtained from $\xi \sim e^{(\tilde{a}/|R_c-R|^2)}$. The values
of the exponents and parameters are: $(\tau +
\sigma\beta\delta-3)/\sigma\nu = -1.9$, $\tilde{a} = 3.4$ ($\tilde{a}$
is not universal), and $R_c \equiv 0$ (by assumption; see previous
paragraph). Notice that this collapse is not as good as the other two; a
better collapse is obtained with $R=0.15$ and $\tilde{a}=2.0$. If this
is the correct scaling form and $R_c=0$, this discrepancy can be due to
finite size effects. The third scaling form is (figure\
\ref{s2_2d_fig}d):
\begin{equation}
{\langle S^2 \rangle}_{int} \sim L^{-(\tau+\sigma\beta\delta-3)/\sigma\nu}\
{\hat {\cal S}}_{int}^{(2)}(L\ e^{-\tilde{b}/|R_c-R|})
\label{2d_equ03}
\end{equation}
which is obtained from $\xi \sim e^{(\tilde{b}/|R_c-R|)}$. The values of
the exponents and parameters are: $(\tau +
\sigma\beta\delta-3)/\sigma\nu = -1.9$, $\tilde{b} = 2.05$ ($\tilde{b}$
is also non-universal), and $R_c= 0.42$. As it is clear from the last
three figures, collapses with these different scaling forms are
comparable. Notice that the exponent $(\tau+\sigma\beta\delta
-3)/\sigma\nu$ is the same for the three collapses, but that $1/\nu$ is
zero for the last two (by assumption) while it is $0.19$ for the first
collapse. Let's now look at the collapses of the integrated avalanche
size distribution curves, which are not finite size scaling
measurements.
\begin{figure}
\centerline{
\psfig{figure=Figures/Non_Span_s2_d2_paper_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/Non_Span_s2_d2_collapse_s2_paper_new_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/Non_Span_s2_d2_collapse_exp_n2_paper_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/Non_Span_s2_d2_collapse_exp_n1_paper_THESIS.ps,width=3truein}
}
\caption[Second moments of the avalanche size distribution in $2$ dimensions]
{(a) {\bf Second moments of the avalanche size distribution in $2$
dimensions,} integrated
over the external field $H$, for several system sizes. The data
points are averages over up to $2200$ random field
configurations. Error bars are
smaller than shown for larger disorders.
(b), (c), and (d) Scaling collapses of the second moments of the
avalanche size distribution in $2$ dimensions, integrated over the
field $H$. The curves that are collapsed are of size:
$50^2$, $100^2$, $300^2$, $500^2$, $1000^2$, $3000^2$, $5000^2$,
$7000^2$, and $30000^2$.
See text for the scaling forms, and the values of the exponents
and parameters. \label{s2_2d_fig}}
\end{figure}
Figure \ref{aval_2d_fig}a shows the integrated avalanche size
distribution curves for a $7000^2$ spin system, at several values of the
disorder $R$. Earlier, in figure\ \ref{bump_345fig}, we saw the fit to
the scaling collapse of such curves, done using the same scaling form as
in $3$, $4$, and $5$ dimensions:
\begin{equation}
D_{int}(S,R)\ \sim\ S^{-(\tau + \sigma\beta\delta)}\
{\bar {\cal D}}^{(int)}_{-}
(S^{\sigma} |r|)
\label{2d_equ1}
\end{equation}
(The $-$ sign indicates that the collapsed curves are for $r<0$, ie.
$R>R_c$.) However, $S^\sigma |r|$ might not be the appropriate scaling
argument in $2$ dimensions. First, from figure\ \ref{bump_345fig}, the
scaling curve in $2$ dimensions differs dramatically from the scaling
curves in higher dimensions for small arguments $X=S^\sigma |r|$. The
mean field scaling function $\bar{\cal D}_{-}^{(int)}(X)$ is a
polynomial for small $X$, and we expected (and found) a similar behavior
in $5$, $4$ and $3$ dimensions (but notice that the scaling function in
$3$ dimensions is starting to look like the curve in $2$ dimensions for
small $X$). In $2$ dimensions, if we collapse our data (figure\
\ref{aval_2d_fig}b) using the scaling form:
\begin{equation}
D_{int}(S,R)\ \sim\ S^{-(\tau + \sigma\beta\delta)}\
{\cal D}^{(int)}_{-}
(S|r|^{1/\sigma})
\label{2d_equ2}
\end{equation}
with $\tau + \sigma\beta\delta=2.04$, $1/\sigma=10$, and $r=(R_c -
R)/R$, we find that the scaling function for small $\tilde
X=S|r|^{1/\sigma}$ looks linear with power one! This might imply that
the scaling function ${\cal D}^{(int)}_{-}(S|r|^{1/\sigma})$ (eqn.\
(\ref{2d_equ2})) is the one that is analytic for small arguments in $2$
dimensions.
\begin{figure}
\centerline{
\psfig{figure=Figures/Aval_histo_d2_L7000_paper_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/Aval_histo_d2_L7000_collapse_paper_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/Aval_histo_d2_L7000_collapse_exp_n2_paper_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/Aval_histo_d2_L7000_collapse_exp_n1_paper_THESIS.ps,width=3truein}
}
\caption[Integrated avalanche size distribution curves in $2$ dimensions]
{(a) {\bf Integrated avalanche size distribution} curves for several
disorders {\bf in $2$ dimensions,}
at the system size $L=7000$. The curves are averages over
$10$ to $20$ random field configurations, and have been smoothed.
(b), (c), and (d) Scaling collapses of the data from (a) using the
three scaling forms and the exponents from the text.
The collapsed curves have disorders:
$0.72$, $0.74$, $0.77$, and $0.80$. The straight
grey line in each of the plots has a slope of one. \label{aval_2d_fig}}
\end{figure}
Second, we conjectured above that the values for $\sigma$ and $1/\nu$
are probably zero in $2$ dimensions, and that only the combination
$\sigma\nu$ is finite ($\sigma\nu$ $=$ $1/2$). It follows that, for the other
two scaling forms we use, the arguments of the scaling function should
be $S e^{-\tilde{a}/\sigma\nu|R-R_c|^2}$ and $S
e^{-\tilde{b}/\sigma\nu|R-R_c|}$, and {\it not} $S^\sigma
e^{-\tilde{a}/\nu|R-R_c|^2}$ and $S^\sigma e^{-\tilde{b}/\nu|R-R_c|}$
respectively. This is analogous to using $S|r|^{1/\sigma}$ in the
scaling form\ (\ref{2d_equ2}). We should mention here that both equation
(\ref{2d_equ1}) and equation (\ref{2d_equ2}) give the same scaling
exponents $\tau+\sigma\beta\delta$ and $\sigma$, and that in all our
scaling collapses we have assumed that the same scaling argument is
valid for small and large $\tilde X$ (and in between). This in general,
does not have to be true.
\begin{figure}
\centerline{
\psfig{figure=Figures/Aval_histo_2d_L30000_R0.650_ave_fit_paper_THESIS.ps,width=3truein}
}
\caption[Linear fits to an integrated avalanche size distribution curve in
$2$ dimensions]
{ {\bf Integrated avalanche size distribution curve in $2$ dimensions},
for a system of $30000^2$ spins, at $R=0.650$. Shown are two linear
fits to the data: one for small sizes and the other for large sizes.
The slope for the fit at small $S$ is $0.90$. The fit was done
for sizes in the range $[10,250]$. The slope differs by less than
$5\%$ when the range is changed ($S$ is never larger than $400$ though)
.
The slope for the fit at large $S$ is $1.78$. The slope differs by
less than $2\%$ when the range is changed ($S$ is never smaller
than $10000$). The conjectured value for $\tau+\sigma\beta\delta$ is
$2$ which is different from $1.78$. This is similar to
the behavior we saw in $3$, $4$, and $5$ dimensions.
On the other hand, for small sizes we expect the exponent
$\tau+\sigma\beta\delta -1=1$ (see text). Again, the two measurements
don't completely agree,
but the slope from our data does seem to indicate
such a behavior. \label{raw_aval_2d_fig}}
\end{figure}
Equation\ (\ref{2d_equ2}) is therefore one of the three scaling forms we
use. The second scaling form is:
\begin{equation}
D_{int}(S,R)\ \sim\ S^{-(\tau + \sigma\beta\delta)}\ {\cal D}^{(int)(2)}_{-}
\Bigl(S e^{-\tilde{a}/\sigma\nu|R_c-R|^2}\Bigr)
\label{2d_equ3}
\end{equation}
shown in figure\ \ref{aval_2d_fig}c, with $\tau +
\sigma\beta\delta=2.04$, $\tilde{a}/\sigma\nu=7.0$ (this implies that
$\sigma\nu=0.49$), and $r=R_c-R$ with $R_c \equiv 0$ by assumption. And
finally, the third scaling form we use is:
\begin{equation}
D_{int}(S,R)\ \sim\ S^{-(\tau + \sigma\beta\delta)}\ {\cal D}^{(int)(1)}_{-}
\Bigl(S e^{-\tilde{b}/\sigma\nu|R_c-R|}\Bigr)
\label{2d_equ6}
\end{equation}
shown in figure\ \ref{aval_2d_fig}d, with $\tau +
\sigma\beta\delta=2.04$, $\tilde{b}/\sigma\nu = 4.0$ (which makes
$\sigma\nu=0.51$), and $r=R_c-R$ with $R_c = 0.42$. Again, not only are
all three collapses comparable, but the exponents extracted from them
are as well. The exponent for the slope of the distribution is
$\tau+\sigma\beta\delta=2.04$ for the three collapses, and the exponent
combination $\sigma\nu$ is around $0.51$ (for the first collapse
$\sigma=0.10$, while $\nu=5.25$ from the equivalent second moment
collapse).
Figures\ \ref{aval_2d_fig}(b-d) show that the scaling function ${\cal
D}^{(int)}_{-}$ seems to be linear with slope one for small arguments
(the grey lines have slope one) and that the constant term in the
polynomial expansion is zero (or close to zero). This leads to a
singular scaling function correction to the avalanche size distribution
exponent $\tau+\sigma\beta\delta$ for small non--zero $\tilde X$:
\begin{equation}
D_{int}(S,R)\ \sim\ S^{-(\tau+\sigma\beta\delta)}\
{\cal D}_{-}^{(int)}(\tilde X)\ \sim\ S^{-(\tau+\sigma\beta\delta)+1}
\label{2d_equ7}
\end{equation}
(Note that we could have used ${\cal D}^{(int)(1)}_{-}$ or ${\cal
D}^{(int)(2)}_{-}$ as well.)
Recall that because of the ``bump'' in the avalanche size distribution
scaling function in $3$, $4$, and $5$ dimensions, and in mean field, the
slope of the raw data curves did not agree with the value of the
exponent $\tau+\sigma\beta\delta$. In $2$ dimensions, this is still
true, but we also find a singular behavior for ${\cal
D}_{-}^{(int)}(\tilde X)$, which changes the slope of the data curve for
small $\tilde X$. In figure\ \ref{raw_aval_2d_fig}, an integrated
avalanche size distribution curve for a system of $30000^2$ spins, at
$R=0.65$, is plotted along with the linear fits to the data for small
and large size $S$. For large $S$, the slope is close to but not equal
to $2$, while for small $S$, the slope is close to one!
The avalanche correlation data (see figure\ \ref{correl_2d_fig}a) is
collapsed with three different scaling forms as well. These forms are
analogous to the ones used for the second moments collapses, but with
the distance $x$ taking the place of the system size $L$. The collapses
and the extracted exponents from these three forms are again very
similar, and only one of the collapses is shown in figure\
\ref{correl_2d_fig}b. The value of $\beta/\nu$ from these collapses is
$0.03 \pm 0.06$. If we compare figure\ \ref{correl_2d_fig}b with the
collapse of the avalanche correlation in $3$ dimensions (fig.\
\ref{correl_collapse_3d_fig}a), we find that the scaling function in $2$
dimensions seems to be singular with slope one for small distances, as
is the integrated avalanche size distribution for small sizes.
The spanning avalanches data are also analyzed using three scaling forms
similar to those used for the second moments of the avalanche size
distribution collapses. The exponent $\theta$ is poorly defined from
these collapses (and is therefore not listed in Table\
\ref{conj_meas_2d_table}), although the data does collapse for the
exact value: $\theta=0$.
The three collapses for all the measurements we have done are very
similar. This is not a surprise: it is always hard to distinguish large
power laws ($\nu$ and $1/\sigma$ are large in the ``linear argument''
scaling form (eqns.\ (\ref{2d_equ01}) and (\ref{2d_equ2}))) from
exponentials. Although some of the exponents have very different values
in the three collapses, the average of the exponents from the three
methods agree within the error bars with each method (see figure\
\ref{exp_compare}) and our conjectures. In conclusion, although we do
not know the correct scaling form for the data in $2$ dimensions, the
possible three scaling forms we mention give exponent values that are
compatible with each other and with our conjectures (see Table\
\ref{conj_meas_2d_table}). (Table\ \ref{conj_2d_table} gives the
conjectured values for the exponents that have not been measured in the
collapses.) Much larger system sizes might be necessary to obtain more
conclusive results.
\begin{figure}
\centerline{
\psfig{figure=Figures/Norm_Correl_d2_L30000_some_paper_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/Norm_Correl_d2_L30000_some_collapse_paper_THESIS.ps,width=3truein}
}
\nobreak
\caption[Avalanche correlation curves in $2$ dimensions]
{(a) {\bf Avalanche correlation function in $2$ dimensions,}
integrated over the external field $H$, for several disorders $R$
and the system size $L=30000$.
Only the curve with the smallest disorder is an average over several
random field configuration.
(b) Scaling collapse of the avalanche correlation curves in $2$
dimensions, for a system of $30000^2$ spins. The exponent values
are: $\nu=5.25$ and $\beta/\nu=0$. The critical disorder is $R_c=0.54$,
and $r=(R-R_c)/R$. Notice that for small $x|r|^\nu$, the scaling
function looks singular with a power close to one (the straight line
has a slope of one). \label{correl_2d_fig}}
\end{figure}
\section{Derivation of the various scaling forms and corrections}
In this paper we make extensive use of scaling collapses. Many
variations are important to us: Widom scaling, finite-size scaling,
singular corrections to scaling, analytic corrections to scaling,
rotating axes, and exponentially diverging correlation length scaling.
The underlying theoretical framework for scaling is given by the
renormalization group, developed by Wilson and Fisher\cite{WilsonFisher}
in the context of equilibrium critical phenomena and by now well
explicated in a variety of texts\cite{Goldenfeld,MaBinney,texts}.
We have discovered that we can derive all the scaling forms and
corrections that have been important to us from two simple hypotheses
(found in critical regions): universality and invariance under
reparameterizations. {\sl Universality} is the statement that two
completely different systems will behave the same near their critical
point\ \cite{note5} (for example, they can have exactly the same kinds
of correlations). {\sl Reparameterization invariance} is the statement
that smooth changes in the units or methods of measurement should not
affect the critical properties. We use these properties to develop the
scaling forms and corrections we use in this paper. Each example we
cover will build on the previous ones while developing a new idea.
For our first example, consider some property $F$ of a system at its
critical point, as a function of a scale $x$. $F$ might be the
spin-spin correlation function as a function of distance $x$ (or it
might be the avalanche probability distribution as a function of size
$x$, etc.) If two different experimental systems are at the same
critical point, their $F$'s must agree. It would seem clear that they
cannot be expected to be equal to one another: the overall scale of $F$
and the scale of $x$ will depend on the microscopic structure of the
materials. The best one could imagine would be that
\begin{equation}
F_1(x_1) = A F_2( B x_2)
\label{single}
\end{equation}
where $A$ would give the ratio of, say, the squared magnetic moment per
domain of the two materials, and $B$ gives the ratio of the domain sizes.
Now, consider comparing a system with itself, but with a different
measuring apparatus. Universality in this self-referential sense would
imply $F(x) = A F(B x)$, for suitable $A$ and $B$. If instead of using
finite constant $A$ and $B$, we arrange for an infinitesimal change in
the measurement of length scales, we find:
\begin{equation}
F(x) = (1-\alpha \epsilon)\ F\Bigl((1-\epsilon) x\Bigr)
\label{small_single}
\end{equation}
where $\epsilon$ is small and $\alpha$ is some constant. Taking the
derivative of both sides with respect to $\epsilon$ and evaluating it at
$\epsilon = 0$, we find $-\alpha F = x F'$, so
\begin{equation}
F(x) \sim x^{-\alpha}.
\label{power}
\end{equation}
The function $F$ is a power--law! The underlying reason why power--laws
are seen at critical points is that power laws look the same at
different scales.
Now consider a new measurement with a distorted measuring apparatus. Now
$F(x) \sim {\cal A}\Bigl[F\Bigl({\cal B}(x)\Bigr)\Bigr]$ where ${\cal
A}$ and ${\cal B}$ are some nonlinear functions. For example, one might
measure the number of microscopic domains $x$ flipped in an avalanche,
or one might measure the total acoustic power ${\cal B}(x)$ emitted
during the avalanche; these two ``sizes'' should roughly scale with one
another, but nonlinear amplifications will occur while the spatial
extent of the avalanche is small compared to the wavelength of sound
emitted: we expand ${\cal B}(x) = B x + b_0 + b_1/x + \ldots$ Similarly,
our microphone may be nonlinear at large sound amplitudes, or the
absorption of sound in the medium may be nonlinear: ${\cal A}(F) = A F +
a_2 F^2 + \ldots$ So,
\begin{eqnarray}
{\cal A}\Bigl[F\Bigl({\cal B}(x)\Bigr)\Bigr]\ \approx\ \nonumber \\
A \Bigl(F(B x) + F'(B x)(b_0 + b_1/x + \ldots)\ +\ \nonumber \\
F''(B x)(\ldots) + \ldots \Bigr)
+\ a_2 F^2(B x)\ + \ldots
\label{nonlinear_single}
\end{eqnarray}
We can certainly see that our assumption of universality cannot hold
everywhere: for large $F$ or small $x$ the assumption of
reparameterization invariance (\ref{nonlinear_single}) prevents any
simple universal form. Where is universality possible? We can take the
power-law form $F(x) \sim x^{-\alpha} = x^{\log A/\log B}$ which is the
only form allowed by linear reparameterizations and plug it into
(\ref{nonlinear_single}), and we see that all these nonlinear
corrections are subdominant ({\it i.e.}, small) for large $x$ and small
$F$ (presuming $\alpha>0$). If $\alpha>1$, the leading correction is
due to $b_0$ and we expect $x^{-\alpha-1}$ corrections to the universal
power law at small distances; if $0<\alpha<1$ the dominant correction
is due to $a_2$, and we expect corrections of order $x^{-2\alpha}$.
Thus our assumptions of universality and reparameterization invariance
both lead us to the power-law scaling forms and inform us as to some
expected deviations from these forms. Notice that the simple rescaling
led to the universal power-law predictions, and that the more complicated
nonlinear rescalings taught us about the dominant corrections: this will
keep happening with our other examples.
For our second example, let us consider a property $K$ of a system, as a
function of some external parameter $R$, as we vary $R$ through the
critical point $R_c$ for the material (so $r=R-R_c$ is small). $K$
might represent the second moment of the avalanche size distribution,
where $R$ would represent the value of the randomness; alternatively $K$
might represent the fractional change in magnetization $\Delta M$ at the
infinite avalanche $\ldots$ If two different experimental systems are
both near their critical points ($r_1$ and $r_2$ both small), then
universality demands that the dependence of $K_1$ and $K_2$ on
``temperature'' $R$ must agree, up to overall changes in scale. Thus,
using a simple linear rescaling $K(r) = (1-\mu \epsilon)
K\Bigl((1-\epsilon) r\Bigr)$ leads as above to the prediction
\begin{equation}
K(r) = r^{-\mu}.
\label{linear_single}
\end{equation}
Now let us consider nonlinear rescalings, somewhat different than the
one discussed above. In particular, the nonlinearity of our measurement
of $K$ can be dependent on $r$. So,
\begin{equation}
{\cal A}_r\Bigl(K(r)\Bigr) = a_0 + a_1 r + a_2 r^2 + \ldots +
a_{01} K(r) + \ldots
\label{nonlinear_k}
\end{equation}
If $\mu>0$, these analytic corrections don't change the dominant power
law near $r=0$. However, if $\mu<0$, all the terms $a_n$ for $n<-\mu$
will be more important than the singular term! Only after fitting them
to the data and subtracting them will the residual singularity be
measurable. For the fractional change in magnetization: $\Delta M \sim
r^\beta$ has $0< \beta < 1$ (at least above three dimensions), so we
might think we need to subtract off a constant term $a_0$, but $\Delta M
= 0$ for $R \ge R_c$, so $a_0$ is zero. On the other hand, in a previous
paper\cite{Dahmen1}, we discussed the singularity in the area of the
hysteresis loop: $Area \sim r^{2-\alpha}$, where $2-\alpha = \beta +
\beta \delta$ is an analogue to the specific heat in thermal systems.
Since $\alpha$ is near zero (slightly positive from our estimates of
$\beta$ and $\delta$ in 3, 4, and 5 dimensions), measuring it would
necessitate our fitting and subtracting three terms (constant, linear,
and quadratic in $r$): we did not measure the area for that reason.
For our third example, let's consider a function $F(x,r)$, depending on
both a scale $x$ and an external parameter $r$. For example, $F$ might
be the probability $D_{int}$ that an avalanche of size $x$ will occur
during a hysteresis loop at disorder $r=R-R_c$. Universality implies
that two different systems must have the same $F$ up to changes in
scale, and therefore that $F$ measured at one $r$ must have the same
form as if measured at a different $r$. To start with, we consider a
simple linear rescaling:
\begin{equation}
F(x,r) = (1 - \alpha \epsilon)\
F\Bigl( (1-\epsilon) x, (1+\zeta \epsilon) r \Bigr).
\label{linear_double}
\end{equation}
Taking the derivative of both sides with respect to $\epsilon$ gives a
partial differential equation that can be manipulated to show $F$ has a
scaling form. Instead, we change variables to a new variable $y =
x^\zeta r$ (which satisfies $y'=y$ to order $\epsilon$). If $\tilde
F(x,y) \equiv F(x,r)$ is our function measured in the new variables,
then
\begin{equation}
F(x,r) = \tilde F(x,y) = (1-\alpha \epsilon)\
\tilde F\Bigl((1-\epsilon)x,y\Bigr)
\label{linear_double_tilde}
\end{equation}
and $-\alpha \tilde F = x \, \partial \tilde F/\partial x$ shows that at
fixed $y$, $F\sim x^{-\alpha}$, with a coefficient ${\cal F}(y)$ which
can depend on $y$. Hence we get the scaling form
\begin{equation}
F(x,r) \sim x^{-\alpha}\ {\cal F}(x^\zeta r).
\label{double_scaling}
\end{equation}
This is just Widom scaling. The critical exponents $\alpha$ and $\zeta$,
and the scaling function ${\cal F}(x^\zeta r)$ are universal (two
different systems near their critical point will have the {\sl same}
critical exponents and scaling functions). We don't need to discuss
corrections to scaling for this case, as they are similar to those
discussed above and below (and because none were dominant in our cases).
Notice that if we sit at the critical point $r=0$, the above result
reduces to equation (\ref{power}) so long as ${\cal F}(0)$ is not zero
or infinity. If, on the other hand, ${\cal F}(y) \sim y^n$ as $y \to 0$,
the two-variable scaling function gives a singular correction to the
power--law near the critical point: $F(x,r) \sim x^{-\alpha}\ {\cal
F}(x^\zeta r) \sim x^{-\alpha + n \zeta}\ $ for $x <\!< r^{-1/\zeta}$:
only when $x \sim r^{-1/\zeta}$ will the power-law $x^{-\alpha}$ be
observed. This is what happened in two dimensions to the integrated
avalanche size distribution (figures\ \ref{aval_2d_fig} and
\ref{raw_aval_2d_fig}) and the avalanche correlation functions (figure\
\ref{correl_2d_fig}b).
For the fourth example, we address finite-size scaling of a property $K$
of the system, as we vary a parameter $r$. If we measure $K(r,L)$ for
a variety of sizes $L$ (say, all with periodic boundary conditions), we
expect (in complete analogy to (\ref{double_scaling}))
\begin{equation}
K(r,L) \sim r^{-\mu}\ {\cal K}(r L^{1/\nu}).
\label{finite_size_scaling}
\end{equation}
Now, suppose our ``thermometer'' measuring $r$ is weakly size-dependent,
so the measured variable is ${\cal C}(r) = r + c/L + c_2/L^2 + \ldots$\
The effects on the scaling function is
\begin{eqnarray}
K\Bigl({\cal C}(r),L\Bigr) \sim r^{-\mu}\ \times \nonumber \\
\Bigl({\cal K}(r L^{1/\nu})\ +\ \nonumber \\
(c L^{1/\nu-1} + c_2 L^{1/\nu-2})\ {\cal K}'(r L^{1/\nu}) +
\ldots \Bigr).
\label{finite_size_corrections_to_scaling}
\end{eqnarray}
In two and three dimensions, $\nu>1$ and these correction terms are
subdominant. In four and five dimensions, we find $1/2 < \nu < 1$, so we
should include the term multiplied by $c$ in equation
(\ref{finite_size_corrections_to_scaling}). However, we believe this
first term is zero for our problem. For a fixed boundary problem (all
spins ``up'' at the boundary) with a first order transition, there is
indeed a term like $c/L$ in $r(L)$\ \cite{finite_size_first_order}. At a
critical transition, the leading correction to $r(L)$ can be $c/L$ or a
higher power of $L$ ($1/L^2$ and so on). This seems to depend on the
model studied, the geometry of the system, and the boundary conditions
(free, periodic, ferromagnetic, $\ldots$)\ \cite{finite_size_critical}.
Furthermore, for the same kind of model, the coefficient $c$ itself
depends on the geometry and boundary conditions, and it can even vanish,
which leaves only higher order corrections. In a periodic boundary
conditions problem like ours, we expect that the correction is smaller
than $c/L$. Our finite-size scaling collapses for spanning avalanches
$N$, the second moments $\langle S^2\rangle$, and the magnetization jump
$\Delta M$, were successfully done by letting $c=0$.
For the fifth example, consider a property $K$ depending on two external
parameters: $r$ (the disorder for example) and $h$ (could be the
external magnetic field $H-H_c$). Analogous to (\ref{double_scaling}),
$K$ should then scale as
\begin{equation}
K(r,h) \sim r^{-\mu}\ {\cal K}(h/r^{\beta\delta}).
\label{double_scaling_2}
\end{equation}
Consider now the likely dependence of the field $h$ on the disorder $r$.
A typical system will have a measured field which depends on the
randomness: $\tilde{\cal C}(h) = h + b\, r + b_2 r^2 + \ldots$
(Corresponding nonlinearities in the effective value of $r$ are
subdominant.) This system will have
\begin{eqnarray}
K\Bigl(r,\tilde{\cal C}(h)\Bigr)\ =\ r^{-\mu}\ \times \nonumber \\
\Bigl( {\cal K}(h/r^{\beta\delta})
+ (b\, r + b_2 r^2)\ r^{-\beta\delta}\
{\cal K}'(h/r^{\beta\delta}) \Bigr).
\label{rotated_double_scaling}
\end{eqnarray}
Now, for our system $1 < \beta \delta < 2$ for dimensions three and
above. This means that the term multiplied by $b$ is dominant over the
critical scaling singularity: unless one shifts the measured $h$ to the
appropriate $h'=h + b\,r$, the curves will not collapse ({\it e.g.}, the
peaks will not line up horizontally). We measure this (non-universal)
constant for our system (Table\ \ref{RH_table}), using the derivative of
the magnetization with field $dM/dH(r,h)$. The magnetization $M(r,h)$
and the correlation length $\xi(r,h)$ should also collapse according to
equation (\ref{double_scaling_2}) (but with $h + b\,r$ instead of $h$);
we don't directly measure the correlation length, and the collapse of
$M(r,h)$ in figure\ \ref{3d_MofH_fig}b includes the effects of the tilt
$b$. In two dimensions, $\beta \delta$ is large (probably infinite), so
in principle we should need an infinite number of correction terms: in
practise, we tried lining up the peaks in the curves (with no correction
terms); because we did not know $\beta$ (which we usually obtained
from $\Delta M$, which gives $\beta/\nu=0$ in two dimensions), we failed
to extract reliable exponents in two dimensions from $dM/dH$.
For the sixth example, suppose $F$ depends on $r$, $h$, and a size $x$.
Then from the previous analysis, we expect
\begin{equation}
F(x,r,h) \sim x^{-\alpha}\ {\cal F}(x^\zeta r,\, h/r^{\beta \delta}).
\label{triple_scaling}
\end{equation}
Notice that universality only removes one variable from the scaling
form. One could in practice do two--variable scaling collapses (and we
believe someone has probably done it), but for our purposes these more
general scaling forms are used by fixing one of the variables. For
example, we measure the avalanche size distribution at various values of
$h$ (binned in small ranges), at the critical disorder $r=0$. We can
make sense of equation (\ref{triple_scaling}) by changing variables from
$h/r^{\beta \delta}$ to $x^{\zeta \beta \delta} h$:
\begin{equation}
F(x,r,h) \sim x^{-\alpha} \tilde{\cal F}(x^\zeta r,\, x^{\zeta \beta \delta} h).
\label{triple_scaling_nice}
\end{equation}
Before we can set $r=0$, we must see what are the possible corrections
to scaling in this case. If the disorder $r$ depends on the field, then
instead of the variable $r$, we must use $r + a h$ (the analysis is
analogous to the one in example five; other corrections are
subdominant). Setting $r=0$ now, leaves $F$ dependent on its first
variable, as well as the second:
\begin{eqnarray}
F(x,r,h) & \sim\ x^{-\alpha}\ \tilde{\cal F}(x^\zeta (a h),\,
x^{\zeta \beta \delta} h)
\approx\ x^{-\alpha}\ \times \nonumber \\
& \Bigl(\tilde{\cal F}(0,\ x^{\zeta \beta \delta} h)
+ \nonumber \\ & a h x^\zeta\
\tilde{\cal F}^{(1,0)}(0,\, x^{\zeta \beta \delta} h) \Bigr),
\label{triple_scaling_reduced_corrections}
\end{eqnarray}
where $\tilde{\cal F}^{(1,0)}$ is the derivative of $\tilde{\cal F}$
with respect to the first variable (keeping the second fixed).
For the binned avalanche size distribution, $x^\zeta$ is $S^\sigma$,
where $0 \le \sigma < 1/2$ as we move from two dimensions to five and
above. Thus, the correction term will only be important for rather large
avalanches, $S > h^{-1/\sigma}$, so long as we are close to the critical
point. Expressed in terms of the scaling variable, important
corrections to scaling occur if the scaling variable $X =
S^{\sigma\beta\delta} h > h^{1-\beta\delta}$. For us, $\beta \delta >
3/2$, and we only use fields near the critical field ($h < 0.08$), so
the corrections will become of order one when $X=4$ for the largest $h$
we use. In $3$ and $4$ dimensions, this correction does not affect our
scaling collapses, while in $5$ dimensions some of the data needs this
correction. We have tried to avoid this problem (since we don't measure
our data such that it can be used in a two--variable scaling collapse)
by concentrating on collapsing the regions in our data curves where this
correction is negligible.
A similar analysis can be done for the avalanche time distribution,
which has two ``sizes'' $S$ and $t$ and one parameter $r$ which is set
to zero; because we integrate over the field $h$ the correction in
(\ref{triple_scaling_reduced_corrections}) does not occur, and other
scaling corrections are small.
Finally, we discuss the unusual exponential scaling forms we developed
to collapse our data in two dimensions. If we assume that the critical
disorder $R_c$ is zero {\it and} that the linear term in the rescaling
of $r$ vanishes ($\zeta \epsilon r$ in equation\ (\ref{linear_double})
vanishes), then from symmetry the correction has to be cubic, and
equation\ (\ref{linear_double}) becomes:
\begin{equation}
F(x,r) = (1 - \alpha \epsilon)\
F\Bigl( (1-\epsilon) x,\, (1+ k \epsilon\, r^2) r \Bigr).
\label{exponential_scaling_cubic}
\end{equation}
with $k$ (which is not universal) and $\alpha$ constants, and $\epsilon$
small.
Taking the derivative of both sides with respect to $\epsilon$ and
setting it equal to zero gives a partial differential equation for the
function $F$. To solve for $F$, we do a change of variable: $(x,r)
\rightarrow (x,y)$ with $y=x\ e^{-a^*/r^2}$. The constant $a^*$ is
determined by requiring that $y$ rescales onto itself to order
$\epsilon$: we find $a^*=1/2\,k$. We then have:
\begin{equation}
0 = -\alpha\ \tilde{F}(x,y) - {\partial{\tilde{F}} \over \partial{x}}\ x
\label{exponential_scaling_cubic_2}
\end{equation}
which gives
\begin{equation}
F(x,r) = x^{-\alpha}\ \tilde{\cal{F}}\Bigl(xe^{-1/2\,k\,r^2}\Bigr).
\label{exponential_scaling_cubic_3}
\end{equation}
This is one of the forms we use in $2$ dimensions for the scaling
collapse of the second moments $\langle S^2 \rangle_{int}$, the
avalanche size distribution $D_{int}$ integrated over the field $H$, the
avalanche correlation $G_{int}$, and the spanning avalanches $N$. We use
another form too which is obtained by assuming that the critical
disorder $R_c$ is not zero but that the linear term in the rescaling of
$r$ still vanishes. Instead of equation\
(\ref{exponential_scaling_cubic}), we have:
\begin{equation}
F(x,r) = (1 - \alpha \epsilon)\
F\Bigl( (1-\epsilon) x,\, (1+ l \epsilon\, r) r \Bigr).
\label{exponential_scaling_square}
\end{equation}
The function $F$ becomes:
\begin{equation}
F(x,r) = x^{-\alpha}\ \tilde{\cal{F}}\Bigl(xe^{-1/l\,r}\Bigr).
\label{exponential_scaling_square_2}
\end{equation}
The corrections to scaling for the last two forms (equations\
(\ref{exponential_scaling_cubic_3}) and
(\ref{exponential_scaling_square_2})) are similar to the ones discussed
above. They are all are subdominant.
\section{Full derivation of the mean field scaling form for the
integrated avalanche size distribution}
The mean field scaling form for the integrated avalanche size
distribution $D_{int}(S,R)$ was obtained in section IV A using the {\it
scaling form} of the avalanche size distribution $D(S,R,H)$. The scaling
form for $D_{int}(S,R)$ can also be obtained by integrating the
avalanche probability distribution $D(S,t)$ (derived originally
in\cite{Sethna}) directly:
\begin{equation}
D_{int} (S,R) = \int_{-\infty}^{+\infty}
\rho(-JM-H)\ D(S,t)\ dH
\label{apA1_eq1}
\end{equation}
where $\rho (-JM-H)$ is the probability distribution for the random
fields, and $\rho (-JM-H)\ dH$ is the probability for a spin to flip
between fields $-JM(H) - H$ and $-JM(H+dH) - (H+dH)$. $D(S,t)$ is the
probability of having an avalanche of size $S$, a small ``distance'' $t
\equiv 2J \rho (-JM-H) - 1$ from the infinite avalanche at $\rho (-JM-H)
= 1/2J$, given that a spin has flipped at $-JM -
H$\cite{Sethna,Dahmen1}. (The scaling form for the non-integrated
avalanche size distribution $D(S,R,H)$ (eqn.\ref{int_aval0}) is obtained
from $D(S,t)$ by expressing $t$ as a function of $R$ and $H$
\cite{Sethna,Dahmen1}). $J$ is the coupling of a spin to all others in
the system, $H$ is the external magnetic field, and $R$ is the disorder.
The advantage of this procedure is that we can find out something about
the scaling function $\bar {\cal D}_{-}^{(int)}$.
The average mean field magnetization $M$ and the avalanche probability
distribution $D(S,t)$ are given by \cite{Sethna,Dahmen1}:
\begin{equation}
M(H,R) = 1 - 2 \int_{-\infty}^{-JM(H)-H} \rho(f)\ df,
\label{apA1_eq2}
\end{equation}
and
\begin{equation}
D(S,t) = {S^{S-2} \over (S-1)!}\ (t+1)^{S-1}\ e^{-S(t+1)}
\label{apA1_eq3}
\end{equation}
To solve equation (\ref{apA1_eq1}), let's define the variable
$y=(-JM-H)/({\sqrt 2}\ R)$ and rewrite the integral as:
\begin{eqnarray}
D_{int} (S,R)\ =\ {\sqrt 2}\ R\ \times \nonumber \\
\Biggl[\int_{-\infty}^{+\infty}
\rho ({\sqrt 2}Ry)\ D\Bigl(S,\ 2J \rho ({\sqrt 2}Ry) - 1\Bigr) \times
\nonumber \\
\ \Bigl(1 - 2J \rho ({\sqrt 2}Ry)\Bigr)\ dy \Biggr],
\label{apA1_eq4}
\end{eqnarray}
where we have used:
\begin{equation}
{dy \over dH} = {1 \over {\sqrt 2}\ R}\ \biggl(-J\ {2\ \rho(-JM-H)
\over {1-2J \rho(-JM-H)}} -1\biggr)
\label{apA1_eq5}
\end{equation}
Since we are interested in the behavior of the integrated avalanche
distribution for large sizes, the factorial in equation (\ref{apA1_eq3})
can be expanded using Stirling's formula. To first order, we have:
\begin{equation}
(S-1)!\ \approx\ {S^S\ {\sqrt {2 \pi}} \over e^S\ {\sqrt S}}
\label{apA1_eq6}
\end{equation}
Substituting this and the random field distribution function $\rho$,
\begin{equation}
\rho ({\sqrt 2}Ry) = {1 \over {\sqrt {2 \pi}} R}\ e^{-y^2},
\label{apA1_eq7}
\end{equation}
in equation (\ref{apA1_eq4}), we obtain:
\begin{eqnarray}
D_{int}(S,R)\ \approx\ C\ \biggl({R_c \over R}\biggr)^S\ \times
\nonumber \\
\int_{-\infty}^{+\infty}
e^{-S\ \bigl(y^2 + {R_c \over R}\ e^{-y^2}\bigr)}
\ \ \biggl(1-{R_c \over R}\ e^{-y^2}\biggr)\ dy
\label{apA1_eq8}
\end{eqnarray}
where $C=S^{-{3 \over 2}}\ e^S\ R_c/(\pi R {\sqrt 2})$, and $S$ is large.
For disorders above but close to the critical disorder $R_c$, we have:
\begin{eqnarray}
\biggl({R_c \over R}\biggr)^S\ =\ e^{S\ log\bigl({R_c \over R}\bigr)}\ \approx\
\nonumber \\
e^{S\ \Bigl(-\ {\bigl(1-{R_c \over R}\bigr) \over 1} -
\ {\bigl(1-{R_c \over R}\bigr)^2 \over 2}
- {\bigl(1 - {R_c \over R}\bigr)^3 \over 3} - ... \Bigr)}
\label{apA1_eq9}
\end{eqnarray}
If we assume that only terms up to $S\ (1-R_c/R)^2$ are important (terms
like $S\ (1-R_c/R)^3$ and $S\ (1-R_c/R)^4$ go to zero as $R \rightarrow
R_c$), and we note that the integrand in equation (\ref{apA1_eq8}) is an
even function of $y$, equation (\ref{apA1_eq8}) becomes:
\begin{eqnarray}
D_{int}(S,R)\ \approx\ 2\ C\ \times \nonumber \\
\Biggl[ \int_{0}^{+\infty}
e^{-S\ \Bigl({\bigl(1-{R_c \over R}\bigr) \over 1} +
{\bigl(1-{R_c \over R}\bigr)^2 \over 2}
+y^2 + {R_c \over R}\ e^{-y^2}\Bigr)} \times \nonumber \\
\ \ \biggl(1-{R_c \over R}\ e^{-y^2}\biggr)\ dy \Biggr]
\label{apA1_eq10}
\end{eqnarray}
The asymptotic behavior of the above integral, as $S \rightarrow
\infty$, is obtained using Laplace's method\cite{BenderOrszag}. The idea
is as follows. The asymptotic behavior as $S \rightarrow \infty$ of
the integral:
\begin{equation}
I(S) = \int_{a}^{b} f(x)\ e^{S \phi (x)}\ dx
\label{apA1_eq11}
\end{equation}
can be found by integrating over a small region $[c-\epsilon,\
c+\epsilon]$ (instead of the interval $[a,b]$) around the maximum of the
function $\phi (x)$ at $x=c$, since in the asymptotic expansion, the
largest contribution to the integral will be from this region. The
corrections will be exponentially small. The maximum of $\phi$ must be
in the interval $[a,b]$, $f(x)$ and $\phi(x)$ are assumed to be real
continuous functions, and $f(c) \ne 0$. $f(x)$ and $\phi (x)$ can now
both be expanded around $x=c$, and the integral solved. Often the
integral is easier to handle if the limit of integration is extended to
infinity. This will add only exponentially small corrections in the
asymptotic limit of $S \rightarrow \infty$.
Let's apply this method to equation (\ref{apA1_eq10}). The function in
the exponential has a maximum at $y=0$. The function $\biggl(1-{R_c
\over R}\ e^{-y^2}\biggr)$ is not zero there if $R \ne R_c$. We can thus
expand both functions in the integral of equation (\ref{apA1_eq10})
around $y=0$. Defining $u=y^2{\sqrt S}$, we obtain:
\begin{eqnarray}
D_{int}(S,R)\ \approx\ C_{1}\ \times \nonumber \\ \Biggl[ \int_{0}^{\epsilon}
e^{-{\sqrt S}\ \Bigl(\bigl(1-{R_c \over R}\bigr) u\ +\
{u^2 \over {2 {\sqrt S}}}{R_c \over R} -\ {u^3 \over {6\ S^2}}{R_c \over R}
+ ...\Bigr)}\ \times \nonumber \\
\biggl(\Bigl(1-{R_c \over R}\Bigr) +
{R_c \over R} {u \over {\sqrt S}} -\ {R_c \over 2R} {u^2 \over S} + ...\biggr)
\ {du \over {\sqrt u}} \Biggr]
\label{apA1_eq12}
\end{eqnarray}
where
\begin{equation}
C_{1} = {1 \over \pi {\sqrt 2}}\ {R_c \over R}\ S^{-{9 \over 4}}
\ \ e^{-{S \over 2}\ \bigl(1-{R_c \over R}\bigr)^2},
\label{apA1_eq13}
\end{equation}
$S$ is large, $R$ is close to but not equal to $R_c$, and only terms up
to $S (1-R_c/R)^2$ are non-vanishing. In the asymptotic limit of $S
\rightarrow \infty$ we can ignore terms with powers of $S$ in the
denominator, and look at the distribution for $R$ close to $R_c$. To
first order in $r = (R_c-R)/R$, $R_c/R \approx 1$ and $1-R_c/R \approx
-r$, which gives:
\begin{eqnarray}
D_{int}(S,R)\ \approx\ {1 \over \pi {\sqrt 2}}\ S^{-{9 \over 4}}
\ e^{-{S \over 2}\ (-r)^2}\ \times \nonumber \\ \int_{0}^{\infty}
e^{\Bigl(-(-r){\sqrt S}\ u\ -\ {u^2 \over 2}\Bigr)}\ \
\Bigl(-r {\sqrt S} + u\Bigr)
\ {du \over {\sqrt u}}
\label{apA1_eq14}
\end{eqnarray}
where we have expanded the integration to infinity. As mentioned above,
this will only add exponentially small corrections in the asymptotic
limit of $S \rightarrow \infty$. Equation\ (\ref{apA1_eq14}) is the
integrated avalanche size distribution in mean field for large sizes
$S$, and finite $S r^2$. We see right away that it gives the correct
scaling form:
\begin{equation}
D_{int}(S,R)\ \sim\ S^{-{9 \over 4}}\
\bar{\cal D}_{\pm}^{(int)} \Bigl({\sqrt S}\ |r|\Bigr)
\label{apA1_eq15}
\end{equation}
where $\pm$ indicates the sign of $r$, the exponent
$\tau+\sigma\beta\delta$ and $\sigma$ are $9/4$ and $1/2$ respectively,
and the scaling function $\bar{\cal D}_{\pm}^{(int)}$ is:
\begin{equation}
\bar{\cal D}_{\pm}^{(int)} \Bigl({\sqrt S}\ |r|\Bigr) =
e^{-{({\sqrt S}\ |r|)^2 \over 2}}\
\bar{\cal F}_{\pm} \Bigl({\sqrt S}\ |r|\Bigr).
\label{apA1_eq16}
\end{equation}
The function $\bar{\cal F}_{\pm} \Bigl({\sqrt S}\ |r|\Bigr)$ is
proportional to the integral in equation\ (\ref{apA1_eq14}). Note that
the above result is equivalent to the one obtained (eqn.\
\ref{int_aval3}) by integrating the {\it scaling form} of $D(S,R,H)$
over the field $H$.
What is the behavior of the scaling function $\bar{\cal
D}_{-}^{(int)}(X)$ for small and large positive arguments $X={\sqrt S}\
(-r) > 0$ ($R > R_c$)? From equations\ (\ref{apA1_eq14}) and
(\ref{apA1_eq16}), for small arguments we have a polynomial in $X$:
\begin{equation}
\bar{\cal D}_{-}^{(int)}(X) \approx A+BX+CX^2+{\cal O}(X^3)
\label{apA1_eq17}
\end{equation}
These parameters can be calculated numerically. We obtain in mean field:
\begin{eqnarray}
{\bar{\cal D}_{-}^{(int)}}\ & \approx\ &
0.232+0.243 X-0.174 X^2- \nonumber \\
& & 0.101 X^3+0.051 X^4
\label{dist_mf2}
\end{eqnarray}
On the other hand, for large arguments we find:
\begin{equation}
\bar{\cal D}_{-}^{(int)}(X) \approx
{\pi}^{1/2} e^{-{X^2 \over 2}}\ {\sqrt X} \Bigl( 1 + {\cal O}
(X^{-2})\Bigr)
\label{apA1_eq18}
\end{equation}
In general, for all dimensions, in equation\ (\ref{apA1_eq18}) the
exponential is of $X^{1/\sigma}$ ($1/\sigma = 2$ in mean field), since
the exponent $\sigma$ gives the exponential cutoff to the power law
distribution for large $X$, and the power of $X$ is $\beta$ ($\beta =
1/2$ in mean field). One can see the latter by expanding the
distribution function $D_{int}(S,R)$ in terms of $1/S$ ($S$ is large),
analogous to\ \cite{Griffiths}:
\begin{equation}
D_{int}(S,R) = \sum_{n=1}^{\infty} f_n (r)\ S^{-n}
\label{apA1_equ18a}
\end{equation}
Since $X=$ $S^\sigma (-r)$, then we can write $S=$ $X^{1/\sigma}\
(-r)^{-1/\sigma}$ and obtain:
\begin{equation}
D_{int}(S,R) = \sum_{n=1}^{\infty} f_n (r)\ X^{-n/\sigma} (-r)^{n/\sigma}
\label{apA1_equ18b}
\end{equation}
The scaling function $\bar{\cal D}_{-}^{(int)}(X)$ scales like
$S^{(\tau+\sigma\beta\delta)}\ \times D_{int}(S,R)$:
\begin{eqnarray}
\bar{\cal D}_{-}^{(int)}(X) \sim\ \Biggl[ \sum_{n=1}^{\infty} f_n (r)\
X^{-n/\sigma} X^{(\tau+\sigma\beta\delta)/\sigma}\ \times \nonumber \\
(-r)^{n/\sigma}
(-r)^{-(\tau+\sigma\beta\delta)/\sigma} \Biggr]
\label{apA1_equ18c}
\end{eqnarray}
and since it is only a function of $X$, it must satisfy:
\begin{equation}
\bar{\cal D}_{-}^{(int)}(X) \sim\ \sum_{n=1}^{\infty} g_n\
X^{-n/\sigma} X^{(\tau+\sigma\beta\delta)/\sigma}
\label{apA1_equ18d}
\end{equation}
where $g_n$ is independent of $r$.
The exponent combination $(\tau+\sigma\beta\delta)/\sigma$ can be
rewritten as:
\begin{equation}
{\tau+\sigma\beta\delta \over \sigma} = {2 \over \sigma} +
{\tau+\sigma\beta\delta - 2 \over \sigma} = {2 \over \sigma} +
\beta
\label{apA1_equ18e}
\end{equation}
where we have used the scaling relation\cite{Dahmen1,Dahmen3}:
$\beta - \beta\delta = (\tau - 2)/\sigma$. Thus we have for the scaling
function $\bar{\cal D}_{-}^{(int)}(X)$:
\begin{equation}
\bar{\cal D}_{-}^{(int)}(X) \sim\ X^\beta \sum_{n=-1}^{\infty} g_n\
X^{-n/\sigma} = X^\beta\ {\cal K}(X^{1/\sigma})
\label{apA1_equ18f}
\end{equation}
which shows (compare to equation\ (\ref{apA1_eq18})) that the power of $X$
is indeed the exponent $\beta$.
We have used the results of the expansion of the mean field scaling
function ${\bar {\cal D}}_{-}^{(int)} (X)$ for small and large
parameters (equations\ (\ref{apA1_eq17}) and (\ref{apA1_eq18})), to
build a fitting function to the integrated avalanche size distribution
scaling functions in $2$, $3$, $4$, and $5$ dimensions, described in
section IV~ B.
Finally, note from equation\ (\ref{apA1_eq17}) that the scaling function:
\begin{equation}
{\cal D}_{-}^{(int)} (S r^2) = e^{-{(S r^2) \over 2}}\
{\cal F}_{-}(S r^2)
\label{apA1_eq19}
\end{equation}
used earlier in reference\cite{Dahmen1}, is not analytic for small
arguments $S r^2$, from which we conclude that the appropriate scaling
variable should be ${\sqrt S}\ (-r)$ and not $S r^2$. (Notice that this
no longer seems true in two dimensions; see section on $2$ dimensional
results.)
\section{Derivation of the mean field scaling form for the spanning avalanches}
We have defined earlier a mean field spanning avalanche to be an
avalanche larger than $\sqrt {S_{mf}}$, where $S_{mf}$ is the {\it
total} size of the system. We want to derive the scaling form for the
number of such avalanches in half of the hysteresis loop (for $H$ from
$-\infty$ to $+\infty$) as a function of the system size $S_{mf}$ and
the disorder $R$. The number of mean field spanning avalanches is
proportional to the probability of having avalanches of size larger than
$\sqrt {S_{mf}}$. Since we want the number of spanning avalanches, we
need to multiply this probability by the total number of avalanches. For
large system sizes, this scales with the system size $S_{mf}$
(corrections are subdominant). We thus obtain by integrating over
equation\ (\ref{apA1_eq15}) (which gives the scaling form for the
probability distribution of avalanches of size $S$ in the hysteresis
loop):
\begin{eqnarray}
N_{mf}(S_{mf},R)\ \sim\
S_{mf}\ \times \nonumber \\
\int_{\sqrt {S_{mf}}}^{\infty}\ S^{-{9 \over 4}}\
e^{-{\bigl({\sqrt S} |r|\bigr)^2 \over 2}}\
\bar{\cal F}_{\pm}\Bigl({\sqrt S} |r|\Bigr)\ dS
\label{apB_eq1}
\end{eqnarray}
Let's define $u={\sqrt S} |r|$, then equation\ (\ref{apB_eq1}) can be
written as:
\begin{eqnarray}
N_{mf}(S_{mf},R)\ \sim\ 2\ S_{mf}\ |r|^{5 \over 2} \times \nonumber \\
\int_{|r| S_{mf}^{1 \over 4}}^{\infty}\ u^{-{7 \over 2}}\
e^{-{u^2 \over 2}}\ \bar{\cal F}_{\pm}(u)\ du
\label{apB_eq2}
\end{eqnarray}
The integral ${\cal I}$ is a function of $S_{mf}^{1 \over 4} |r|$ only,
and we can write it as:
\begin{equation}
{\cal I} = \Bigl(S_{mf}^{1 \over 4} |r|\Bigr)^{-{5 \over 2}}\
{\cal N}_{\pm}^{mf}\Bigl(S_{mf}^{1 \over 4} |r|\Bigr)
\label{apB_eq3}
\end{equation}
to obtain the scaling form for the number $N_{mf}$ of mean field
spanning avalanches:
\begin{equation}
N_{mf}(S_{mf},R)\ \sim\
S_{mf}^{3 \over 8}\
{\cal N}_{\pm}^{mf}\Bigl(S_{mf}^{1 \over 4} |r|\Bigr)
\label{apB_eq4}
\end{equation}
From this scaling form, we can extract the exponents $\tilde \theta
=3/8$, and $1/\tilde \nu=1/4$. This form is used for collapses of the
spanning avalanche curves in mean field (see mean field section).
\subsubsection{Avalanche Size Distribution}
\paragraph{Integrated Avalanche Size Distribution}
In our model the spins often flip in avalanches, which are collective
flips of neighboring spins at a constant external field $H$. These
avalanches come in different sizes. The integrated avalanche size
distribution is the size distribution of all the avalanches that occur
in one branch of the hysteresis loop (for $H$ from $-\infty$ to
$\infty$). Figure\ \ref{aval_3fig}\cite{Perkovic} shows some of the raw
data (thick lines) in $3$ dimensions. Note that the curves follow a
power law behavior over several decades. Even $50\%$ away from
criticality (at $R=3.2$), there are still two decades of scaling, which
implies that the critical region is large. In experiments, a few decades
of scaling could be interpreted in terms of self-organized criticality
(SOC). However, our model and simulation suggest that several decades of
power law scaling can still be present rather ${\it far}$ from the
critical point (note that the size of the critical region is
non--universal). In the figure, the cutoff in the power law which
diverges as the critical disorder $R_c$ is approached ($R_c=2.16$ in $3$
dimensions), is a signature that the system is away from criticality,
and that a parameter can be tuned (here $R$) to bring it to the
transition. This cutoff scales as $S \sim |r|^{-1/\sigma}$, where $S$ is
the avalanche size and $r=(R_c-R)/R$ is the reduced disorder.
The power law for the curves of figure\ \ref{aval_3fig} can be obtained
through scaling collapses. A plot is shown in the inset of figure\
\ref{aval_3fig}. The scaling form is (see mean field section)
\begin{equation}
D_{int}(S,R) \sim S^{-(\tau+\sigma\beta\delta)}\
\bar{{\cal D}}^{(int)}_{-} (S^{\sigma}|r|)
\label{aval_3d_eq}
\end{equation}
where $\bar{{\cal D}}_{-}^{(int)}$ is the scaling function (the $-$ sign
indicates that the collapsed curves are for $R > R_c$). The critical
exponents $\tau+\sigma\beta\delta=2.03$ and $\sigma=0.24$ are obtained
from collapses and linear extrapolation of the extracted values to
$R=R_c$ (figures\ \ref{3d_aval_expfig}a and \ref{3d_aval_expfig}b), as
was done in mean field. (Although the ``real'' scaling variables are
$r^{\prime}$ and $h^{\prime}$, when integrating over the field $H$ we
recover the same form as in mean field; see appendix A.) Table\
\ref{measured_exp_table} lists all the exponents extracted from scaling
collapses, and extrapolated to $R \rightarrow R_c$ and $1/L \rightarrow
0$.
\begin{figure}
\centerline{
\psfig{figure=Figures/Aval_histo_d3_L320_paper_norm_THESIS.ps,width=3truein}
}
\caption[Integrated avalanche size distribution curves in $3$ dimensions]
{{\bf Avalanche size distribution integrated over the field $H$
in $3$ dimensions}, for $320^3$
spins and disorders $R=4.0, 3.2$, and $2.6$.
The last curve is at $R=2.25$, for
a $1000^3$ spin system. The $320^3$ curves are averages
over up to $16$
initial random field configurations. All curves are
smoothed by $10$ data points
before they are collapsed. The inset shows the scaling collapse
of the integrated avalanche size distribution curves
in $3$ dimensions, using $r=(R_c-R)/R$, $\tau+\sigma\beta\delta=2.03$,
and $\sigma=0.24$,
for sizes $160^3$, $320^3$, $800^3$, and $1000^3$, and
disorders ranging from $R=2.25$ to $R=3.2$ ($R_c = 2.16$).
The two top curves in the collapse, at
$R=3.2$, show noticeable corrections to scaling. The
thick dark curve through the collapse is the fit to the data (see
text). In the
main figure, the distribution curves obtained from the fit to the
collapsed data are plotted (thin lines) alongside the raw data
(thick lines). The straight dashed line is the expected asymptotic
power law behavior: $S^{-2.03}$, which does not agree with the
measured slope of the raw data due to the shape of the scaling
function (see text). \label{aval_3fig}}
\end{figure}
We have mentioned earlier that the mean field scaling function
$\bar{\cal D}_{-}^{(int)}(X)$, with $X=S^{\sigma}|r|$ and $r<0$, is a
polynomial for small $X$ and gives an exponential in $X^{1/\sigma}$
($1/\sigma=2$ in mean field) multiplied by $X^\beta$ ($\beta=1/2$ in
mean field) for large $X$ (see mean field section and appendix B). As we
have done in mean field, we can try to fit the scaling function
$\bar{\cal D}_{-}^{(int)}$ in dimensions $5$ and below with a product of
a polynomial and an exponential function. This is done in $3$ dimensions
in the inset of figure\ \ref{aval_3fig} (thick black line through the
data). The {\it phenomenological} fit is:
\begin{eqnarray}
\bar{{\cal D}}_{-}^{\it (int)}(X)\ =\ e^{-0.789X^{1/\sigma}}\ \times
\nonumber \\
(0.021+0.002X+0.531X^2-0.266X^3+0.261X^4)
\label{aval_fit_3d}
\end{eqnarray}
with $1/\sigma=4.20$ which is obtained from scaling collapses. The
distribution curves obtained using the above fit are plotted (thin lines
in figure\ \ref{aval_3fig}) alongside the raw data (thick lines). They
agree remarkably well even far above $R_c$. We should recall though,
from the mean field discussion (see figure\ \ref{mf_aval_fit}), that the
fitted curve to the collapsed data can differ from the ``real'' scaling
function even for large sizes and close to the critical disorder (in
mean field the error was about $10\%$). We expect a similar behavior in
finite dimensions.
\begin{figure}
\centerline{
\psfig{figure=Figures/Tau_Sigma_Beta_Delta_d3_L320_paper_over_R_Rc_legend_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/Sigma_d3_L320_paper_over_R_Rc_legend_THESIS.ps,width=3truein}
}
\caption[Integrated avalanche size distribution exponents
$\tau+\sigma\beta\delta$ and $\sigma$ in $3$ dimensions]
{(a) and (b) $\tau+\sigma\beta\delta$ and $\sigma$ respectively,
from collapses of the {\bf integrated
avalanche size distribution curves} for a $320^3$ spin system.
The data is plotted
as in mean field. The two closest points to $|r|_{\it avg}=0$ are for
a $800^3$ system, for a collapse using curves with disorder:
$2.26$, $2.28$, $2.30$, $2.32$, $2.34$, and $2.36$.
The extrapolation to $|r|_{\it avg}=0$ gives:
$\tau + \sigma \beta \delta = 2.03$ and $\sigma = 0.24$.
\label{3d_aval_expfig}}
\end{figure}
The scaling function in the inset of figure\ \ref{aval_3fig} has a
peculiar shape: it grows by a factor of ten before cutting off. The
consequence of this shape is that in the simulations, it takes many
decades in the size distribution for the slope to converge to the
asymptotic power law. This can be seen from the comparison between a
straight line fit through the $R=2.25$ ($1000^3$!) curve in figure\
\ref{aval_3fig} and the asymptotic power law $S^{-2.03}$ obtained from
scaling collapses and the extrapolation (thick dashed straight line in
the same figure). A similar ``bump'' exists in other dimensions and mean
field as well. Figure\ \ref{bump_345fig} shows the scaling functions in
different dimensions and in mean field. In this graph, the scaling
functions are normalized to one and the peaks are aligned (the scaling
forms allow this). The curves plotted in figure\ \ref{bump_345fig} are
not raw data but fits to the scaling collapse in each dimensions, as was
done in the inset of figure\ \ref{aval_3fig}. The mean field and $3$
dimensions curves are given by equations\ (\ref{int_aval3a}) and
(\ref{aval_fit_3d}) respectively. For $5$, $4$, and $2$ dimensions, we
have respectively:
\begin{eqnarray}
\bar{{\cal D}}_{-}^{\it (int)}(X)\ =\ e^{-0.518 X^{1/\sigma}}\ \times
\nonumber \\
(0.112 + 0.459 X - 0.260 X^2 + 0.201 X^3 - 0.050 X^4)
\label{aval_fit_5d}
\end{eqnarray}
\begin{eqnarray}
\bar{{\cal D}}_{-}^{\it (int)}(X)\ =\ e^{-0.954 X^{1/\sigma}}\ \times
\nonumber \\
(0.058 + 0.396 X + 0.248 X^2 - 0.140 X^3 + 0.026 X^4)
\label{aval_fit_4d}
\end{eqnarray}
\begin{eqnarray}
\bar{{\cal D}}_{-}^{\it (int)}(X)\ =\ e^{-1.076 X^{1/\sigma}}\ \times
\nonumber \\
(0.492 - 4.472 X + 14.702 X^2 - \nonumber \\
20.936 X^3 + 11.303 X^4)
\label{aval_fit_2d}
\end{eqnarray}
with $1/\sigma=2.35, 3.20$, and $10.0$. The errors in the fits are in
the same range as for the mean field simulation data (see figure\
\ref{mf_aval_fit}). The $2$ dimensional fit plotted in grey will be
covered further in the next section.
\begin{figure}
\centerline{
\psfig{figure=Figures/curly_D_all_norm_shifted_chopped_paper_THESIS.ps,width=3truein}
}
\caption[Integrated avalanche size distribution scaling functions in $2$,
$3$, $4$, and $5$ dimensions, and mean field]
{{\bf Integrated avalanche size distribution scaling functions in
$2$, $3$, $4$, and $5$ dimensions, and mean field.} The curves are
fits (see text) to the scaling collapses done with exponents
from Table\ \protect\ref{measured_exp_table} and\
\protect\ref{conj_meas_2d_table}. The peaks are
aligned to fall on (1,1). Due to the ``bump'' in the scaling function
the power law exponent can not be extracted from a linear fit to
the raw data. \label{bump_345fig}}
\end{figure}
From figure\ \ref{bump_345fig} we can conclude that in each dimension
(and in mean field!), a straight line fit to the integrated avalanche size
distribution data is going to give the {\it wrong} critical exponent,
and that only by doing scaling collapses and an extrapolation the
asymptotic value can be found. This is shown for $3$ dimensions in
figure\ \ref{aval_3fig}, and was found to be true in other dimensions as
well. We will next see that this is different for the {\it binned}
avalanche size distribution. The value for the slope obtained from a
linear fit to the data agrees very well with the value obtained from the
scaling collapses.
\begin{figure}
\centerline{
\psfig{figure=Figures/Bin_H_norm_L80_Hc1.265_paper_legend_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/Bin_H_norm_L80_Hc1.265_above_and_below_collapse_paper_THESIS.ps,width=3truein}
}
\caption[Binned in $H$ avalanche size distribution curves in $4$ dimensions]
{(a) {\bf Binned in $H$ avalanche size distribution in $4$ dimensions}
for a system of $80^4$ spins at $R=4.09$ ($R_c=4.10$). The
critical field is $H_c=1.265$. The curves are averages over
close to $60$ random field configuration. Only a few curves are shown.
(b) Scaling collapse of the binned avalanche size distribution
for $H<H_c$ (upper collapse) and $H>H_c$ (lower collapse). The
critical exponents are
$\tau=1.53$ and $\sigma\beta\delta=0.54$, and the critical field is
$H_c=1.265$. The bins are at fields: $1.162$, $1.185$, $1.204$,
$1.220$, $1.234$, $1.245$, $1.254$, $1.276$, $1.285$, $1.296$,
$1.310$, $1.326$, $1.345$, and
$1.368$. Notice that the two scaling functions do not have a ``bump''
(see text). \label{bin_aval_4dfig}}
\end{figure}
\paragraph{Binned in $H$ Avalanche Size Distribution}
The avalanche size distribution can also be measured at a field $H$ or
in a small range of fields centered around $H$. We have measured this
${\it binned}$ in $H$ avalanche size distribution for systems at the
critical disorder $R_c$ ($r=0$). To obtain the scaling form, we start
from the distribution of avalanches at field $H$ and disorder $R$
(eqn. \ref{int_aval0}):
\begin{equation}
D(S,R,H) \sim S^{-\tau}\ {\cal D}_{\pm}(S^\sigma |r|, |h|/|r|^{\beta\delta})
\label{aval_distr1}
\end{equation}
where as before ${\cal D}_{\pm}$ is the scaling function and $\pm$
indicates the sign of $r$. (For most of our data, we can ignore the
corrections due to the ``rotation'' of axis as explained in appendix A.)
The scaling function can be rewritten as ${\hat {\cal
D}}_{\pm}\Bigl(S^\sigma |r|, (S^\sigma |r|)^{\beta\delta}
|h|/|r|^{\beta\delta}\Bigr)$, where ${\hat D}_{\pm}$ is a new scaling
function. Letting $R \rightarrow R_c$, the scaling for the avalanche
size distribution at the field $H$, measured at the critical disorder
$R_c$ is:
\begin{equation}
D(S,H) \sim S^{-\tau}\ {\hat {\cal D}}_{\pm}(|h| S^{\sigma\beta\delta})
\label{aval_distr2}
\end{equation}
\begin{figure}
\centerline{
\psfig{figure=Figures/Tau_norm_d4_L80_paper_legend_new_THESIS.ps,width=3truein}
}
\caption[Exponent $\tau$ from the binned in $H$ avalanche size distribution
curves in $4$ dimensions]
{{\bf Values for the exponent
$\tau$ extracted from the binned in $H$ avalanche size
distribution curves in $4$ dimensions}, for a $80^4$ spin system
at $R=4.09$ ($R_c=4.10$). The critical field is $H_c=1.265$.
The exponent $\tau$ is found from this linear extrapolation to
$\Delta H_{avg} = 0$. The exponent $\sigma\beta\delta$ is
calculated from the value of $\tau+\sigma\beta\delta$,
extracted from the
integrated avalanche size distribution, and the value of $\tau$
from this plot. \label{bin_aval_4dfigc}}
\end{figure}
Figure\ \ref{bin_aval_4dfig}a shows the binned in $H$ avalanche size
distribution curves in $4$ dimensions, for values of $H$ below the
critical field $H_c$. (The curves and analysis are similar in $3$ and
$5$ dimensions; results in $4$ dimensions are used here for variety.)
The simulation was done at the best estimate of the critical disorder
$R_c$ ($4.1$ in $4$ dimensions). The binning in $H$ is logarithmic and
started from an approximate critical field $H_c$ obtained from the
magnetization curves; better estimates of $H_c$ are obtained from the
binned distribution data curves and their collapses. Our best estimate
for the critical field $H_c$ in $4$ dimensions is $1.265 \pm 0.007$. The
scaling form for the logarithmically binned data is the same as in
equation\ (\ref{aval_distr2}), if the log-binned data is normalized by
the size of the bin. Figure\ \ref{bin_aval_4dfig}b shows the scaling
collapse for our data, both below {\it and} above the critical field
$H_c$. The ``top'' collapse gives the shape of the ${\hat {\cal D}_{-}}$
($H<H_c$) function, while the ``bottom'' collapse gives the ${\hat {\cal
D}_{+}}$ ($H>H_c$) function. Above the critical field $H_c$, there are
spanning avalanches in the system\ \cite{note4}. These are not included
in the binned avalanche size distribution collapse shown in figure\
\ref{bin_aval_4dfig}b.
The exponent $\tau$ which gives the power law behavior of the binned
avalanche size distribution is obtained from an extrapolation similar to
previous ones (figure\ \ref{bin_aval_4dfigc}), but with the field $H$
($\Delta H_{avg}$ in figure\ \ref{bin_aval_4dfigc} is the algebraic
average of $H-H_c$ for three curves collapsed together) as the variable
instead of the disorder $R$. The exponent $\sigma\beta\delta$ is found
to be very sensitive to $H_c$, while $\tau$ is not. We have therefore
used the values of $\tau + \sigma\beta\delta$ and $\sigma$ from the
integrated avalanche size distribution collapses, and $\tau$ from the
binned avalanche size distribution collapses to further constrain $H_c$
(by constraining $\sigma\beta\delta$), and to calculate $\beta\delta$.
The latter is then used to obtain collapses of the magnetization curves.
We should mention here that $H_c$ in all the dimensions is difficult to
find and that it is influenced by finite sizes. The values listed in
Table\ \ref{RH_table} are the best estimates obtained from the largest
system sizes we have. Nevertheless, systematic errors for $H_c$ could be
larger than the errors given in Table\ \ref{RH_table}. This implies
possible systematic errors for $\sigma\beta\delta$ which depends on
$H_c$, and for $\beta\delta$ which is calculated from
$\sigma\beta\delta$. These could also be larger than the errors listed
in Table\ \ref{calculated_exp_table}.
\begin{figure}
\centerline{
\psfig{figure=Figures/Bin_H_norm_L80_Hc1.265_paper_linear_fit_THESIS.ps,width=3truein}
}
\caption[Linear fit to a binned in $H$ avalanche size distribution curve in
$4$ dimensions]
{{\bf Binned avalanche size distribution curve (dashed line)
in $4$ dimensions}, for a system of $80^4$ spins at $R_c=4.09$.
The magnetic
field is $H=1.265$. The straight solid line is a linear fit
to the data for $S < 13,000$ spins. The slope from the fit
is $1.55$ (this varies by not more than $3\%$ as the range
over which the fit is done is changed),
while the exponent $\tau$ obtained from the collapses
and the extrapolation in figure\ \protect\ref{bin_aval_4dfigc}
is $1.53 \pm 0.08$. \label{bin_aval_fit_4dfig}}
\end{figure}
From figure\ \ref{bin_aval_4dfig}b, we see that the two binned avalanche
size distribution scaling function do not have a ``bump'' as does the
scaling function for the integrated avalanche size distribution (inset
in figure \ref{aval_3fig}). Therefore, we expect that the exponent
$\tau$ which gives the slope of the distribution in figure\
\ref{bin_aval_4dfig}a can also be obtained by a linear fit through the
data curve closest to the critical field. Figure\
\ref{bin_aval_fit_4dfig} shows the curve for the $H=1.265$ bin (dashed
curve) as well as the linear fit. The slope from the linear fit is
$1.55$ while the value of $\tau$ obtained from the collapses and the
extrapolation in figure\ \ref{bin_aval_4dfigc} is $1.53 \pm 0.08$.
Fitting the binned distribution curves with a straight line to extract
the exponent $\tau$ is also possible in other dimensions and mean field
as well.
\section{Comparison with the analytical results}
We have compared the simulation results with the renormalization group
analysis of the same system\ \cite{Dahmen1,Dahmen2}. According to the
renormalization group the upper critical dimension (UCD), at and above
which the critical exponents are equal to the mean field values, is six.
Close to the UCD, it is possible to do a $6-\epsilon$ expansion
($\epsilon$ is small and greater than $0$), and obtain estimates for the
critical exponents and the magnetization scaling function, which can
then be compared with our numerical results. Furthermore, at dimension
eight there is a prediction for another transition. Below eight
dimensions, there is a discontinuity in the slope of the magnetization
curve as it approaches the ``jump'' in the magnetization ($R < R_c$),
while above eight dimensions the approach is smooth.
\begin{figure}
\centerline{
\psfig{figure=Figures/Exponents_paper_new_THESIS.ps,width=3truein}
}
\caption[Comparison between the critical exponents from the simulation
and the $\epsilon$ expansion]
{Numerical values (filled symbols) of the exponents $\tau +
\sigma\beta\delta$, $\tau$, $1/\nu$, $\sigma\nu z$, and $\sigma\nu$
(circles, diamond, triangles up, squares, and triangle left) in $2$, $3$, $4$,
and $5$ dimensions. The empty symbols are values for these exponents in
mean field (dimension 6). Note that the value of $\tau$ in $2$d is the
conjectured value: we have not extracted $\tau$ from scaling collapses
(see text).
We have simulated sizes up to $30000^2$, $1000^3$, $80^4$, and $50^5$,
where for $320^3$ for example, more than $700$ different random field
configurations were measured.
The long-dashed lines are the $\epsilon$
expansions to first order for the exponents $\tau + \sigma\beta\delta$,
$\tau$, $\sigma\nu z$, and $\sigma\nu$. The short-dashed lines are Borel
sums\protect\cite{LeGuillou-Kleinert} for $1/\nu$. The lowest is the
variable-pole Borel sum from LeGuillou {\it et
al.}\protect\cite{LeGuillou-Kleinert}, the middle uses the method of
Vladimirov {\it et al.} to fifth order, and the upper uses the method of
LeGuillou {\it et al.}, but without the pole and with the correct fifth
order term.
The error bars denote systematic errors in finding the
exponents from extrapolation of the values obtained from collapses of curves
at different disorders $R$.
Statistical errors are smaller. \label{exp_compare}}
\end{figure}
Figure\ \ref{exp_compare} shows the numerical and analytical results for
five of the critical exponents obtained in dimensions two to six (in six
dimensions, the values are the mean field ones). The other exponents can
be obtained from scaling relations\cite{Dahmen1,Dahmen3}. The exponent
values in figure\ \ref{exp_compare} are obtained by extrapolating the
results of scaling collapses to either $R \rightarrow R_c$ or $1/L
\rightarrow 0$ (see section on simulation results). In two dimensions,
which is possibly the lower critical dimension, the plotted values are
averages from the three different scaling forms used to collapse the
data and extract the exponents. The error bars shown span all three {\it
ans\"atze}, and are compatible with our conjectures from the previous
section. The long-dashed lines are the $\epsilon$ expansions to first
order for $\tau + \sigma\beta\delta$, $\tau$, $\sigma\nu z$, and
$\sigma\nu$. The three short-dashed lines\cite{Dahmen1} are Borel
sums\protect\cite{LeGuillou-Kleinert} for $1/\nu$. The lowest is the
variable-pole Borel sum from LeGuillou {\it et
al.}\protect\cite{LeGuillou-Kleinert}, the middle uses the method of
Vladimirov {\it et al.} to fifth order, and the upper uses the method of
LeGuillou {\it et al.}, but without the pole and with the correct fifth
order term\ \cite{Dahmen1}. Notice that the numerical values converge
nicely to the mean field predictions, as the dimension approaches six,
and that the agreement between the numerical values and the $\epsilon$
expansion is quite impressive.
\begin{figure}
\centerline{
\psfig{figure=Figures/dMdH_d5_L30_dmdh_epsilon_scaled_paper_THESIS.ps,width=3truein}
}
\caption[Comparison between simulated $dM/dH$ curves in $5$ dimensions, and
the $dM/dH$ curve obtained from the $\epsilon$ expansion]
{Comparison between six simulation curves (thin lines)
and the $dM/dH$ curve (thick dashed line)
obtained from a parametric form\ \protect\cite{Zinn-Justin}
to third order
in $\epsilon$. The six curves are for a system of $30^5$ spins
at disorders: $7.0, 7.3,$ and $7.5$ ($R_c=5.96$ in $5$ dimensions),
and for a system of $50^5$ spins at disorders: $6.3, 6.4$, and $6.5$
(for larger fields, these are closer to the dashed line in the
figure). All the curves have been
stretched/shrunk in the horizontal and vertical direction to lie
on each other, and shifted horizontally. \label{5d_dmdh_fig}}
\end{figure}
The $\epsilon$ expansion can be an even more powerful tool if it can
predict the scaling functions. This has been done for the magnetization
scaling function of the pure Ising model in $4-\epsilon$ dimensions\
\cite{DombWallace,Zinn-Justin}. Since the $\epsilon$ expansion for our
model is the same as the one for the {\it equilibrium} RFIM\
\cite{Dahmen1}, and the latter has been mapped to {\it all} orders in
$\epsilon$ to the corresponding expansion of the regular Ising model in
two lower dimensions\ \cite{Dahmen1,Aharony,Parisi}, we can use the
results obtained in\ \cite{DombWallace,Zinn-Justin}. This is done in
figure\ \ref{5d_dmdh_fig}, which shows the comparison between the
$dM/dH$ curves obtained in $5$ dimensions at $R=6.3, 6.4, 6.5, 7.0, 7.3,
7.5$ ($R_c=5.96$) (the curves have been stretched/shrunk to lie on top
of each other, and shifted horizontally so that the peaks align), and
the parametric form (thick dashed line) for the scaling function of
$dM/dH$, to third order in $\epsilon$, where $\epsilon =1$ in $5$
dimensions (see\ \cite{Zinn-Justin}). As we see, the agreement is very
good in the scaling region (close to the peaks).This brings up the
possibility of using the $\epsilon$ expansion for the scaling function
to extract the critical exponents from simulation or experimental data.
So far though, only the scaling function for the magnetization has been
obtained.
\begin{figure}
\centerline{
\psfig{figure=Figures/MofH_579d_paper_THESIS.ps,width=3truein}
}
\caption[Magnetization curves showing the approach to the ``infinite jump''
in $5$, $7$, and $9$ dimensions]
{{\bf Magnetization curves in $5$, $7$, and $9$ dimensions.}
The disorders for these curves are $R=3.3$, $4.7$, and $6.0$ for
$30^5$, $10^7$, and $5^9$ size systems respectively. The dashed
lines represent the ``jump'' in the magnetization. Notice that in
$9$ dimensions the approach to the ``jump'' seems to be continuous.
\label{transition_8d}}
\end{figure}
As another check between the simulation and the renormalization group,
we have looked for the predicted transition in eight dimensions. Figure\
\ref{transition_8d} shows the magnetization curves in $5$, $7$, and $9$
dimensions (system sizes: $30^5$, $10^7$, and $5^9$) for values of the
disorder equal to ${2 \over 3}d$, where $d$ is the dimension. These
values of disorder are below the critical disorder in dimensions below
six, and are expected to be below for dimensions $7$ and $9$ as well.
For $5$d and $7$d, the approach to the ``jump'' in the magnetization is
discontinuous. Above the eight dimension, the approach is continuous
(see close ups in figure\ \ref{closeup_8d}). This is as expected from
the renormalization group\ \cite{Dahmen1}. We have also looked at
$dM/dH$, which appears clearly to diverge in $d=9$ and not in $d=7$
(figure\ \ref{slope_8d}).
\section{Conclusion}
We have used the zero temperature random field Ising model, with a
Gaussian distribution of random fields, to model a random system that
exhibits hysteresis. We found that the model has a transition in the
shape of the hysteresis loop, and that the transition is critical. The
tunable parameters are the amount of disorder $R$ and the external
magnetic field $H$. The transition is marked by the appearance of an
infinite avalanche in the thermodynamic system. Near the critical point,
($R_C$, $H_C$), the scaling region is quite large: the system can
exhibit power law behavior for several decades, and still not be near
the critical transition. This is important to keep in mind whenever
experimental data are analyzed. If a tunable parameter can be found, a
system that appears to be SOC, might in reality have a disorder induced
critical point.
\begin{figure}
\centerline{
\psfig{figure=Figures/MofH_79d_closeup_paper_THESIS.ps,width=3truein}
}
\caption[Closeup of magnetization curves with the ``infinite jump''
in $7$ and $9$ dimensions]
{(a) and (b) Closeup of the {\bf magnetization curves in $7$ and $9$
dimensions} respectively from figure\ \protect\ref{transition_8d}.
In $8$ dimensions, there is a prediction from the renormalization
group\ \protect\cite{Dahmen1} that there is a transition
in the way the jump is approached (see text). \label{closeup_8d}}
\end{figure}
We have extacted critical exponents for the magnetization, the
avalanche size distribution (integrated over the field and binned in the
field), the moments of the avalanche size distribution, the avalanche
correlation, the number of spanning avalanches, and the distribution of
times for different avalanche sizes. These values are listed in Table\
\ref{measured_exp_table} and Table\ \ref{conj_meas_2d_table}, and were
obtained as an average of the extrapolation results (to $R \rightarrow
R_c$ or $L \rightarrow \infty$) from several measurements. For example,
the correlation length exponent $\nu$ is the average value from three
different collapses: the correlation function, the spanning avalanches,
and the second moments of the avalanche size distribution, while the
critical disorder $R_c$ is estimated from both the spanning avalanches
collapses and the collapses of the moments of the avalanche size
distribution. As shown earlier, the numerical results compare well with
the $\epsilon$ expansion\ \cite{Dahmen1,Dahmen2}. Furthermore, the
renormalization group work predicts another transition in eight
dimensions, which we find in the simulation as well. Comparisons to
experimental Barkhausen noise measurements\ \cite{Perkovic} are very
encouraging, and a more comprehensive review of possible experiments
that exhibit disorder--driven critical phenomena similar to our model
is under way\ \cite{Dahmen3}.
\begin{figure}
\centerline{
\psfig{figure=Figures/dMdH_579d_paper_THESIS.ps,width=3truein}
}
\caption[$dM/dH$ for magnetization curves with the ``infinite jump'',
in $7$ and $9$ dimensions]
{Derivative of the magnetization with respect to the field $H$, for
the curves in figure\ \protect\ref{transition_8d}.
The approach to the ``infinite jump'' seems to be continuous
in $9$ dimensions. Note that the
vertical axis is logarithmic. \label{slope_8d}}
\end{figure}
Finally, we should mention that there are other models for avalanches in
disordered magnets. There is a large body of work on depinning
transitions and the motion of the single
interface\cite{depinning,SameDepinning}. In these models, avalanches
occur only at the growing interface. Our model though, deals with many
interacting interfaces: avalanches can grow anywhere in the system.
Similar models exist with random bonds\cite{RandomBonds,Vives} and
random anisotropies. In the random bonds model, the interaction
$J_{ij}$ between neighboring spins $i$ and $j$ is random.
The zero temperature random bond Ising model\ \cite{RandomBonds,Vives}
also exhibits a critical transition in the shape of the hysteresis loop,
where the mean bond strength is analogous to our disorder $R$. It has
been argued numerically\ \cite{Vives} and analytically\ \cite{Dahmen1},
that as long as there are no long-range forces\cite{Urbach} and
correlated disorder, the random bond and the random field Ising model
are in the same universality class. A comparison between our simulation
and the results in reference\ \cite{Vives} show that the $3$ dimensional
results agree quite nicely. However, in $2$ dimensions, there are large
differences, which we believe occur because of the small system sizes
used by the authors for their simulation (only up to $L=100$). We have
seen that our results (see section on the $2$ dimensional simulation)
are very size dependent. Looking back for example at figure\
\ref{span_2d_fig}, we find that for a system of $L=100$ spins, a
``good'' estimate for the critical disorder $R_c$ would indeed be $0.75$
as was found in\ \cite{Vives}. However, we find after increasing the
system size that the critical disorder $R_c$ is $0.54$ or lower.
\subsubsection{Avalanche Correlation}
The avalanche correlation function $G(x,R,H)$ measures the probability
that a flipping spin will trigger, through an avalanche of spins,
another spin a distance $x$ away. From the renormalization group
description\cite{Dahmen1,Dahmen2}, close to the critical point and for
large distances $x$, the correlation function is given by (corrections
are subdominant as explained in appendix A):
\begin{equation}
G(x,R,H) \sim {1 \over {x^{d-2+\eta}}}\ {\cal G}_{\pm}(x/{\xi(r,h)})
\label{correl_equ1}
\end{equation}
where $r$ and $h$ are respectively the reduced disorder and field,
${\cal G_{\pm}}$ ($\pm$ indicates the sign of $r$) is the scaling
function, $d$ is the dimension, $\xi$ is the correlation length, and
$\eta$ is called the ``anomalous dimension''. The correlation length
$\xi (r,h)$ is a macroscopic length scale in the system which is roughly
on the order of the mean linear extent of the avalanches for a system
away from the critical point.
\begin{figure}
\centerline{
\psfig{figure=Figures/Norm_correl_d3_L320_paper_THESIS.ps,width=3truein}
}
\caption[Avalanche correlation curves in $3$ dimensions]
{{\bf Avalanche correlation function integrated over the field $H$
in $3$ dimensions}, for $L=320$. The curves are averages of up to $19$
random field configurations. The critical disorder $R_c$ is $2.16$.
\label{correl_3d_fig}}
\end{figure}
{\noindent At the critical field $H_c$ (h=0) and near
$R_c$, the correlation length scales like $\xi \sim |r|^{-\nu}$, while
for small field $h$ it is given by $\xi \sim |r|^{-\nu}\ {\cal
Y}_{\pm}(h/|r|^{\beta\delta})$ where ${\cal Y}_{\pm}$ is a universal
scaling function. The avalanche correlation function should not be
confused with the cluster or ``spin-spin'' correlation which measures
the probability that two spins a distance $x$ away have the same value.
(The algebraic decay for this other, spin-spin correlation function at
the critical point ($r=0$ and $h=0$), is $1/{x^{d-4+{\tilde
\eta}}}$\cite{Dahmen1}.)}
\begin{figure}
\centerline{
\psfig{figure=Figures/Norm_correl_d3_L320_collapse_paper_THESIS.ps,width=3truein}
}
\nobreak
\nobreak
\centerline{
\psfig{figure=Figures/Nu_d3_L320_correl_paper_THESIS.ps,width=3truein}
}
\caption[Scaling collapse of the avalanche correlation curves in $3$
dimensions, and exponent $\nu$]
{(a) Scaling collapse of the {\bf avalanche correlation function
integrated over the field $H$, in $3$ dimensions}
for $L=320$. The values of the disorders
range from $R=2.35$ to $R=3.0$, with $R_c=2.16$. The exponents are:
$\nu=1.39 \pm 0.20$ and $d + \beta/\nu = 3.07 \pm 0.30$.
(b) Exponent $\nu$ extracted from collapses of avalanche
correlation curves
(see (a)).
The extrapolated value at $|r|_{avg}=0$ is
$1.37 \pm 0.18$. \label{correl_collapse_3d_fig}}
\end{figure}
We have measured the avalanche correlation function integrated over the
field $H$, for $R>R_c$. For every avalanche that occurs between
$H=-\infty$ and $H =+\infty$, we keep a count on the number of times a
distance $x$ occurs in the avalanche. To decrease the computational time
not every pair of spins is selected; instead we do a statistical average
for $S$ pairs where $S$ is the size of the avalanche. Our simulation
seems to indicate that the difference between this statistical average
and the exact measurement is less than the fluctuations obtained from
measurements of the correlation function for different realizations of
the random field distribution. The data is saved in ``distance'' bins
separated by $0.5$ and starting at a distance of $1.0$ (the self
correlation is not included), and is normalized by the number of
neighbors at each distance. The spanning avalanches are not included in
our correlation measurement. Figure\ \ref{correl_3d_fig} shows several
avalanche correlation curves in $3$ dimensions for $L=320$. The scaling
form for the avalanche correlation function integrated over the field
$H$, close to the critical point and for large distances $x$, is
obtained by integrating equation (\ref{correl_equ1}):
\begin{equation}
G_{\it int}(x,R) \sim \int {1 \over {x^{d-2+\eta}}}\
{\cal G}_{\pm}\Bigl(x/{\xi(r,h)}\Bigr)\ dh
\label{correl_equ2}
\end{equation}
Near the critical point $\xi(r,h) \sim |r|^{-\nu} {\cal Y}_{\pm}
(h/|r|^{\beta\delta})$.
Defining $u=h/|r|^{\beta\delta}$, equation (\ref{correl_equ2}) becomes:
\begin{equation}
G_{\it int}(x,R) \sim |r|^{\beta\delta}{x^{-(d-2+\eta)}} \int
{\cal G}_{\pm}\Bigl(x/|r|^{-\nu} {\cal Y}_{\pm} (u)\Bigr)\ du
\label{correl_equ3}
\end{equation}
The integral ($\cal I$) in equation (\ref{correl_equ3}) is a function of
$x|r|^{\nu}$ and can be written as:
\begin{equation}
{\cal I} =
(x|r|^{\nu})^{-\beta\delta/\nu}\ {\widetilde {\cal G}}_{\pm}(x|r|^{\nu})
\label{correl_equ4}
\end{equation}
to obtain the scaling form:
\begin{equation}
G_{\it int}(x,R) \sim {1 \over x^{d+\beta/\nu}}\
{\widetilde {\cal G}}_{\pm}(x|r|^{\nu})
\label{correl_equ5}
\end{equation}
where we have used the scaling relation $(2-\eta)\nu=\beta\delta-\beta$
(see\ \cite{Dahmen1,Dahmen3} for the derivation).
\begin{figure}
\centerline{
\psfig{figure=Figures/Norm_Correl_and_Aniso_d3_L320_paper_THESIS.ps,width=3truein}
}
\caption[Avalanche correlation function anisotropies in $3$ dimensions]
{{\bf Anisotropies in the avalanche correlation function}. The curves
are for a system of $320^3$ spins at $R=2.35$. Four curves are
shown on the graph: one is the avalanche correlation function integrated
over the field $H$ (as in figure\ \protect\ref{correl_3d_fig}),
while the other three are measurements of the correlation along
the three axis, the six face diagonals, and the four body diagonals.
Avalanches involving more than four spins show no noticeable
anisotropy: the critical point appears to have spherical symmetry.
The same result is found in $2$ dimensions.
\label{correl_aniso_3d_fig}}
\end{figure}
Figure\ \ref{correl_collapse_3d_fig}a shows the integrated avalanche
correlation curves collapse in $3$ dimensions for $L=320$ and $R>R_c$.
The exponent $\nu$ is obtained from such collapses by extrapolating to
$R = R_c$ (figure\ \ref{correl_collapse_3d_fig}b) as was done for other
collapses. The exponent $\beta/\nu$ can be obtained from these collapses
too, but it is much better estimated from the magnetization
discontinuity covered below. The value of $\beta/\nu$, listed in Table\
\ref{measured_exp_table} alongside all the other exponents, is derived
from the magnetization discontinuity collapses only.
We have also looked for possible anisotropies in the integrated
avalanche correlation function in $2$ and $3$ dimensions. The
anisotropic integrated avalanche correlation functions are measured
along ``generalized diagonals'': one along the three axis, the second
along the six face diagonals, and the third along the four body
diagonals. We compare the integrated avalanche correlation function and
the anisotropic integrated avalanche correlation functions to each
other, and find no anisotropies in the correlation, as can be seen from
figure\ \ref{correl_aniso_3d_fig}.
\section{Introduction}
The increased interest in real materials in condensed matter physics has
brought disordered systems into the spotlight. Dirt changes the free
energy landscape of a system, and can introduce metastable states with
large energy barriers\ \cite{barrier}. This can lead to extremely slow
relaxation towards the equilibrium state. On long length scales and
practical time scales, a system driven by an external field will move
from one metastable local free-energy minimum to the next. The
equilibrium, global free energy minimum and the thermal fluctuations
that drive the system toward it, are in this case irrelevant. The state
of the system will instead depend on its history.
The motion from one local minima to the next is a collective process
involving many local (magnetic) domains in a local region - {\it an
avalanche}. In magnetic materials, as the external magnetic field $H$ is
changed continuously, these avalanches lead to the magnetic noise: the
Barkhausen effect\cite{Jiles,McClure}. This effect can be picked up as
voltage pulses in a coil surrounding the magnet. The distribution of
pulse (avalanche) sizes is found\
\cite{McClure,Barkhausen,Urbach,SOC_example} to follow a power law with
a cutoff after a few decades, and was interpreted by some\
\cite{SOC_example} to be an example of self-organized criticality\
\cite{SOC}. (In SOC, a system organizes itself into a critical state
without the need to tune an external parameter.) Other systems can
exhibit avalanches as well. Several examples where disorder may play a
part are: superconducting vortex line avalanches\ \cite{Field},
resistance avalanches in superconducting films\ \cite{Wu}, and capillary
condensation of helium in Nuclepore\ \cite{Nuclepore}.
The history dependence of the state of the system leads to hysteresis.
Experiments with magnetic tapes\ \cite{Berger} have shown that the shape
of the hysteresis curve changes with the annealing temperature. The
hysteresis curve goes from smooth to discontinuous as the annealing
temperature is increased. This transition can be explained in terms of a
{\it plain old critical point} with two tunable parameters: the
annealing temperature and the external field. At the critical
temperature and field, the correlation length diverges, and the
distribution of pulse (avalanche) sizes follows a power law.
We have argued earlier\ \cite{Perkovic} that the Barkhausen noise
experiments can be quantitatively explained by a model\ \cite{Sethna}
with two tunable parameters (external field and disorder), which
exhibits {\it universal}, non-equilibrium collective behavior. The model
is athermal and incorporates collective behavior through nearest
neighbor interactions. The role of {\it dirt} or disorder, as we call
it, is played by random fields. This paper presents the results and
conclusions of a large scale simulation of that model: the
non-equilibrium zero-temperature Random Field Ising Model (RFIM), with a
deterministic dynamics. The results compare very well to our $\epsilon$
expansion\cite{Dahmen1,Dahmen2}, and to experiments in Barkhausen
noise\cite{Perkovic}. A more detailed comparison to experimental systems
is in process\cite{Dahmen3}.
The paper is divided as follows. Section II quickly reviews the model.
Section III explains the simulation method that we use. Section IV
explains the data analysis and shows results for the simulation in $2$,
$3$, $4$, and $5$ dimensions, as well as in mean field. Section V gives
a comparison between the simulation and the $\epsilon$ expansion
exponents, and a comparison between the shape of the magnetization
curves in $5$, $7$, and $9$ dimensions, and the predicted shape from the
$\epsilon$ expansion. Section VI summarizes the results. This is
followed by three appendices that cover derivations that were omitted in
the text for continuity.
\subsection{Mean Field Simulation}
The mean field simulation shows how well the results for the critical
exponents, obtained close to $R_c$ and for finite size systems (finite
number of spins), extrapolate to the calculated values for a system in
the thermodynamic limit, at the critical disorder. Thus, we will omit in
this section some details that are only relevant for understanding the
non-mean field simulation results. We start with the curves for the
magnetization as a function of the field for different values of the
disorder, which we find are not useful for extracting critical
exponents. We then go on to measurements of spin avalanche sizes and
their moments. Avalanches that span the system from one ``side'' to
another will also be mentioned although since in mean field there are no
``sides'', we will define what we mean by a mean field spanning
avalanche. Since distances are irrelevant in mean field, we do not have
any correlation measurements, but we can still apply what we learn from
other collapses in mean field to the correlation measurement data in
$2$, $3$, $4$, and $5$ dimensions.
Figure\ \ref{mf_mofhfiga} shows the magnetization curves, and figure
\ref{mf_mofhfig}a shows a scaling collapse for a $10^6$ mean field spin
system and $r<0$ ($R>R_c$). As mentioned earlier, near the critical
point ($R_c = {\sqrt {2/\pi}}$ for $J=1$, in mean field), the
magnetization scales like\cite{Sethna,Dahmen1}
\begin{equation}
M(H,R) - M_c(H_c,R_c) \sim |r|^\beta\ {\cal M}_{\pm}(h/|r|^{\beta\delta})
\label{mf_mofh_equ1}
\end{equation}
where $\pm$ refers to the sign of the reduced disorder $r=(R_c-R)/R$,
and $h=(H-H_c)$. The mean field critical exponents are $\beta = 1/2$ and
$\beta\delta = 3/2$. Notice in figure\ \ref{mf_mofhfig}a that the
scaling region around $M_c=0$ and $H_c=0$ is very small; figure\
\ref{mf_mofhfig}b shows that a substantially different set of critical
exponents leads to a similar looking collapse. In general, the critical
field $H_c$ and the critical magnetization $M_c$ are not zero as in mean
field, and $M_c$ is not well determined numerically. In dimensions that
we simulate ($2$ through $5$), the critical region is not only small but
it is also poorly defined, which does not sufficiently constrain the
values of the exponents. This makes the magnetization function $M(H,R)$
a poor choice for extracting critical exponents.
\begin{figure}
\centerline{
\psfig{figure=Figures/MofH_S1000000_all_paper_THESIS.ps,width=3truein}
}
{\caption[Mean field magnetization curves]
{{\bf Mean field magnetization} curves for $10^6$ spins.
The critical disorder is $R_c= 0.79788456$.
The curves are averages of $6$ to $10$ different initial realizations
of the random field distribution. \label{mf_mofhfiga}}}
\end{figure}
The critical magnetization $M_c$ can be removed from the scaling form if
we look at the first derivative of the magnetization with respect to the
field instead. $dM/dH$ scales like:
\begin{equation}
{dM \over dH} (H,R) \sim |r|^{\beta-\beta\delta}\
\dot{\cal M}_{\pm}(h/|r|^{\beta\delta})
\label{dmdh_mf_equ1}
\end{equation}
where $\dot{\cal M}_{\pm}$ denotes the derivative of the scaling
function ${\cal M}_{\pm}$ with respect to its argument
$h/|r|^{\beta\delta}$. The $dM/dH$ mean field curves and collapses are
shown in figure~\ref{mf_dmdh_figa} and figures~\ref{mf_dmdh_fig}(a--b).
Notice that the incorrect exponents $\beta=0.4$ and $\beta\delta=1.65$
give a better collapse (fig.\ \ref{mf_dmdh_fig}b). Figure\
\ref{mf_dmdh_figd} shows a close up of figure\ \ref{mf_dmdh_fig}a,
alongside with three (thin) curves for disorders: $0.80,0.81,$ and
$0.82$. These are not measured in the simulation (the finite number of
mean field spins we use give rise to finite size effects near $R_c$ as
we will see soon); instead they are numerically calculated from the mean
field implicit equation for the magnetization\ \cite{Sethna,Dahmen1}:
\begin{equation}
M(H) = 1 - 2 \int_{-\infty}^{-J^{*}M(H)-H} \rho(f)\ df
\label{mofh_implicit}
\end{equation}
where $J^{*}$ denotes the coupling of one spin to {\it all} the other
spins in the system, and $\rho(f)$ is the random field distribution
function.
\begin{figure}
\centerline{
\psfig{figure=Figures/MofH_S1000000_collapse_R_all_paper_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/MofH_S1000000_collapse_R_all_paper_1.4_THESIS.ps,width=3truein}
}
\caption[Scaling collapse of mean field magnetization curves]
{(a) {\bf Scaling collapse for the mean field magnetization curves} at
disorders $R=0.912$, $0.974$, $1.069$, $1.165$, $1.197$, and $1.460$.
(These values of disorder were chosen relative to $R_c=0.79788456$,
to match some of the values we measured in $3$ dimensions (see figure\
\protect\ref{3d_MofH_fig}). The value of the critical disorder $R_c$ in
$3$ dimensions has since been modified, and there is no correspondence
anymore.) The
exponents are $\beta=1/2$ and $\beta\delta=3/2$. $m$ is defined as
$M-M_c$, and in mean field both $M_c$ and $H_c$ are zero. The inset
shows a closeup of the critical region.
(b) Scaling collapse of the same curves as in (a) but with the
(wrong) exponents
$\beta=0.4$ and $\beta\delta =1.65$. The two collapses are very
similar. The inset is a closeup. \label{mf_mofhfig}}
\end{figure}
The scaling collapse converges to the expected scaling
function (dashed thick line) as we get closer to the critical disorder.
The expected scaling form is also obtained from an analytic expression
derived in mean field\ \cite{Sethna,Dahmen1}. It is given by the
smallest real root $g(y)$ of the cubic equation:
\begin{equation}
g^3 + {12 \over \pi} g - {12 \sqrt{2} \over \pi^{3/2} R_c} y = 0.
\label{g_eqn}
\end{equation}
\begin{figure}
\centerline{
\psfig{figure=Figures/dMdH_S1000000_less2_points_paper_THESIS.ps,width=3truein}
}
\caption[Mean field $dM/dH$ curves]{{\bf Mean field $dM/dH$ curves}
for $10^6$ spins and disorders $R=0.912$,
$0.974$, and $1.069$ (from largest to smallest peak).
The original data is the same as in figure\ \protect\ref{mf_mofhfiga}.
The critical disorder is $R_c = 0.79788456$. \label{mf_dmdh_figa}}
\end{figure}
We again find that the critical exponents and $R_c$, obtained from the
$dM/dH$ curves, are ill-determined. In finite dimensions, that is even
more true since we have another parameter to fit: $H_c$. For dimensions
$3$, $4$, and $5$, we extract $\beta$, $\beta\delta$, $H_c$, and $R_c$
by other means and simply show the resulting collapse of the $M(H)$ and
$dM/dH$ curves as a check.
As mentioned earlier, the spins flip in avalanches of varying sizes. The
distribution of ${\it all}$ the avalanches that occur at a disorders $R$
while the external field $H$ is raised adiabatically from $-\infty$ to
$+\infty$ is plotted in figure\ \ref{mf_aval_collapfigaa}. The curves in
this plot are normalized by the number of spins in the system, and
therefore represent the probability {\it per spin} for an avalanche of
size $S$ to occur in the hysteresis loop, at disorder $R$. The curves
can be normalized to one if they are divided by the total number of
avalanches in the loop, and multiplied by the number of spins in the
system.
Often in experiments, the $\it binned$ avalanche size distribution,
which contains only avalanches that occur in a small range of fields
around a particular value of the field $H$, is measured instead. The
scaling form for this distribution\cite{note2} is\
\cite{Sethna,Dahmen1}:
\begin{equation}
D(S,R,H) \sim S^{-\tau}\ {\bar {\cal D}_{\pm}}
(S^\sigma |r|, h/|r|^{\beta\delta})
\label{int_aval0}
\end{equation}
where $S$ is the size of the avalanche and is large, and $r$ and $h$ are
small. In mean field, $\sigma=1/2$ and $\tau=3/2$. The scaling form for
the integrated avalanche size distribution is obtained by integrating
the above form over all fields:
\begin{equation}
D_{\it int}(S,R) \sim \int S^{-\tau}\ {\bar {\cal D}_{\pm}} (S^\sigma |r|,
h/|r|^{\beta\delta})\ dh
\label{int_aval1}
\end{equation}
\begin{figure}
\centerline{
\psfig{figure=Figures/dMdH_S1000000_less2_points_smooth5_collapse_paper_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/dMdH_S1000000_less2_points_smooth5_collapse_paper_good_THESIS.ps,width=3truein}
}
\caption[Scaling collapse of mean field $dM/dH$ curves]
{(a) {\bf Scaling collapse of mean field $dM/dH$ curves} from figure
\protect\ref{mf_dmdh_figa}. The exponents are
$\beta=1/2$ and $\beta\delta=3/2$ (mean field values). The curves are
smoothed over $5$ data points (using a running average)
to show the collapse better. The curves
do not collapse well for large and small $h/r^{\beta\delta}$,
unless we get very close to the critical disorder (see figure
\protect\ref{mf_dmdh_figd}).
(b) Scaling collapse of data in (a) but with exponents
$\beta=0.4$ and $\beta\delta=1.65$. The collapse is better, although
the exponents are wrong. \label{mf_dmdh_fig}}
\end{figure}
With the change of variable $u=h/|r|^{\beta\delta}$, equation\
(\ref{int_aval1}) becomes:
\begin{equation}
D_{\it int}(S,R) \sim S^{-\tau}\ |r|^{\beta\delta} \int {\bar {\cal D}_{\pm}}
(S^\sigma |r|,u)\ du
\label{int_aval2}
\end{equation}
The integral in equation\ (\ref{int_aval2}) is a function of $S^\sigma
|r|$ only,so we can write it as:
\begin{equation}
(S^\sigma |r|)^{-\beta\delta}\ {\bar {\cal D}_{\pm}}^{(int)}
(S^\sigma |r|)
\label{int_aval2a}
\end{equation}
to obtain the scaling form for the integrated avalanche size
distribution:
\begin{equation}
D_{\it int} (S,R) \sim S^{-(\tau + \sigma\beta\delta)}\
{\bar {\cal D}_{\pm}}^{(int)} (S^\sigma |r|)
\label{int_aval3}
\end{equation}
To obtain equation\ (\ref{int_aval2a}), we have assumed that the
integral in (\ref{int_aval2}) converges. This is usually safe to do
since the distribution curves near the critical point drop off
exponentially for large arguments. The same kind of argument can be made
for the integrals of other measurements as well.
\begin{figure}
\centerline{
\psfig{figure=Figures/dMdH_S1000000_less2_points_smooth5_meanfield_collapse_paper_THESIS.ps,width=3truein}
}
\caption[Close--up of mean field $dM/dH$ curves collapse]
{{\bf Close-up of the mean field $dM/dH$ curves collapse}
in figure \protect\ref{mf_dmdh_fig}a.
Also plotted are
three curves (thin lines) calculated using the mean field
analytic solution to $M(H)$ (see text).
These are for $R=0.80$, $0.81$, and $0.82$. We see that the
scaling collapse, at the mean field exponents, of the $dM/dH$ curves
converges to the expected mean field scaling function
(thick dashed line), as $R \rightarrow R_c$. \label{mf_dmdh_figd}}
\end{figure}
\begin{figure}
\centerline{
\psfig{figure=Figures/Aval_histo_S1000000_paper_norm_THESIS.ps,width=3truein}
}
\caption[Mean field integrated avalanche size distribution curves]
{{\bf Mean field integrated avalanche size distribution curves} for
$10^6$ spins and disorders $R=0.912$, $0.974$, $1.069$, $1.197$, and
$1.460$
(from right to left). The straight line is the slope of the power law
behavior in mean field: $\tau + \sigma\beta\delta=9/4$.
\label{mf_aval_collapfigaa}}
\end{figure}
Figures\ \ref{mf_aval_collapfiga}a and \ref{mf_aval_collapfiga}b show
two collapses with different critical exponents of the curves from
figure\ \ref{mf_aval_collapfigaa}, using the scaling form in equation\
(\ref{int_aval3}). Notice that the collapse with the incorrect exponents
$\tau + \sigma\beta\delta =2.4$ and $\sigma = 0.44$ is better than the
collapse with the mean field exponents $\tau + \sigma\beta\delta = 9/4$
and $\sigma = 1/2$. Although the distribution curves in figures\
\ref{mf_aval_collapfiga}a and \ref{mf_aval_collapfiga}b have disorders
that are far from the critical disorder $R_c=0.79788456$, the curves
collapse but with the {\it wrong} exponents.
\begin{figure}
\centerline{
\psfig{figure=Figures/Aval_histo_S1000000_collapse_sigma_R_paper_norm_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/Aval_histo_S1000000_collapse_sigma_R_paper_norm_good_THESIS.ps,width=3truein}
}
\caption[Scaling collapse of the mean field integrated avalanche
size distribution curves]
{(a) {\bf Scaling collapse of three integrated avalanche
size distribution curves in mean field}, for disorders:
$1.069$, $1.197$, and $1.460$.
The curves are smoothed
over $5$ data points before they are collapsed. The collapse
is done using the mean field values of the exponents $\sigma$ and
$\tau + \sigma\beta\delta$ ($1/2$ and $9/4$ respectively),
and $r = (R_c-R)/R$.
(b) Same curves and scaling form as in (a), but with the exponents
$\sigma=0.44$ and $\tau + \sigma\beta\delta = 2.4$. The collapse is
better for the incorrect exponents! We use this ``best'' collapse
to extract exponents for figures\ \protect\ref{mf_aval_expfig}a
and \protect\ref{mf_aval_expfig}b, and then extrapolate to $R=R_c$
to obtain the correct mean field exponents. \label{mf_aval_collapfiga}}
\end{figure}
It is surprising that these curves collapse at all since the scaling
form is correct only for $R$ close to $R_c$. Corrections to scaling
become important away from the critical point, but it seems that the
scaling form has enough ``freedom'' that collapses are possible even far
from $R_c$. In the limit of $R \rightarrow R_c$, we expect that the
exponents obtained from such collapses will converge to the mean field
value, and that the extrapolation will remove the question of scaling
corrections. To test this, we have collapsed three curves at a time, and
plotted the values of the exponents extracted from such collapses
against the average of the reduced disorder $|r|$ for the three curves,
which we call $|r|_{\it avg}$ (figures\ \ref{mf_aval_expfig}a and
\ref{mf_aval_expfig}b). In these figures, notice two things. First, the
linear extrapolation to $|r|_{\it avg}=0$ agrees quite well with the
mean field exponent values, and second, the points obtained by doing
collapses using $r=(R_c-R)/R$ either converge faster to the mean field
exponents or do as well as the points obtained from collapses done with
$r=(R_c-R)/R_c$. This is true for all the extrapolations that we have
done in mean field. Other models (see for example\ \cite{Robbins})
exhibit this behavior, and experimentalists seem to have known about
this for a while\ \cite{Souletie}.
\begin{figure}
\centerline{
\psfig{figure=Figures/Tau_sigma_beta_delta_S1000000_paper_over_R_Rc_legend_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/Sigma_S1000000_paper_over_R_Rc_legend_THESIS.ps,width=3truein}
}
\caption[Mean field exponents $\tau + \sigma\beta\delta$ and $\sigma$,
from the integrated avalanche size distribution]
{(a) $\tau + \sigma \beta \delta$ from collapses of {\bf mean field
integrated avalanche size distribution curves}
for $10^6$ spins. The two points closest
to $|r|_{\it avg}=0$ are for a system of $10^7$ spins.
$|r|_{\it avg}$ is the average reduced disorder $|r|$
for the three curves collapsed together (see text).
(b) $\sigma$ from collapses of integrated avalanche size
distribution curves for the system in (a). Again, the two closest
points to $|r|_{\it avg}=0$ are for a system of $10^7$ spins. The mean
field values are calculated analytically. \label{mf_aval_expfig}}
\end{figure}
In dimensions $2$ to $5$, we obtain the exponents
$\tau+\sigma\beta\delta$ and $\sigma$ in the limit $R \rightarrow R_c$,
using the above linear extrapolation method. For other collapses, if the
two extrapolation results differ substantially, we ``bias'' our result
towards the $r=(R_c-R)/R$ extrapolated value of the exponent.
Notice in figures\ \ref{mf_aval_collapfiga}a and
\ref{mf_aval_collapfiga}b that the scaling function ${\bar {\cal
D}_{-}^{(int)}}$ has a ``bump'' (the $-$ sign indicates that the
collapse is for curves with $R > R_c$). Although we will come back to
this point when we talk about the results in $5$ and lower dimensions,
it is interesting to know what the shape of the scaling function ${\bar
{\cal D}}_{-}^{(int)}$ is. In appendix B, we calculate the mean field
scaling function for $r<0$ (equations\ \ref{apA1_eq14} and
\ref{apA1_eq15}):
\begin{eqnarray}
{\bar {\cal D}}_{-}^{(int)}(-r S^{\sigma})\ =\ {e^{-(-r S^{\sigma})^2 \over 2}
\over \pi {\sqrt 2}}\ \times \nonumber \\
\int_0^{\infty}
e^{\Bigl(-(-r) S^{\sigma}\ u - {u^2 \over 2}\Bigr)}\
\Bigl(-r S^{\sigma} + u\Bigr)\ {du \over {\sqrt u}}
\label{int_aval3b}
\end{eqnarray}
where $\sigma=1/2$.
\begin{figure}
\centerline{
\psfig{figure=Figures/Aval_histo_mf_fit_paper_THESIS.ps,width=3truein}
}
\caption[Mean field scaling function for the integrated avalanche size
distribution]
{Scaling collapse of the {\bf mean field integrated avalanche size
distribution curves} (dashed lines),
for $S=10^6$ spins and $R=0.912,0.974$, and
$S=10^7$ spins and $R=0.854, 0.878, 0.912$. The critical exponents
are: $\tau+\sigma\beta\delta =9/4$ and $\sigma =1/2$. The thick black
line is the best fit to the data using a function that is the product
of a polynomial and an exponential (eqn.\ (\protect\ref{int_aval3a})).
The thick grey line is the ``real''
mean field scaling function (see text).\label{mf_aval_fit}}
\end{figure}
A closed analytic form can not be obtained, but we can find the behavior
of this function for small and large arguments $-rS^\sigma$. For small
arguments $X=-rS^\sigma$, the scaling function is a polynomial in $X$
(\ref{apA1_eq17}), while for large arguments, the scaling function is
given by the product of an exponential decay in $X^2$ and the square
root of $X$ (\ref{apA1_eq18}). We can then try to fit our data (the
scaling collapse) with a function that will incorporate a polynomial
and an exponential decay (as an approximation to the real function). We
obtain:
\begin{eqnarray}
e^{-{X^2 \over 2}}\ &
(0.204 + 0.482 X - 0.391 X^2 + \nonumber \\
& 0.204 X^3 - 0.048 X^4)
\label{int_aval3a}
\end{eqnarray}
This form has the expected exponential behavior at large $X$, but the
wrong pre-factor. On the other hand, for small $X$, the above function
is analytic. A better approach might be to use a parametric
representation\ \cite{Schofield}, which we have not yet tried.
Equation (\ref{int_aval3a}) can be compared with the curve obtained by
numerically integrating the scaling function ${\bar {\cal
D}}_{-}^{(int)}$ in equation (\ref{int_aval3b}). Figure\
\ref{mf_aval_fit} shows the fit in black (equation (\ref{int_aval3a}))
to the collapsed data, for curves (dashed lines) of different disorder,
and system size $S=10^6$ and $S=10^7$ spins. The grey curve is the
``real'' scaling function obtained from the numerical integration of
equation (\ref{int_aval3b}). Notice that the scaling collapse (done with
the mean field values of the exponents $\tau+\sigma\beta\delta$ and
$\sigma$) of even a system of $10^7$ spins and within $7\%$ of $R_c$
(ie. $R=0.854$) (this is the curve with the smallest peak in the graph)
is not close to the ``real'' scaling function (the thick grey curve).
The error is within $5\%$ for this curve (within $10\%$ for the fit).
However, as $R \rightarrow R_c$, the avalanche size distribution curves
seem to be approaching the ``real'' scaling function (grey curve). It is
important to keep in mind when analyzing experimental or numerical data
as we will in $5$ and lower dimensions, that the scaling collapse most
likely does not give the limiting curve one would obtain for $1/L
\rightarrow 0$ and $R \rightarrow R_c$, even for what seems like a large
size, and close to the critical disorder.
\begin{figure}
\centerline{
\psfig{figure=Figures/Span_aval_paper_collapse_MF_legend_THESIS.ps,width=3truein}
}
\caption[Spanning avalanches in mean field]
{{\bf Number of mean field spanning avalanches}
$N_{mf}$ as a function
of the disorder $R$. Curves at sizes $125$ and
$343$ are not plotted. All the
error bars are not shown for clarity. The ones that are shown are
representative for the peaks. The error bars are smaller for
larger disorders.
About 26 points are used for each curve; each point being an average
between $250$ (for size $512000$) and $2500$ (for size $1000$)
random field configurations. The inset shows the collapse
of the three largest size curves using the mean field (calculated)
exponents ${\tilde \theta}= 3/8$ and $1/{\tilde \nu}=1/4$.
\label{span_aval_mfa}}
\end{figure}
The avalanches in the avalanche size distribution are finite, by which
we mean that they don't span the system. We have mentioned earlier that
due to the finite size of a system, close to the critical disorder
$R_c$, the largest avalanche or avalanches will span the system from one
side to another. We will talk about spanning avalanches in more details
later, but for now we just need to know that the number $N$ of spanning
avalanches scales as $N(L,R) \sim L^\theta\ {\cal N}_{\pm}(L^{1/\nu}
|r|)$ where ${\cal N}_{\pm}$ is a scaling function ($\pm$ indicates the
sign of $r$), $L$ is the linear size of the system, $\theta$ is the
exponent that arises from the existence of more than one spanning
avalanche, and $\nu$ is the correlation length ($\xi$) exponent: $\xi
\sim |r|^{-\nu}$.
\begin{figure}
\centerline{
\psfig{figure=Figures/1_over_Nu_span_aval_paper_MF_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/Theta_paper_MF_THESIS.ps,width=3truein}
}
\caption[Mean field exponents $1/{\tilde \nu}$
and ${\tilde \theta}$ from the spanning avalanches]
{(a) and (b) {\bf $1/{\tilde \nu}$ and $\tilde \theta$ respectively,
extracted from the mean field spanning avalanches collapses},
as a function of the geometric average of $1/S_{mf}$ for three curves
collapsed together (see text).
The extrapolation (non-linear for $\tilde \theta$)
to $1/S_{mf} \rightarrow 0$ agrees with the calculated
values for the two exponents. \label{span_aval_mf}}
\end{figure}
As was mentioned earlier, in mean field there is no meaning to distance
or lattice, and thus there are no ``sides''. Purely for the purpose of
testing our extrapolation method for finite size scaling collapses in
the mean field simulation, we have defined a mean field ``spanning
avalanche'' to be one with more than $\sqrt {S_{mf}}$ spins flipping at
a field $H$, where $S_{mf}$ is the total number of spins in the system.
(Note that the mean field exponents are valid for dimensions $6$ and
above, but that in those dimensions distances do have a meaning.) Using
the above definition of a mean field spanning avalanche, it can be shown
(see appendix C) that the scaling form for their number is:
\begin{equation}
N_{mf}(S_{mf},R) \sim S_{mf}^{\tilde
\theta}\ {\cal N}_{\pm}^{mf} (S_{mf}^{1/{\tilde \nu}} |r|)
\label{span_aval_mf_eqn1}
\end{equation}
and that the values of the exponents ${\tilde \theta}$ and $1/{\tilde
\nu}$ are $3/8$ and $1/4$ respectively. $N_{mf}$ is the number of mean
field spanning avalanches, while ${\cal N}_{\pm}^{mf}$ is a universal
scaling function. The exponents ${\tilde \theta}$ and $1/{\tilde \nu}$
are defined by the arbitrary definition for a spanning avalanche.
Because of how they are defined, their values are different from the
mean field values of $1/\nu = 2$ and $\theta=1$, obtained from the
renormalization group\cite{Dahmen1,Dahmen2} and the exponent scaling
relation $1/\sigma=(d-\theta)\nu-\beta$\cite{Dahmen1,Dahmen3}.
Figure\ \ref{span_aval_mfa} shows the number of mean field spanning
avalanches as a function of disorder, for several sizes, as well as the
scaling collapse of the data. Note that the number of spanning
avalanches close to the critical disorder $R_c=\sqrt {2/\pi}$ increases
with the size $S_{mf}$ of the system, and that the peaks are getting
narrower. The scaling collapse in the inset, shows only the three
largest curves. For smaller sizes, the peaks do not collapse well with
the larger size systems presumably due to finite size effects. The
extrapolation plots for $\tilde \theta$ and $1/{\tilde \nu}$ are shown
in figures\ \ref{span_aval_mf}a and \ref{span_aval_mf}b. On the
horizontal axis of these two plots is the geometric mean of $1/S_{mf}$
for the three curves that are collapsed together, analogous to the
extrapolation method used for the integrated avalanche size
distribution. Note that the extrapolation to $1/S_{mf} \rightarrow 0$
for $\tilde \theta$ does not seem to be linear, and that the value of
$1/{\tilde \nu}$ from the linear extrapolation of the $r=(R_c-R)/R$ data
agrees better with the mean field value than the value obtained from the
linear extrapolation of the $r=(R_c-R)/R_c$ data.
Note that we measure the avalanche size distribution only for disorders
at which there are no ``mean field spanning avalanches'' (for a $10^6$
system, that is for $R \ge 0.912$), since that's what we do in
dimensions $2$ through $5$ (finite dimensions) to avoid large finite
size effects. For the second moments of the avalanche size distribution
measurements (see below), the spanning avalanches were removed (same as
in finite dimensions).
We have also measured the change in the magnetization $\Delta M$ due to
all the spanning avalanches, as a function of the disorder $R$ (figure\
\ref{deltaM_mofhfiga}a). This gives us an independent measurement of the
exponent $\beta$. In the thermodynamic limit above the critical
disorder, there are no spanning avalanches so the change in the
magnetization $\Delta M$ will be zero, while for small disorders the
change in the magnetization will converge to one. Close and below the
critical disorder $R_c$, at the critical field, the scaling form for the
change in the magnetization due to the spanning avalanches will be (from
equation\ (\ref{mf_mofh_equ1})):
\begin{equation}
\Delta M (H=H_c,R) \sim |r|^\beta.
\label{deltaM_mofheq1}
\end{equation}
For finite size systems, as shown in the figure, the change
in the magnetization is not zero above the critical disorder: the data
has to be analyzed using finite size scaling.
\begin{figure}
\centerline{
\psfig{figure=Figures/DeltaM_mf_paper_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/DeltaM_mf_collapse_paper_THESIS.ps,width=3truein}
}
\caption[Magnetization change due to spanning avalanches in mean field]
{(a) {\bf Change in the magnetization}
due to spanning avalanches as a
function of disorder $R$. The data is for several mean field
system sizes.
The critical disorder is
$R_c =0.79788456$. The statistical errors are not larger than
$0.005$ (in units of $\Delta M$).
(b) Mean field scaling collapse of the change in the magnetization
curves for sizes $S_{mf}=1000, 8000, 64000, 512000$. The
exponents are $1/{\tilde \nu} = 0.25$ and $\beta/{\tilde \nu}= 0.125$
and $r=(R_c-R)/R$. The part of the curve that is collapsed is for
$R>R_c$. \label{deltaM_mofhfiga}}
\end{figure}
{\noindent The dependence on the
system size $S_{mf}$ can be brought in through a scaling function (see
references\ \cite{Goldenfeld,Barber}) that we call $\Delta {\cal
M}_{\pm}$:}
\begin{equation}
\Delta M(S_{mf},R) \sim |r|^\beta\
\Delta {\cal M}_{\pm}(S_{mf}^{1/{\tilde \nu}} |r|)
\label{deltaM_mofheq2}
\end{equation}
where $\tilde \nu$ is defined above, and $\pm$ refers to the sign of
$r$. We are free to define the scaling function $\Delta {\cal M}_{\pm}$
as:
\begin{equation}
\Delta {\cal M}_{\pm}(S_{mf}^{1/{\tilde \nu}} |r|)
\equiv\ \Bigl(S_{mf}^{1/{\tilde \nu}} |r|\Bigr)^{-\beta}\
{\Delta \widetilde {\cal M}_{\pm}} (S_{mf}^{1/{\tilde \nu}} |r|),
\label{deltaM_mofheq3}
\end{equation}
where ${\Delta \widetilde {\cal M}_{\pm}}$ is now a different scaling
function. The scaling form for the change of the magnetization $\Delta
M$ then becomes:
\begin{equation}
\Delta M(S_{mf},R) \sim S_{mf}^{-\beta/{\tilde \nu}}\
{\Delta \widetilde {\cal M}_{\pm}} (S_{mf}^{1/{\tilde \nu}} |r|).
\label{deltaM_mofheq4}
\end{equation}
Figure\ \ref{deltaM_mofhfiga}b shows a collapse of the data using this
scaling form. The collapse is done for disorders close to and above the
critical disorder, that is, for $r<0$. The scaling function in figure\
\ref{deltaM_mofhfiga}b, in the range of the collapse, is therefore
${\Delta \widetilde {\cal M}_{-}}$.
\begin{figure}
\centerline{
\psfig{figure=Figures/1_over_Nu_deltaM_paper_MF_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/Beta_over_Nu_deltaM_paper_MF_THESIS.ps,width=3truein}
}
\caption[Mean field exponents $1/{\tilde \nu}$ and $\beta/{\tilde \nu}$
from the magnetization change due to spanning avalanches]
{(a) and (b) {\bf Mean field exponents $1/{\tilde \nu}$ and
$\beta/{\tilde \nu}$ respectively,} from
collapses of the magnetization change due to spanning avalanches
(see text). The extrapolation to
$(1/S_{mf})_{gm} = 0$ agrees with the calculated values.
\label{deltaM_mofhfigb}}
\end{figure}
Values for the exponents $1/{\tilde \nu}$ and $\beta/{\tilde \nu}$
extracted from such collapses at several {\it geometric average}
reciprocal sizes are shown in figures\ \ref{deltaM_mofhfigb}a and
\ref{deltaM_mofhfigb}b. (These plots are done the same way as for the
spanning avalanches exponents.) The linear extrapolation to $1/S_{mf} =
0$ is in very good agreement with the calculated values. Note that the
extrapolation for $1/\tilde \nu$ of the $r=(R_c-R)/R$ data gives again a
better agreement with the calculated value than the extrapolation using
the $r=(R_c-R)/R_c$ data. The exponent $\beta$ in $3$, $4$, and $5$
dimensions is calculated from $\beta/\nu$, which is extracted from the
above kind of collapse. The obtained value is used to check the collapse
of the $M(H)$ and $dM/dH$ data curves.
\begin{figure}
\centerline{
\psfig{figure=Figures/Non_span_s2_paper_and_collapse_MF_THESIS.ps,width=3truein}
}
\caption[Second moments of the avalanche size distribution in
mean field]
{{\bf Mean field second moments} of the avalanche size distribution
integrated over the field $H$, for several different sizes.
More than $20$ points are used for each curve; each point being an
average of a few to several hundred random field configurations.
The error bars for the $S_{mf}=1,000$ curve are too small to be
shown. Curves at $S_{mf}=125$ and $343$ are not shown.
The inset shows the collapse of these
four curves at
$\iota = -(\tau +\sigma\beta\delta - 3)/\sigma{\tilde \nu}
= 3/8$ and $1/{\tilde \nu} = 1/4$,
which are the mean field calculated values.
\label{s2_mf_figa}}
\end{figure}
Another quantity that is related to the avalanches is the moment of the
size distribution. We have measured the second, third, and fourth moment,
and we will show how the second moment scales and collapses in mean
field. The second moment is defined as:
\begin{equation}
\langle S^2 \rangle = \int S^2\ D(S,R,H,S_{mf})\ dS
\label{s2_mf1}
\end{equation}
where $D(S,R,H,S_{mf})$ is the avalanche size distribution mentioned
above, but with the system size $S_{mf}$ included as a variable since we
are looking for the finite size scaling form, as is clear from the data
in figure\ \ref{s2_mf_figa}. Recall that only non-spanning avalanches
are included in the distribution function $D(S,R,H,S_{mf})$. Equation\
\ref{s2_mf1} can be written in terms of the scaling form for large sizes
$S$ of the avalanche size distribution $D$:
\begin{equation}
\langle S^2 \rangle \sim \int S^2\ S^{-\tau}\ {\bar {\cal D}_\pm}
(S^\sigma |r|, h/|r|^{\beta\delta}, S_{mf}^{1/\tilde \nu} |r|)\ dS
\label{s2_mf2}
\end{equation}
As we have seen before, the dependence on the system size in the scaling
function ${\bar {\cal D}_{\pm}}$ is given by $S_{mf}^{1/\tilde \nu} |r|$
where $\tilde \nu$ is defined above through the definition of a mean
field spanning avalanche. If we define
\begin{eqnarray}
{\bar {\cal D}_\pm}
(S^\sigma |r|,h/|r|^{\beta\delta}, S_{mf}^{1/\tilde \nu} |r|)\ =\ \nonumber \\
(S^\sigma |r|)^{-(2-\tau) \over \sigma}\ {\widetilde {\cal D}}_\pm
(S^\sigma |r|, h/|r|^{\beta\delta}, S_{mf}^{1/\tilde \nu} |r|)
\label{s2_mf2a}
\end{eqnarray}
where ${\widetilde {\cal D}}_\pm$ is a different scaling function, and
let $u = S|r|^{1/\sigma}$, we obtain:
\begin{equation}
\langle S^2 \rangle\ \sim\ |r|^{(\tau-3)/\sigma} \int {\widetilde {\cal D}}_\pm
(u^{\sigma}, h/|r|^{\beta\delta}, S_{mf}^{1/\tilde \nu} |r|)\ du\
\label{s2_mf3}
\end{equation}
The integral in equation (\ref{s2_mf3}) is a function of
$h/|r|^{\beta\delta}$ and $S_{mf}^{1/\tilde \nu} |r|$ only, so we can
write:
\begin{equation}
\langle S^2 \rangle\ \sim
|r|^{(\tau-3)/\sigma}\ {\cal S}_{\pm}^{(2)}( h/|r|^{\beta\delta},
S_{mf}^{1/\tilde \nu} |r|)
\label{s2_mf3a}
\end{equation}
which is the second moment scaling form, and ${\cal S}_{\pm}^{(2)}$ is a
universal scaling function.
\begin{figure}
\centerline{
\psfig{figure=Figures/1_over_nu_paper_MF_non_span_s2_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/Tau_sigma_beta_delta_nu_paper_MF_THESIS.ps,width=3truein}
}
\caption[Mean field exponents $1/{\tilde \nu}$ and
$(\tau +\sigma\beta\delta - 3)/\sigma{\tilde \nu}$ from the
second moments of the avalanche size distribution]
{(a) and (b) {\bf Values for $1/{\tilde \nu}$ and
$(\tau +\sigma\beta\delta - 3)/\sigma{\tilde \nu}$
respectively,} extracted from the collapses of the
second moments of the avalanche
size distribution. The exponents are plotted as a function
of the geometric average of $1/S_{mf}$ for
three curves collapsed at a time (see text).
The extrapolation to large sizes
agrees with the calculated values for these exponents.
\label{s2_mf_fig}}
\end{figure}
In the simulation, we have measured the second moment of the
distribution integrated over the field $H$, whose scaling form can be
obtained by integrating the result of equation\ (\ref{s2_mf3a}):
\begin{equation}
\langle S^2 {\rangle}_{\it int} \sim |r|^{(\tau-3)/\sigma}
\int {\cal S}_{\pm}^{(2)}(h/|r|^{\beta\delta},
S_{mf}^{1/\tilde \nu} |r|)\ dh
\label{s2_mf4}
\end{equation}
As was done previously, we define $u=h/|r|^{\beta\delta}$, and call the
remaining integral:
\begin{eqnarray}
\int {\cal S}_{\pm}^{(2)}(h/|r|^{\beta\delta}, S_{mf}^{1/\tilde \nu} |r|)\ dh\
=\ \nonumber \\
(S_{mf}^{1/\tilde \nu} |r|)^{-(\tau + \sigma\beta\delta-3)/\sigma}\
{\widetilde {\cal S}}_{\pm}^{(2)}(S_{mf}^{1/\tilde \nu} |r|)
\label{s2_mf4a}
\end{eqnarray}
to obtain the second moment of the avalanche size distribution
integrated over the magnetic field $H$:
\begin{equation}
\langle S^2 {\rangle}_{\it int} \sim S_{mf}^{-(\tau +
\sigma\beta\delta-3)/{\sigma{\tilde \nu}}}\
{\widetilde {\cal S}}_{\pm}^{(2)}(S_{mf}^{1/\tilde \nu} |r|)
\label{s2_mf5}
\end{equation}
where ${\widetilde {\cal S}}_{\pm}^{(2)}$ is a universal scaling
function ($\pm$ indicates the sign of $r$). The mean field value for
$-(\tau + \sigma\beta\delta-3)/{\sigma{\tilde \nu}}$ is $3/8$.
Figure\ \ref{s2_mf_figa} shows the integrated second moments of
non-spanning mean field avalanches for several system sizes, and a
collapse using the scaling form in equation (\ref{s2_mf5}). Figures
\ref{s2_mf_fig}a and \ref{s2_mf_fig}b show the values for $1/{\tilde
\nu}$ and $-(\tau + \sigma\beta\delta-3)/{\sigma{\tilde \nu}}$
respectively, for several {\it geometric average} reciprocal sizes, and
show how well they linearly extrapolate to $1/S_{mf} \rightarrow 0$.
These plots are done the same way as for the mean field spanning
avalanches. Notice that for $1/\tilde \nu$, the linear extrapolation of
the data using $r=(R_c-R)/R$ gives a much better agreement with the
calculated value than the linear extrapolation of the data obtained
using $r=(R_c-R)/R_c$.
To summarize this section, we have shown that the values of the critical
exponents extracted from our mean field simulation by scaling collapses,
extrapolate to the expected (calculated) values for $R \rightarrow R_c$
and $1/S_{mf} \rightarrow 0$. Thus corrections to scaling due to finite
sizes as well as finite size effects near the critical point seem to be
avoided by extrapolation. The same extrapolation method is therefore
used for extracting exponents in $3$, $4$, and $5$ dimensions, which we
will see next. The results in $2$ dimensions will be shown last.
\section{The Model}
To model the long-range, far from equilibrium, collective behavior
mentioned in the previous section, we define\cite{Sethna} spins $s_i$ on
a hypercubic lattice, which can take two values: $s_i = \pm 1$. The
spins interact ferromagnetically with their nearest neighbors with a
strength $J_{ij}$, and are sitting in a uniform magnetic field $H$
(which is directed along the spins). Dirt is simulated by a random field
$h_i$, associated with each site of the lattice, which is given by a
gaussian distribution function $\rho (h_i)$:
\begin{equation}
\rho (h_i) = {1 \over {\sqrt {2\pi}} R}\ e^{-{h_i}^2 \over 2R^2}
\label{model_equ1}
\end{equation}
of width proportional to $R$ which we call the disorder parameter, or
just disorder. The hamiltonian is then
\begin{equation}
{\cal H} = - \sum_{<i,j>} J_{ij} s_i s_j - \sum_{i} (H + h_i) s_i
\label{model_equ2}
\end{equation}
For the analytic calculation, as well as the simulation, we have set the
interaction between the spins to be independent of the spins and equal
to one for nearest neighbors, $J_{ij}=J=1$, and zero otherwise.
The dynamics is deterministic, and is defined such that a spin $s_i$
will flip only when its local effective field $h^{ef\!f}_i$:
\begin{equation}
h^{ef\!f}_i = J \sum_{j} s_j + H + h_i
\label{model_equ3}
\end{equation}
changes sign. All the spins start pointing down ($s_i=-1$ for all $i$).
As the field is adiabatically increased, a spin will flip. Due to the
nearest neighbor interaction, a flipped spin will push a neighbor to
flip, which in turn might push another neighbor, and so on, thereby
generating an avalanche of spin flips. During each avalanche, the
external field is kept constant. For large disorders, the distribution
of random fields is wide, and spins will tend to flip independently of
each other. Only small avalanches will exist, and the magnetization
curve will be smooth. On the other hand, a small disorder implies a
narrow random field distribution which allows larger avalanches to
occur. As the disorder is lowered, at the disorder $R=R_c$ and field
$H=H_c$, an infinite avalanche in the thermodynamic system will occur
for the first time, and the magnetization curve will show a
discontinuity. Near $R_c$ and $H_c$, we find critical scaling behavior
and avalanches of all sizes. Therefore, the system has two tunable
parameters: the external field $H$ and the disorder $R$. We found from
the mean field calculation\ \cite{Dahmen1,Dahmen2} and the simulation
that a discontinuity in the magnetization exists for disorders $R \le
R_c$, at the field $H_c(R) \ge H_c(R_c)$, but that only at $(R_c, H_c)$,
do we have critical behavior. For finite size systems of length $L$, the
transition occurs at the disorder $R_c^{ef\!f}(L)$ near which avalanches
first begin to span the system in one of the {\it d} dimensions
(spanning avalanches). The effective critical disorder $R_c^{ef\!f}(L)$
is larger than $R_c$, and $R_c^{ef\!f}(L) \rightarrow R_c$ as $L
\rightarrow \infty$.
\subsection{Simulation Results in $3$, $4$, and $5$ Dimensions}
\subsubsection{Magnetization Curves}
The magnetization as a function of the external field $H$ is measured
for different values of the disorder $R$. Initially all the spins are
pointing down ($s_i = -1$ for all $i$). The field is then slowly raised
from a large negative value, until a spin flips. When the first spin has
flipped, the external field is held constant while all the spins in the
avalanche are flipping. The change in the magnetization due to this
avalanche is just twice the size of the avalanche.
Figure \ref{3d_MofH_fig}a shows the magnetization curves obtained from
our simulation in $3$ dimensions for several values of the disorder $R$.
Similar plots can be obtained in $4$ and $5$ dimensions\cite{mosaic}. As
the disorder $R$ is decreased, a discontinuity (``jump'') in the
magnetization curve appears. The critical disorder $R_c$ is the value of
the disorder at which this discontinuity appears for the first time as
the amount of disorder is decreased, for a system in the thermodynamic
limit. For finite size systems, like the ones we use in our simulation,
the ``jump'' will occur earlier. The effective critical disorder for a
system of size $L$ is larger than the critical disorder $R_c$ ($1/L
=0$). The critical disorder $R_c$ is found from finite size scaling
collapses of the spanning avalanches and second moments of the avalanche
size distribution which will be covered later. The values are
listed in Table\ \ref{RH_table}.
We have seen in mean field that the magnetization curves near the
transition scale as
\begin{equation}
m(H,R) \sim |r|^\beta\ {\cal M}_{\pm}(h/|r|^{\beta\delta})
\label{mofh_3d_eq1}
\end{equation}
where $m=M(H,R)-M_c(H_c,R_c)$, $h=H-H_c$, and ${\cal M}_{\pm}$ is the
corresponding scaling function. The critical magnetization $M_c$ and
critical field $H_c$ are not universal quantities: in our mean field
simulation and the hard--spin mean field model for our system\
\cite{Dahmen1}, both are zero; however they are non--zero quantities in
a soft--spin model\ \cite{Dahmen1}.
In general, the scaling variables in\ (\ref{mofh_3d_eq1}) need not be
$r$ and $h$, but can instead be some ``rotated'' variables $r^\prime$
and $h^\prime$\ \cite{SouthAfrica} which to first approximation can be
written as:
\begin{equation}
r^\prime = r +a h
\label{mofh_3d_eq2}
\end{equation}
and:
\begin{equation}
h^\prime = h + br
\label{mofh_3d_eq3}
\end{equation}
(See appendix A for these and other corrections.) The constants $a$ and
$b$ are not universal and the critical exponents do not depend on them
(for the mean field data $a=b=0$).
In equation\ (\ref{mofh_3d_eq1}), the scaling variables $r$ and $h$
should be replaced by the ``rotated'' variables $r^\prime$ and
$h^\prime$, but since the measurements in our simulation are in terms of $r$
and $h$, we rewrite the scaling form in terms of those.
We find that in the leading order of scaling behavior, the
magnetization scales like:
\begin{equation}
M(H,R)-M_c\ \sim\ |r|^{\beta}\
{\widetilde {\cal M}}_{\pm}\Bigl((h+br)/|r|^{\beta\delta}\Bigr).
\label{mofh_3d_eq4}
\end{equation}
The correction $b\,r$ is dominant for $R \rightarrow R_c$, and can not
be ignored. The opposite is true for $a\,h$ (see appendix A).
From the previous equation, the parameters that need to be fitted are
$M_c$, $H_c$, $\beta$, $\beta\delta$, and the ``tilting'' constant $b$.
These should be found by collapsing the magnetization curves onto each
other. As in mean field, we find that collapses of magnetization curves
in $3$, $4$, and $5$ dimensions do not define well the value of the
critical magnetization $M_c$. Furthermore, we observe strong
correlations between the parameters, which lead to weak constraints on
their values.
\begin{figure}
\centerline{
\psfig{figure=Figures/MofH_3d_L320_paper_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/MofH_3d_L320_collapse_paper_THESIS.ps,width=3truein}
}
\caption[Magnetization curves in $3$ dimensions]
{(a) {\bf Magnetization curves in $3$ dimensions}
for size $L=320$, and three
values of disorder. The curves are averages
of up to $48$ different random field configurations. Note the
discontinuity in the magnetization for $R=2.20$. In finite size
systems, the discontinuity in the magnetization curve occurs even for
$R>R_c$ ($R_c=2.16$ in $3$ dimensions).
(b) ``Tilted'' scaling collapse (see text) of the magnetization curves
in $3$ dimensions for size $L=320$. The disorders range from $R=2.35$ to
$R=3.20$ ($R>R_c$).
The critical magnetization is chosen as $M_c=0.9$ from an analysis
of the magnetization curves and is kept fixed during the collapse. The
exponents are $\beta=0.036$, $\beta\delta=1.81$, and the critical
field and disorder are $1.435$ and $2.16$ respectively. The ``tilting"
parameter $b$ is $0.39$. \label{3d_MofH_fig}}
\end{figure}
To remove the dependence on the critical magnetization $M_c$, we can
look at the collapse of $dM/dH$ which scales like:
\begin{equation}
{dM \over dH}(H,R)\ \sim\ |r|^{\beta-\beta\delta}\ {\widetilde {\cal M}}_{\pm}
\Bigl((h+br)/|r|^{\beta\delta}\Bigr)
\label{mofh_3d_eq5}
\end{equation}
Although $M_c$ does not appear in the above form, the other parameters
are still not uniquely defined by the collapse. We find that we need to
extract $\beta$ from the magnetization discontinuity ($\Delta M$)
collapses, and $\beta\delta$ and $H_c$ from the binned avalanche size
distribution collapses rather than from the magnetization curves
themselves. Using the values obtained from these collapses, and the
value of $R_c$, the ``tilting'' constant $b$ is then found from
magnetization curve collapses (figure\ \ref{3d_MofH_fig}b).
\begin{figure}
\centerline{
\psfig{figure=Figures/dMdH_d3_L320_smooth10_less2_points_paper_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/dMdH_d3_L320_smooth10_less2_points_collapse_paper_THESIS.ps,width=3truein}
}
\caption[$dM/dH$ curves in $3$ dimensions]
{(a) {\bf Derivative with respect to the field $H$ of the magnetization}
$M$, for disorders $R=$ $2.35$, $2.4$, $2.45$, $2.5$, $2.6$, $2.7$,
$2.85$, $3.0$,
and $3.2$ (highest to lowest peak), {\bf in $3$ dimensions}.
The curves are smoothed by $10$ data points before they are
collapsed.
(b) Scaling collapse of the data in (a) with $\beta =0.036$,
$\beta\delta = 1.81$, $b=0.39$, $H_c= 1.435$, and $R_c=2.16$.
While the curves are not collapsing onto a single curve, neither
did they for the mean field theory curves
(figure\ \protect\ref{mf_dmdh_fig}a). This is because the curves
are still far from the critical disorder $R_c$. \label{dMdH_3d_fig}}
\end{figure}
Figure\ \ref{dMdH_3d_fig}a shows the curves for the derivative of the
magnetization with respect to the field $H$, and figure\
\ref{dMdH_3d_fig}b shows the scaling collapse using the same exponent
and parameter values as in figure\ \ref{3d_MofH_fig}b. The collapsed
curves have disorders larger than the critical disorder: below $R_c$,
the fluctuations are larger and the collapses are less reliable.
Since we found that $b \neq 0$ ($b=0.39$ in $3$d), the scaling variables
are indeed some $r^\prime$ and $h^\prime$, and not the variables we
measure: $r$ and $h$. Therefore, the scaling functions will in general
be functions of a different combination of scaling variables from the
ones we used in mean field, where the scaling variables are $r$ and $h$.
However, we find in appendix A that the measurements that are integrated
over the external field $H$ remove the ``tilt'' parameter $b$ (other
analytic corrections might still be important though). This is true for
the integrated avalanche size distribution, the avalanche correlation
(integrated over the field), the number of spanning avalanches, the
moments of the avalanche size distribution, and the time distribution of
avalanche sizes. In the sections that treat these measurements, we will
ignore the ``rotation'' of axis to simplify the presentation. Note that
the change in the magnetization $\Delta M$ due to the spanning
avalanches is integrated over only a small range of external fields
(wherever there are spanning avalanches). On the other hand, the
binned avalanche size distribution is not integrated over the field $H$,
and we therefore examine this measurement more carefully.
\subsubsection{Moments of the Avalanche Size Distribution}
The second moment of the avalanche size distribution was defined earlier
(see the mean field simulation section). We found that the scaling form
of the integrated over $H$ second moment is (equation\ \ref{s2_mf5}):
\begin{equation}
\langle S^2 {\rangle}_{\it int} \sim L^{-(\tau+\sigma\beta\delta
- 3)/\sigma\nu}\
{\widetilde {\cal S}}_{\pm}^{(2)}(L^{1/\nu}|r|)
\label{s2_5d}
\end{equation}
where $L$ is the linear size of the system, $r$ is the reduced disorder,
$\widetilde {\cal S}_{\pm}^{(2)}$ is the scaling function, and $\nu$ is
the correlation length exponent. The corrections are subdominant
(appendix A). We can similarly define the third and fourth moment, with
the exponent $-(\tau+\sigma\beta\delta - 3)/\sigma\nu$ replaced by
$-(\tau+\sigma\beta\delta-4)/\sigma\nu$ and $-(\tau+\sigma\beta\delta
-5)/\sigma\nu$ respectively. Figures\ \ref{s2_5d_fig}a and
\ref{s2_5d_fig}b show the second moments data in $5$ dimensions for
sizes $L=5, 10, 20,$ and $30$, and a collapse (again, results in $3$ and
$4$ dimensions are similar and we have chosen to show the curves in $5$
dimensions for variety). The curves are normalized by the average
avalanche size integrated over all fields $H$: $\int_{-\infty}^{+\infty}
\int_{1}^{\infty} S\ D(S,R,H,L)\ dS\ dH$. The spanning avalanches and
the infinite avalanche are not included in the calculation of the
moments. The collapse does not include the $L=5$ curve because, due to
finite size effects, this curve does not collapse well with the larger
size curves. Table\ \ref{s2_5d_exp_table} shows the values of the
exponents and $R_c$ from the collapses. The exponents for the third and
fourth moment can be calculated from this table, and we find that they
agree with the values obtained from their respective collapses.
\section{Algorithm}
There are several methods that can be used to simulate the above model.
The simplest but most time and space (memory) consuming method starts by
assigning a random field to each spin on the hypercubic lattice. At the
beginning of the simulation, all the spins are pointing down. The
external field $H$ is then increased by small increments, starting from
a large negative value. After each increase of the field, all the spins
are checked to find if one of them should flip (a spin flips when its
effective field changes sign). If a spin flips, its neighbors are
checked, and so on until no spins are left that can flip. Then, the
external field is further increased, and the process repeated. Since the
external magnetic field is increased in equal increments, a large amount
of time is spent searching the lattice for spins that can flip. The
increments have to be big enough to avoid searching the lattice when
there are no spins that can flip, but small enough so that two or more
spins far apart don't flip at the same field. This is the method used
experimentally, but it is suited only for ``that kind of'' massively
parallel computing.
A variation on the above method, removes the searching through the
lattice that is done even if there are no spins that can flip. It
involves looking at all the spins, finding the next one that will flip
and {\it then} increasing the external field so that it does. The
average searching time for a flip is decreased, but is still very large.
Far from the critical point, where spins will tend to flip independently
of each other, the time for searching scales like $N^2$ where $N$ is the
number of spins in the system.
The search time can be further decreased if the random fields are
initially ordered in a list. The first spin that will flip is the one on
``top'' of the list. The external field is increased until the effective
field of the top spin become zero, and the spin flips. We then check its
nearest neighbors, and so on, while keeping the external field constant.
When no spins are left to flip, the external field needs to be increased
again. The change in the external field $\Delta H$, necessary to flip
the next spin, is found by looking for the spin whose random field $h_i$
satisfies:
\begin{equation}
h_i \ge - (H_{old}+\Delta H) - (2{n_{\uparrow}}-z)J
\label{num_equ1}
\end{equation}
where $H_{old}$ is the field at which the previous spins have flipped,
$z$ is the coordination number, and ${n_{\uparrow}}$ is the number of
nearest neighbors pointing up ($s_j = +1 $) for spin $s_i$. In general,
there will be a minimum of $z+1$ spins to check from the list, since
${n_{\uparrow}}$ can have the integer value between zero and $z$. The
spin for which equation (\ref{num_equ1}) is satisfied for the smallest
$\Delta H$, and for which the number of up neighbors is
${n_{\uparrow}}$, will flip. In general, more than $z+1$ spins will need
to be checked because a spin can satisfy equation (\ref{num_equ1}) for
some value of ${n_{\uparrow}}$ but might not have that number of up
neighbors, or the spin might have already flipped. This algorithm
decreases the searching time since not all the spins need to be checked
to find the next spin that will flip. Our early simulation work
\cite{Sethna,Dahmen2} used this method. In practice, about half of the
time was spent for the $N\ log_2 N$ initial sorting of the list of
random field numbers, where $N$ is the total number of spins in the
system. The big drawback of this method (as for the ones mentioned
above) is the huge amount of storage space needed to store the random
fields, the positions of each spin, and the values of the spins. This
becomes particularly important when larger size systems are simulated.
The results in this paper use a more sophisticated algorithm which
removes the need for a large storage space. It revolves around the idea
that the change $\Delta H$ in the external field, between two
avalanches, follows a probability distribution since the random fields
$h_i$ are given by a Gaussian distribution. The increments $\Delta H$ in
the external field should be chosen according to that distribution. The
probability distribution itself is not known explicitly, but its
integral from $0$ to some finite $\Delta H$ is. It is the probability,
$P_{all}^{none}(\Delta H)$, that {\it no} spin will flip in the whole
system during a field change less than $\Delta H$. It is given by:
\begin{equation}
P_{all}^{none}(\Delta H) = \Pi_{n_{\uparrow}}\ P_{n_{\uparrow}}^{none}(\Delta H)
\label{simul_equ1}
\end{equation}
where the product is over $n_{\uparrow}=0,1,...,z$, and
$P_{n_{\uparrow}}^{none}(\Delta H)$ is the probability for a down spin with
$n_{\uparrow}$ up nearest neighbors not to flip when the external field
changes by less than $\Delta H$:
\begin{eqnarray}
P_{n_{\uparrow}}^{none}(\Delta H)\ =\ \nonumber \\
\biggl(1 - {\int_{0}^{H_{local}(n_{\uparrow})}
\rho (f)\ df - \int_{0}^{H_{local}^{new}(n_{\uparrow})} \rho (f)\ df \over
P_{n_{\uparrow}}^{noflip} \Bigl(H_{local}(n_{\uparrow})\Bigr)}
\biggr)^{N_{n_{\uparrow}}}
\label{simul_equ2}
\end{eqnarray}
The function $\rho (f)$ is the random field distribution function, and
$H_{local}(n_{\uparrow})$ and $H_{local}^{new}(n_{\uparrow})$ are
defined respectively as:
\begin{equation}
H_{local}(n_{\uparrow}) = - H - (2n_{\uparrow} - z)J
\label{simul_equ4}
\end{equation}
and
\begin{equation}
H_{local}^{new} (n_{\uparrow}) = -(H+\Delta H) - (2n_{\uparrow} - z)J.
\label{simul_equ5}
\end{equation}
$P_{n_{\uparrow}}^{noflip}(H_{local})$ gives the probability that a spin
with $n_{\uparrow}$ up nearest neighbors has not flipped {\it before}
the field has reached the external magnetic field value $H$:
\begin{equation}
P_{n_{\uparrow}}^{noflip}\Bigl(H_{local}(n_{\uparrow})\Bigr) =
{1 \over 2} + \int_{0}^{H_{local}(n_{\uparrow})}
\rho (f)\ df,
\label{simul_equ6}
\end{equation}
and $N_{n_{\uparrow}}$ is the number of {\it down} spins that have
$n_{\uparrow}$ up neighbors.
A field increment $\Delta H$ that has the required probability
distribution is found by choosing a uniform random number between zero
and one and solving for $\Delta H$ from equation\ (\ref{simul_equ1}), by
setting the probability $P_{all}^{none}(\Delta H)$ equal to the value of
the random number. Once the increment $\Delta H$ is known, we can find
the next spin that will flip. We first calculate\ \cite{note1} the
probability $P^{flip}(n_{\uparrow})$ for a down spin with $n_{\uparrow}$
up neighbors to flip at the new field $H + \Delta H$:
\begin{equation}
P^{flip}(n_{\uparrow}) = {R_{n_{\uparrow}} \over R_{tot}}
\label{simul_equ61}
\end{equation}
where
\begin{equation}
R_{n_{\uparrow}} = {N_{n{\uparrow}}\
\rho \Bigl(H_{local}^{new}(n_{\uparrow})\Bigr) \over
P_{n_{\uparrow}}^{noflip} \Bigl(H_{local}^{new}(n_{\uparrow})\Bigr)}
\label{simul_equ62}
\end{equation}
is the rate at which down spins with $n_{\uparrow}$ up neighbors would
flip, and $R_{tot}$ is the sum of the rates $R_{n_{\uparrow}}$ for all
$n_{\uparrow}$. The spin that flips will have $k$ up neighbors, which is
found by satisfying the following inequality:
\begin{equation}
\Sigma_{n_{\uparrow}=0}^{k}\ P^{flip}(n_{\uparrow}) > C >
\Sigma_{n_{\uparrow}=0}^{k-1}\ P^{flip}(n_{\uparrow})
\label{simul_equ63}
\end{equation}
where the cutoff $C$ is a random number between $0$ and $1$. Once $k$ is
known, a spin is then randomly picked from the list of down spins with
$k$ up neighbors.
After the first spin has flipped, its neighbors are checked. The
probability for one of the neighbors, with ($n_{\uparrow} + 1$) up
nearest neighbors, to flip at $H + \Delta H$, given that it has not yet
flipped, is:
\begin{equation}
P_{next}(n_{\uparrow}, H+\Delta H) = 1 - {{{1 \over 2}\ +\
\int_{0}^{H_{local}^{new} (n_{\uparrow}+1)}
\rho (f)\ df} \over {{1 \over 2}\ +\
\int_{0}^{H_{local}^{new}(n_{\uparrow})} \rho (f)\ df}}
\label{simul_equ7}
\end{equation}
When all the neighbors have been checked, the size of the avalanche is
stored, as well as all the other measurements. The external magnetic
field $H$ is then incremented again by finding the next $\Delta H$,
starting back with equation\ (\ref{simul_equ1}).
The important characteristic of this method is that the random fields
are not assigned to the spins at the beginning of the simulation, which
for large system sizes decreases memory requirements tremendously
(asymptotically, we use one bit per spin). This method has allowed us to
simulate system sizes of up to $30000^2$, $1000^3$, $80^4$, and $50^5$
spins. The majority of the data analysis was performed on systems of
sizes $7000^2$, $320^3$, $80^4$, and $30^5$. The SP1 and SP2
supercomputers at the Cornell Theory Center, and IBM RS6000 model 560
and J30 workstations were used for the simulation. Using this new
algorithm, close to the critical disorder, one run (for a particular
random field configuration) for a $320^3$ system took more than $1$ CPU
hour on a SP1 node at the Cornell Theory Center, while it took close to
$37$ CPU hours for a $800^3$ system on an IBM RS6000 model 560
workstation. Far above the critical disorder $R_c$, the simulation time
increases substantially: $40\%$ above the critical disorder, for the
$320^3$ system, the simulation time was six times longer than for the
simulation at $10\%$ above $R_c$.
\section{The Simulation Results}
The following measurements were obtained from the simulation as a function
of disorder R: \par
$\bullet$ the magnetization $M(H,R)$ as a function of the \par external
field $H$. \par
$\bullet$ the avalanche size distribution integrated over the \par
field $H$: $D_{int}(S,R)$. \par
$\bullet$ the avalanche correlation function integrated over the \par
field $H$:
$G_{int}(x,R)$. \par
$\bullet$ the number of spanning avalanches $N(L,R)$ as a \par function of the
system length $L$, integrated over the \par field $H$. \par
$\bullet$ the discontinuity in the magnetization $\Delta M (L,R)$ as \par a
function of the system length $L$. \par
$\bullet$ the second $\langle S^2 \rangle_{int}(L,R)$, third
$\langle S^3 \rangle_{int}(L,R)$, and \par
fourth $\langle S^4 \rangle_{int}(L,R)$
moments of the avalanche size \par
distribution as a function of the system length $L$, \par
integrated over the field $H$. \par
{\noindent In addition, we have measured:} \par
$\bullet$ the avalanche size distribution $D(S,H,R)$ as a \par function of the
field $H$ and disorder $R$. \par
$\bullet$ the distribution of avalanche times $D_{t}^{(int)}(S,t)$
as a \par function of the avalanche size $S$, at
$R=R_c$, integrated \par over the field $H$. \par
The data obtained from the simulation was used to find and describe the
critical transition. It was analyzed using {\bf scaling collapses}. The
mean field calculation\cite{Sethna,Dahmen1} for our model shows
that near the critical point, the magnetization curve has the scaling
form
\begin{equation}
M(H,R) - M_c(H_c,R_c) \sim |r|^\beta\ {\cal M}_{\pm}(h/|r|^{\beta\delta})
\label{model_equ4}
\end{equation}
where $M_c$ is the critical magnetization (the magnetization at $H_c$,
for $R=R_c$), $r=(R_c-R)/R$ and $h=(H-H_c)$ are the reduced disorder and
reduced field respectively, and ${\cal M}_{\pm}$ is a universal scaling
function ($\pm$ refers to the sign of $r$). Both $r$ and $h$ are small.
The critical exponent $\beta$ gives the scaling for the magnetization at
the critical field $H_c$ ($h=0$). Its mean field value is $1/2$, and the
mean field value of $\beta\delta$ is $3/2$. (Appendix A gives a short
review on why scaling and scaling functions occur near a critical point,
and why they have the form they do).
The significance of scaling for experimental and numerical data is as
follows\cite{Goldenfeld}. If the magnetization data, for example, is
plotted against the field $H$, there would be one data curve for each
disorder $R$ (fig.\ \ref{mf_mofhfig}a). While if we plot $|r|^{-\beta}
M(H,R)$ against $h/|r|^{\beta\delta}$, all the curves close to $R_c$ and
$H_c$ will {\bf collapse} (fig. \ref{mf_mofhfig}b) onto either one of
two curves: one for $r<0$ (${\cal M}_{-}$), and one for $r>0$ (${\cal
M}_{+}$). The functions ${\cal M}_{\pm}$ depend only on the combination
$h/|r|^{\beta\delta}$ and not on the field $H$ and disorder $R$
separately, and are therefore {\it universal}. Usually, the exponents
are unknown and scaling or data collapses are used to obtain them: the
exponents are varied until all the curves lie on top of each other. This
method is useful for analyzing numerical as well as experimental data,
and is often preferred to ``data fitting'', as we will show.
Numerical simulations and experiments are done on finite size systems.
Often the properties of the system will depend on the linear size $L$.
Functions that depend on the system's length are analyzed using {\bf
finite size collapses}\cite{Goldenfeld,Barber}. An example is the number $N$ of
spanning avalanches: $N(L,R) \sim L^{\theta}\ {\cal N}(L^{1/\nu} |r|)$ (to
be explained later). If $N$ is plotted against $R$, there would be one
data curve for each length $L$. The exponents $\theta$ and $\nu$ are
obtained by plotting $L^{-\theta}N(L,R)$ against $L^{1/\nu} |r|$ onto one
curve (the collapse), and extracting the exponents.
The scaling forms we use for the collapses do not include corrections
that are present when the data is {\it not taken} in the limit $R
\rightarrow R_c$ and $L \rightarrow \infty$ (see appendix A for
corrections that exist in those limits). On the other hand, finite size
effects close to $R_c$ become important. It is thus necessary to
extrapolate to $R \rightarrow R_c$ and $L \rightarrow \infty$ to obtain
the correct exponents. We have done a mean field simulation to test our
extrapolation method. The mean field exponents can be calculated
analytically\ \cite{Sethna,Dahmen1}, but it is useful to check that the
numerical results from the mean field simulation, for disorders away
from $R_c$ and for finite sizes, extrapolate to the analytical values at
$R=R_c$ and $1/L=0$. We will see that this indeed occurs, and we will
use the same extrapolation method in $3$, $4$, and $5$ dimensions.
The mean field simulation was done with the same code, but with some
changes. In mean field, the interactions between spins are infinite in
range (each spin interacts when every spin in the system with the same
interaction). This means that distances and positions are not relevant,
and therefore we don't need to keep track of the spins and their
neighbors; we just need to know the total number of flipped spins, and
the value of the external field $H$. The following section will show the
results of the mean field simulation and explain the extrapolation
method. Then, we will turn to results in $3$, $4$, and $5$ dimensions.
And finally, we will cover the more subtle scaling behavior in two
dimensions.
\section{Acknowledgments}
We acknowledge the support of DOE Grant \#DE-FG02-88-ER45364 and NSF
Grant \#DMR-9419506. We would like to thank Sivan Kartha and Bruce W.
Roberts for their initial ideas on the "probabilities'' algorithm.
Furthermore, we would like to thank M. E. J. Newman, J. A. Krumhansl, J.
Souletie, and M. O. Robbins for helpful conversations. This work was
conducted on the SP1 and SP2 at the Cornell National Supercomputing
Facility (CNSF), funded in part by the National Science Foundation, by
New York State, and by IBM, and on IBM 560 workstations and the IBM J30
SMP system (both donated by IBM). We would like to thank CNSF and IBM
for their support. Further pedagogical information using Mosaic is
available at http://www.lassp.cornell.edu/sethna/hysteresis.
\input Appendices
\subsubsection{Spanning Avalanches}
The critical disorder $R_c$ was defined earlier as the disorder $R$ at
which an ${\it infinite}$ avalanche first appears in the system, in the
thermodynamic limit, as the disorder is lowered. At that point, the
magnetization curve will show a discontinuity at the magnetization
$M_c(R_c)$ and field $H_c(R_c)$. For each disorder $R$ below the
critical disorder, there is ${\it one}$ infinite avalanche that occurs
at a critical field $H_c(R)>H_c(R_c)$\ \cite{Dahmen1,Dahmen2}, while
above $R_c$ there are only finite avalanches. This is the behavior for
an infinite size system. In a finite size system far below and above
$R_c$ the above picture is still true, but close to the critical
disorder, as we approach the transition, the avalanches get larger and
larger, and we expect that one of them will be on the order of the
system size and span the system from one ``side'' to another in at least
one direction. This avalanche is not the infinite avalanche; it is only
the largest avalanche that occurs close to the critical point. If the
system was larger, this avalanche would be non--system spanning. Such an
avalanche (which spans the system) we call a spanning avalanche.
In our numerical simulation, we find that for finite sizes $L$, there
are not one but ${\it many}$ such avalanches in $4$ and $5$ dimensions
(and maybe $3$), and that their number increases as the system size
increases. Figures\ \ref{span_aval_345fig}(a-c) show the number of
spanning avalanches as a function of disorder $R$, for different sizes
and dimensions. In $4$ and $5$ dimensions, the spanning avalanche curves
become more narrow as the system size is increased. Also, the peaks
shift toward the critical value of the disorder ($4.1$ and $5.96$
respectively), and the number of spanning avalanches at $R_c$ increases.
This suggests that in $4$ and $5$ dimensions, for $L \rightarrow
\infty$, there will be one infinite avalanche below $R_c$, none above,
and an infinite number of spanning avalanches at the critical disorder
$R_c$. (These spanning avalanches are infinite avalanches for $L
\rightarrow \infty$.) In $3$ dimensions, the results are not conclusive,
which can be noticed from figure\ \ref{span_aval_345fig}a, but also from
the value of the spanning avalanche exponent $\theta = 0.15 \pm 0.15$
defined below (a value of $0$ implies only one infinite or spanning
avalanche at $R_c$ as $L \rightarrow \infty$).
\begin{figure}
\centerline{
\psfig{figure=Figures/Span_aval_paper_d3_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/Span_aval_paper_d4_legend_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/Span_aval_paper_d5_legend_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/Span_aval_d4_collapse_paper_THESIS.ps,width=3truein}
}
\caption[Spanning avalanches in $3$, $4$, and $5$ dimensions]
{(a) {\bf Number of spanning avalanches $N$ in $3$ dimensions,}
occurring in the system between
$H= -\infty$ to $H=\infty$, as a function of the disorder $R$,
for linear sizes $L$: $20$ (dot-dashed), $40$ (long
dashed), $80$ (dashed), $160$ (dotted), and $320$ (solid). The critical
disorder $R_c$ is at $2.16$. The error bars for each curve tend to be
smaller than the peak error bar for disorders above the peak and larger
for disorders below the peak. They are not given here for clarity.
Note that the number of avalanches
increases only slightly as the size is increased.
(b) {\bf Number of spanning avalanches in $4$
dimensions.} The critical disorder is $4.1$.
(c) {\bf Number of spanning avalanches in $5$ dimensions.}
The critical disorder is $5.96$. Both in $4$ and $5$ dimensions,
the peaks grow and shift towards $R_c$ as the size of the
system is increased.
(d) Collapse of the spanning avalanche curves in $4$ dimensions
for linear sizes $L=20,40$, and $80$.
The exponents are $\theta = 0.32$ and
$\nu = 0.89$, and the critical disorder is
$R_c = 4.10$. The collapse is done using $r = (R_c-R)/R$.
\label{span_aval_345fig}}
\end{figure}
In percolation, a similar multiplicity of infinite clusters\
\cite{Arcangelis,Stauffer} (as the system size is increased) is found
for dimensions above $6$ which is the upper critical dimension (UCD).
The UCD is the dimension at and above which the mean field exponents are
valid. Below $6$ dimensions, there is only one such infinite cluster.
The existence of a diverging number of infinite clusters in percolation
is associated with the breakdown of the hyperscaling relation above $6$
dimensions. Since a hyperscaling relation is a relation between critical
exponents that includes the dimension $d$ of the system, it is always
only satisfied up to and including the upper critical dimension. In our
system, the upper critical dimension is also $6$, but we find spanning
avalanches in dimensions even below that. In a comment by Maritan {\it
{et al.}}\cite{Maritan}, it was suggested that our system should satisfy
the hyperscaling relation: $d\nu-\beta = 1/\sigma$ which is also the one
found in percolation\ \cite{Stauffer}. But since our system has spanning
avalanches below the upper critical dimension, this hyperscaling
relation breaks down below $6$ dimensions. Due to the existence of many
spanning avalanches near $R_c$, the new ``violation of hyperscaling''
relation for dimensions $3$ and above becomes\ \cite{Dahmen1,Dahmen3}:
\begin{equation}
(d-\theta)\nu - \beta = 1/\sigma
\label{span_aval_eqn1}
\end{equation}
where $\theta$ is the ``breakdown of hyperscaling'' or spanning
avalanches exponent defined below. One can check that our exponents in
$3$, $4$, and $5$ dimensions and mean field satisfy this equation (see
Tables~\ref{measured_exp_table} and~ \ref{calculated_exp_table}).
For the simulation, we define a spanning avalanche to be an avalanche
that spans the system in one direction. We average over all the
directions to obtain better statistics. Depending on the size and
dimension of the system and the distance from the critical disorder, the
number of spanning avalanches for a particular value of disorder $R$ is
obtained by averaging over as few as $5$ to as many as $2000$ different
random field configurations. We define the exponent $\theta$ such that
the number $N$ of spanning avalanches, at the critical disorder $R_c$,
increases with the linear system size as: $N \sim L^{\theta}$ ($\theta >
0$). The finite size scaling form\cite{Goldenfeld,Barber} for the number of
spanning avalanches close to the critical disorder is:
\begin{equation}
N(L,R) \sim L^{\theta}\ {\cal N}_{\pm}(L^{1/\nu}|r|)
\label{span_aval_eqn2}
\end{equation}
where $\nu$ is the correlation length exponent and ${\cal N}_{\pm}$ is
the corresponding scaling function ($\pm$ indicates the sign of $r$).
The corrections to scaling are subdominant as explained in appendix A.
The collapse is shown in figure\ \ref{span_aval_345fig}d. The values for
$\theta$ and $\nu$ from collapses of curves of sizes $L=20, 30, 40,$ and
$80$ in $4$ dimensions, are shown in Table\ \ref{span_exp_4d_table}. (We
show the results and collapses in $4$ dimensions here since the
existence of spanning avalanches in $3$ dimensions is not conclusive.)
These values are used along with the results from other collapses to
obtain Table\ \ref{measured_exp_table}. In the analysis of the avalanche
size distribution, magnetization, and correlation functions for $R>R_c$,
how close we chose to come to the critical disorder $R_c$ was determined
by the spanning avalanches: we include no values $R$ below the first
value which exhibited a spanning avalanche.
\subsubsection{Magnetization Discontinuity}
We have mentioned earlier that in the thermodynamic limit, at and below
the critical disorder $R_c$, there is a critical field $H_c(R)>H_c(R_c)$
at which the infinite avalanche occurs. Close to the critical
transition, for $r$ small and $H=H_c(R)$, the change in the
magnetization due to the infinite avalanche scales as (equation
(\ref{mofh_3d_eq1})):
\begin{equation}
\Delta M(R) \sim\ r^{\beta}
\label{deltaM_eq1}
\end{equation}
where $r=(R_c-R)/R$, while above the transition, there is no infinite
avalanche.
\begin{figure}
\centerline{
\psfig{figure=Figures/DeltaM_d4_paper_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/DeltaM_d4_collapse_paper_THESIS.ps,width=3truein}
}
\caption[Magnetization change due to spanning avalanches, in $4$ dimensions]
{(a) {\bf Change in the} {\bf magnetization} {\bf due to the}
{\bf spanning} {\bf avalanches in} {\bf $4$ dimensions,}
for several linear sizes $L$,
as a function of the disorder $R$.
(b) Scaling collapse of the curves in (a) using $r=(R_c-R)/R$.
The exponents are
$1/\nu = 1.12$ and $\beta/\nu = 0.19$, and the critical disorder is
$R_c = 4.1$. \label{deltaM_4d_fig}}
\end{figure}
{\noindent In finite size systems, the transition is not as sharp: we
have spanning avalanches above the critical disorder. If we measure the
change in the magnetization due to all the spanning avalanches (the
infinite avalanche is included too), the scaling form for that quantity
is going to depend on the system size $L$ analogous to the scaling of
the number of spanning avalanches:}
\begin{equation}
\Delta M(L,R) \sim\ |r|^{\beta}\ \Delta {\cal M}_{\pm}(L^{1/\nu}|r|)
\label{deltaM_eq2}
\end{equation}
where $\Delta {\cal M}_{\pm}$ is a universal scaling function. (Since
$\Delta M(L,R)$ is measured at $h^{\prime}=0$, corrections to scaling
are subdominant; see also appendix A.) Defining a new universal scaling
function $\Delta \widetilde {\cal M}_{\pm}$:
\begin{equation}
\Delta {\cal M}_{\pm}(L^{1/\nu}|r|) \equiv\ (L^{1/\nu}|r|)^{-\beta}\
\Delta \widetilde {\cal M}_{\pm}(L^{1/\nu}|r|)
\label{deltaM_eq3}
\end{equation}
we obtain the scaling form:
\begin{equation}
\Delta M(L,R) \sim\ L^{-\beta/\nu}\ \Delta {\widetilde {\cal M}_{\pm}}
(L^{1/\nu}|r|)
\label{deltaM_eq4}
\end{equation}
Figures\ \ref{deltaM_4d_fig}a and \ref{deltaM_4d_fig}b show the change
in the magnetization due to the spanning avalanches in $4$ dimensions,
and a scaling collapse of that data (similar results exist in $3$ and
$5$ dimensions). Notice that as the system size increases, the curves
approach the $|r|^{\beta}$ behavior. The exponents $1/\nu$ and
$\beta/\nu$ are extracted from scaling collapses (figure\
\ref{deltaM_4d_fig}b) and are listed in table\ \ref{deltaM_4d_table}.
The value of $\beta$ is calculated from $\beta/\nu$ and the knowledge of
$\nu$, and is the value used for collapses of the magnetization curves
(see earlier).
\begin{figure}
\centerline{
\psfig{figure=Figures/non_span_d5_paper_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/non_span_d5_collapse_paper_THESIS.ps,width=3truein}
}
\caption[Second moments of the avalanche size distribution
in $5$ dimensions]
{(a) {\bf Second moments}
of the avalanche size distribution integrated over
the field $H$, {\bf in $5$ dimensions.}
Error bars are largest for smaller
disorders (shown on the curves).
The curves have between $24$ and $50$ points, and the value of the
second moment for each
disorder is averaged over $3$ to $100$ different random field
configurations.
(b) Scaling collapse of the $L=10, 20$, and $30$ curves from (a) using
$r=(R_c-R)/R$.
The exponents are $1/\nu = 1.47$ and $\rho = -(\tau+\sigma\beta\delta
- 3)/\sigma\nu = 2.95$,
and the critical disorder is $R_c = 5.96$. \label{s2_5d_fig}}
\end{figure}
\subsubsection{Avalanche Time Measurement}
The exponents we have measured so far are static scaling exponents: they
do not depend on the dynamics of the model. If we measure the time an
avalanche takes to occur, we are making a dynamical measurement. The
time measurement in the numerical simulation is done by increasing the
time ``meter'' by one for each shell of spins in the avalanche; it
corresponds to a synchronous dynamics, where, when all unstable spins
are flipped, time is incremented by one, and the new list of unstable
spins is generated. The scaling relation between the time $t$ it takes
an avalanche to occur and the size $S$ of that avalanche for small
disorder $r$ can be found by noting that the characteristic duration of
an avalanche is proportional to the correlation length $\xi$ to the
power $z$\ \cite{dynamicz,MaBinney}:
\begin{equation}
t \sim \xi^z
\label{time_equ1}
\end{equation}
The exponent $z$ is known as the dynamical critical exponent. Equation\
(\ref{time_equ1}) gives the scaling for the time it takes for a spin to
``feel'' the effect of another a distance $\xi$ away. Since the
correlation length $\xi$ scales like $r^{-\nu}$ close to the critical
disorder, and the characteristic size $S$ as $r^{-1/\sigma}$, the time
$t$ then scales with large sizes as:
\begin{equation}
t \sim S^{\sigma \nu z}
\label{time_equ2}
\end{equation}
\begin{figure}
\centerline{
\psfig{figure=Figures/Aval_Time_3d_L800_paper_THESIS.ps,width=3truein}
}
\nobreak
\centerline{
\psfig{figure=Figures/Aval_Time_3d_L800_paper_collapse_THESIS.ps,width=3truein}
}
\caption[Avalanche time distribution curves in $3$ dimensions]
{(a) {\bf Avalanche time distribution curves in $3$ dimensions,}
for avalanche size bins
from about $2000$ to $40000$ spins (from upper left to lower right
corner). The system size is $800^3$ at $R=2.26$. The curves
are from only one random field configuration.
(b) Scaling collapse of curves in (a). The values of the exponents
are $\sigma\nu z = 0.57$ and $(\tau+\sigma\beta\delta+
\sigma\nu z)/\sigma\nu z = 4.0$. \label{time_3d_fig}}
\end{figure}
In our simulation, we measure the distribution of times for each
avalanche size $S$. The distribution of times $D_t(S,R,H,t)$ for an
avalanche of size $S$ close to the critical field $H_c$ and critical
disorder $R_c$ is
\begin{equation}
D_t(S,R,H,t) \sim S^{-q}\ {\bar {\cal D}}_{\pm}^{(t)} (S^{\sigma}|r|,
h/|r|^{\beta\delta}, t/S^{\sigma\nu z})
\label{time_equ3}
\end{equation}
where $q=\tau +\sigma\nu z$, and is defined such that
\begin{eqnarray}
\int_{-\infty}^{+\infty} \!\! \int_{1}^{\infty} D_t(S,R,H,t)\ dH\ dt\ =\
\nonumber \\
S^{-(\tau + \sigma\beta\delta)}\ {\bar {\cal D}}_{\pm}^{(int)}(S^{\sigma}|r|)
\label{time_equ4}
\end{eqnarray}
where ${\bar {\cal D}}_{\pm}^{(int)}$ was defined in the integrated
avalanche size distribution section. The avalanche time distribution
integrated over the field $H$, at the critical disorder ($r=0$) is:
\begin{equation}
D_t^{(int)}(S,t)\ \sim\
t^{-{(\tau + \sigma\beta\delta + \sigma\nu z) /\sigma\nu z}}\
{\cal D}_t^{(int)}(t/S^{\sigma\nu z})
\label{time_equ5}
\end{equation}
which is obtained from equation (\ref{time_equ3}) in a derivation
analogous to the one for the integrated avalanche size distribution
scaling form.
Figures\ \ref{time_3d_fig}a and \ref{time_3d_fig}b show the avalanche
time distribution integrated over the field $H$ for different avalanche
sizes, and a collapse of these curves using the above scaling form, for
a $800^3$ system at $R=2.260$ (just above the range where spanning
avalanches occur). The data is saved in logarithmic size bins, each
about $1.2$ times larger than the previous one. The time is also
measured logarithmically (next bin is $1.1$ times larger than the
previous one). The extracted value for $z$ in $3$ dimensions is $1.68
\pm 0.07$. The results for other dimensions are listed in Table\
\ref{measured_exp_table}.
| proofpile-arXiv_065-414 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
Extensive attention has been lavished on the overscreened multichannel
Kondo model after the discover of its non-fermi liquid (NFL) behavior by
Nozi\'{e}res and Blandin (NB) \cite{blandin}.
NB also pointed out that lattice effects
in real metals will cause the anisotropy between the two flavor channels
and that in the low temperature limit,
the impurity is totally screened by the strong coupling channel
with the weak coupling channel unaffected. Using Numerical
Renormalization Group (NRG), Ref.\cite{cox2} confirmed NB's conjecture.
Using Conformal Field Theory (CFT), Ref.\cite{affleck}
found a relevant dimension 1/2 operator in the flavor sector near
the 2 channel Kondo (2CK) fixed point and
suggested the system flows to the Fermi-liquid (FL) fixed point
pointed out by NB. Using Yuval-Anderson's approach, Ref.\cite{gogolin}
found a solvable line and calculated the exact crossover free energy
function from the 2CK fixed point to the FL fixed point
along this solvable line.
It is known that in the large $ U $ limit, the ordinary one channel symmetric
Anderson impurity model(AIM) can be mapped to the one channel Kondo model. However, as shown
by Ref.\cite{com,coleman}, if the original $ O(4) $ symmetry of the AIM
is broken to $ O(3) \times O(1) $, in the strong coupling limit, the AIM is mapped to
the one channel compactified Kondo model (1CCK) where the impurity spin couples
to both the spin and the isospin(charge) currents of the one channel conduction electrons.
Recently, Andrei and Jerez \cite{andrei}, using Bethe Ansatz,
reinvestigated the 2CFAK and conjectured that the 2CFAK flow to some
new NFL fixed points. Coleman and Schofield \cite{coleman}, using strong
coupling method, reinvestigated the 1CCK and
claimed the system flows to another kind of non-Fermi liquid fixed point
which, similar to 1-dim Luttinger liquid,
has the same thermodynamics as fermi liquid but different excitation
spectrum. Moreover, they claimed that the 1CCK has exactly the same low energy excitations as
those of the 2CFAK, therefore concluded that their results also apply
to the 2CFAK.
So far, Bethe Ansatz can only calculate thermodynamic quantities
of multichannel Kondo models, the correlation functions are needed to resolve if the fixed
points are NFL or FL.
It is important to point out that the charge degrees of freedom of the original
model being removed, the 1CCK in Ref.\cite{com,coleman}
has completely different transport properties, correlation functions and excitation
spectrums than the original 2CFAK, although it do
share the same thermodynamic properties as the 2CFAK.
As emphasized by AL \cite{review}, although the boundary interactions
only happen in the spin sector;
the spin, flavor and charge degree of freedoms
are {\em not} totally decoupled, there is a constraint( or gluing condition)
to describe precisely how these degree of freedoms are combined at different
boundary fixed points, the finite size spectrum is determined by this
gluing condition. The boundary operator contents and the scaling
dimensions of all the boundary operators are also given by the
gluing condition. However, in order to find the gluing conditions at the
intermediate coupling fixed points, the fusion rules should be identified which are usually
difficult in Non-Abelian bosonization approach.
For 4 pieces of bulk fermions, the non-interacting theory possesses $ SO(8) $ symmetry,
Maldacena and Ludwig (MS) \cite{ludwig} showed that finding the gluing conditions at
the fixed points are exactly equivalent
to finding the boundary conditions of the fermions at the fixed points;
the CFT describing the fixed points are simply free chiral bosons with the boundary conditions.
In Ref.\cite{powerful}, the author developed a simple and powerful method to study
certain class of quantum impurity models. The method can quickly identify all the
possible boundary fixed points and their {\em maximum } symmetry, therefore avoid
the difficulty of finding the fusion rules, it can also
demonstrate the physical picture at the boundary explicitly. In this paper, we apply
the method to study the two models. All {\em the possible} fixed points and their symmetries
are identified; the finite size spectra, the electron conductivity and pairing
susceptibility are calculated. All the leading and subleading
irrelevant operators are identified, their corrections to the correlation functions
are evaluated. In section II, {\em Taking all the degrees of freedom into
account}, We show that the only NFL fixed point of the 2CFAK is
the NFL fixed point of the 2CK with the symmetry $ O(3) \times O(5) $.
Any flavor anisotropies between the two channels drive the system to the
fermi-liquid (FL) fixed point
with the symmetry $ O(4) \times O(4) $ where one of the two channels suffers the
phase shift $ \pi/2 $ and the other remains free.
The conventional wisdom about the 2CFAK is rigorously shown to be correct.
In section III, we repeat the same program to the 1CCK. We find that
the NFL fixed point of the 1CCK has the symmetry $ O(3) \times O(1) $ and has the same
thermodynamics as the NFL fixed point of the 2CK. The finite size spectrum
is listed and compared with that of the 2CK. However, {\em in contrast to} the 2CK,
its conductivity shows $ T^{2} $ bahaviour and
there is {\em no} pairing susceptibility enhancement.
Any anisotropies between the spin and isospin sectors drive the system to the
FL fixed point with the symmetry $ O(4) $ where the electrons suffer the
phase shift $ \pi/2 $. The finite size spectrum of this FL fixed point is also listed and
compared with that of the 2CFAK. In section IV, we conclude and propose some
open questions. Finally,
in the appendix, we study the stable FL fixed point of the 2CFAK using Non-Abelian bosonization
and compare with the Abelian bosonization calculations done in section II.
\section{The two channel flavor anisotropic Kondo model}
The Hamiltonian of the 2CFAK is:
\begin{eqnarray}
H &= & i v_{F} \int^{\infty}_{-\infty} dx
\psi^{\dagger}_{i \alpha }(x) \frac{d \psi_{i \alpha }(x)}{dx}
+ \sum_{a=x,y,z} \lambda^{a} ( J^{a}_{1}(0)+J^{a}_{2}(0) ) S^{a}
+ \sum_{a=x,y,z} \alpha^{a}
(J^{a}_{1}(0)-J^{a}_{2}(0)) S^{a} \nonumber \\
& + & h ( \int dx J^{z}_{s}(x) + S^{z} )
\label{kondob}
\end{eqnarray}
where $ J^{a}_{i}(x) =\frac{1}{2} \psi^{\dagger}_{i \alpha }(x)
\sigma^{a}_{\alpha \beta} \psi_{i \beta }(x) $
are the spin currents of the channel $ i=1,2 $ conduction electrons respectively.
$ \alpha^{a}
=0, \pm \lambda^{a} $ correspond to the 2CK and the one channel
Kondo model respectively.
If $ \lambda^{a} =\lambda, \alpha^{a} =\alpha \neq 0 $ , the above Hamiltonian
breaks $ SU(2)_{s} \times SU(2)_{f} \times U(1)_{c} $ symmetry of
the 2CK to $ SU(2)_{s} \times U(1)_{f} \times U(1)_{c} $ ( or equivalently
$ SU(2)_{s} \times U(1)_{c1} \times U(1)_{c2}$, because we have two
independent U(1) charge symmetries in the channel 1 and the channel 2 ).
In this section, for simplicity, we take
$ \lambda^{x}=\lambda^{y}=\lambda, \lambda^{z} \neq \lambda;
\alpha^{x}=\alpha^{y}=\alpha, \alpha^{z} \neq \alpha $.
The symmetry in the spin sector is reduced to $ U(1) \times Z_{2}
\sim O(2) $ \cite{semi}. In the following, we closely follow
the notations of Emery-Kivelson \cite{emery}. Abelian-bosonizing
the four bulk Dirac fermions separately:
\begin{equation}
\psi_{i \alpha }(x )= \frac{P_{i \alpha}}{\sqrt{ 2 \pi a }}
e ^{- i \Phi_{i \alpha}(x) }
\label{first}
\end{equation}
Where $ \Phi_{i \alpha} (x) $ are the real chiral bosons satisfying
the commutation relations
\begin{equation}
[ \Phi_{i \alpha} (x), \Phi_{j \beta} (y) ]
= \delta_{i j} \delta_{\alpha \beta} i \pi sgn( x-y )
\end{equation}
The cocyle factors have been chosen as: $ P_{1 \uparrow}= P_{1 \downarrow}
= e^{i \pi N_{1 \uparrow} }, P_{2 \uparrow}= P_{2 \downarrow}
= e^{i \pi ( N_{1 \uparrow} + N_{1 \downarrow} + N_{2 \uparrow} ) } $.
It is convenient to introduce the following charge, spin, flavor,
spin-flavor bosons:
\begin{eqnarray}
\Phi_{c} & = & \frac{1}{2} ( \Phi_{1 \uparrow }+ \Phi_{1 \downarrow }+
\Phi_{2 \uparrow }+ \Phi_{2 \downarrow } ) \nonumber \\
\Phi_{s} & = & \frac{1}{2} ( \Phi_{1 \uparrow }- \Phi_{1 \downarrow }+
\Phi_{2 \uparrow }- \Phi_{2 \downarrow } ) \nonumber \\
\Phi_{f} & = & \frac{1}{2} ( \Phi_{1 \uparrow }+ \Phi_{1 \downarrow }-
\Phi_{2 \uparrow }- \Phi_{2 \downarrow } ) \nonumber \\
\Phi_{sf}& = & \frac{1}{2} ( \Phi_{1 \uparrow }- \Phi_{1 \downarrow }-
\Phi_{2 \uparrow }+ \Phi_{2 \downarrow } )
\label{second}
\end{eqnarray}
The spin currents $ J^{a}(x) = J^{a}_{1}(x) + J^{a}_{2}(x) $
and $ \tilde{J}^{a}(x) = J^{a}_{1}(x) - J^{a}_{2}(x) $ can be expressed
in terms of the above chiral bosons
\begin{eqnarray}
J_{x}= \frac{1}{\pi a} \cos \Phi_{s} \cos \Phi_{sf},~
J_{y}= \frac{1}{\pi a} \sin \Phi_{s} \cos \Phi_{sf},~
J_{z}= -\frac{1}{ 2 \pi} \frac{ \partial \Phi_{s}}{\partial x} \nonumber \\
\tilde{J}_{x}=- \frac{1}{\pi a} \sin \Phi_{s} \sin \Phi_{sf},~
\tilde{J}_{y}= \frac{1}{\pi a} \cos \Phi_{s} \sin \Phi_{sf},~
\tilde{J}_{z}= -\frac{1}{ 2 \pi} \frac{ \partial \Phi_{sf}}{\partial x}
\label{current}
\end{eqnarray}
After making the canonical transformation $ U= \exp [ i S^{z} \Phi_{s}(0)] $
and the following refermionization
\begin{eqnarray}
S^{x} &= & \frac{ \widehat{a}}{\sqrt{2}} e^{i \pi N_{sf}},~~~
S^{y}= \frac{ \widehat{b}}{\sqrt{2}} e^{i \pi N_{sf}},~~~
S^{z}= -i \widehat{a} \widehat{b} \nonumber \\
\psi_{sf} & = & \frac{1}{\sqrt{2}}( a_{sf} - i b_{sf} ) =
\frac{1}{\sqrt{ 2 \pi a}} e^{i \pi N_{sf}} e^{-i \Phi_{sf} } \nonumber \\
\psi_{s,i} & = & \frac{1}{\sqrt{2}}( a_{s,i} - i b_{s,i} )=
\frac{1}{\sqrt{ 2 \pi a}} e^{i \pi( d^{\dagger}d + N_{sf})} e^{-i \Phi_{s} }
\label{refer}
\end{eqnarray}
The transformed Hamiltonian $ H^{\prime}= U H U^{-1} =
H_{sf} + H_{s} + \delta H $ can be written in terms of
the Majorana fermions \cite{atten}:
\begin{eqnarray}
H_{sf} &= & \frac{ i v_{F} }{2} \int dx (a_{sf}(x) \frac{ \partial a_{sf}(x)}
{\partial x} + b_{sf}(x) \frac{ \partial b_{sf}(x)} {\partial x} )
-i \frac{ \lambda }{\sqrt{ 2 \pi a}} \widehat{a} b_{sf}(0)
+i \frac{ \alpha }{\sqrt{ 2 \pi a}} \widehat{b} a_{sf}(0)
\nonumber \\
H_{s}& = & \frac{ i v_{F} }{2} \int dx (a_{s}(x) \frac{ \partial a_{s}(x)}
{\partial x} + b_{s}(x) \frac{ \partial b_{s}(x)} {\partial x} )
-i h \int dx a_{s}(x) b_{s}(x) \nonumber \\
\delta H &= & -\lambda_{z}^{\prime} \widehat{a} \widehat{ b} a_{s}(0)
b_{s}(0)
-\alpha_{z} \widehat{a} \widehat{b} a_{sf}(0) b_{sf}(0)
\label{anderson}
\end{eqnarray}
where $ \lambda_{z}^{\prime} = \lambda^{z} - 2 \pi v_{F} $.
It is instructive to compare the above equation with Eq.3 in
Ref.\cite{sf}. They looks very similar:
{\em half} of the impurity spin coupled to half of the
spin-flavor electrons, {\em another half} of the impurity spin coupled
to {\em another} half of the spin-flavor electrons.
However {\em the two canonical transformations employed in the two models
are different}.
This fact make the boundary conditions of this model rather different
from that of the two channel spin-flavor Kondo model (2CSFK) discussed in Ref.\cite{sf}.
The above Hamiltonian was first derived by Ref.\cite{gogolin}
using Anderson-Yuval's approach. They found the solvable line $
\lambda^{z} =2 \pi v_{F}, \alpha^{z}=0 $ and calculated the exact crossover
function of free energy
along this solvable line. Using EK's method, We rederived
this Hamiltonian \cite{trivial}. The huge advantage of EK's method over Anderson-Yuval's
approach is that the {\em boundary conditions} at different boundary
fixed points can be identified \cite{powerful}.
By using the Operator Product Expansion (OPE) of the various
operators in Eq.\ref{anderson} \cite{cardy},
we get the RG flow equations near the weak coupling fixed
point $\lambda_{z}=2 \pi v_{F}, \lambda=\alpha=\alpha_{z}=0 $
\begin{eqnarray}
\frac{ d \lambda}{d l} & = &\frac{1}{2} \lambda+ \alpha \alpha_{z}
\nonumber \\
\frac{ d \alpha}{d l} & = &\frac{1}{2} \alpha -
\lambda \alpha_{z} \nonumber \\
\frac{ d \alpha_{z}}{d l} & = & -\lambda \alpha
\label{danger}
\end{eqnarray}
The fact that we find {\em two} relevant operators in the above equations
may indicate there are {\em two} intermediate coupling fixed points.
However, in the following, the two intermediate coupling fixed points are
shown to be the same.
The {\em original} impurity spin in $ H $ are related to those in $ H^{\prime} $ by
\begin{eqnarray}
S^{H}_{x} &= & U S_{x} U^{-1} = S_{x} \cos \Phi_{s}(0) - S_{y} \sin \Phi_{s}(0) \nonumber \\
S^{H}_{y} &= & U S_{y} U^{-1} = S_{x} \sin \Phi_{s}(0) + S_{y} \cos \Phi_{s}(0) \nonumber \\
S^{H}_{z} &= & U S_{z} U^{-1} = S_{z}
\label{change}
\end{eqnarray}
Using the refermionization Eq.\ref{refer}, the {\em original} impurity spin in $ H $ can
be written in terms of fermions
\begin{eqnarray}
S^{H}_{x} &= & i( \widehat{b} a_{s,i}+\widehat{a} b_{s,i} ) \nonumber \\
S^{H}_{y} &= & i( \widehat{b} b_{s,i}-\widehat{b} a_{s,i} ) \nonumber \\
S^{H}_{z} &= & -i \widehat{a} \widehat{b}
\label{imp}
\end{eqnarray}
At $ \lambda^{\prime}_{z}=0 $, the spin boson $ \Phi_{s} $ completely decouples
from the impurity in $ H^{\prime} $, therefore $ \chi_{imp} =0 $.
Because the canonical transformation $ U $ is a boundary
condition changing operator \cite{boundary,powerful}, at $ \lambda^{\prime}_{z} =0 $,
this leads to
\begin{equation}
a^{s}_{L}(0)=-a^{s}_{R}(0), ~~ b^{s}_{L}(0)=-b^{s}_{R}(0)
\label{bound1}
\end{equation}
Following Ref.\cite{powerful}, in order to identify the fixed points along the solvable line
$ \lambda^{\prime}_{z}=0, \alpha_{z}=0 $ (we also set $ h=0 $),
we write $ H_{sf} $ in the action form
\begin{eqnarray}
S &= & S_{0} + \frac{\gamma_{1}}{2} \int d \tau \widehat{a}(\tau)
\frac{\partial \widehat{a}(\tau)}{\partial \tau}
+ \frac{\gamma_{2}}{2} \int d \tau \widehat{b}(\tau)
\frac{\partial \widehat{b}(\tau)}{\partial \tau}
\nonumber \\
& - & i \frac{ \lambda }{\sqrt{ 2 \pi a}}
\int d \tau \widehat{a}(\tau) b_{sf}(0, \tau)
+i \frac{ \alpha }{\sqrt{ 2 \pi a}}
\int d \tau \widehat{b}(\tau) a_{sf}(0,\tau)
\label{action}
\end{eqnarray}
When performing the RG analysis of the action $ S $, we keep \cite{above}
1: $ \gamma_{2}=1, \lambda $ fixed,
2: $ \gamma_{1}=1, \alpha $ fixed,
3: $ \lambda, \alpha $ fixed;
three fixed points of Eq.\ref{anderson} can be identified
\subsection{ Fixed point 1}
This fixed point is located at $ \gamma_{1}=0, \gamma_{2}=1 $
where $ \widehat{b} $ decouples, but $ \widehat{a} $ loses its
kinetic energy and becomes a Grassmann Lagrangian multiplier.
Integrating $\widehat{a} $ out leads to
the following boundary conditions \cite{trick}:
\begin{equation}
b^{sf}_{L}(0)=-b^{sf}_{R}(0)
\label{bound2}
\end{equation}
Eqs.\ref{bound1},\ref{bound2} can be expressed in terms of bosons:
\begin{equation}
\Phi_{s,L}(0)=\Phi_{s,R}(0)+\pi, ~~~ \Phi_{sf,L}(0)=-\Phi_{sf,R}(0)+\pi
\end{equation}
This is just the non-fermi liquid fixed point of the 2CK.
The three Majorana fermions in the spin sector being twisted, this fixed point
possesses the symmetry $ O(3) \times O(5) $. The finite size spectrum
of this fixed point was listed in Ref.\cite{powerful}.
The local correlation functions at the 2CK fixed point are \cite{powerful}:
\begin{equation}
\langle \widehat{a}( \tau ) \widehat{a}(0) \rangle =\frac{1}{\tau},~~~~
\langle b_{sf}( \tau ) b_{sf}(0) \rangle =\frac{\gamma^{2}_{1}}{\tau^{3}}
\label{dimension}
\end{equation}
From the above equation, we can read the scaling dimensions of the various
fields $ [\widehat{b}]=0, [\widehat{a}]=[a_{s}]=[b_{s}]=[a_{sf}]=1/2, [b_{sf}]=3/2 $.
As shown in Ref.\cite{powerful}, at the fixed point, the impurity degree of freedoms
completely disappear: $\widehat{b} $ decouples and $ \widehat{a} $ turns into the
{\em non-interacting } scaling field at the fixed point \cite{care}
\begin{equation}
\widehat{a} \sim b_{sf}(0,\tau)
\end{equation}
Using Eq.\ref{imp}, the impurity spin turns into
\begin{eqnarray}
S^{H}_{x}(\tau) &= & i( \widehat{b} a_{s,i}(0,\tau)+b_{sf}(0,\tau) b_{s,i}(0,\tau) ) \nonumber \\
S^{H}_{y}(\tau) &= & i( \widehat{b} b_{s,i}(0,\tau)-b_{sf}(0,\tau) a_{s,i}(0,\tau) ) \nonumber \\
S^{H}_{z}(\tau) &= & i \widehat{b} b_{sf}(0,\tau)
\end{eqnarray}
Using the relation
\begin{equation}
\psi^{H}_{s}(x)= U \psi_{s}(x) U^{-1}=i sgnx \psi_{s,i}(x)
\label{reverse}
\end{equation}
We get \cite{cut}
\begin{eqnarray}
S_{x}(\tau) &= & i( -\widehat{b} b_{s}(0,\tau)+b_{sf}(0,\tau) a_{s}(0,\tau) ) \nonumber \\
S_{y}(\tau) &= & i( \widehat{b} a_{s}(0,\tau)+b_{sf}(0,\tau) b_{s}(0,\tau) ) \nonumber \\
S_{z}(\tau) &= & i ( \widehat{b} b_{sf}(0,\tau) + a_{s}(0,\tau) b_{s}(0,\tau) )
\label{add}
\end{eqnarray}
The impurity spin-spin correlation function $ \langle S^{a}(\tau) S^{a}(0) \rangle
=\frac{1}{\tau} $.
The above equations \cite{add} are consistent with the CFT identifications \cite{line}
\begin{equation}
\vec{S} \sim \vec{\phi} + \vec{ J} + \cdots
\end{equation}
The 2CK fixed point is unstable, because there is a dimension 1/2 relevant
operator $ \widehat{b} a_{sf} $, the OPE of $ a_{sf} $ with itself will
generate the dimension 2 energy momentum tensor of this Majorana fermion
$ T(\tau)= \frac{1}{2} a_{sf}(0,\tau) \frac{ \partial a_{sf}(0,\tau)}{\partial \tau} $,
The OPE of the energy momentum tensor with the primary field $ a_{sf} $ is
\begin{equation}
T(\tau_{1}) a_{sf}(\tau_{2})= \frac{ \frac{1}{2} a_{sf}(\tau_{2})}{ (\tau_{1}-\tau_{2})^{2}}
+ \frac{ L_{-1} a_{sf}(\tau_{2})}{\tau_{1}-\tau_{2}} + L_{-2} a_{sf}(\tau_{2}) + \cdots
\end{equation}
First order descendant field of this primary field $ L_{-1} a_{sf}(0,\tau)=
\frac{ \partial a_{sf}(0,\tau)}{\partial \tau} $ with dimension 3/2 is generated.
$ \lambda^{\prime}_{z} $ term in $ \delta H $ has scaling
dimension 3/2, it will generate a dimension 2 operator
$ a_{s}(0,\tau) \frac{ \partial a_{s}(0,\tau)}{\partial \tau} +
b_{s}(0,\tau) \frac{ \partial b_{s}(0,\tau)}{\partial \tau} $. $\gamma_{2} $ term
has dimension 2 also.
From Eq.\ref{dimension}, we can see $ \alpha_{z} $ term has scaling dimension 5/2,
it can be written as
\begin{equation}
:\widehat{a}(\tau) \frac{\partial \widehat{a}(\tau)}{\partial \tau}: a_{sf}(0,\tau)
= :b_{sf}(0,\tau) \frac{\partial b_{sf}(0,\tau)}{\partial \tau}: a_{sf}(0,\tau)
\end{equation}
The bosonized form of this operator is
\begin{equation}
:( \cos 2\Phi_{sf}(0,\tau)-\frac{1}{2} (\partial \Phi_{sf}(0,\tau))^{2}): \sin \Phi_{sf}(0,\tau)
\end{equation}
Using CFT, Ref.\cite{affleck} predicted a dimension 1/2 relevant operator
$ \phi^{3}_{f} $ in the flavor sector. Ref.\cite{line} classified all the
first order descendants of the primary operator in the spin sector.
In the flavor sector, the same classification apply,
$ \vec{J}_{-1} \cdot \vec{\phi}_{f} $ is Charge-Time Reversal (CT) odd,
therefore is not allowed,
but $ L_{-1} \phi^{3}_{f} $ is CT even. The CFT analysis is completely
consistent with the above EK's solution.
In order to make this fixed point stable, we have to tune $ \alpha
=\alpha_{z}=0 $, namely the channel anisotropy is strictly prohibited.
If $\alpha=0 $, but $ \alpha_{z} \neq 0 $, because
$ \alpha_{z} $ is highly irrelevant, it {\em seems} the 2CK
fixed point is stable. However, this is not true. From the RG flow Eq.\ref{danger}, it
is easy to see that even initialy $ \alpha=0 $, it will be generated,
$ \alpha_{z} $ is 'dangerously' irrelevant.
\subsection{ Fixed point 2}
This fixed point is located at $ \gamma_{1}=1, \gamma_{2}=0 $
where $ \widehat{a} $ decouples, but $ \widehat{b} $ loses its
kinetic energy and becomes a Grassmann Lagrangian multiplier.
Integrating $\widehat{b} $ out leads to
the following boundary conditions:
\begin{equation}
a^{sf}_{L}(0)=-a^{sf}_{R}(0)
\label{dual}
\end{equation}
Eqs.\ref{bound1},\ref{dual} can be expressed in terms of bosons:
\begin{equation}
\Phi_{s,L}(0)=\Phi_{s,R}(0)+\pi, ~~~ \Phi_{sf,L}(0)=-\Phi_{sf,R}(0)
\end{equation}
This fixed point also possesses the symmetry $ O(3) \times O(5) $.
In fixed points 1 and 2, $\widehat{a} $ and
$\widehat{b} $, $ b_{sf} $ and $ a_{sf} $ exchange roles.
As discussed in the fixed point 1, $ \alpha_{z} $ is 'dangerously' irrelevant.
In order to make this fixed point stable, we have to tune $ \lambda
=\alpha_{z}=0 $. This fixed point
is actually {\em the same} with the 2CK fixed point. This can be seen most
clearly from the original Eq.\ref{kondob}: if $\lambda=\alpha_{z}
=0 $, under the $ SU(2) $ transformation on the channel 2 fermions
$ \psi_{2 \uparrow} \rightarrow i \psi_{2 \uparrow},
\psi_{2 \uparrow} \rightarrow -i \psi_{2 \uparrow} $, the spin currents
of channel 2 transform as
$ J^{x}_{2} \rightarrow -J^{x}_{2}, J^{y}_{2} \rightarrow -J^{y}_{2},
J^{z}_{2} \rightarrow J^{z}_{2} $, Eq.\ref{kondob} is transformed back to
the 2 channel flavor symmetric Kondo model. This can also be seen from
Eq.\ref{current}, $ \tilde{J}_{x}, \tilde{J}_{y}, J_{z} $ also satisfy
the $ \widehat{SU}_{2}(2) $ algebra.
\subsection{ Fixed point 3}
This fixed point is located at $ \gamma_{1}=\gamma_{2}=0 $
where both $\widehat{a} $ and $ \widehat{b} $
lose their kinetic energies and become
two Grassmann Lagrangian multipliers. Integrating them out leads to
the following boundary conditions:
\begin{equation}
b^{sf}_{L}(0)=-b^{sf}_{R}(0), ~~ a^{sf}_{L}(0)=-a^{sf}_{R}(0)
\label{bound3}
\end{equation}
Eqs.\ref{bound1}, \ref{bound3} can be expressed in term of bosons:
\begin{equation}
\Phi^{s}_{L}=\Phi^{s}_{R} + \pi, ~~~ \Phi^{sf}_{L}=\Phi^{sf}_{R} + \pi
\end{equation}
Substituting the above equation to Eqs. \ref{first} \ref{second} and paying
attention to the {\em spinor} nature of the representation \cite{hopping},
it is easy to see that depending on the sign of $\alpha$,
{\em one} of the two channels suffer $\frac{\pi}{2} $ phase shift,
{\em the other} remains free. The four Majorana fermions being twisted,
this fixed point has the symmetry $ O(4) \times O(4) $ with $ g=1 $.
The finite size spectrum of this fixed point is listed in Table \ref{flavor},
it is the sum of
that with phase shift $ \pi/2 $ and that of free electrons.
This scenario is completely consistent with NRG results of Ref.\cite{cox2}.
The local correlation functions at the FL fixed point are \cite{powerful}:
\begin{eqnarray}
\langle \widehat{a}( \tau ) \widehat{a}(0) \rangle =\frac{1}{\tau},~~~~
\langle b_{sf}( \tau ) b_{sf}(0) \rangle =\frac{\gamma^{2}_{1}}{\tau^{3}} \nonumber \\
\langle \widehat{b}( \tau ) \widehat{b}(0) \rangle =\frac{1}{\tau},~~~~
\langle a_{sf}( \tau ) a_{sf}(0) \rangle =\frac{\gamma^{2}_{2}}{\tau^{3}}
\end{eqnarray}
From the above equation, We can read the scaling dimensions of the various fields:
$[\widehat{a}]=[\widehat{b}]=[a_{s}]=[b_{s}]=1/2, [a_{sf}]=[b_{sf}]=3/2 $.
At the fixed point, the impurity degree of freedoms
completely disappear: $\widehat{a}, \widehat{b} $ turn into the
{\em non-interacting } scaling fields at the fixed point
\begin{equation}
\widehat{a} \sim b_{sf}(0,\tau),~~~ \widehat{b} \sim a_{sf}(0,\tau)
\end{equation}
Using Eqs.\ref{imp}, \ref{reverse}, the impurity spin turns into
\begin{eqnarray}
S_{x}(\tau) &= & i( -a_{sf}(0,\tau) b_{s}(0,\tau)+b_{sf}(0,\tau) a_{s}(0,\tau) ) \nonumber \\
S_{y}(\tau) &= & i( a_{sf}(0,\tau) a_{s}(0,\tau)+b_{sf}(0,\tau) b_{s}(0,\tau) ) \nonumber \\
S_{z}(\tau) &= & i ( a_{sf}(0,\tau) b_{sf}(0,\tau) + a_{s}(0,\tau) b_{s}(0,\tau) )
\end{eqnarray}
The impurity spin-spin correlation function show typical FL behavior
\begin{equation}
\langle S^{z}(\tau) S^{z}(0) \rangle =\frac{1}{\tau^{2}}
\end{equation}
Using the fermionized form of the Eq.\ref{current} and paying attention to
the {\em spinor} nature of the representation,
it is easy to see the impurity spin renormalizs into either $ \vec{J}_{1} (0,\tau) $
or $ \vec{J}_{2}(0,\tau) $ depending on the sign of $\alpha $. This is consistent
with the CFT analysis in the Appendix.
There are 4 leading irrelevant operators with dimension 2 in
the action $ S $ : $ \gamma_{1} $ and $ \gamma_{2} $ terms,
$\lambda_{z}^{\prime} $ term and
$ a_{s}(0,\tau) \frac{ \partial a_{s}(0,\tau)}{\partial \tau} +
b_{s}(0,\tau) \frac{ \partial b_{s}(0,\tau)}{\partial \tau} $ which
will be generated by the $\lambda_{z}^{\prime} $ term.
The $ \alpha_{z} $ term has dimension 4, it can be written as $
:\widehat{a}(\tau) \frac{\partial \widehat{a}(\tau)}{\partial \tau}:
:\widehat{b}(\tau) \frac{\partial \widehat{b}(\tau)}{\partial \tau}: $.
The bosonized forms of the 4 leading irrelevant operators are \cite{another}
\begin{eqnarray}
\widehat{a}(\tau) \frac{\partial \widehat{a}(\tau)}{\partial \tau}
& = & \cos 2\Phi_{sf}-\frac{1}{2} (\partial \Phi_{sf}(0))^{2} \nonumber \\
\widehat{b}(\tau) \frac{\partial \widehat{b}(\tau)}{\partial \tau}
& = & -\cos 2\Phi_{sf}-\frac{1}{2} (\partial \Phi_{sf}(0))^{2} \nonumber \\
\widehat{a}\widehat{b}a_{s}(0)b_{s}(0) & = & \partial\Phi_{sf}(0,\tau)
\partial\Phi_{s}(0,\tau) \nonumber \\
a_{s}(0,\tau) \frac{ \partial a_{s}(0,\tau)}{\partial \tau} & + &
b_{s}(0,\tau) \frac{ \partial b_{s}(0,\tau)}{\partial \tau}
= (\partial \Phi_{s}(0,\tau))^{2}
\label{four}
\end{eqnarray}
Following the method developed in Ref.\cite{powerful}, their contributions
to the single particle Green functions can be calculated.
The first order correction
to the single particle L-R Green function ( $ x_{1}>0, x_{2}<0 $ )
due to the first operator in the above Eq. is
\begin{eqnarray}
&\langle & \psi_{1 \uparrow}( x_{1},\tau_{1} ) \psi^{\dagger}_{1 \uparrow}( x_{2},\tau_{2} ) \rangle =
\int d\tau
\langle e^{-\frac{i}{2} \Phi_{c}( x_{1}, \tau_{1} )} e^{\frac{i}{2} \Phi_{c}( x_{2}, \tau_{2} )}\rangle
\nonumber \\
& \times &\langle e^{-\frac{i}{2} \Phi_{s}( x_{1}, \tau_{1} )}
e^{\frac{i}{2} ( \Phi_{s}( x_{2}, \tau_{2} ) + \pi )} \rangle
\langle e^{-\frac{i}{2} \Phi_{f}( x_{1}, \tau_{1} )} e^{\frac{i}{2} \Phi_{f}( x_{2}, \tau_{2} )}\rangle
\nonumber \\
& \times & \langle e^{-\frac{i}{2} \Phi_{sf}( x_{1}, \tau_{1} )}
(:\cos2 \Phi_{sf}( 0, \tau ): -\frac{1}{2} : (\partial \Phi_{sf}( 0,\tau) )^{2} :)
e^{\frac{i}{2} ( \Phi_{sf}( x_{2}, \tau_{2} ) +\pi )}\rangle \nonumber \\
& \sim & (z_{1}-\bar{z}_{2} )^{-2}
\label{single}
\end{eqnarray}
Where $ z_{1}=\tau_{1}+i x_{1} $ is in the upper half plane,
$ \bar{z}_{2} =\tau_{2}+i x_{2} $ is in the lower half plane.
By using the following OPE:
\begin{eqnarray}
: e^{-\frac{i}{2} \Phi_{sf}( z_{1} )}: : e^{\frac{i}{2} \Phi_{sf}( z_{2})}: =
(z_{1}-z_{2})^{-1/4}-\frac{i}{2}(z_{1}-z_{2})^{3/4} :\partial \Phi_{sf}(z_{2}):
\nonumber \\
-\frac{i}{4}(z_{1}-z_{2})^{7/4} :\partial^{2} \Phi_{sf}(z_{2}):
-\frac{1}{8}(z_{1}-z_{2})^{7/4} : (\partial \Phi_{sf}(z_{2}) )^{2}: + \cdots
\label{ope}
\end{eqnarray}
It is ease to see that the primary field $ :\cos2 \Phi_{sf}( 0, \tau ): $ makes {\em no} contributions
to the three point function.
It was shown by the detailed calculations
in Ref.\cite{conductivity} that only the part of the self-energy which is both {\em imaginary}
and {\em even} function of $ \omega $ contributes to the conductivity.
Although the energy momentum tensor $ : (\partial \Phi_{sf}( 0,\tau) )^{2} : $
do make $\sim \omega $ contribution to the self-energy in the first order \cite{conn},
because it is a {\em odd} function, it does {\em not} contribute to the electron conductivity in this order.
Same arguments apply to the other operators in Eq.\ref{four}.
Second order perturbations in these operators lead to the generic $ T^{2} $ fermi
liquid bahaviour of the electron conductivity.
The results of this section were applied to a two level tunneling system with slight
modifications in Ref.\cite{hopping}. The universal scaling functions in the presence
of external magnetic field which breaks the channel symmetry were also discussed there.
\section{Compactified one channel Kondo Model}
Assuming Particle-Hole symmetry, the {\em non-interacting} one channel
Kondo model has two commuting $ SU(2) $ symmetry,
one is the usual spin symmetry with the generators $ J^{a} (a=x,y,z) $
another is the isospin symmetry with the generators $ I^{a} (a=x,y,z) $.
\begin{eqnarray}
J_{x} & = & \frac{1}{2}( \psi^{\dagger}_{\uparrow} \psi_{\downarrow}
+ \psi^{\dagger}_{\downarrow} \psi_{\uparrow} ), ~~
J_{y}=\frac{1}{2i}( \psi^{\dagger}_{\uparrow} \psi_{\downarrow}
- \psi^{\dagger}_{\downarrow} \psi_{\uparrow} ), ~~
J_{z}=\frac{1}{2}( \psi^{\dagger}_{\uparrow} \psi_{\uparrow}
- \psi^{\dagger}_{\downarrow} \psi_{\downarrow} ) \nonumber \\
I_{x} & = & \frac{1}{2}( \psi^{\dagger}_{\uparrow} \psi^{\dagger}_{\downarrow}
+ \psi_{\downarrow} \psi_{\uparrow} ), ~~
I_{y}=\frac{1}{2i}( \psi^{\dagger}_{\uparrow} \psi^{\dagger}_{\downarrow}
- \psi_{\downarrow} \psi_{\uparrow}), ~~
I_{z}=\frac{1}{2}( \psi^{\dagger}_{\uparrow} \psi_{\uparrow}
+ \psi^{\dagger}_{\downarrow} \psi_{\downarrow} )
\label{si}
\end{eqnarray}
The diagonal and off-diagonal components of the isospin currents represent respectively
the charge and pairing density at the site $ x$.
The one channel compactified model proposed by Ref.\cite{coleman} is a model
where the impurity spin couples to both the spin and the isospin currents of the
one channel conduction electrons
\begin{eqnarray}
H_{c} &= & i v_{F} \int^{\infty}_{-\infty} dx
\psi^{\dagger}_{ \alpha }(x) \frac{d \psi_{ \alpha }(x)}{dx}
+ \sum_{a=x,y,z} \lambda^{a}( I^{a}(0)+ J^{a}(0) ) S^{a}
+ \sum_{a=x,y,z} \alpha^{a}
(I^{a}(0)-J^{a}(0)) S^{a} \nonumber \\
& + & h ( \int dx (I^{z}(x) + J^{z}(x)) + S^{z} )
\label{com}
\end{eqnarray}
The ordinary symmetric Anderson impurity model in a one dimensional lattice is
\begin{eqnarray}
H & = & i t \sum_{n,\alpha} ( \psi^{\dagger}_{\alpha}(n+1) \psi_{\alpha}(n) - h. c. ) \nonumber \\
& + & i V \sum_{\alpha} ( \psi^{\dagger}_{\alpha}(0) d_{\alpha} -h.c.)
+ U(n_{d \uparrow}-\frac{1}{2})(n_{d \downarrow}-\frac{1}{2})
\label{aim}
\end{eqnarray}
The $ O(4) $ symmetry of the AIM can be clearly displayed in terms of the
Majorana fermions
\begin{eqnarray}
\psi_{\uparrow}(n) & = &\frac{1}{\sqrt{2}} ( \chi_{1}(n) -i \chi_{2}(n) ), ~~~
d_{\uparrow}=\frac{1}{\sqrt{2}}( d_{1}-i d_{2} ) \nonumber \\
\psi_{\downarrow}(n) & = & \frac{1}{\sqrt{2}} ( -\chi_{3}(n) -i \chi_{0}(n) ), ~~~
d_{\downarrow}=\frac{1}{\sqrt{2}}( -d_{3}-i d_{0} )
\end{eqnarray}
Breaking the symmetry from $ O(4) $ to $ O(3) \times O(1) $ in the hybridization \cite{ising},
the Hamiltonian \ref{aim} becomes:
\begin{eqnarray}
H & = & i t \sum_{n} \sum^{3}_{\alpha=0} \chi_{\alpha}(n+1) \chi_{\alpha}(n) +i V_{0} \chi_{0}(0) d_{0}
\nonumber \\
& + & i V \sum^{3}_{ \alpha=1} \chi_{\alpha}(0) d_{\alpha} + U d_{1} d_{2} d_{3} d_{0}
\label{break}
\end{eqnarray}
In the large $ U $ limit, projecting out the excited impurity states, we can map
the Hamiltonian \ref{break} to the 1CCK Hamiltonian \ref{com} with
\begin{equation}
\lambda= \frac{ 2 V^{2}}{U},~~~ \alpha= -\frac{ 2 V_{0} V }{ U }
\end{equation}
If $ V_{0}=V $, Eq.\ref{break} comes back to the original $ O(4) $ symmetric AIM. In the strong
coupling limit, it becomes the one channel Kondo model where the impurity only couples to
the spin currents (or isospin currents) of the conduction electrons \cite{exchange}.
If $ V_{0} =0 $, then $ \alpha=0 $, Eq.\ref{break} becomes the isotropic 1CCK where the impurity couples to
the spin and isospin currents with equal strength.
If we define the P-H transformation $ \psi_{\uparrow} \rightarrow \psi_{\uparrow},
\psi_{\downarrow} \rightarrow \psi^{\dagger}_{\downarrow} $, then spin and isospin currents
transform to each other $ I^{a} \rightarrow J^{a}, J^{a} \rightarrow I^{a} $.
The Hamiltonian \ref{com} has the P-H symmetry if $\alpha=0 $.
In the following, parallel to the discussions on the 2CFAK, we take
$ \lambda^{x}=\lambda^{y}=\lambda, \lambda^{z} \neq \lambda;
\alpha^{x}=\alpha^{y}=\alpha, \alpha^{z} \neq \alpha $.
We bosonize the spin $\uparrow $ and spin $\downarrow $ electrons separately
\begin{equation}
\psi_{ \alpha }(x )= \frac{P_{ \alpha}}{\sqrt{ 2 \pi a }}
e ^{- i \Phi_{ \alpha}(x) }
\label{one}
\end{equation}
The cocyle factors have been chosen as $ P_{ \uparrow}= P_{ \downarrow}
= e^{i \pi N_{ \uparrow} } $.
The bosonized form of the spin and isospin currents in Eq.\ref{si} are
\begin{eqnarray}
J_{x} & = & \frac{1}{2\pi a} \cos \sqrt{2} \Phi_{s},~~~
J_{y}=\frac{1}{2\pi a} \sin \sqrt{2} \Phi_{s},~~~
J_{z}=-\frac{1}{4\pi } \frac{\partial}{\partial x} \sqrt{2} \Phi_{s} \nonumber \\
I_{x} & = & \frac{1}{2\pi a} \cos \sqrt{2} \Phi_{c},~~~
I_{y}=\frac{1}{2\pi a} \sin \sqrt{2} \Phi_{c},~~~
I_{z}=-\frac{1}{4\pi } \frac{\partial}{\partial x} \sqrt{2} \Phi_{c}
\end{eqnarray}
where $ \Phi_{c}, \Phi_{s} $ are charge and spin bosons:
\begin{eqnarray}
\Phi_{c} = \frac{1}{\sqrt{2}}( \Phi_{\uparrow}+\Phi_{\downarrow}),~~~~~
\Phi_{s} = \frac{1}{\sqrt{2}}( \Phi_{\uparrow}-\Phi_{\downarrow})
\label{cs}
\end{eqnarray}
The sum $ J_{s}^{a}(x) = I^{a}(x) + J^{a}(x) $
and the difference $ J_{d}^{a}(x) = I^{a}(x) - J^{a}(x) $ can be expressed
in terms of the chiral bosons
\begin{eqnarray}
J^{s}_{x}= \frac{1}{\pi a} \cos \Phi_{\uparrow} \cos \Phi_{\downarrow},~
J^{s}_{y}= \frac{1}{\pi a} \sin \Phi_{\uparrow} \cos \Phi_{\downarrow},~
J^{s}_{z}= -\frac{1}{ 2 \pi} \frac{ \partial \Phi_{\uparrow}}{\partial x} \nonumber \\
J^{d}_{x}=- \frac{1}{\pi a} \sin \Phi_{\uparrow} \sin \Phi_{\downarrow},~
J^{d}_{y}= \frac{1}{\pi a} \cos \Phi_{\uparrow} \sin \Phi_{\downarrow},~
J^{d}_{z}= -\frac{1}{ 2 \pi} \frac{ \partial \Phi_{\downarrow}}{\partial x}
\label{iso}
\end{eqnarray}
Compare Eq.\ref{current} with Eq.\ref{iso}, we immediately realize that
the mapping between the 2CFAK and the 1CCK
is $ \Phi_{s} \rightarrow \Phi_{\uparrow}, \Phi_{sf} \rightarrow \Phi_{\downarrow} $,
therefore $ \psi_{s} \rightarrow \psi_{\uparrow},
\psi_{sf} \rightarrow \psi_{\downarrow} $. The following fixed point structure
can be immediately borrowed from the corresponding discussions on the 2CFAK.
\subsection{ Fixed point 1 }
The boundary conditions are
\begin{equation}
\psi_{\uparrow,L} = -\psi_{\uparrow,R},~~~
\psi_{\downarrow,L} =\psi^{\dagger}_{\downarrow,R}
\end{equation}
It is easy to see that the above boundary conditions respect the P-H symmetry,
they can be expressed in terms of bosons
\begin{equation}
\Phi_{\uparrow,L} =\Phi_{\uparrow,R} + \pi,~~~
\Phi_{\downarrow,L} = -\Phi_{\downarrow,R} + \pi
\end{equation}
Spin $\uparrow $ electrons suffer a $ \frac{\pi}{2} $ phase shift, however,
spin $\downarrow $ electrons are scattered into holes and vice-versa.
The one particle S-matrix are $ S_{\uparrow}=-1, S_{\downarrow}=0 $. The residual
conductivity of the spin $\uparrow$ electron takes unitary limit, but that of the spin
$\downarrow $ is half of the unitary limit.
The isotropic 1CCK has the same thermodynamic behaviors as the 2CK, but its
fixed point has the local KM symmetry $ \widehat{O}_{1}(3) \times \widehat{O}_{1}(1) $.
The finite size spectrum of this NFL fixed point is listed in
Table \ref{compactnfl}. Comparing this finite size spectrum with that of the
NFL fixed point of the 2CK listed in Ref.\cite{powerful}, it is easy to see that
it has the {\em same } energy levels as those of the 2CK, but the corresponding
degeneracy is {\em much smaller}. This is within the expectation, because
the central charge $ c=2 $ and the fixed point symmetry of the isotropic 1CCK
is smaller than that of the 2CK.
This fixed point is stable only when $\alpha=\alpha_{z}=0 $ where
the Hamiltonian \ref{com} has P-H symmetry.
Away from the fixed point, there is only one dimension 3/2 operator
\begin{equation}
\widehat{a} \widehat{b} \partial \Phi_{\uparrow}(0) \sim
\cos \Phi_{\downarrow}(0) \partial \Phi_{\uparrow}(0)
\end{equation}
The first order correction to the single particle L-R Green function ( $ x_{1}>0, x_{2}<0 $ )
due to this operator is
\begin{eqnarray}
\int d\tau \langle e^{- i \Phi_{\uparrow}( x_{1}, \tau_{1} )} \partial \Phi_{\uparrow}(0,\tau)
e^{ i ( \Phi_{\uparrow}( x_{2}, \tau_{2} ) +\pi )}\rangle
\langle :\cos \Phi_{\downarrow}( 0, \tau ): \rangle =0 \nonumber \\
\int d\tau \langle e^{- i \Phi_{\downarrow}( x_{1}, \tau_{1} )} \cos \Phi_{\downarrow}(0,\tau)
e^{- i \Phi_{\downarrow}( x_{2}, \tau_{2} )}\rangle
\langle \partial \Phi_{\uparrow}( 0, \tau ) \rangle =0
\end{eqnarray}
By Wick theorem, it is easy to see that any {\em odd} order corrections vanish.
Second order correction goes as $ \sim \omega $ which is a {\em odd} function, therefore
does not contribute to the electron conductivity. The fourth order
makes $ T^{2} $ contributions.
There are two dimension 2 operators:
\begin{eqnarray}
a_{\uparrow}(0,\tau) \frac{ \partial a_{\uparrow}(0,\tau)}{\partial \tau} & + &
b_{\uparrow}(0,\tau) \frac{ \partial b_{\uparrow}(0,\tau)}{\partial \tau} =
(\partial \Phi_{\uparrow}(0,\tau))^{2} \nonumber \\
\widehat{a}(\tau) \frac{\partial \widehat{a}(\tau)}{\partial \tau}
& = & \cos 2\Phi_{\downarrow}-\frac{1}{2} (\partial \Phi_{\downarrow}(0))^{2}
\label{cool}
\end{eqnarray}
The first order correction to the spin $\uparrow $ electron L-R Green function
due to the first operator in Eq.\ref{cool} is
\begin{eqnarray}
\int d\tau \langle e^{- i \Phi_{\uparrow}( x_{1}, \tau_{1} )} :( \partial \Phi_{\uparrow}(0,\tau) )^{2}:
e^{ i ( \Phi_{\uparrow}( x_{2}, \tau_{2} ) +\pi )}\rangle \sim ( z_{1}-\bar{z}_{2} )^{-2}
\label{many}
\end{eqnarray}
As pointed out in the last section, the energy momentum tensor $ : (\partial \Phi_{\uparrow}( 0,\tau) )^{2} : $
makes $\sim \omega $ contribution to the self-energy in the first order, therefore
does not contribute to the electron conductivity.
Second order perturbation in this operator leads to $ T^{2} $ contributions.
Adding the contributions from all the leading irrelevant operators, we get
\begin{equation}
\sigma_{\uparrow}(T) \sim \sigma_{u}(1+ T^{2} + T^{4} + \cdots)
\end{equation}
The first order correction to the spin $\downarrow $ electron L-R Green function
due to the 2nd operator in Eq. \ref{cool} is
\begin{eqnarray}
\int d\tau \langle e^{- i \Phi_{\downarrow}( x_{1}, \tau_{1} )} \cos 2\Phi_{\downarrow}(0,\tau)
e^{- i \Phi_{\downarrow}( x_{2}, \tau_{2} )}\rangle \nonumber \\
-\frac{1}{2} \int d\tau \langle e^{- i \Phi_{\downarrow}( x_{1}, \tau_{1} )}
( \partial \Phi_{\downarrow}(0,\tau) )^{2}
e^{- i \Phi_{\downarrow}( x_{2}, \tau_{2} )}\rangle
\end{eqnarray}
By using the following OPE:
\begin{equation}
: e^{-i \Phi_{\downarrow}( z_{1} )}: : e^{-i \Phi_{\downarrow}( z_{2})}: =
(z_{1}-z_{2}): e^{-i 2 \Phi_{\downarrow}( z_{2} )}:
-i(z_{1}-z_{2})^{2} : e^{-i 2 \Phi_{\downarrow}( z_{2} )} \partial \Phi_{\downarrow}(z_{2}): +\cdots
\end{equation}
It is ease to see that the {\em second} integral vanishes, but the {\em first} becomes
\begin{equation}
\frac{1}{ (z_{1}-\bar{z}_{2} )^{-1} } \int d \tau \frac{1}{ (z_{1}-\tau)^{2} (\tau-\bar{z}_{2} )^{2} }
\sim (z_{1}-\bar{z}_{2} )^{-2}
\end{equation}
Putting $ \Delta=1 $ in Eq. (3.52) of Ref.\cite{conductivity}, we find the imaginary and
real parts of self-energy go as $ Im \Sigma(\omega, T=0)=0, Re \Sigma(\omega, T=0)
\sim \omega $, therefore the first order perturbation does not contribute to the spin $\downarrow $
electron conductivity. Second order perturbation
yields a $ T^{2} $ contributions.
Adding the contributions from all the leading irrelevant operators, we get
\begin{equation}
\sigma_{\downarrow}(T) \sim 2 \sigma_{u}(1+ T^{2}+ T^{4} + \cdots )
\end{equation}
The total conductivity is the summation of the two spin components \cite{bhatt}
\begin{equation}
\sigma(T)= \sigma_{\uparrow}(T)+ \sigma_{\downarrow}(T)
\sim 3 \sigma_{u}(1+ T^{2}+ T^{4} + \cdots )
\end{equation}
The boundary OPE of the spin and density of the {\em conduction electrons} are
\begin{eqnarray}
\psi^{\dagger}_{\uparrow}(z_{1}) \psi_{\uparrow}( \bar{z}_{2} ) & = &
( z_{1}-\bar{z}_{2} )^{-1} +i \partial \Phi_{\uparrow} + \cdots \nonumber \\
\psi^{\dagger}_{\downarrow}(z_{1}) \psi_{\downarrow}( \bar{z}_{2} ) & = &
0 + \cdots \nonumber \\
\psi^{\dagger}_{\uparrow}(z_{1}) \psi_{\downarrow}( \bar{z}_{2} ) & = & e^{i \sqrt{2} \Phi_{c}(0) }
+ \cdots \nonumber \\
\psi^{\dagger}_{\downarrow}(z_{1}) \psi_{\uparrow}( \bar{z}_{2} ) & = & e^{-i \sqrt{2} \Phi_{s}(0) }
+ \cdots
\label{sdnfl}
\end{eqnarray}
The boundary OPE of the spin singlet and triplet pairing operators are
\begin{eqnarray}
\psi_{\uparrow}(z_{1}) \psi_{\uparrow}( \bar{z}_{2} ) & = & 0
+ \cdots \nonumber \\
\psi_{\downarrow}(z_{1}) \psi_{\downarrow}( \bar{z}_{2} ) & = & ( z_{1}-\bar{z}_{2} )^{-1}
-i \partial \Phi_{\downarrow} + \cdots \nonumber \\
\psi_{\uparrow}(z_{1}) \psi_{\downarrow}( \bar{z}_{2} ) & = & e^{-i \sqrt{2} \Phi_{s}(0) }
+ \cdots \nonumber \\
\psi_{\downarrow}(z_{1}) \psi_{\uparrow}( \bar{z}_{2} ) & = & - e^{-i \sqrt{2} \Phi_{c}(0) }
+ \cdots
\label{pairingnfl}
\end{eqnarray}
The P-H symmetry interchanges the pairing and spin operators in
the $ \uparrow \downarrow $ and $ \downarrow \uparrow $ channels.
From Eq.\ref{pairingnfl}, we can identify the pairing operators
\begin{equation}
{\cal O}_{s}= e^{-i \sqrt{2} \Phi_{s}(0) },~~~ {\cal O}_{c}= e^{-i \sqrt{2} \Phi_{c}(0)},~~~
{\cal O}_{\downarrow}= \partial \Phi_{\downarrow}(0)
\end{equation}
The paring operators in all the channels except in the $ \uparrow \uparrow $ channel
have scaling dimension 1, therefore their correlation functions decay as
$ \tau^{-2} $. Comparing these pairing operators with those at the FL fixed point
( Eq. \ref{pairingfl} ) to be discussed in the following, we find the pairng
susceptibility in $ \downarrow \downarrow $ channel is enhanced. However,
in contrast to the 2CK fixed point \cite{powerful}, the enhancement is so weak that
there is {\em no} pairing susceptibility
{\em divergence} at the impurity site in {\em any spin channel}.
This result is somewhat surprising. Naively, we expect pairing
susceptibility divergence because the impurity interacts with the pairing density of the
conduction electrons at the impurity site. However, the above explicit calculations showed
that this is {\em not} true if there is only {\em one } channel of conduction electrons.
Naively, we do {\em not} expect pairing susceptibility divergence in the 2CK, because
the impurity spin interacts only with the total {\em spin } currents of channel 1 and 2,
{\em no } isospin currents of channel 1 and 2 are involved in the interaction. However,
the explicit calculation of the 2CK showed that the pairing operator in the spin and flavor
singlet channel has dimension 1/2 ( however, the pairing operators in flavor singlet and spin triplet channel
has dimension 3/2 ), therefore the spin and flavor singlet pairing susceptiblity at the impurity site
is {\em divergent} \cite{powerful}. This indicates that we can achieve the pairing susceptiblity
divergence without a pairing source term. We conclude that {\em more than } one channel of conduction
electrons are needed to achieve the pairing susceptipility {\em divergence}.
\subsection{ Fixed pointed 2}
The boundary conditions are
\begin{equation}
\psi_{\uparrow,L} = -\psi_{\uparrow,R},~~~
\psi_{\downarrow,L} = -\psi^{\dagger}_{\downarrow,R}
\end{equation}
The above boundary conditions can be expressed in terms of bosons
\begin{equation}
\Phi_{\uparrow,L} =\Phi_{\uparrow,R} + \pi,~~~
\Phi_{\downarrow,L} = -\Phi_{\downarrow,R}
\end{equation}
This fixed point is stable only when $\lambda=\alpha_{z}=0 $.
If we define the P-H transformation $ \psi_{\uparrow} \rightarrow \psi_{\uparrow},
\psi_{\downarrow} \rightarrow -\psi^{\dagger}_{\downarrow} $, then the spin and isospin currents
transform as $ I^{x} \rightarrow -J^{x}, I^{y} \rightarrow -J^{y}, I^{z} \rightarrow J^{z};
J^{x} \rightarrow -I^{x}, J^{y} \rightarrow -I^{y}, J^{z} \rightarrow I^{z}$.
The Hamiltonian \ref{com} has this P-H symmetry if $\lambda=\alpha_{z}=0 $.
This is the same fixed point as fixed point 1.
\subsection{ Fixed pointed 3}
The boundary conditions are
\begin{equation}
\psi_{\uparrow,L} =-\psi_{\uparrow,R},~~~
\psi_{\downarrow,L} = -\psi_{\downarrow,R}
\end{equation}
The above boundary conditions can be expressed in terms of bosons
\begin{equation}
\Phi_{\uparrow,L} =\Phi_{\uparrow,R} + \pi,~~~
\Phi_{\downarrow,L} = \Phi_{\downarrow,R} + \pi
\end{equation}
Both spin $\uparrow $ and $\downarrow $ electrons suffer $ \frac{\pi}{2} $
phase shift. The physical picture is that the impurity spin is either totally
screened by the spin current or the isospin current of conduction
electrons depending on which coupling is stronger \cite{exchange}.
This is a FL fixed point with $ O(4) $ symmetry. The finite size spectrum
is listed in Table \ref{compactfl}.
The bosonized forms of the 4 leading irrelevant operators are \cite{another}
\begin{eqnarray}
\widehat{a}(\tau) \frac{\partial \widehat{a}(\tau)}{\partial \tau}
& = & \cos 2\Phi_{\downarrow}-\frac{1}{2} (\partial \Phi_{\downarrow}(0))^{2} \nonumber \\
\widehat{b}(\tau) \frac{\partial \widehat{b}(\tau)}{\partial \tau}
& = & -\cos 2\Phi_{\downarrow}-\frac{1}{2} (\partial \Phi_{\downarrow}(0))^{2} \nonumber \\
\widehat{a}\widehat{b}a_{\uparrow}(0)b_{\uparrow}(0) & = & \partial\Phi_{\downarrow}(0,\tau)
\partial\Phi_{\uparrow}(0,\tau) \nonumber \\
a_{\uparrow}(0,\tau) \frac{ \partial a_{\uparrow}(0,\tau)}{\partial \tau} & + &
b_{\uparrow}(0,\tau) \frac{ \partial b_{\uparrow}(0,\tau)}{\partial \tau}
= (\partial \Phi_{\uparrow}(0,\tau))^{2}
\label{last}
\end{eqnarray}
The first order correction to the spin $\uparrow $ electron L-R Green function due to the 4th
operator in Eq.\ref{last} is also given by Eq.\ref{many}. The correction due to the 3rd operator
in Eq.\ref{last} can be similarly evaluated. We get the low temperature expansion
of the spin $\uparrow $ electron conductivity
\begin{equation}
\sigma_{\uparrow}(T) \sim \sigma_{u}(1+ T^{2} + T^{4} + \cdots)
\label{phyup}
\end{equation}
The first order correction to the spin $\downarrow $ electron L-R Green function
due to the first operator in Eq. \ref{last} is
\begin{eqnarray}
\int d\tau \langle e^{- i \Phi_{\downarrow}( x_{1}, \tau_{1} )} \cos 2\Phi_{\downarrow}(0,\tau)
e^{ i ( \Phi_{\downarrow}( x_{2}, \tau_{2} ) +\pi) }\rangle \nonumber \\
-\frac{1}{2} \int d\tau \langle e^{- i \Phi_{\downarrow}( x_{1}, \tau_{1} )}
( \partial \Phi_{\downarrow}(0,\tau) )^{2}
e^{ i ( \Phi_{\downarrow}( x_{2}, \tau_{2} ) +\pi)}\rangle
\end{eqnarray}
By using the following OPE:
\begin{eqnarray}
: e^{-i \Phi_{\downarrow}( z_{1} )}: : e^{i \Phi_{\downarrow}( z_{2})}: =
(z_{1}-z_{2})^{-1}-i :\partial \Phi_{\downarrow}(z_{2}):
\nonumber \\
+\frac{z_{1}-z_{2}}{2} :\partial^{2} \Phi_{\downarrow}(z_{2}):
-\frac{z_{1}-z_{2}}{2} : (\partial \Phi_{\downarrow}(z_{2}) )^{2}: + \cdots
\end{eqnarray}
It is ease to see that the first integral vanishes and the second are
the same as Eq.\ref{many}. The corrections due to the 2nd and the 3rd operators in Eq.\ref{last}
can be similarly evaluated, the low temperature expansion
of the spin $\downarrow $ electron conductivity follows
\begin{equation}
\sigma_{\downarrow}(T) \sim \sigma_{u}(1+ T^{2} + T^{4} + \cdots)
\label{phydown}
\end{equation}
Note that only at the FL fixed point, the spin $ SU(2) $ symmetry is restored,
therefore the expansion coefficients in Eqs.\ref{phyup},\ref{phydown}
are {\em different}.
The total conductivity is the summation of the two spin components
\begin{equation}
\sigma(T)= \sigma_{\uparrow}(T)+ \sigma_{\downarrow}(T)
\sim 2 \sigma_{u}(1+ T^{2}+ T^{4} + \cdots )
\end{equation}
The boundary OPE of the spin and density of the {\em conduction electrons} are
\begin{eqnarray}
\psi^{\dagger}_{\uparrow}(z_{1}) \psi_{\uparrow}( \bar{z}_{2} ) & = &
( z_{1}-\bar{z}_{2} )^{-1} +i \partial \Phi_{\uparrow} + \cdots \nonumber \\
\psi^{\dagger}_{\downarrow}(z_{1}) \psi_{\downarrow}( \bar{z}_{2} ) & = &
( z_{1}-\bar{z}_{2} )^{-1} +i \partial \Phi_{\downarrow} + \cdots \nonumber \\
\psi^{\dagger}_{\uparrow}(z_{1}) \psi_{\downarrow}( \bar{z}_{2} ) & = & e^{i \sqrt{2} \Phi_{s}(0) }
+ \cdots \nonumber \\
\psi^{\dagger}_{\downarrow}(z_{1}) \psi_{\uparrow}( \bar{z}_{2} ) & = & e^{-i \sqrt{2} \Phi_{s}(0) }
+ \cdots
\label{sdfl}
\end{eqnarray}
The boundary OPE of the spin singlet and triplet pairing operators are
\begin{eqnarray}
\psi_{\uparrow}(z_{1}) \psi_{\uparrow}( \bar{z}_{2} ) & = & 0
+ \cdots \nonumber \\
\psi_{\downarrow}(z_{1}) \psi_{\downarrow}( \bar{z}_{2} ) & = & 0 +\cdots \nonumber \\
\psi_{\uparrow}(z_{1}) \psi_{\downarrow}( \bar{z}_{2} ) & = & -e^{-i \sqrt{2} \Phi_{c}(0) }
+ \cdots \nonumber \\
\psi_{\downarrow}(z_{1}) \psi_{\uparrow}( \bar{z}_{2} ) & = & e^{-i \sqrt{2} \Phi_{c}(0) }
+ \cdots
\label{pairingfl}
\end{eqnarray}
The above equations should be compared with the corresponding Eqs.\ref{sdnfl} and \ref{pairingnfl}
at the NFL fixed point.
\section{Conclusions}
By the detailed discussions on the low temperature properties of the two related, but
different single impurity models, we clarify the confusing conjectures and
claims made on these two models. In evaluating the single
particle Green functions and pairing susceptibilities, all the degree of freedoms
have to be taken into account, even though some of them decouple from the interactions
with the impurity.
We explicitly demonstrate that different quantum impurity models are simply
free chiral bosons with different boundary conditions. In Ref.\cite{sf}, the author
studied another single impurity model where the impurity couples to both
the spin and the flavor currents of the two channel electrons ( 2CSFK). In Ref.\cite{hopping},
the author solved a two level tunneling model which can also mapped to a single impurity model.
As shown in Ref.\cite{twoimp},
finite number of impurity models can always mapped to a single impurity model.
From the results of this paper and Refs.\cite{sf,hopping},
we conclude that in clean, finite number of impurity models (1) FL behaviors
are extremely robust, any perturbation in the flavor sectors will destroy the NFL
behaviors.(2) due to the phase space arguments given in this paper and
in Refs.\cite{sf,hopping}, it is very
unlikely to find the NFL linear $ T $ bahaviour of the electron conductivity
which was observed in the certain heavy fermion systems \cite{linear} and in the normal
state of high- $T_{c} $ cuprate superconductors. There are three possible ways to explain
this experimental observation (1) disorder effects \cite{bhatt} (1) Kondo lattice model \cite{lattice}
(3) near to some quantum phase transitions \cite{millis}, for example, near the phase transition between
the metallic spin-glass and disordered metal \cite{phase,sachdev}.
\centerline{\bf ACKNOWLEDGMENTS}
We thank D. S. Fisher, B. Halperin, A. Millis, N. Read for helpful discussions.
This research was supported by NSF Grants Nos. DMR 9630064, DMR9416910 and Johns Hopkins University.
| proofpile-arXiv_065-415 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction and results}
In this paper we investigate the problem of finding the
mapping of a three-dimensional free fermion theory with non-abelian
symmetry onto an equivalent bosonic quantum field theory. This
mapping, commonly called bosonization, has been already
established along the lines of the present investigation
for the case of abelian symmetry \cite{FS}-\cite{LNS} and
it has been also discussed in the non-abelian case,
using a related method, in \cite{B}. In all these
investigations we employ an approach
close to that put forward in
a series of very interesting works on smooth
bosonization and
duality bosonization
\cite{DNS1}-\cite{DNS5}. Other related or alternative
approaches to
bosonization in $d>2$ dimensions have been also developed
\cite{TSSG}-\cite{BBG}.
The advantage of the bosonization method
that we employ here lies on the fact that it provides
a systematic procedure for deriving $d \ge 2$ bosonization recipes both
for abelian and non-abelian symmetries. In this way, it gives an
adequate framework for obtaining the bosonic equivalent of the original
fermionic theory, the recipe for mapping fermionic
and bosonic currents as well as the current commutation relations
which are at the basis of bosonization.
The approach we follow starts from the path-integral
defining the generating functional
for a theory of free fermions (including sources
for fermionic currents) and ends with
the generating functional for an equivalent
bosonic theory. This allows to identify, {\it exactly},
the bosonization recipe for fermion currents
independently of the number of space-time dimensions.
Interestingly enough,
one follows a series of steps which
are the same for any space-time dimension and both
for abelian and non-abelian symmetries.
Of course, apart from the two-dimensional
case and from the
large fermion mass limit in
the three dimensional case,
one only achieves in general a partial bosonization in the sense
that one cannot compute exactly the fermion path-integral in order to
derive a local bosonic Lagrangian.
Extending the three dimensional
abelian bosonization approach discused
in \cite{FS}-\cite{LNS}
we derive
in the present paper
three dimensional
non-abelian bo\-so\-niza\-tion. Concerning
the fermion current, our method
allows to derive the exact bosonization recipe
\begin{equation}
\bar \psi^i \gamma_\mu t^a_{ij} \psi^j \to
\pm \frac{i}{8\pi}\varepsilon_{\mu \nu \alpha}
\partial_\nu A^a_\alpha \,\, ,
\label{Ssss}
\end{equation}
where $i,j=1,\cdots,N$, $a=1,\cdots,{\rm dim}G$,
$t^a$ are the generators and $f^{a b c}$ the
structure constants of the symmetry group G.
Finally $A_\mu$ is a vector field taking values in the
Lie algebra of $G$.
The knowledge of the bosonic action accompanying
this bosonization rule is necessarily
approximate since it implies the evaluation
of the $d=3$ determinant for fermions
coupled to a vector field. We then consider
the case
of very massive free fermions
showing that, within this approximation,
the fermion lagrangian bosonizes to
the non-abelian Chern-Simons term,
\begin{equation}
\bar\psi^i \left(i\gamma^{\mu} \partial_{\mu}+m \right) \psi^i \to
\mp \frac{i}{8 \pi} \epsilon_{\mu \nu\alpha} \left(
A_{\mu}^a \partial_{\nu}A^a_{\alpha} + 1/3 f^{a b c}
A_{\mu}^a A_{\nu}^b A_{\alpha}^c \right) ~,
\label{aq11}
\end{equation}
This result, advanced in \cite{B} using a completely
different approach, is the natural extension of the
abelian result \cite{FS}-\cite{LNS}. In this last
case the bosonic theory corresponds, in the large fermion mass
limit, to a Chern-Simons theory while in the massless case
it coincides with the abelian
non-local action discussed in \cite{Mar},\cite{BFO}. In
fact one should expect that an analysis similar to that
in \cite{BFO} can be carried out leading to an
explicit (although complicated) bosonic action valid in all
fermion mass regimes.
The results above are derived in section 3. As a warming up
exercise, we rederive in section 2 the bosonization rules for
two-dimensional non-abelian models. Indeed, following
an approach related to
that developed in \cite{BQn} for two-dimensional
bosonization we arrive
to the Wess-Zumino-Witten action
and the well-known bosonization recipe for
fermion currents
\begin{equation}
j_+ \to \frac{i}{4\pi}a^{-1} \partial_+ a
\label{ele1}
\end{equation}
\begin{equation}
j_- \to \frac{i}{4\pi}a \partial_- a^{-1}
\label{ele2}
\end{equation}
with $a$ the group-valued bosonic field with
dynamics governed by the Wess-Zumino-Witten
action.
Now, these rules, as well as the Polyakov-Wiegmann identity
that we repeatedly use in its derivation,
deeply relay on the holomorphic properties of
two-dimensional theories \cite{PWI}\cite{Wi0}. Strikingly,
we found that in
the three dimensional case, a BRST symmetry structure underlying the
bosonic version of the fermionic generating functional
plays a similar role and allows to end, at least
in the large mass limit, with a simple bosonization rule.
This BRST symmetry is highly related to that used in
\cite{DNS1}-\cite{DNS3}, \cite{DNS4}-\cite{TSSG},
and is
analogous to that arising
in topological field theories \cite{BS}-\cite{BRT},
its origin being related to the
way the originally ``trivial'' bosonic field enters into play.
\section{Warming up: $d=2$ non-abelian bosonization}
Non-abelian bosonization in two dimensional space time was formulated
first by Witten \cite{Wi} by comparing the current algebra for free fermions
and for a bosonic sigma model with a Wess-Zumino term. Afterwards,
different approaches rederived and discussed
the bosonization recipe \cite{dVR}-\cite{GR} but in
general they were not constructive in the sense that the bosonic
theory was not obtained from the fermionic one by following a
series of steps that could be generalized to other cases, in particular
to a possible higher-dimensional bosonization. More recently, using the
the duality technique \cite{BQ}-\cite{BLQ} which has as
starting point the
smooth bosonization approach \cite{DNS1}-\cite{DNS3},
\cite{DNS4}-\cite{DNS5}
the recipe for
non-abelian bosonization was obtained by Burgess and Quevedo \cite{BQn}
in a way which is more adaptable to generalizations to higher dimensions.
The approach to two-dimensional bosonization that we present in this section
is related to that in \cite{BQn} and we think that it
is worthwhile to describe it here in detail since it provides many of the
clues which allow us to derive the
non-abelian bosonization recipe for 3-dimensional fermions.
Following Witten \cite{Wi}
one can see that the bosonic picture for a theory of
$N$ free massless Dirac fermions corresponds
to a bosonic field $a \in SU(N)$
with a Wess-Zumino-Witten action and a real bosonic field $\phi$
with a free scalar field action
\cite{dVR}-\cite{GO}. Since we shall be mainly interested
in the specific non-abelian aspect of bosonization, we will not discuss
the $\phi$ sector of the corresponding bosonic theory (although
the method can trivially take into account the $U(1)$ sector
associated with it).
We start from the (Euclidean) Lagrangian for free massless Dirac fermions
in $2$ dimensions
\begin{equation}
L = \bar\psi (\id) \psi
\label{L}
\end{equation}
where fermions are in the fundamental representation of some group $G$.
The corresponding
generating functional reads
\begin{equation}
Z_{fer}[s]
= \int D\bar \psi D\psi
\exp[-\int d^2x \bar\psi (\id + /\kern-.52em s) \psi]
\label{Zs}
\end{equation}
with $s_\mu = s_\mu^a t^a$ an external source
taking values in the Lie algebra of
$SU(N)$.
Our derivation of bosonization rules both in $2$ and $3$ dimensions
heavily relies on the invariance of the measure under local transformations
of the fermion variables,
$ \psi \to h(x) \psi$, $\bar \psi \to \bar \psi h(x)^{-1}$ with $h \in G$.
As a consequence
the generating functional (\ref{Zs}) is automatically invariant under
local transformations of the source
\begin{equation}
s_\mu \to s_\mu^h = h^{-1} s_\mu h + i h^{-1} \partial_\mu h
\label{inva}
\end{equation}
\begin{equation}
Z[s^h] = Z[s]
\label{idiss}
\end{equation}
In view of this, if we perform the change of variables
\[
\psi =g(x) \psi'
\]
\begin{equation}
\bar \psi = \bar \psi' g^{-1}(x) .
\label{chas22}
\end{equation}
$Z_{fer}[s]$ becomes
\begin{equation}
Z_{fer}[s] = \int {\cal{D}} \bar \psi {\cal{D}} \psi {\cal{D}} g
\exp[-\int d^2x \bar \psi (\id + /\kern-.52em s ^g ) \psi]
\label{n2s}
\end{equation}
where an integration over $g$ has been included since it just amounts
to a change of normalization. Integrating out fermions we have
\begin{equation}
Z_{fer}[s] = \int {\cal{D}} g \;\det(\id + {/\kern-.52em s}^g)
\label{cambios}
\end{equation}
Now, posing
\begin{equation}
{s}_\mu^g = b_\mu
\label{nuezz}
\end{equation}
and using
\begin{equation}
f_{\mu \nu}[b] = g^{-1} f_{\mu \nu}[s] g
\label{rela}
\end{equation}
we shall trade the $g$ integration for an integration over
connections $b$ satisfying condition (\ref{rela}).
To this end we shall use the $d=2$ identity (proven in the Appendix)
\begin{equation}
\int {\cal{D}} b_\mu {\cal H}[b]
\delta[\varepsilon_{\mu \nu} (f_{\mu\nu}[b] - f_{\mu\nu}[s])]
= \int {\cal{D}} g {\cal H}[s^g]
\label{idle}
\end{equation}
Here ${\cal H}$ is a gauge invariant function. Identity (\ref{idle})
allows us to write eq.(\ref{cambios}) in the form
\begin{equation}
Z_{fer} = \int {\cal{D}} b_\mu \Delta \delta(b_+ - s_+)
\delta\left[\varepsilon_{\mu\nu}(f_{\mu \nu}[b]- f_{\mu \nu}[s])\right]
\det(\id + /\kern-.52em b) \, .
\label{n3ss}
\end{equation}
For convenience, we have chosen to fix the gauge using the
condition $b_+ = s_+$ being $\Delta$ the corresponding
Faddeev-Popov determinant.
We
now introduce a Lagrange multiplier ${\hat a}$
(taking values in the Lie algebra of $G$) to enforce the
delta function condition
\begin{eqnarray}
Z_{fer}[s] & = &
\int {\cal{D}} {\hat a}{\cal{D}} b_\mu \,\Delta \delta(b_+ - s_+)
\det[\id +/\kern-.52em b] \times \nonumber \\
& &
\exp \left(-\frac{C}{8\pi}
tr \int d^2x {\hat a} \,\varepsilon _{\mu \nu }
(f_{\mu \nu }[b]
- f_{\mu \nu }[s])\right)
\label{copia}
\end{eqnarray}
with $C$ a constant to be conveniently adjusted.
We now write sources and the $b_\mu$ field in terms of group-valued
variables,
\begin{equation}
s_+ = i {\tilde s}^{-1} \partial_+ \tilde s
\label{s+}
\end{equation}
\begin{equation}
s_- = i s \partial_- s^{-1}
\label{s-}
\end{equation}
\begin{equation}
b_+ = i (\tilde b \tilde s)^{-1} \partial_+ (\tilde b \tilde s)
\label{b+}
\end{equation}
\begin{equation}
b_- = i (sb) \partial_- (sb)^{-1}
\label{ga}
\end{equation}
so that the fermion determinant can be related to the
Wess-Zumino-Witten action \cite{PWI},
\begin{equation}
\det[\id +/\kern-.52em b] = \exp (W[\tilde b \tilde s s b])
\label{deter}
\end{equation}
In writing eq.(\ref{deter}) a gauge-invariant regularization
is assumed so that the left and right-handed sectors
enter in gauge invariant
combinations.
In this way, gauge transformations of
the source $s_\mu$, which as stated before,
should leave the generating functional invariant,
do not change the determinant. This will be
the criterion we shall adopt each time
determinants (always needing a regularization)
has to be computed.
Concerning the Jacobian for passing from the $b_\mu$ variable to the
$b, \tilde b$ one can easily show that
\begin{equation}
\int \Delta \delta(b_+ - s_+){\cal{D}} b_\mu = \int \exp(\kappa W[\tilde b
\tilde s s b])
\delta( \tilde b - I) {\cal{D}} \tilde b {\cal{D}} b
\label{jaci}
\end{equation}
so that $Z_{fer}[s]$ becomes
\begin{eqnarray}
Z_{fer}[s] & = &
\int {\cal{D}} {\hat a} {\cal{D}} b \exp \left(i\frac{C}{4\pi}
tr \int d^2x
( D_+[\tilde s] {\hat a}) \, s b ( \partial_- b^{-1}) s^{-1}\right)
\times \nonumber\\
& &
\exp \left((1+\kappa)W[ \tilde s s b] \right) \, .
\label{copiag}
\end{eqnarray}
A convenient change of variables to pass from
integration over the algebra valued
Lagrange multiplier ${\hat a}$ to
a group valued variable $a$ is the one defined through
\begin{equation}
D_+[\tilde s] {\hat a} = {\tilde s}^{-1} (a^{-1}\partial_+a) \tilde s
\label{nuez}
\end{equation}
Calculation of the
corresponding jacobian $J_{L} $
\begin{equation}
{\cal{D}} \hat a = J_{L} {\cal{D}} a
\end{equation}
should be carefully done. Indeed, in
the present approach
to bosonization, it is the group valued variable $a$ who plays
the role of the boson field equivalent to the original fermion field.
Since for the latter (a free fermi field) there was no local symmetry,
the former should not be endowed with this symmetry.
With this in mind, we shall
maintain $a$ unchanged under local transformations $g$
while transforming $\hat a$,
$\hat a \to g^{-1} \hat a g$,
so that eq.(\ref{nuez}) changes covariantly when
one simultaneously changes $\tilde s \to \tilde s g$.
One can easily prove that
\begin{equation}
J_L = \exp (\kappa W[a \tilde s s] - \kappa W[\tilde s s])
\label{nuezi}
\end{equation}
so that the generating functional reads
\begin{eqnarray}
Z_{fer}[s] & =
\int {\cal{D}} a {\cal{D}} b \exp((1 +\kappa)W[\tilde s s b]) \times
\exp \left(\kappa (W[a \tilde s s]- W[\tilde s s])\right)
\times \nonumber\\
&\exp(-\frac{C}{4\pi} tr \int d^2x {\tilde s}^{-1}
(a^{-1} \partial_+ a) \tilde s s (b \partial_- b^{-1}) s^{-1} ) \, .
\label{copg}
\end{eqnarray}
If one repeatedly uses Polyakov-Wiegmann identity
and chooses the up to now arbitrary
constant $C$ as
\begin{equation}
-C = 1 + \kappa
\label{D}
\end{equation}
one can write $Z_{fer}[s]$ in the form
\begin{equation}
Z_{fer}[s] = \int {\cal{D}} a {\cal{D}} b \;
\exp (W[\tilde s s] - W[ a \tilde s s])
\exp\left((1+\kappa)W{[a \tilde s s b ]} \right)
\label{fifi}
\end{equation}
Now, the $b$ integration can be
trivially factorized this leading to
\begin{equation}
Z_{fer}[s] =
\int {\cal{D}} a \;\exp(-W[a \tilde s s] + W[\tilde s s] ) \,.
\label{su}
\end{equation}
or, after the shift
$ a {\tilde s} s \to \tilde s a s$
\label{ayy}
\begin{eqnarray}
Z_{fer}[s_+]& = &
\int {\cal{D}} a \;\exp(-W[a] + \frac{i}{4\pi} tr \int d^2x (s_+ a \partial_- a^{-1}
+ \nonumber \\
& & s_- a^{-1} \partial_+ a) ) \times
\exp(\frac{1}{4\pi} tr \int d^2x (a^{-1} s_+ a s_- - s_+ s_-))
\label{chuchi}
\end{eqnarray}
We have then arrived to the identity
\begin{equation}
Z_{fer}[s] = Z_{bos}[s]
\label{cg}
\end{equation}
where $Z_{bos}[s]$ is
the generating function for a Wess-Zumino-Witten model. Differentiation with
respect to any one of the
two sources gives correlation functions in a given chirality sector.
The answer corresponds to Witten's bosonization recipe \cite{Wi}
\begin{equation}
\bar \psi t^a \gamma_+ \psi \to \frac{i}{4\pi} a^{-1} \partial_+ a
\label{rec1}
\end{equation}
\begin{equation}
\bar \psi t^a \gamma_- \psi \to \frac{i}{4\pi} a \partial_- a^{-1} \, ,
\label{rec2}
\end{equation}
the l.h.s. to be computed in a free fermionic model, the r.h.s. in
a Wess-Zumino-Witten model.
\section{ $d=3$ non-abelian bosonization}
Contrary to the case of two-dimensional massless
fermions,
one cannot compute exactly the Dirac operator determinant
for $d > 2$
in the presence of an arbitrary gauge field $b_\mu$,
neither in the massless nor in the massive case. This
implies the necessity of making approximations at some stage of
our bosonization procedure to render calculations feasible.
In the $d=3$ Abelian case
one can handle these approximations in a very general framework
\cite{BFO},\cite{LNS}. Being the non-Abelian case far more complicated
than the Abelian one, we shall only discuss the limiting case
of very massive fermions, for which
the fermion determinant is related to the Chern-Simons (CS)
action \cite{NS}-\cite{Red}.
A second problem arising when one tries to extend the non-Abelian
boson-fermion mapping from $d=2$ to $d=3$
concerns the central role that plays the Polyakov-Wiegman identity,
related to the holomorphic
properties of the two-dimensional model \cite{Wit}. In principle, a
$3$-dimensional analogue of this identity is not available and
this forbids a trivial extension to $d=3$ of the procedure
described in the precedent section for two-dimensional
bosonization. However, as we shall see,
once one introduces the auxiliary field $b_\mu$, a BRST
invariance of the kind arising in topological
field theories \cite{BS}-\cite{BRT} can be unraveled.
The use of BRST technique for bosonization of fermion models
was initiated in the developement of the smooth bosonization
approach \cite{DNS1}-\cite{DNS3},
\cite{DNS4}-\cite{DNS5},
closely related to bosonization duality and
to the present
treatment. In the present case, it
allows to factor out the auxiliary field
in the same way Polyakov-Wiegmann identity did
the job in $d=2$.
The resulting bosonization action coincides with that obtained using
a completely different approach \cite{B},
based in the use of an interpolating Lagrangian \cite{VN}-\cite{vN}.
The advantage of the present method lies in the fact that the BRST
symmetry can be formulated in arbitrary dimensions while the
interpolating Lagrangian, which replaces the role of this
symmetry in decoupling
auxiliary and bosonic fields is in principle applicable only in odd-dimensional spaces.
We consider $N$ massive Dirac fermions in $d=3$ Euclidean dimensions
with Lagrangian
\begin{equation}
L = \bar\psi (\id + m ) \psi
\label{3L}
\end{equation}
The corresponding generating functional reads
\begin{equation}
Z_{fer}[s] = \int {\cal{D}} \bar \psi {\cal{D}} \psi
\exp[-\int d^3x \bar \psi (\id + /\kern-.52em s + m ) \psi ]
\label{41}
\end{equation}
Again, we introduce an auxiliary vector field
$b_\mu$ and use the $d=3$ identity
(proven in the Appendix)
\begin{equation}
Z_{fer}[s] = X[s]^{-1}\!\!
\int \!\!Db_\mu X[b]
\det(2\varepsilon_{\mu \nu \alpha} D_\nu[b])
\delta( {^*\!\!f}_\mu[b] - {^*\!\!f}_\mu[s] )
\det(\id + m + /\kern-.52em b) \label{IX}
\end{equation}
Here
\begin{equation}
^*\!\!f_\mu = \varepsilon_{\mu \nu \alpha}
f_{\nu\alpha}
\label{II}
\end{equation}
Concerning $X[b]$, it is an arbitrary functional which can be introduced
in order to control the issue of symmetries at each stage of our
derivation.
Indeed, bosonization
of three dimensional very massive fermions ends with a
bosonic field with dynamics governed by a
Chern-Simons action. As explained in \cite{FAS}, an
appropriate choice
of $X$ allows to end with the natural gauge connection
transformation law for this bosonic field. Following
\cite{FAS}, we choose $X$ in the form
\begin{equation}
X[b] = \exp(\mp \frac{i}{24\pi} \varepsilon_{\mu\nu\alpha}
tr \int d^3x b_\mu b_\nu b_\alpha )
\label{x}
\end{equation}
We can see at this point how an exact bosonization rule for the fermion
current can be derived independently of the fact that one cannot
calculate exactly the fermion determinant for $d > 2$.
Indeed, if we introduce a Lagrange
multiplier $A_\mu$ to represent the delta function, we can write $Z_{fer}$
in the form
\begin{equation}
Z_{fer}[s] = X[s]^{-1}
\int {\cal{D}} A_\mu \exp \left( \mp \frac{i}{16\pi}
tr \int d^3x A_\mu {^*\!\!f}_\mu[s] \right) \times
\exp(-S_{bos}[A]) \label{ches}
\end{equation}
where we have defined the bosonic action $S_{bos}[A]$
as
\begin{eqnarray}
\exp(-S_{bos}[A]) & = & \int {\cal{D}} b_\mu \det(\id + m + /\kern-.52em b) X[b] \times
\nonumber \\
& & \det(2\varepsilon_{\mu \nu \alpha}D_\nu[b])
\exp\left( \pm \frac{i}{16\pi}
tr \int d^3x A_\mu {^*\!\!f}_\mu[b] \right)
\label{jeje}
\end{eqnarray}
With the choice (\ref{x}) one indeed has gauge invariance
of $S_{bos}[A]$, $A_\mu$ and $b_\mu$ both transforming as gauge fields,
and one also explicitely verifies eq.(\ref{idiss}).
Then, from eq.(\ref{ches}) we have
\begin{equation}
\bar \psi \gamma_\mu t^a \psi \to
\pm \frac{i}{8\pi}\varepsilon_{\mu \nu \alpha}
\partial_\nu A^a_\alpha \,\, .
\label{Xsss}
\end{equation}
In writing eq.(\ref{Xsss}) we have ignored terms quadratic
and cubic in the source
which, as in $d=2$, are irrelevant for the current algebra.
Correlation functions of currents pick a contribution
from these terms, as already discussed
in other approaches to bosonization \cite{dVR}-\cite{F}. Having
these terms local support, they do not contribute
to the current commutator algebra. (That this is so
can be easily seen using for example the
Bjorken-Johnson-Low method).
We insist that our result (\ref{Xsss}) does not imply any kind
of approximation. However, to achieve
a complete bosonization, one needs an explicit local
form for the bosonic
action and it is at this point where
approximations have to be envisaged so
as to evaluate the fermion determinant. In $d=3$ dimensions
this determinant cannot be
computed exactly. However, all approximation approaches and
regularization schemes have shown the occurrence
of a parity violating Chern-Simons term together with
parity conserving terms which can be computed approximately.
We shall use the result obtained by making an expansion
in inverse powers of the fermion mass \cite{Red},
\begin{equation}
\ln \det (\id + m + /\kern-.52em b) =
\pm \frac{i}{16\pi} S_{CS}[b] +
I_{PC}[b] +
O(\partial^2/m^2) ,
\label{9f}
\end{equation}
where the Chern-Simons action $S_{CS}$ is given by
\begin{equation}
S_{CS}[b] = \int\!d^3\!x\,
\varepsilon_{\mu\nu\lambda} \mathop{\rm tr}\nolimits \int\!d^3\!x\,
(
f_{\mu \nu} b_{\lambda} -
\frac{2}{3} b_{\mu}b_{\nu}b_{\lambda}
) .
\end{equation}
Concerning the parity conserving contributions, one has
\begin{equation}
I_{PC}[b] =
- \frac{1}{24\pi m} \mathop{\rm tr}\nolimits\int\!d^3\!x\, f^{\mu\nu} f_{\mu\nu}
+ \cdots ,
\label{8f}
\end{equation}
We can then write, up to corrections of order $1/m$, the bosonic
action $S_{bos}[A]$
in the form
(From here on we shall omit to indicate the trace {\it tr}
for notation simplicity)
\begin{eqnarray}
\exp(-S_{bos}[A]) & = & \int {\cal{D}} b_\mu \, X[b]
\exp(\pm \frac{i}{16\pi} S_{CS}[b]) \times
\nonumber \\
& & \det(2\varepsilon_{\mu \nu \alpha}D_\nu[b])
\exp\left( \pm \frac{i}{16\pi}
\int d^3x A_\mu {^*\!\!f}_\mu[b] \right)
\label{ufin}
\end{eqnarray}
We shall now introduce ghost fields ${\bar c}_\alpha$ and
$c_\alpha$ to write the determinant in the r.h.s. of eq.(\ref{ufin}).
With this, $Z_{fer}[s]$ takes the form
\begin{eqnarray}
Z_{fer}[s] & = & X[s]^{-1}
\int {\cal{D}} b_\mu {\cal{D}} {\bar c}_\alpha {\cal{D}} c_\alpha {\cal{D}} A_\mu
\exp \left( \mp\frac{i}{16\pi}
\int d^3x A_\mu {^*\!\!f}_\mu[s] \right) \times
\nonumber \\
& & \exp(-S_{eff}[b,A,\bar c, c])
\label{312}
\end{eqnarray}
with
\begin{eqnarray}
S_{eff}[b,A,\bar c, c] & = &
\mp \frac{i}{16\pi} S[b] \nonumber \\
& & \mp \frac{i}{8\pi} \varepsilon_{\mu \nu \alpha} \int d^3x
( A_\mu (\partial_\nu b_\alpha + b_\nu b_\alpha)
- {\bar c}_\mu D_\nu [b] c_\alpha
)
\label{312S}
\end{eqnarray}
and
\begin{equation}
S[b] = 2 \varepsilon_{\mu \nu \alpha} \int d^3x
b_\mu(\partial_\nu b_\alpha +\frac{1}{3} b_\nu b_\alpha)
\label{sssx}
\end{equation}
At this point we have arrived to an exact bosonization recipe for the fermion
current, eq.(\ref{Xsss}), but we still need
an explicit formula for the
bosonic action as a functional of $A_\mu$. This requires integration over
the auxiliary
fields $b_\mu$, $\bar c_\mu$ and $c_\mu$ of the complicated effective action
$S_{eff}$ as defined by eq.(\ref{312S}). In the two-dimensional case, this
last step was possible because Polyakov-Wiegmann identity allowed us to
decouple the auxiliary fields from the bosonic field ($A_\mu$). In
the present case, integration will be possible because of the existence
of an underlying BRST invariance that can be made apparent in $S_{eff}$.
In order to directly get an {\it off-shell} nilpotent set of BRST
transformations leaving invariant the effective action, we shall
introduce additional auxiliary fields \cite{bas}, thus writing
\begin{eqnarray}
Z_{fer}[s] & = & X[s]^{-1}
\int {\cal{D}} b_\mu {\cal{D}} {\bar c}_\alpha {\cal{D}} c_\alpha {\cal{D}} A_\mu {\cal{D}} h_\mu {\cal{D}} l
{\cal{D}} \bar \chi
\nonumber \\
& & \exp \left( \mp\frac{i}{16\pi}
\int d^3x A_\mu {^*\!\!f}_\mu[s] \right)
\exp(-{\tilde S}_{eff}[b,A,\bar c, c, h, l, \bar \chi])
\label{barra}
\end{eqnarray}
with $\tilde S_{eff}$ defined as
\begin{eqnarray}
& & {\tilde S}_{eff}[b,A,\bar c, c, h, l, \chi] =
\mp \frac{i}{16\pi} S[b-h]
\mp \frac{i}{16\pi} \int d^3x
(l h_\mu h_\mu - 2 \bar \chi h_\mu c_\mu ) \nonumber \\
& & \mp \frac{i}{8\pi} \varepsilon_{\mu \nu \alpha} \int d^3x
( A_\mu (\partial_\nu b_\alpha + b_\nu b_\alpha)
- {\bar c}_\mu D_\nu [b] c_\alpha
)
\label{312SS}
\end{eqnarray}
Integration over the auxiliary field $l$ makes $h_\mu = 0$
this showing the equivalence of eq.(\ref{barra}) and eq.(\ref{312}).
Now, the effective action ${\tilde S}_{eff}$ is invariant
under BRST transformations defined as
\[
\delta {\bar c}_\alpha = A_\alpha
\;\;\;\;\;\;
\delta A_\alpha = 0
\]
\[
\delta b_\alpha = c_\alpha
\;\;\;\;\;\;
\delta c_\alpha = 0
\]
\begin{equation}
\delta h_\alpha = c_\alpha
\;\;\;\;
\delta \bar \chi = l
\;\;\;\;
\delta l = 0
\label{trb}
\end{equation}
This
BRST transformations are
related to those employed
in the smooth bosonization \cite{DNS1}-\cite{DNS3},
\cite{DNS4}-\cite{DNS5} approach and
resemblant of those arising
in topological field theories \cite{BS}-\cite{BRT}. For example, in
$d=4$ topological
Yang-Mills theory the invariance of the starting classical action
(the Chern-Pon\-trya\-gin topological charge)
under the most general transformation of the gauge field,
$b_\mu \to b_\mu + \epsilon_\mu$, leads to a BRST
transformation for $b_\mu$ of the form $\delta b_\mu = c_\mu$, which
corresponds to that in formula (\ref{trb}) \cite{Wi0}-\cite{BS}.
Closer to our model are the so-called
Schwartz type topological theories which
include the Chern-Simons theory and the
BF model
analyzed in detail in refs.\cite{hor}-\cite{BRT}. It should be
stressed that the topological character of the
effective action (\ref{312S}) exclusively concerns
the large fermion mass regime where the fermion determinant can be
written in terms of the CS action.
Now, using transformations (\ref{trb}), $\tilde S_{eff}$
can be compactly written in the form
\begin{equation}
\tilde S_{eff}[b,A,\bar c, c] =
\mp \frac{i}{16\pi} S[b-h] \mp
\frac{i}{8\pi} \int d^3x \, \delta{\cal F}[\bar c,b,h,\bar \chi]
\label{sss}
\end{equation}
with
\begin{equation}
{\cal F} = \varepsilon_{\mu \nu \alpha}
\bar c_\mu(\partial_\nu b_\alpha + b_\nu b_\alpha)
+ \frac{1}{2} \bar \chi h_\mu h_\mu
\label{off}
\end{equation}
At this point, an arbitrary functional ${\cal G}$
may be added to ${\cal F}$ without changing
the partition function since it will enter
in $Z_{fer}$ as an exact BRST form. The
idea is to choose ${\cal G}$ so as to decouple
the auxiliary field $b_\mu$ (to be integrated out
afterwards) from the vector field $A_\mu$
which will be the bosonic counterpart of the original
fermion field.
We shall then consider
\begin{equation}
{\cal F} \to {\cal F} + {\cal G}
\label{s1}
\end{equation}
with
\begin{eqnarray}
{\cal G} & = &
\frac{1}{2} \varepsilon _{\mu \nu \alpha }\bar c_\mu \;
([b_\nu,A_\alpha] + [A_\nu,A_\alpha] +C[b_\nu,h_\alpha]
+(1+C)[A_\nu,h_\alpha] \nonumber \\
& & - (C+1) [h_\nu,h_\alpha] +2\partial_\nu b_\alpha
+4\partial_\nu A_\alpha +2C \partial_\nu h_\alpha )
\label{el1}
\end{eqnarray}
Here $C$ is an arbitrary constant.
The addition of $\delta {\cal G}$ allows us to make
contact at this point with the effective action discussed in
refs.\cite{B},\cite{vN}. Indeed,
after the shift
\begin{equation}
b_\mu \to 2b_\mu - A_\mu + h_\mu
\label{LG}
\end{equation}
(the new $b_\mu$ transforms again as
a gauge connection, with $h_\mu$ transforming covariantly)
the Lagrange multiplier $A_\mu$
(which will play the role of the bosonic field in our bosonization
approach, as identified by the
source term)
completely decouples for $h_\mu = 0$, so that
integrating out auxiliary fields we end with
\begin{equation}
Z_{fer}[s] = {\cal N}X[s]^{-1}
\int {\cal{D}} A_\mu
\exp ( \mp\frac{i}{16\pi}
\int d^3x A_\mu {^*\!\!f}_\mu[s] ) \times
\exp(\pm \frac{i}{16\pi}S_{CS}[A]).
\label{chesi}
\end{equation}
Here ${\cal N}$ is a constant (i.e. it
is independent of the source) resulting from integration
of auxiliary, ghosts and the $b$ field,
\begin{equation}
{\cal N} = \int {\cal{D}} b_\mu \,\,
det\left( 2(2+C)\varepsilon_{\mu\nu\alpha} D_\nu[b] \right)
\exp(\pm \frac{i}{4\pi} S_{CS}[b])
\label{final}
\end{equation}
We have then the bosonization result
\begin{equation}
Z_{fer} [s] \approx Z_{CS}[s]
\label{uli}
\end{equation}
where $\approx$ means that our result is valid up to $1/m$ corrections
since we used a result for the fermion
determinant which is valid up to this order.
We then see that we have ended with a Chern-Simons action as the bosonic
equivalent of the original free fermion action with a coupling
to the external source $s_\mu$ of the form $ A_\mu {^*\!\!f}_\mu[s] $.
In considering fermion current bosonization
within the $1/m$ approximation, the following facts
should be taken into account.
It is at the lowest order in $1/m$ that
the resulting bosonic action is topological and a
large BRST invariance is unraveled.
Now, using
the freedom to
modify the action by BRST exact forms, one
could think of adding to the topological bosonic action
terms of the form $ \delta{\cal H}$ with
\begin{equation}
{\cal H} = \int d^3x \ \varepsilon_{\mu \nu \alpha}
{\bar c}_{\mu} {\cal H}_{\nu \alpha}[s] \, ,
\label{ad1}
\end{equation}
with ${\cal H}_{\mu \nu}[s]$ an arbitrary functional of the external
source $s_{\mu}$. In particular, choosing adequately ${\cal H}$ one
could think of changing or even
eliminating, to this order in $1/m$, the source dependence from
$Z_{fer}[s]$.
Now, this is a characteristic of
Schwarz like topological models \cite{BRT}. In particular,
the phase space of the Chern-Simons theory is
the moduli space of flat connections on the given space manifold.
So, if one looks {\it up to this order in $1/m$},
at the
generating functional of current Green
functions, one has, from Eqs.(\ref{41}) and (\ref{9f}) that the generating functional of connected Green functions is
precisely a Chern-Simons action for the source $s$,
\begin{equation}
W[s]=- log Z[s] = \mp \frac{i}{16\pi} S_{CS}[s].
\label{ad3}
\end{equation}
Thus, making functional derivatives in the above expression with respect
to the source and then putting the sources to zero, all
current vacuum expectation values vanish identically up to contact terms
(these terms, derivatives of the Dirac delta function, also vanish if we
regularize appropriately the product of operators at coincident points.
Moreover our results are valid in the $m\to\infty$ limit where the deep
ultraviolet region is excluded).
Non-vanishing observables are in fact
topological objets, non local functionals
of $A_\mu$ (Wilson loops) that
are in correspondance to knots polynomial invariants.
Hence the bosonization recipe
(\ref{Xsss}) when used
to this order in $1/m$
makes sense if one is to calculate vacuum expectation
values of fermion
objects leading for example to holonomies in terms of the bosonic
field $A_\mu$. This calculation was discussed at length in \cite{B}.
\section{Summary}
We have shown in this paper that the path-integral bosonization
approach developed in previous investigations
\cite{FS}-\cite{B} is well-suited to study fermion models
in $d \ge 2$ dimensions when a non-Abelian symmetry
is present.
We have started by reobtaining
in Section 2 the well-honoured non-Abelian bosonization
recipe for two dimensional
massless fermions. Although
well-known, this result allowed us to identify the point
in which the non-Abelian character of the symmetry
makes difficult the factorization of the path-integral
which will represent the partition function of the
resulting bosonic model. In two dimensions this
factorization can be seen as a result of the existence
of the Polyakov-Wiegmann identity for Wess-Zumino-Witten actions,
and this can {\it a priori} put some doubts on the possibility of
extending
the approach to $d>2$.
That also in $d=3$ one can obtain very simple bosonization
rules for the non-Abelian case is the main result of
section 3.
Concerning the fermion current, we obtained an
exact bosonization result which is the
natural extension of the Abelian case.
In respect with the bosonization recipe for the fermion action,
we considered
the case of very massive fermions
for which the fermion determinant
is related to the non-Abelian Chern-Simons action.
In this case the factorization of the auxiliary
and Lagrange multiplier fields is achieved after
discovering a BRST invariance reminiscent of that at the
root of topological models and
related to that exploited in the
smooth bosonization approach \cite{DNS1}-\cite{DNS3},
\cite{DNS4}-\cite{DNS5}.
Addition of BRST exact terms
allows us to extract the partition function for the
boson counterpart of the original fermion fields.
Our bosonization method starts by introducing
in the fermionic generating functional an
auxiliary field as it is done in the
smooth bosonization and duality
approaches to bosonization \cite{DNS1}-\cite{mar}. It becomes
clear in our approach that, for non-Abelian
symmetries, it is crucial to include the
``Faddeev-Popov'' like determinant which
accompanies the delta function imposing a condition
on the auxiliary field curvature. In fact, the BRST symmetry
which allowed to arrive to the correct bosonic generating functional
can be seen as a result of this fact and related to
the way in which BRST symmetry can be unraveled
by a change of variables as advocated in ref.\cite{bas}.
It should be stressed that the only approximation in our
approach stems from the
necessity of evaluating the fermion determinant which, in
$d > 2$, implies some kind of expansion. In the
present work we have used a result valid for very massive
fermions but one can envisage approximations which
can cover other regimes, in particular the massless case.
This was considered for the abelian case in \cite{BFO} and
the corresponding bosonization analysis thoroughly
discussed in \cite {LNS}. We expect that a similar analysis
can be done in the non-abelian case and we hope to report
on it in a future paper.
\newpage
\section*{Appendix}
\subsection*{ $d=2$}
We shall prove identity (\ref{idle}) used in our derivation
of $d=2$ bosonization rules,
\begin{equation}
\int {\cal{D}} b_\mu {\cal H}[b]
\delta\left[\varepsilon_{\mu \nu} (f_{\mu\nu}[b] - f_{\mu\nu}[s])\right]
=\int {\cal{D}} g {\cal H}[s^g]
\label{idlex}
\end{equation}
where ${\cal H}$ is a gauge-invariant functional. Note
that in eq.(\ref{idlex}) it is implicit that $b_\mu$ should
be treated as a gauge field and hence a gauge fixing is
required.
A convenient gauge choice
is
\begin{equation}
b_+ = s_+
\label{otrav}
\end{equation}
so that identity (\ref{idlex}) takes the form
\begin{equation}
\int {\cal{D}} b_+ {\cal{D}} b_- \;\Delta \delta(b_+ - s_+){\cal H}[b_+,b_-]
\delta[\varepsilon_{\mu \nu} (f_{\mu\nu}[b] - f_{\mu\nu}[s]]
= \int {\cal{D}} g {\cal H}[s^g]
\label{idleg}
\end{equation}
with $\Delta$ the Faddeev-Popov determinant for gauge condition
(\ref{otrav}),
\begin{equation}
\Delta = det D^{Adj}_+[s_+]
\label{FP}
\end{equation}
We now prove eq.(\ref{idleg}).
Let us start from the l.h.s. of eq.(\ref{idleg})
performing first the $b_+$ trivial integration and then the
$b_-$ one
\begin{eqnarray}
& & \int {\cal{D}} b_+ {\cal{D}} b_- \Delta \delta(b_+ - s_+){\cal H}[b_+,b_-]
\delta(\varepsilon_{\mu \nu} (f_{\mu\nu}[b] - f_{\mu\nu}[s]) = \nonumber\\
& &
\Delta[s_+] \int {\cal{D}} b_- {\cal H}[s_+,b_-]
\delta( D_+[s_+] b_- - D_+[s_+]s_- ) = \nonumber\\
& & \frac{\Delta[s_+]}{det D^{Adj}_+[s_+]} \int {\cal{D}} b_- {\cal H}[s_+,b_-]
\delta(b_- - s_-)
= {\cal H}[s_+,s_-]
\label{largui}
\end{eqnarray}
In the last line we have used the explicit form of the Faddeev-Jacobian to
cancel out both determinants.
Being ${\cal H}[s_+,s_-]$ gauge independent, we can rewrite
(\ref{largui}) in the form (appart from a gauge group volume factor)
\begin{equation}
\int {\cal{D}} b_+ {\cal{D}} b_- \Delta \delta(b_+ - s_+){\cal H}[b_+,b_-]
\delta \left[\varepsilon_{\mu \nu} (f_{\mu\nu}[b] - f_{\mu\nu}[s])
\right] =
\int {\cal{D}} g {\cal H}[s_+^g,s_-^g]
\label{uno}
\end{equation}
Identity (\ref{idlex}) is then proven.
\subsection*{ $d=3$}
The
proof of identity (\ref{IX}), the analogous in the $d=3$ case of
(\ref{idlex}), is very simple. One
wishes to prove that
the generating functional $Z_{fer}[s]$ in the presence of a source
$s_\mu$,
\begin{equation}
Z_{fer}[s] = \det (\id + m + /\kern-.52em s) = F[s]
\label{leg16}
\end{equation}
can be written in the form
\begin{equation}
Z_{fer}[s] = {X[s]}^{-1} \int {\cal{D}} b_\mu \, X[b]F[b]
\,
\det( 2\varepsilon_{\mu \nu \alpha} D_\nu[b] ) \,
\delta\left( \varepsilon_{\mu \nu \alpha} (f_{\nu \alpha}[b]
- f_{\nu \alpha}[s])\right)
\label{leg17}
\end{equation}
Here $X[b]$ is an arbitrary functional of $b_\mu$ satisfying
$X[0^g]=1$. As advocated in \cite{FAS}, its introduction
allows to end with a model in which the bosonic field transforms
as a connection, this being consistent with the fact
its dynamics is governed by a Chern-Simons action.
The proof of eq.(\ref{leg17}) is based on the
well-known identity
\begin{equation}
\delta(H[b]) = [\det (\delta H/\delta b)]^{-1} \delta(b - b^*)
\end{equation}
with $H[b^*] = 0$ and the fact that
the equation
$f_{\mu\nu}[b] = f_{\mu\nu}[s]$
has the unique solution $ b_\mu = s_\mu$.
Let us end by noting that if one compares formula (\ref{leg17})
in $d=3$ dimensions with the corresponding one in $d=2$ (for example
the identity (\ref{idlex})), one sees that a determinant equivalent to that
appearing in the former is absent in the latter. This is due
to the fact that the curvature condition requires three delta functions
in $d=3$ dimensions but only one in $d=2$. Handling these
delta functions leaves a jacobian
in three dimensions while no jacobian remains in two dimensions.
\vspace{1 cm}
\underline{Acknowledgements}: The authors wish to thank Matthias Blau,
Daniel Cabra, Fran\c{c}ois Delduc, C\'esar Fosco, Eric Ragoucy
and Frank Thuillier for helpful discussions
and comments. F.A.S. and
C.N. are partially suported
by Fundacion Antorchas, Argentina and a
Commission of the European Communities
contract No:C11*-CT93-0315.
| proofpile-arXiv_065-416 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
Since the discovery of high temperature superconductors (HTSC), the pairing
mechanism and the symmetry of the order parameter in these compounds are
key questions at stake \cite{Dyn94,Sch94}. There are several experimental
techniques which are able to address this problem. The experiments on
quasiparticle tunneling \cite{Val91}, the linear temperature dependence of
the penetration depth \cite{Har93}, the NMR and NQR measurements
\cite{Kit93,Sli93}, and angular resolved photoemission experiments in
Bi$_2$Sr$_2$CaCu$_2$O$_8$ \cite{She93,Ma} have yielded results consistent
with $d$-wave pairing. On the other hand, quasiparticle tunneling,
the exponential temperature dependence of the penetration depth, as well
as the measurements of the electronic Raman scattering in
Nd$_{2-x}$Ce$_x$CuO$_4$
are consistent with s-wave pairing \cite{Tra91,Mao94,Hac}.
The measurements of the magnetic field dependence of the dc-SQUID
(YBaCuO-Au-Pb arrangement) \cite{Woh94} clearly indicated d-wave behavior,
while the experiments on single Josephson-junction Pb-YBCO \cite{Sun94}
showed s-type behavior. So while the experimental evidence in favor of
d-wave symmetry of the order parameter continuously grows, there
is still no final consensus about it.
Raman scattering is a powerful tool to address the problem of the symmetry of
the order parameter. It allows to probe the symmetry of the scattering
tensor by simply choosing different polarization directions of the incident
and scattered light.
From the investigations of the Raman scattering in conventional
superconductors it is known that the superconducting transition manifests
itself in a renormalization of the electronic Raman scattering intensity
below T$_c$. It was found for Nb$_3$Sn and V$_3$Si \cite{Hac83,Die83} that
normalized Raman spectra of these compounds show for temperatures below T$_c$
a peak associated with the pair breaking process at the energy 2$\Delta$,
together with a strong decrease of the scattering intensity at frequencies
lower than 2$\Delta$. In high temperature superconductors, the first
measurements of electronic Raman scattering were reported in
Refs.\onlinecite{Lyo87,Baz87,Hac88,Coo88}. But in this case the behavior of
the electronic scattering differs from that in conventional superconductors:
A pair breaking peak develops in the spectra below T$_c$, but the scattering
intensity at frequencies below 2$\Delta$ does not show the usual sharp
decrease. Instead, a monotonic decrease toward zero frequency is found.
Moreover, for different symmetry components (A$_{1g}$, B$_{1g}$ and B$_{2g}$)
the renormalization of the scattering intensity for T$<$T$_c$ is different
and they exhibit peaks at different frequencies
\cite{Hac88,Coo88,Hof94,Sta92,Hof95,Nem93,Che92,Che94,Dev94,Dev95,Kra94,Kra95}.
These facts have been explained by Devereaux et al. \cite{Dev94,Dev95} in
terms of d$_{x^2-y^2}$-wave pairing. Their calculations of the scattering
cross-section have been performed for a cylindrical single-sheeted Fermi
surface in the framework of the kinetic equation approach. The symmetry of the
crystal was taken into account through calculating the Raman vertex,
which was expanded in terms of a complete set of crystal harmonics defined
on the Fermi surface. It was found that nontrivial coupling between the
Raman vertex and an assumed strongly anisotropic energy gap leads to the
strong symmetry dependence of the scattering intensity. The calculations
\cite{Dev94,Dev95} predict specific symmetry dependencies of the low
frequency scattering as well as the peak positions for the different
symmetry components of the electronic Raman scattering at temperatures
below T$_c$. The A$_{1g}$ peak position is sensitive to the parameters of
the model calculation. It will appear below
B$_{1g}$ peak position while with some parameters it may also appear at the
B$_{1g}$ peak position. Nevertheless there is one set of parameters
which can perfectly reproduce the experimental data\cite{Priv}.
Later on this model was criticized by Krantz and Cardona
\cite{Kra94,Kra95}. Their calculations \cite{Kra95} are based on the general
description of the Raman scattering cross-section through the inverse effective
mass tensor. In case of the multisheeted Fermi surface,(e.g. several CuO$_2$
planes per unit cell in HTSC) polarization dependent Raman efficiencies are
determined by the averages of the corresponding effective mass fluctuations.
The authors of Ref. \onlinecite{Kra95} used the effective masses from LDA band
structure calculations for YBa$_2$Cu$_3$O$_7$ to determine the Raman scattering
cross-section. They found that it contradicts the experimental results if one
uses only d-wave pairing for a multisheeted Fermi surface of YBa$_2$Cu$_3$O$_7$.
An explanation was given by assuming different types of the order parameter on
different sheets of the Fermi surface. For a single-sheeted Fermi
surface (i.e. one CuO$_2$-plane per unit cell) no mass fluctuations should
occur. Therefore in the framework of the effective mass fluctuation approach,
the A$_{1g}$ scattering component will be nearly totally screened and should
peak at the same position where the B$_{1g}$ scattering component
(2$\Delta_{max}$).
Therefore straihgtforward measurements of the electronic Raman scattering in
single-CuO$_2$ layered high-temperature superconductors (Tl-2201, La-214,
Bi-2201, (Nd,Ce)-214) should clarifay this controversial point.
Tl-2201 has the highest T$_c$ (up to 110K) \cite{Kol92} among the above
mentioned single-CuO$_2$ layered compounds. Therefore all effects due to the
gap opening are expected in the range of 300-600cm$^{-1}$, and they should
not be obscured due to the Rayleigh scattering at small wavenumbers. Nevertheless,
Raman measurements in only one pure scattering geometry (B$_{1g}$) are known
\cite{Nem93,Blu94} for this compound, which showed \cite{Nem93} becides a
T$_c$=80K two additional transitions at 100 and 125K, which can be indicative
of the Tl-2212 and Tl-2223 phases.
These facts lead us to reinvestigate the electronic Raman scattering in the
Tl-based high temperature superconductor Tl-2201 (with different oxygen
content) with one CuO$_2$-plane per unit cell. The comparison with the
results of electronic Raman scattering experiments reported for the high
temperature superconductors with several CuO$_2$-planes should clarify
whether the multiband scattering is indeed important. We should mention that
similar experiments on the single layered compound (La-214) were already
carried out \cite{Che94}. Nevertheless, in the framework of comparison of the
compounds with different number of CuO$_2$-planes the measurements on Tl-2201
are more favourable due to its high T$_c$.
\section{EXPERIMENT}
The investigated single crystals of Tl$_2$Ba$_2$CuO$_{6+\delta}$ (Tl-2201)
had the shape of rectangular platelets with the size of approximately
2x2x0.2 mm$^3$.
The two crystals investigated were characterized by a SQUID magnetometer.
T$_c$ was found to be 90$\pm$3 K and 80$\pm$5 K. The crystals are slightly
underdoped. It is known \cite{Shi92} that
differences in T$_c$ in Tl-2201 compound originate from different oxygen
concentrations. These crystals can be over- as well as underdoped. The
heavily oxygen doped crystals show a metallic type of conductivity
\cite{Shi92}
and do not show a superconducting transition. The orientation of the
tetragonal crystals was controlled by X-ray diffraction.
Raman measurements were performed on "as grown" surfaces of the freshly
prepared crystals. This is very important, because the crystal surface of
Tl-based superconductors as well as of all high temperature superconductors
is very sensitive to long exposure to air and especially to humid
atmosphere. For the Raman measurements a DILOR XY triple spectrometer
combined with a nitrogen cooled CCD detector was used. All Raman data
were obtained at nearly backscattering geometry. The photon excitation was
provided by the 488-nm line of an Ar$^+$ ion laser with laser power equal to
15W/cm$^2$. The estimated additional heating did not exceed 5K.
\section{EXPERIMENTAL RESULTS}
All measurements were performed with the polarization of the incident and
scattered light parallel to the basal plane of the crystal, i.e. the
CuO$_2$-planes. It was possible to measure the A$_{1g}$, B$_{1g}$, and B$_{2g}$
symmetry components of the Raman scattering cross-section. In addition to the
previously published phonon peaks ($\approx$123cm$^{-1}$,
$\approx$169cm$^{-1}$, $\approx$490cm$^{-1}$, $\approx590$cm$^{-1}$,
$\approx$610cm$^{-1}$) \cite{Gas89,Kall94} we have detected some
additional phonons ($\approx$240cm$^{-1}$, $\approx$300cm$^{-1}$,
$\approx$330cm$^{-1}$, $\approx375$cm$^{-1}$)
which we believe are the defect induced infrared active phonons. For all
scattering geometries the spectra for temperatures well below T$_c$ were divided
by the spectra just above T$_c$ in order to emphasize the redistribution of
the scattering intensity in the superconducting state compared to the normal
state.
The results of the electronic Raman scattering in the crystals of Tl-2201
(T$_c$=80K,90K) are shown on Figs. \ref{f1}-\ref{f5}.
In the crystal with
T$_c$=80K the B$_{1g}$ scattering component measured in X'Y' configuration
shows a well-defined peak at 430$\pm$15 cm$^{-1}$ (Fig. 1a). The X' and Y'
axes are rotated by 45$^\circ$ with respect to the crystal X and Y axes,
respectively, which are parallel to the crystallographic axes. The B$_{2g}$
scattering component in Fig. \ref{f1}b is less intense, but shows also a broad
maximum with an average frequency of 380$\pm$35cm$^{-1}$. Raman spectra in the
X'X' and XX geometries are presented in Fig. \ref{f2}a and b, showing spectra
of A$_{1g}$ + B$_{2g}$ and A$_{1g}$ + B$_{1g}$ scattering components,
respectively. In order to evaluate the A$_{1g}$ scattering component we
subtracted the B$_{1g}$ and B$_{2g}$ components (see Fig. \ref{f1}a,b) from
the XX and X'X' spectra, respectively. As one can see from Fig. \ref{f3}a,b
the A$_{1g}$ scattering component peaks in both cases, at 345$\pm$20cm$^{-1}$. For
the crystal with T$_c$=90K we found peaks of the B$_{1g}$, A$_{1g}$ and
B$_{2g}$ scattering components at 460$\pm$15cm$^{-1}$,
350$\pm$20cm$^{-1}$, and 400$\pm$35cm$^{-1}$, respectively
(Fig. \ref{f4}, lower panel).
Another very important observation is that the low frequency behavior of
the electronic Raman scattering exhibits strong anisotropy with respect
to the symmetry components. One can see in Fig.4a (upper and lower panel)
that the intensity decrease of the B$_{1g}$ scattering component toward lower
frequencies fits the $\omega^3$ law predicted by Devereaux et al. \cite{Dev95}.
For the A$_{1g}$ and B$_{2g}$ scattering components in Fig. \ref{f4}b and c,
respectively, there is a linear decrease, which also agrees with the predictions
by Deveraux et al. \cite{Dev95}.
A summary of our results on Tl-2201 is presented in Table
\ref{t1}.
In order to follow the temperature behavior of the superconductor gap,
we have measured the temperature dependence of the electronic Raman
scattering. Following Devereaux et. al.\cite{Dev94,Dev95}, we assume that
the peak in the B$_{1g}$ component of the electronic scattering corresponds
to the value of 2$\Delta_{max}$. In Fig. \ref{f5}a and b, respectively, we
show the B$_{1g}$ and A$_{1g}$ scattering component of Tl-2201 (T$_c$=80K) at
different temperatures between 10K and T$_c$ divided by the spectrum at 100K.
The experiments for the 90-K crystal yielded a similar behavior.
With increasing temperature the intensity of the peak in Fig. \ref{f5}a
associated with the pair breaking process decreases and the maximum shifts
slightly to lower frequencies. Obviously, the temperature dependence of the
superconductor gap does not follow the BCS behavior.
In other words, upon cooling below T$_c$ the gap opens more abruptly than
predicted by BCS theory. These results are similar to results reported
for underdoped Bi-2212 \cite{Hof95}. Because the peak position of the A$_{1g}$
scattering component in Fig. \ref{f5}b has larger error bars compared to the
B$_{1g}$ component, one cannot definitely say whether the data fit the BCS
behavior or not.
We also searched for superconductivity-induced changes in frequency and
linewidth of the optical phonons. With the resolution of 1cm$^{-1}$ we have
not observed such changes. Upon heating from 10K up to 200K the frequencies
of all phonons decreased and the linewidths increased monotonically.
\section {DISCUSSION}
The Raman scattering intensity can be written in terms of the
differential scattering cross section\cite{Dev95}:
\begin{equation}
\frac{\partial^2\sigma}{\partial\omega\partial\Omega}=
\frac{\omega_s}{\omega_i}r_0^2S_{\gamma\gamma}(\vec{q},\omega)
\end{equation}
with
\begin{equation}
S_{\gamma\gamma}(\vec{q},\omega)=-\frac{1}{\pi}\left[1+n(\omega)\right]\Im
m\chi_{\gamma\gamma}(\vec{q},\omega)
\end{equation}
Here r$_0=e^2/mc^2$ is the Thomson radius, $\omega_i (\omega_s)$ is the
frequency of incident (scattered) photon, $\hbar$ and k$_B$ were set to 1.
S$_{\gamma\gamma}$ is the generalized structure function, which is connected
to the imaginary part of the Raman response function $\chi_{\gamma\gamma}$
through the fluctuation-dissipation theorem;
$n(\omega)=1/[\exp (\omega/T)-1]$ is the Bose-Einstein distribution function.
The Raman response function can be written as \cite{Kle84}:
\begin{equation}\label{rares}
\chi_{\gamma\gamma}(\vec{q},\omega)=\langle\gamma^2_{\vec{k}}\lambda_{\vec{k}}
\rangle -\frac{\langle\gamma_{\vec{k}}\lambda_{\vec{k}}\rangle^2}
{\langle\lambda_{\vec{k}}\rangle}
\end{equation}
with the Raman vertex $\gamma_{\vec{k}}$ written as
\begin{equation}
\gamma_{\vec{k}}(\omega_i,\omega_s)=\sum_L\gamma_L(\omega_i,\omega_s)
\Phi_L(\vec{k}),
\end{equation}
where $\Phi_L(\vec{k})$ are either Brillouin zone or Fermi surface harmonics
\cite{Dev95} which transform according to point group transformations of the
crystal and $\lambda_{\vec{k}}$ is the Tsuneto function:
\begin{equation}
\lambda_{\vec{k}}\propto\frac{\left|\Delta_{\vec{k}}\right|^2}
{\omega\sqrt{\omega^2-4\left|\Delta_{\vec{k}}\right|^2}}.
\end{equation}
The brackets $\langle\cdots\rangle$ in Eq. \ref{rares} denote an average
of the momentum $\vec{k}$ over the Fermi surface.
As is obvious, Raman scattering probes only $\left|\Delta\right|^2$.
Therfore it is not possible to determine whether the gap function changes sign
for different directions of $\vec{k}=(k_x,k_y)$ or not. But nevertheless
the symmetry of the gap function can be inferred from the specific spectral
features of each symmetry component of the electronic Raman scattering.
For the gap of d-wave symmetry ($\Delta_{\vec{k}}=\Delta_{max}\cdot\cos
2\phi$, where $\phi$ is an angle between $\vec{k}$ and the a-axis),
calculations \cite{Dev94,Dev95} predict different low frequency
behavior for the different symmetry components.
For B$_{2g}$ and A$_{1g}$ scattering components it should show a linear
dependence in $\omega$, but for B$_{1g}$ it should be $\sim\omega^3$. The
appearance of a power law in the low frequency scattering characterizes an
energy gap which vanishes on lines on the Fermi surface. An appearance of
$\omega^3$ law in B$_{1g}$ scattering component is specific for
d$_{x^2-y^2}$-wave pairing \cite{Dev95}. The predicted values of the
peak maxima for the B$_{1g}$, B$_{2g}$ and A$_{1g}$ scattering components are
$\sim 2\Delta_{max}$, $\sim 1.6\Delta_{max}$ and $\sim 1.2\Delta_{max}$,
respectively. These above mentioned peculiarities appear in our data. Indeed,
the low frequency behavior of the B$_{1g}$ scattering component
definitely differs from a linear behavior as seen in Fig. \ref{f4}a, whereas
for the A$_{1g}$ and B$_{2g}$ scattering components it is linear in $\omega$
(see Fig. \ref{f4}b,c). For both crystals, the B$_{1g}$ scattering component
peaks at a higher frequency than the B$_{2g}$, which in turn peaks at a higher
frequency than the A$_{1g}$ component.
Since Raman scattering does not probe the phase of the order parameter it is
important to take into consideration other types of the pairing which can
also have nodes on the Fermi surface, but do not change the sign, i.e.
$s+id$-pairing, or strongly anisotropic s-pairing.
For the $s+id$-pairing \cite{Dev95}
$(\Delta (k)=\Delta_s + i\Delta_d\cos 2\phi)$ one gets the threshold at
$\omega = 2\Delta_s$ (minimum pair breaking energy). While A$_{1g}$ and
B$_{2g}$ scattering components exhibit a jump at this frequency, the
B$_{1g}$ scattering component shows a continuous rise from zero and up to the
peak at $\omega = 2\Delta_{max} =2\sqrt{(\Delta_s^2+\Delta_d^2)}$. The A$_{1g}$
and B$_{2g}$ scattering components also show broad maxima as in the case of
pure d$_{x^2-y^2}$-wave pairing, but these maxima will be cut-off toward lower
frequencies due to the strong jump at 2$\Delta_s$. Thus one should observe a
low-frequency cutt-off in both A$_{1g}$ and B$_{2g}$ scattering components,
which, however, is not observed in our data.
For anisotropic s-pairing, showing the minimum of the gap on the diagonals
of the two-dimensional Brillouin zone, ($\Delta (k) = \Delta_0 +
\Delta_1\cos^42\phi$) one gets a single threshold on 2$\Delta_0$ for all
scattering components as well as a peak at $\omega = 2\Delta_{max} =
2(\Delta_0 + \Delta_1)$ for the B$_{1g}$ scattering component. Therefore we
will expect a picture which is very similar to the case of $s+id$-pairing,
with one exception. The B$_{1g}$ scattering component should show an
additional shoulder at the same position where the A$_{1g}$ and B$_{2g}$
scattering components show peaks \cite{Dev95}. This is also not the case for
our data. In principle one can assume $\Delta_0$ to be very small or even
zero. In this case one gets peaks at 2$\Delta_{max}$, 0.6$\Delta_{max}$ and
0.2$\Delta_{max}$ for the B$_{1g}$, B$_{2g}$ and A$_{1g}$ scattering
components \cite{Dev95}, respectively. In addition, the low frequency behavior
of the B$_{1g}$ scattering component will be linear. This also contradicts
our results.
Recently the model calculations of Devereaux et al. were criticized by Krantz
and Cardona \cite{Kra94,Kra95}. The main argument against this theoretical
model is that the realistic electronic band structure of the crystal is
important, but that the one-sheeted Fermi surface approximation used by
Devereaux et al.\cite{Dev95} is inappropriate. The authors of
Ref.\onlinecite{Kra95} used a numerical model based on the LDA band
structure calculations for YBaCuO in order to take into account the
multisheeted Fermi surface of the superconductors with several CuO$_2$ -planes.
It was pointed out that for the $\Delta = \Delta_0 cos 2\phi$ (d-wave pairing)
and a multisheeted Fermi surface the calculations lead to a contradiction
with the experiment, i.e. the A$_{1g}$ and B$_{1g}$ scattering components
peak at the same position 2$\Delta_0$. In order to get consistency with the
experiment, different types of the order parameter on different sheets of the
Fermi surface were proposed. Only in this case the calculations in
Ref.\onlinecite{Kra95} were able to get different positions of the maxima of
the B$_{1g}$, A$_{1g}$, and B$_{2g}$ scattering components. For a one-sheeted
Fermi surface the authors of Ref.\onlinecite{Kra95} found identical positions
of the maxima for the A$_{1g}$ and B$_{1g}$ components, but a different
position for the B$_{2g}$ component. Hence it was concluded that any
difference in peak position of the A$_{1g}$ and B$_{1g}$ component is only
consistent with multiband scattering of a multisheeted Fermi surface and
different gap symmetries for each of the sheets. For superconductors with
one CuO$_2$-plane, a multisheeted Fermi
surface is invoked originating from Tl-like s-states \cite{Kra95}
(Tl-2201) or from Sr-doping\cite{Kra} (La$_{2-x}$Sr$_x$CuO$_4$) in
order to yield a difference in peak position for the A$_{1g}$ and B$_{1g}$
components. However, no experimental proof for such a Fermi surface
contribution exists so far. Moreover, the calculations in
Ref.\onlinecite{Kra95}
failed in explaining of the symmetry-dependent low-frequency dependence of the
Raman scattering intensity, whereas this important experimental fact was
observed not only in our experiments, but also in Bi-Sr-Ca-Cu-O
\cite{Sta92,Hof95,Dev94}, Y-Ba-Cu-O\cite{Hac88,Coo88} and
La-Sr-Cu-O \cite{Che94} systems. In addition, it is obvious that all
superconductors with different crystal structures have a different
electronic structure. Hence, if the multiband scattering model would be
crucial we would expect absolutely different behavior for the different
superconductors which is actually in contradiction with existing experimental
results. Even if one compares the superconductors with the same number of
CuO$_2$ planes,
one finds that, while the interplanar distance (distance between Cu atoms in
different CuO$_2$-planes) is quite similar, the dimpling (in-plane Cu-O-Cu
angle) differs very much from compound to compound (see Table \ref{t2}).
YBa$_2$Cu$_3$O$_7$ exhibits the largest dimpling compared to other compounds.
Strong dimpling should lift the degeneracy of otherwise identical
CuO$_2$-planes or Fermi surface sheets. This dimpling can strongly affect the
LDA calculations because the interplanar interaction should depend on this
parameter.
And finally on top of that, use of the effective mass approach is very much
questionable in case of high temperature superconductors, because
this approach can be used only for nonresonant Raman scattering \cite{Abr}.
In high temperature superconductors we, however, are always in the regime of
the resonant scattering. Moreover, the electron correlation effects in HTSC are
not treated sufficiently by LDA.
In contrast to the conclusion of Ref. \onlinecite{Kra95} our experiments show that the
one-CuO$_2$-plane compound Tl-2201 shows very similar behavior compared to
compounds with several CuO$_2$-planes, such as Tl-2223, Bi-2212 and YBaCuO
\cite{Hac88,Coo88,Hof94,Sta92,Hof95,Nem93,Che92,Che94,Dev94,Dev95,Kra94,Kra95}.
We also found that the frequency of the B$_{1g}$ maximum scales
with T$_c$, and it corresponds to the value 2$\Delta_{max}/k_BT_c=7.6\pm 0.4$.
This value is very close to the values (with the exception of
Nd$_{2-x}$Ce$_x$CuO$_4$ \cite{Hac}) found for other high temperature
superconductors as shown in Table \ref{t3}.
The temperature dependence of the gap (B$_{1g}$ component in Fig. \ref{f5}a)
in our experiment differs from the BCS behavior, i.e. upon cooling the gap
opens more abruptly than predicted by BCS theory.
This is consistent with the spin fluctuation theory of high temperature
superconductivity \cite{Mon92}, favoring d$_{x^2-y^2}$-wave pairing. The
model considers pair binding as well as pair breaking effects due to the
spin fluctuations. Gap opening leads to a suppression of low-frequency spin
fluctuations and therefore to reduced pair-breaking. Therefore in underdoped
crystals (we consider our Tl-2201 crystals as underdoped) this effect
will lead to a more abrupt opening of the gap upon cooling below T$_c$
compared to BCS behavior.
In conclusion, we presented measurements of the electronic Raman scattering on
high-T$_c$ Tl-2201 single crystals with one CuO$_2$-plane per unit cell.
The peculiarities of the electronic Raman scattering, i.e. the power law
frequency dependence of the diferent scattering components at low frequencies,
their different peak positions as well as the values of
2$\Delta_{max}/k_BT_c=7.6\pm 0.4$ are found to be very similar in compounds
with one and several CuO$_2$ planes.
All nearly optimally doped high-T$_c$ superconductors (with the exception of
(Nd,Ce)-214 \cite{Hac}) show a very similar behavior of the electronic Raman
scattering consistent with the d$_{x^2-y^2}$- wave symmetry of the underlying
order parameter.
\section {ACKNOWLEDGMENTS}
This work was supported by DFG through SFB 341 and by BMBF FKZ 13 N 6586.
One of the authors L.V.G acknowledges support from the Alexander von Humboldt
Foundation and expresses his gratitude for the hospitality at the
2.Physikalisches Institut RWTH-Aachen.
| proofpile-arXiv_065-417 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
\label{intro}
Granular fluids are dense media composed of elementary elements
of macroscopic size, undergoing collisions
in which their {\it macroscopic} energy is not conserved.
In the last decade, the flow of this granular fluids
has received a great attention from the physics
community, both because these media offer a "simple"
example of dissipative systems and because of their
numerous industrial applications. Two factors make the behaviour of
granular fluids very different from that of molecular fluids.
Firstly, the macroscopic size of the particles
implies that external fields (boundaries or gravity) have a much
stronger effect on granular fluids. Secondly, the energy of the
granular fluid is not a conserved variable, since the heat
dissipated in collisions can be considered as lost as
far as the flow is concerned. These two effects are often difficult
to disentangle in experiments, and also in the numerical simulations
that aim at a realistic modelling of these experiments \cite{HERRMAN}.
A different type of numerical simulations was initiated
by several groups \cite{Goldhirsch, Young1,Young3,Young2,Bernu}. In this approach
external fields are ignored, and only the dissipation is
taken into account. This dissipation is moreover modelled in a
very simplistic way, by describing the particles as monodisperse
rigid hard spheres undergoing inelastic collisions. The dissipation is
entirely specified by the restitution coefficient $r$, where $1-r$
is the
fraction of the kinetic energy lost in a collision
(in the center of mass frame).
The aim of these simulations is not to model actual experiments
involving granular fluids. Instead, the models are used to assess
the difference in behaviour induced by the dissipation between a
granular fluid and its "atomic" counterpart ($r=1$, the hard sphere
fluid.) In particular, the simulations can be used to investigate
the validity of the hydrodynamic equations frequently used to
describe granular flow in more realistic situations
\cite{Haff,Campbell}. As for atomic fluids, they also
provide a direct way of measuring the equation of state
and transport coefficients that enter these equations. These
transport
coefficients can then be compared to those obtained using
"granular kinetic theory" \cite{Jenkins}.
A particularly simple and instructive situation that can easily
be studied in numerical simulations is the 2-dimensional
"cooling problem"
first studied in \cite{Goldhirsch, Young2}. In their simulations,
these authors start from an equilibrium configuration of an
{\it elastic} ($r=1$) hard disc fluid. The behaviour of this fluid
after introducing a nonzero restitution coefficient is followed
using the standard molecular dynamics method for hard bodies
\cite{AT87}, with a collision rule that takes the
dissipation into account. The kinetic temperature (average kinetic
energy per particle) decreases due to the inelasticity of the
collisions, so that the fluid cools down as time increases.
It was shown in \cite{Goldhirsch, Young2} that, depending on the
system size, on the restitution coefficient and density,
this cooling follows different routes. In the simplest case,
(small systems or small dissipation)
the fluid remains homogeneous at all temperatures. In larger
systems or for larger dissipations, either the velocity
field or the density field in the fluid develop
instabilities and become inhomogeneous. It was also found by the
same authors that the occurence of such instabilities is
in qualitative agreement with the predictions of a linear stability
analysis of the hydrodynamic equations for granular
fluids \cite{Haff}. Finally, it was discovered in \cite{Young1,Young2}
that in some cases, the cooling ends at a finite time due to
a singularity in the system dynamics, which was shown to correspond
to an infinite number of collisions within a finite time. This
singularity, observed both in 1 and 2 dimensions, was described
as an "inelastic collapse" of the system.
In this paper, a detailed and quantitative analysis
of the instability of homogeneous cooling of granular fluids
is attempted. Our aim is to compare quantitatively the
predictions of granular kinetic theory and granular hydrodynamics
to the results of molecular dynamics simulations of the cooling
of an inelastic hard disk fluid. The paper is organized as follows.
The main predictions of granular hydrodynamics concerning the
cooling problem are briefly recalled. Computational details
concerning the simulation are given in section 3. The different
regimes occuring during the cooling are analyzed in section
4, and compared with the theoretical predictions. Our main focus
will be on the growth rate of the density instability, that
can be computed by monitoring the structure factor
of the system as a function of time. Finally, the problem of the
"inelastic collapse"
is adressed in section 5, where a possible method for avoiding this
singularity in the system dynamics is proposed.
\section{Hydrodynamic analysis of the cooling problem}
The hydrodynamic equations that have been
proposed to describe granular flow \cite{Haff}
are based on mass and momentum conservation, and
are very similar to the usual Navier-Stokes equations. The only
modification is the appearance of a new term in the energy
(or temperature)
equation, accounting for the loss of energy in the collisions.
These equations can be compactly written in the
form
\begin{eqnarray}
\Dhyd{\ensuremath {\rho}}{t} & = & -\ensuremath {\rho} \Div{ \ensuremath {\Vect{v}}} \\
\ensuremath {\rho} \Dhyd{\ensuremath {\Vect{v}}}{t} & = & -\Div{\Tens{P}} \\
\ensuremath {\rho} \Dhyd{T}{t} & = & -\Div{\Vect{Q}}-tr \left( \Tens{P}\,\Tens{D} \right )-\gamma T^\frac{3}{2}
\end{eqnarray}
where
$D/Dt$ is the hydrodynamic derivative, $\Tens{D} $ the symmetrized
velocity gradient tensor, \Tens{P} the stress tensor, \Vect{Q} the heat flux
and $\gamma$ represented the rate of energy lost due to inelastic collisions.
For a hard disk fluid, the equation of state
and the expression of the various transport coefficients
can be obtained from Jenkins and Richman kinetic theory \cite{Jenkins}.
These expressions are recalled in Appendix A.
The energy sink term, $\gamma T^{3/2}$, has also been
written in the form appropriate for hard disks. $\gamma$ in that case
is a function of the density and the restitution coefficient,
which at least in the low density limit must
be proportionnal to $ \ensuremath {\rho} $ and $(1-r)$. This can be understood
from the following reasoning: the kinetic energy loss
per particle per unit time is proportional to the collision
frequency (i.e. to $\ensuremath {\rho} T^{1/2}$) and to the
energy loss per collision $(1-r) T$.
A trivial solution of the cooling problem formulated above corresponds
to an homogeneously cooling fluid, with a uniform density,
a vanishing velocity field, and a uniform temperature
with an algebraic time decay
\begin{eqnarray}
\label{temperature}
T(t)=T_0{\left(1+\frac{t}{t_0}\right)}^{-2}
\end{eqnarray}.
Here $t_0=2 \ensuremath {\rho}_0/(\gamma_0 {T_0}^{1/2})$ sets the time scale for temperature decay in the fluid.
The linear stability of this homogeneous solution
has been investigated in references \cite{Goldhirsch, McNamara}.
For completness, the main steps of this analysis
will be repeated here. The linearized equations
that describe the evolution of a sinusoidal
perturbation
\begin{eqnarray}
\ensuremath {\delta \rho}=\ensuremath {{\delta \rho}_{\Vect{k}}} \exp \left ( i \ensuremath {\Vect{k}} \cdot r \right ) \\
\ensuremath {\delta \V}=\ensuremath {{\delta \V}_{\Vect{k}}} \exp \left ( i \ensuremath {\Vect{k}} \cdot r \right ) \\
\ensuremath {\delta T}=\ensuremath {{\delta T}_{\Vect{k}}} \exp \left ( i \ensuremath {\Vect{k}} \cdot r \right )
\end{eqnarray}
around the homogeneous solution are
\begin{eqnarray}
\label{cont1}
\Der{\ensuremath {{\delta \rho}_{\Vect{k}}}}{t} & = & -i\ensuremath {\rho}_0 \left ( \ensuremath {\Vect{k}} \cdot \ensuremath {{\delta \V}_{\Vect{k}}} \right ) \\
\label{eq:vitesseParral2}
\ensuremath {\rho}_0 \Der{\ensuremath {\Vect{k}} \cdot \ensuremath {{\delta \V}_{\Vect{k}}}}{t} & = & -ik^2 \left [ \ensuremath {\rho}_0 p'( \nu_0)\ensuremath {{\delta T}_{\Vect{k}}}+T_0 \left (p'(\nu_0)+\nu_0 \left ( \Der {p'}{\nu} \right )_0 \right )\ensuremath {{\delta \rho}_{\Vect{k}}} \right ] \nonumber\\
& & \mbox{ } - \mu_0 \left [ k^2 \left ( \ensuremath {\Vect{k}} \cdot \ensuremath {{\delta \V}_{\Vect{k}}} \right ) \right ] \\
\label{eq:vitessePerp2}
\ensuremath {\rho}_0 \Der{\ensuremath {\Vect{k}}_\perp \cdot \ensuremath {{\delta \V}_{\Vect{k}}}}{t} & = & - \mu_0 \left [ k^2 \left ( \ensuremath {\Vect{k}}_\perp \cdot \ensuremath {{\delta \V}_{\Vect{k}}} \right ) \right ] \\
\label{eq:temperature2}
\left [ \ensuremath {{\delta \rho}_{\Vect{k}}} \Der{T_0}{t}+\ensuremath {\rho}_0 \Der{\ensuremath {{\delta T}_{\Vect{k}}}}{t} \right ] & = & -\kappa_0 k^2 \ensuremath {{\delta T}_{\Vect{k}}}
-{p_h}_0\left ( i \ensuremath {\Vect{k}} \cdot \ensuremath {{\delta \V}_{\Vect{k}}} \right) \nonumber\\
& & \mbox{ } - \frac{3}{2} \gamma_0 {T_0}^\frac{1}{2} \ensuremath {{\delta T}_{\Vect{k}}}
- T^{\frac{3}{2}} \nu {\left( \Der{\gamma}{\nu} \right )}_0 \frac{\ensuremath {{\delta \rho}_{\Vect{k}}}}{\ensuremath {\rho}_0} \end{eqnarray}
As for usual fluids, the transverse part of the velocity
field completely decouples from the longitudinal part, and decays
with time as $ \left ( 1 + {t}/{t_0} \right )^{-k^2 T_0^{1/2}
{t_0}/{\ensuremath {\rho}_0}}
$. The longitudinal part of the velocity field,
the temperature and the density are coupled, and give rise
to three modes that have an algebraic time dependance
\begin {eqnarray}
\ensuremath {{\delta \rho}_{\Vect{k}}} & = & \ensuremath { \delta \widetilde{\rho}}_{\Vect{k}} { \left [ 1+ t/t_0 \right ]}^\ensuremath {\xi} \\
\ensuremath {{\delta \V}_{\Vect{k}}} & = & \ensuremath { \delta \widetilde{\V}}_{\Vect{k}} { \left [ 1+ t/t_0 \right ]}^{\ensuremath {\xi}-1} \\
\ensuremath {{\delta T}_{\Vect{k}}} & = & \ensuremath { \delta \widetilde{T}}_{\Vect{k}} { \left [ 1+ t/t_0 \right ]}^{\ensuremath {\xi}-2}
\end{eqnarray}.
The exponents $\xi(k)$ for the three modes are the three
roots of the determinant of the following set of equations
\begin{eqnarray*}
\ensuremath { \delta \widetilde{\rho}}_{\Vect{k}} \left (\frac{\ensuremath {\xi}(\ensuremath {\xi}-1)}{t_0^2}+ \ensuremath {\Vect{k}}^2 T_0 \left [p'(\nu_0)+\nu_0 \left ( \Der {p'}{\nu} \right )_0 \right ] + \frac{\mu_0 k^2} {\ensuremath {\rho}_0 t_0} \ensuremath {\xi} \right )
+ \ensuremath { \delta \widetilde{T}}_{\Vect{k}} \left [ k^2 \ensuremath {\rho}_0 p'(\nu_0) \right ] & = & 0 \\
\ensuremath { \delta \widetilde{\rho}}_{\Vect{k}} \left [ \frac{\ensuremath {T}_0}{t_0} \left (-2
+ T_0^{\frac{1}{2}} \nu {\left( \Der{\gamma}{\nu}\right )}_0 \frac{t_0}{\ensuremath {\rho}_0}
- p'(\nu_0) \ensuremath {\xi} \right ) \right ]
+ \ensuremath { \delta \widetilde{T}}_{\Vect{k}} \left [ (\ensuremath {\xi}+1) \frac{\ensuremath {\rho}_0}{t_0} + \kappa_0 k^2 \right ] & = & 0
\end{eqnarray*}
A typical plot of the wavevector dependance
of these three roots, together with the growth rate
of the velocity perturbations, is shown in figure 1.
It must be emphasized that the stability of velocity
disturbances is determined by the comparison
between the growth exponent of the disturbance with
the value $-1$ that characterizes the decay of the
{\it thermal} velocity. Hence a growth exponent larger than
$-1$ for the transverse or longitudinal velocity
fields is indicative of an instability of the macroscopic
velocity. If the growth exponent of the longitudinal
velocity field is larger than $-1$, a corresponding instability
in the density field will follow from equation \ref{cont1}.
This analysis yields to the prediction of three different
possible behaviours of the system, depending on the
value of the parameters and on the system size, that
introduces a lower wavevector cutoff. If this lower cutoff
corresponds to the line $C$ of figure 1, the homogeneous solution
will be linearly stable. This regime will be described
as the homogeneous kinetic regime. If the lower cutoff moves
to the abscissa indicated by line $B$ in figure 1, the transverse
velocity field will become unstable while the system remains
homogeneous. In this "shearing" regime, first observed in
reference \cite{Goldhirsch}, a shearing flow will
develop in the system. Finally, for a lower cutoff
corresponding to abscissa $A$, an instability of
the longitudinal velocity field and the corresponding
instability in the density field will take
place together with the shearing instability.
In this "clustering" regime, the growth
of density disturbances will yield to the formation
of dense clusters of particles, as first observed in
reference \cite{Hopkins}.
All three situations have already been observed
in numerical simulations of the cooling in two dimensional
granular fluids. The aim of the next sections
will be to attempt a quantitative analysis of the
behaviour of a cooling granular fluid, and to compare the results
to the predictions summarized above.
\section{Computational details}
The model simulated in this work is in all respects similar
to that studied in \cite{Goldhirsch, Young2}. The system is made up of
$N$ hard inelastic disks of diameter $\sigma$, in a square cell of
size $L$ with periodic boundary conditions. The cell size
$L$ sets the lower cutoff in wavevector space, $k_{min}=2\pi/L$.
A standard cell-linked Molecular Dynamics algorithm for hard bodies
\cite{AT87} is used. In a first step, the system is equilibrated
with a coefficient $r$ equal to unity. At time
$t=0$, inelasticity is switched on and cooling starts,
with an initial
temperature $T_0$. The restitution coefficient
enters through a simple modification of
the standard collision rule between hard disks, the velocities
of the two disks after a collision being given by
\begin{eqnarray}
\label{coll1}
\Vect {u_1}'=\Vect{u_1}-\frac{1}{2}(1+r)
[\hat{\Vect{n}} \cdot (\Vect{ u_1}-\Vect {u_2}) ] \hat{\Vect{n}} \\
\label{coll2}
\Vect {u_2}'=\Vect{u_2}+\frac{1}{2}(1+r)
[\hat{\Vect{n}} \cdot (\Vect{ u_1}-\Vect {u_2}) ] \hat{\Vect{n}}
\end{eqnarray}
where the primes denote the quantities after collision and $ \hat{{\Vect{n}}} $
is a unit vector along the centers line from particle 1 towards particle 2.
The natural units in this problem are the particle mass $m$
and diameter,
and the thermal energy at $t=0$,
i.e. $T_0$. The corresponding time unit
is $\tau=(m/T)^{1/2}\sigma$. The state of the system
is defined by three dimensionless numbers,
which are the reduced size $L/\sigma$ (or equivalently
the reduced cutoff $k_{min}^*= k_{min}\sigma$), the reduced density
$\rho*=\sigma^2 N/L^2 $, and the restitution coefficient $r$.
The state of the fluid during the cooling was
monitored by a systematic computation
of coarse-grained (hydrodynamic) density and velocity fields.
The coarse graining is obtaining here from a division of
the system into 100 square subcells. Besides, statistical quantities
characterizing the state of the system have
also been systematically computed. These quantities are
the momenta of the velocity distribution of
individual particles, the pair correlation function $g(r)$
for interparticle distance, and the structure factor
\begin{eqnarray}
S(\Vect{k}) & = & \frac{1}{N} \rho_\Vect{k} \rho_\Vect{-k}
\end{eqnarray}
This structure factor can be computed
for all wavevectors compatible with
the periodic boundary condition, of the form $(n_x,n_y) k_{min}$.
As the system is not in a stationary state, these
quantities are time dependant. A large enough system
is thus necessary to obtain reasonable statistics
without time averaging. The values of $N$ investigated in this work
vary from $N=1600$ to $N=10000$.
\section{results}
\subsection{Kinetic regime}
According to the analysis of section \ref{intro},
the kinetic regime corresponding to
a stable homogeneous cooling will be observed
(at a given density and restitution coefficient)
for small enough systems. Such a situation allows
a clear testing of some of the hypothesis
of the kinetic theory description of the granular fluid.
In particular, the pair correlation function and
velocity distribution can be compared to that of
an elastic hard disk fluid throughout the cooling
process. The temperature decay can be monitored
and compared to the theoretical prediction
(equation \ref{temperature}), and the decay time $t_0$ (or equivalently the
coefficient $\gamma(\rho)$) compared to the prediction
of kinetic theory.
The pair correlation of an homogeneously
cooling granular fluid after the temperature
has dropped by a factor of 10 is shown in figure 2. This comparison
shows that the local structure of the cooling
granular medium (which determines
its equation of state) remains essentially identical to that
of an equilibrium fluid. The study of the
velocity distribution function shows that this distribution remains
maxwellian throughout the cooling.
This similarity between the
structure and velocity distribution of the
granular fluid and the usual hard disk fluid
suggests that the kinetic theory of Jenkins \cite{Jenkins} is applicable.
This expectation is borne out by the study of the time dependance of
the fluid temperature. As shown in figure 3, the temperature decay
is perfectly described by equation \ref{temperature}. The density dependance
of the decay time $t_0$ is compared in figure 4 to the prediction
of kinetic theory (see appendix B). The agreement is extremely good,
and suggests that all the transport
coefficients appearing in the hydrodynamic
equations can be estimated using this kinetic theory.
\subsection{Shearing regime}
If the restitution coefficient $r$ decreases
or if the size of the system increases, the hydrodynamic theory
predicts a regime in which transverse fluctuations of
the velocity field are unstable. This regime is indeed
observed in the simulations, as shown in figure 5.
A shear flow that corresponds to the smallest wavevector
compatible with the periodic boundary conditions develops
in the system. In this regime, the total kinetic energy of
the system (which in that case is not the temperature, since
the system has developped an ordered flow pattern) appreciably deviates
from equation \ref{temperature}, as shown in figure 6.
\subsection{Clustered regime}
For even larger systems
or smaller restitution coefficients, the cooling granular fluid
becomes inhomogeneous, as shown in figure 7.
this spontaneous formation of density inhomogeneities
(or clusters) was first observed in the simulations of the cooling problem
by Goldhirsch and Zanetti and Young and McNamara \cite{Goldhirsch, Young2}.
Two different explanations have been put forward to explain this
cluster formation. The first one, found in \cite{Goldhirsch}, is to consider this
cluster formation as a secondary instability of the shearing regime,
due to the developpment of temperature and pressure gradients
in the shearing regime. The second possible explanation is
that cluster formation is directly related to the linear instability of the
density modes predicted by hydrodynamic theory.
In order to characterize
quantitatively this clustering regime, the structure factor
$S(k,t)$ of the system has been computed as a function of time
and wavevector. The corresponding data is shown in figure
8. The growth of the density inhomogeneities results
in the appearance of a low wavevector peak in the
structure factor, that rapidly increases with time.
According to hydrodynamics, the time dependance
of $S(k,t)$ should be algebraic, i.e.
\begin{eqnarray}
S(\ensuremath {\Vect{k}},t) & = & S(\ensuremath {\Vect{k}},0) {\left ( 1 + \frac{t}{t_0} \right )}^{2\xi(\ensuremath {\Vect{k}})}
\end{eqnarray}
so that the ratio
\begin{eqnarray}
\frac{\ln (S(\ensuremath {\Vect{k}},t))-\ln (S(\ensuremath {\Vect{k}},0))}{ln \left ( 1 + \frac{t}{t_0} \right )} & =
& 2\xi(\ensuremath {\Vect{k}})
\end{eqnarray}
should be independent of time. This ratio is plotted in figure 9 as
a function of wavevector for different times. $2\xi(k)$ seems to be
reasonably independent of time, and its low wavevector
value appears to be consistent with the prediction of linearized
hydrodynamics. Hence the density instability
can be interpreted as resulting from
a linear instability of the homogeneous solution
of the hydrodynamic equations. Note that it was recently observed
by McNamara and Young that the "clustering" fluid
eventually develops for long times into an ordered flow pattern
of the "shearing" type. This is also consistent with hydrodynamics,
since the growth rate of the transverse velocity modes is
positive. The description of the formation of this shearing flow
in an inhomogeneous system, however, is beyond the possibilities
of linearized hydrodynamics.
\section{Inelastic collapse and how to avoid it}
The inelastic collapse singularity was first observed
by \cite{Bernu, Young1} in simulations of unidimensional
inelastic system. This collapse can be described as the
appearance of an infinite number of correlated collisions between
a few particles, taking place in a finite time. The same phenomenon
was observed in two dimensions by \cite{Young2}. It was shown
that in that case the correlated collisions take place
between a small number of essentially aligned particles,
so that the unidimensional situation is practically
reproduced.
In order to avoid this inelastic collapse, a slightly
modified collision rule between the particles can be introduced.
At each collision, the relative velocity
of the two particles is first computed according to the
usual rule (equations \ref{coll1} and \ref{coll2}), then rotated by a small
(less than 5 degrees) random angle.
This can be justified by invoking the unavoidable
roughness of actual solid particles, conservation
of angular momentum being (virtually) ensured by a transfer to
the internal degrees of freedom of the particles.
As to inelastic collapse, the aim of this modified collision
rule is to hinder the formation of correlated particle lines
that cause this singularity. Indeed, inelastic collapse
was not observed in the simulations where this "random"
collision rule was used, while under the same conditions
a system following the "deterministic" collision
rule always underwent inelastic collapse (figure 10).
Hence inelastic collapse appears to be a pathology related to the
use of purely specular collision rule
between particles, rather than a characteristic of inelastic
fluids.
\section{Conclusion and perspectives}
The main objective of this work
was to assess the validity of the hydrodynamic
description of granular fluids originally proposed
by \cite{Haff}, and of the kinetic theory calculation of
the associated transport coefficients. The study of the
particularly simple "cooling fluid" case
and of the associated instabilities provides an ideal
benchmark for this description. The comparison between numerical
simulations and theoretical predictions in this simple case
shows that the theory is quantitatively accurate.
A similar conclusion was also reached in a recent study by McNamara and
Young \cite{preprint}, who showed that the transitions between the
different cooling regimes were correctly predicted by the theory.
The description of the inelastic collapse phenomenon observed
by McNamara and Young is obviously beyond
the possibilities of kinetic theory or hydrodynamics. It was
shown that this phenmenon can easily be avoided by introducing a small amount of
randomness in the collisions between particles, similar to what would
be caused by the natural roughness of granular particles.
Obviously, a correct description of granular fluid
cannot be achieved without a knowledge of the boundary conditions
that must be used for the hydrodynamic equations. These conditions, and
in particular those that correspond to the very important case of vibrating
solid walls, are not known. Their determination, through the quantitative
comparison of numerical simulation and theory, will be the subject of future
work.
\section*{Ackowledgments}
This work was supported by the Pole Scientifique de Mod\'elisation
Num\'erique at ENS-Lyon.
\pagebreak
| proofpile-arXiv_065-418 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{ Introduction}
\label{Introduction}
Relativistic wind models of Gamma Ray Bursts (GRB) are a recent development
of the dissipative relativistic fireball scenario which is a natural
consequence of observationally set requirements irrespective of the
GRB distance scale (e.g.~\cite{me95}).
The observed bursts are expected to be produced in optically
thin shocks in the later stages of the fireball expansion.
Two types of shocks have been considered, based on different
interpretations of the burst duration and variability. First,
the ``external'' deceleration shocks (\cite{rm92}; \cite{mr93})
that develop at $r_{dec}$, due to the unavoidable interaction of the relativistic ejecta with the
surrounding medium, give bursts whose duration
is determined (in the ``impulsive'' regime) by the relativistic
dynamic timescale $t_{dec} \sim r_{dec}/c \Gamma^2$, where $\Gamma$ is
the Lorentz factor of the expansion (e.g. \cite{mr93}; \cite{ka94};
Sari, Narayan \& Piran 1996).
Second, the
``internal'' dissipation shocks in an unsteady relativistic wind
outflow (\cite{rm94} [RM94]; \cite{paxu94}), where both the burst duration
$t_w$ and time
variability $t_{var} \leq t_w$, are
characteristic of the progenitor mechanism (e.g. a disrupted disk accretion
timescale, dynamic or turbulent timescales, etc.). Simple ``external'' shocks
tend to produce fairly smooth light
curves with a fast rise followed by an exponential decay (FRED), as observed in
some bursts. However, many bursts have multi-peaked, irregular light curves;
those may find a more natural explanation (Fenimore, Madras \& Nayakshin 1996)
in the relativistic wind scenario mentioned above.
While the spectral properties of impulsive ``external'' shocks have been studied
in some detail (\cite{mrp94} [MRP94]; \cite{sa96}), so far, only rough estimates have
been made for wind ``internal'' shock spectra (\cite{mr94};
\cite{th94}).
It is of great interest to explore the spectral properties of both types of shock scenarios,
particularly in view of the increasingly sophisticated analyses of both GRB
light curves (e.g. \cite{fe96}; \cite{nor96}) and spectral characteristics (e.g.
\cite{band93}).
In this letter, we investigate the spectral properties of ``internal''
shocks for representative parameter values and discuss those properties in the light of
anticipated results from HETE.
\section{ Physical Model and Shock Parameters}
\label{sec:model}
In the unsteady relativistic wind model, the GRB event is caused by
the release of an amount of energy $E=10^{51}E_{51}$ erg, inside a volume of
typical dimension $r_{\ell}= 3 \times 10^{10} t_{var}$ cm, over a timescale
$t_w$ (RM94). The average dimensionless entropy per particle $\eta=
\langle E/(M c^2) \rangle - 1 = \langle L/({\dot M} c^2)\rangle$ determines
the character of the overall flow. The bulk Lorentz factor of the flow
increases as $\Gamma \propto r$ at first and later saturates to a value
$\Gamma = \Gamma_s$ at $r \mathrel{\mathpalette\simov >} r_s \approx r_{\ell} \Gamma_s $,
where $\Gamma_s \sim \eta $ for the values of $\eta$ that we consider
here (eq. [\ref{eta_lim}]). It is assumed that $\eta$ varies substantially,
$\Delta \eta \sim \eta$, on a timescale $t_{var} $. As an example,
we consider shells of matter with two different $\eta$ values ejected at
time intervals $t_{var}$. After saturation these will coast with bulk Lorentz
factors $\Gamma_f \sim q_f \Gamma_s $ and $\Gamma_r \sim q_r \Gamma_s$,
where $ q_r < q_f \sim 1$. Shells of different $\eta$ catch up with each other
at a dissipation radius
$r_{d} \approx \Gamma_{f} \Gamma_{r} c t_{var} \approx 3 \times 10^{14}\;
q^2 \Gamma_{s_2}^{2} \; t_{var} \; $ cm (see RM94, eq. [4]),
where $q =\sqrt{q_f q_r}$ and the subscript 2 (3) stands for quantities
measured in units of $10^{2}$ ($10^3$).
This gives rise to a $\em{forward}$ and a $\em{reverse}$ shock which
are mirror images of each other in the contact discontinuity (center of
momentum, CoM) frame. The CoM moves in the lab frame with
$\Gamma_{b}\approx \sqrt{ \Gamma_{f} \Gamma_{r}}$, for $\Gamma_s \gg 1$.
In the CoM frame the two shocks move in opposite directions
with small Lorentz factors $\Gamma'_{sh} -1 \approx (1/2)
\;[({\Gamma_{f}}/{\Gamma_{r}}) + ({\Gamma_{r}}/{\Gamma_{f}})]^{1/2}-1
\ge 10^{-2}$, for $\Gamma_{f}, \Gamma_{r} \gg 1$, while in the lab frame both
shocks move forward with Lorentz factors $\Gamma_{f}$ and $ \Gamma_{r} $
respectively. The shocks compress the gas to a comoving frame
( primed quantities refer to the $\em{comoving}$ frame, which is the CoM frame )
density
\begin{equation}
n\acute{}\; (r_{d}) \approx \frac{8 \times 10^{9}}{q^4}
\frac{E_{51}}{\theta ^{2} t_{w} t_{var}^{2}}
\frac{1} { \eta_{2} \Gamma_{s_2}^{5} } \; \; \; \rm{cm^{-3}},
\end{equation}
where $\theta^{2}$ is the normalized opening solid angle of the wind ($1$ if
spherical). The bulk kinetic energy carried by the flow is randomized in the
shocks with a mechanical efficiency
$\varepsilon_{sh} \approx (q_f + q_r - 2 q)/ (q_f+ q_r)$ (eq. [1] in RM94).
In order for the shocks to produce a non-thermal spectrum and radiate a
substantial portion of the wind energy, the dissipation must occur beyond both
the wind photosphere (see section 2.2 in RM94) and the saturation radius. This restricts $\eta$ to the
range
\begin{equation}
33 \; \left( \frac{E_{51}}{\theta^{2} t_{w} t_{var}} \right)^{1/5} \mathrel{\mathpalette\simov <} \eta
\mathrel{\mathpalette\simov <} 80 \; \left( \frac{E_{51}}{\theta^{2} t_w t_{var}}\right)^{1/4}.
\label{eta_lim}
\end{equation}
The collisionless ``internal'' shocks
accelerate particles to relativistic energies with a nonthermal distribution.
We
parameterize the post-shock electron energy by a factor $\kappa$, so that the
average
random electron Lorentz factor is $\gamma_e \sim \kappa\Gamma\acute{}_{sh} \sim
\kappa \le m_p/m_e $ (with the protons' being $\gamma_p \sim
\Gamma\acute{}_{sh}$). An injection fraction $\zeta$ of the post-shock electrons
is accelerated to a power-law
distribution ($ d N(\gamma) \sim \gamma^{-p} d\gamma$, for $\gamma_{min}
\leq \gamma $).
We take
$p \ge 3$, so that most of the energy is carried by the
low energy part of the electron spectrum ($\gamma_{min} \approx ((p-2)/(p-1))
\kappa$). A measure of the efficiency of the transfer of energy between protons
and electrons behind the shock is given by $\varepsilon_{pe} \simeq
\zeta \kappa/(m_p/m_e) \sim \zeta\kappa_3$. Here,
we will use $\zeta=1$ and $\kappa\le 10^3$
( but see \cite{byme96}) and $p=3$ which is consistent with the fits
to observed spectra (\cite{hanlon94}).
The electrons get
accelerated at the expense of the randomized proton energy behind the shocks,
which, for an individual shock (of which there are $t_w/t_{var}$), represents a
fraction $ ({t_{var}}/{t_w}) \varepsilon_{pe} \varepsilon_{sh} $ of the total
energy $E_0$.
Magnetic fields in the shocks can be due to a frozen-in component from the
progenitor, or may build up by turbulence behind the shocks.
We parameterize the strength of the magnetic field in the shocks
by the fraction $\lambda$ of the magnetic energy density to the particle
random energy density ($u\acute{}_{\scriptscriptstyle B} = \lambda
n\acute{} m_p c^2$). The comoving magnetic field therefore is
\begin{equation}
B\acute{} \approx
\frac{1.72\times 10^{4}}{ q^2 \Gamma_{s_2}^2} \;
\sqrt{\frac{ E_{51}\varepsilon_{sh}}
{\theta^{2} t_w t_{var}^{2}} \;
\frac{\lambda} {\eta_2 \Gamma_{s_2}}} \quad \hbox{G}.
\end{equation}
The relativistic electrons will lose energy due to synchrotron radiation,
and inverse Compton (IC) scattering of this radiation. The respective radiative efficiencies are
$\varepsilon_{sy} = t_{sy}^{'-1}/(t_{sy}^{'-1}
+t_{ic}^{'-1} +t_{ex}^{'-1})$, and
$\varepsilon_{ic} = t_{ic}^{'-1}/(t_{sy}^{'-1}
+t_{ic}^{'-1} +t_{ex}^{'-1}) $ . The timescales are defined below.
The comoving synchrotron timescale is determined by the least energetic
electrons :
\begin{equation}
t'_{sy} \approx 8 \times 10^8 /\gamma_{min} B^{'2} =
5.2 \; \displaystyle q^4 \;
\frac{\theta^2 t_w t_{var}^2}{E_{51} \varepsilon_{sh}}\;
\frac{\eta_2 \Gamma_{s_2}^5}{\lambda \kappa_{sy_3}}
\; \; \; \hbox{ms},
\end{equation}
where the subscript {\em sy} ({\em ic}) refers to the synchrotron (IC) emitting
shock.
The IC cooling depends on, and competes with, synchrotron cooling.
IC cooling dominates if the magnetic field is weak, a large fraction
of electrons are accelerated and share the protons' momentum very efficiently
(i.e. $\lambda /(\zeta \kappa_{ic_3} )\ll 1$);
while synchrotron cooling dominates in the opposite case.
The IC timescale ($t'_{ic} \approx 3 \times
10^7/u\acute{}_{sy} \gamma_{min}$) for the two limiting cases is
\begin{equation}
t'_{ic} \approx
\left\{ \begin{array}{ll}
9.6 \;\displaystyle \frac{q^4}{\zeta \varepsilon_{sh}}
\frac{\theta^2 t_w t_{var}^2}{E_{51}}
\frac{\eta_2 \Gamma_{s_2}^5}{<\gamma_3^2>}
\frac{\gamma_{*_3}}{\kappa_{ic_3}}
\left[1 +\frac{t\acute{}_{sy}(\gamma_*)}{t\acute{}_{ex}}\right]
& \mbox{ms $\;\;\;$if IC dominates}\\
\label{t_ic_ic}
\\
4.95\; \displaystyle \;\frac{q^4}{\varepsilon_{sh}}
\frac{\theta^2 t_w t_{var}^2}{E_{51}}\;
\frac{\eta_2 \Gamma_{s_2}^5}
{\sqrt{\zeta \lambda \kappa_{ic_3} <\gamma_3^2>}}
& \mbox{ms \quad if synchr. dominates}
\label{eq:t_ic_sy}.
\end{array}
\right.
\end{equation}
Here $\gamma_* $ is the electron Lorenz factor that corresponds
to the peak emission frequency,
$\gamma_* =max \left[\gamma_{min}, \gamma_{abs} \right]$,
$\gamma_{abs}$ is defined below equation (\ref{v_abs}) and
$ <\gamma^2_3> = 2\times 10^{-6}
\displaystyle \gamma_{min}^2
\;ln(\gamma_{max}/{\gamma_*})$.
The lab frame shell width (shocked region) is $\Delta r \sim r_d \Gamma_{b}^{-2}
\sim \alpha c t_{var}$, and the comoving crossing time
$t_{ex}' \approx 10^{2} (\alpha q^2) \; \Gamma_{s_2} \;t_{var}$ s
provides an estimate for the adiabatic loss time.
The spectrum of an ``internal'' shock burst consists of two
synchrotron and four IC components (coming from all shocks' combinations).
The synchrotron components are characterized by up to three {\it break
frequencies}, given below in the lab frame:
i) The $\gamma_{min}$ break frequency $\nu' = 8 \times 10^5 B' \gamma_{min}^{2}$, which in the lab
frame gets blue-shifted by $\Gamma_{f}$ ($\Gamma_r$) for the forward
(reverse) shock. Using $q_{sh}=[q_f,q_r]$ to refer to the shock, we have
\begin{equation}
h\nu_{min} \approx 1.45\; \displaystyle
\;\frac{q_{sh}}{q^2} \; \frac{\kappa_{sy_3}^2}{ \Gamma_{s_2} }
\;\sqrt{\frac{E_{51} \varepsilon_{sh}}{\theta^2 t_w t_{var}^2}
\frac{\lambda}{\eta_2 \Gamma_{s_2}}} \quad \hbox{keV}.
\label{v_sy_min}
\end{equation}
ii) The self absorption frequency $\nu_{abs}$. If the electron power law
starts at low enough energies, the radiation field becomes optically thick
at a frequency determined by $\frac{3}{2} m_{e} \gamma_{abs} \nu_{abs}^{'2} =
F'_{\nu_{abs}}$, and is obtained by solving a non-linear algebraic
equation; in the limit $\nu_{abs} \gg \nu_{min}$
it is
\begin{equation}
h \nu_{abs} \approx 18.4 \;\displaystyle \frac{q_{sh}}{q^{10/7}}\;
\frac{1}{\Gamma_{s_2}} \left( \frac{\lambda}{\eta_2 \Gamma_{s_2}} \right)^{1/14}
\left ( \frac{E_{51}\varepsilon_{sh} }
{\theta^{2} t_{w} t_{var}^{2}} \right)^{5/14}\;
(\zeta \varepsilon_{sy} \kappa_{sy_3})^{2/7}
\mbox{ eV $\; \; \;$ for $\gamma_{min} < \gamma_{abs}$},
\label{v_abs}
\end{equation}
where $\gamma_{abs} = 9 \times 10^3 \sqrt{\nu_{abs}/(q_{sh}
\Gamma_{s_2} B\acute{})}$.
iii) The frequency where the photon spectrum of the minimum energy electrons
becomes optically thick ($\nu_{\scriptscriptstyle{RJ}}$), and below
which it assumes a
Rayleigh-Jeans spectrum slope. This happens at
$\nu_{min}$ when $\gamma_{min}< \gamma_{abs}$; if $\gamma_{min} \gg
\gamma_{abs}$ it is
\begin{equation}
h\nu_{\scriptscriptstyle{RJ}} \approx
0.15 \; \displaystyle \frac{q_{sh}}{q^{4/5}}
\left(\frac{E_{51} \varepsilon_{sh}}
{\theta^{2} t_{w} t_{var}^{2}} \right)^{1/5}
\left(\frac{\eta_2}{\lambda} \right)^{2/5}
\left(\frac{\zeta \varepsilon_{sh}}{\Gamma_{s_2}}\right)^{3/5}
\frac{1}{\kappa_{sy_3}^{8/5}} \; \; \hbox{eV}.
\label{eq:v_RJ}
\end{equation}
iv) A cutoff is expected at
$\nu_{max} =
(\gamma_{max}/\gamma_{min})^{2} \; \nu_{min}$,
where $\gamma_{max}$ is the electron energy in each shock at which the electron
power law cuts off due to radiative or adiabatic losses. It is determined by
$t_{acc}(\gamma_{max}) \approx
t'_{cool}(\gamma_{max})$, where $t_{acc}$ is the electron acceleration timescale in
the shocks (a multiple $10 \times A_{10}$ of the inverse gyro-synchrotron frequency),
$t_{acc} \approx {3.57 \times 10^{-6} A_{10}}{B'(r_{d})^{-1}} \gamma$ s,
and $t_{cool}$ is the minimum of all the radiation and adiabatic cooling
timescales involved,
$t'_{cool_{r,f}} = min \left\{ t_{ex}', t'_{sy_{f,r}}, t'_{IC_{r}, }
t'_{IC_{f}} \right \}$ .
Each synchrotron component can be characterized by three frequencies in ascending
order, $\nu_{sy,j}^{i}$, where $i=1,..,3$ and $j=1$ for the reverse and
$j=2$ for the forward shock. Similarly, the pure and combined IC spectra are
characterized by the frequencies $\nu_{ic,j}^{i} \approx ({4}/{3})
\;\kappa_{ic}^{2} \nu_{sy,l}^{i}$ where, $j=1,..,4$
(1 corresponds to {\it IC reverse} , 2 to {\it IC forward}, 3 to {\it IC
reverse-forward} and 4 to {\it IC forward-reverse}), and if j is odd,
$l=1$, otherwise $l=2$. The shape of each component depends on the
relationship between the relevant $\gamma_{min}, \gamma_{abs} $ and $\gamma_{max}$.
In a power per logarithmic frequency interval plot, the fluence of
each component exhibits a peak of
$ S_{i} = 1.6 \times 10^{-6} (E_{51}/(\theta D_{28})^{2}) \;
\varepsilon_{sh}
\varepsilon_{i} \zeta \kappa_{i_3} \; \; {\rm erg/cm}^{2}$,
where $i=1 (2)$ for synchrotron (IC), and $ D_{28}$ is the luminosity
distance corresponding to $z \approx 1$ for a flat Universe, with $H_{o} \approx
80$ ($D_{L} = (2 c/H_{o}) (1+ z - \sqrt{1+z} \approx 3 \; \hbox{Gpc})$).
The spectrum is obtained by adding up these six components. In practice, the
forward and reverse components have values very close to each other and they
essentially merge. The resultant spectrum is then checked for the effects of pair
production; the $\gamma\gamma$ optical depth is calculated for
each comoving frequency above $m_e c^2 /k$ using the number density of photons
above the corresponding threshold, as obtained from the initial spectrum;
finally the spectrum is modified accordingly.
We note that most of the scattering in our spectra occurs in the Thomson regime,
unlike in ~\cite{sa96}, who consider large $\gamma$ in the framework of
impulsive shock models. ( Klein-Nishina corrections may become relevant
in a few $\kappa \mathrel{\mathpalette\simov <} 10^2$ cases, but at frequencies
which lie in the $\gamma\gamma$ absorbed part of the spectrum).
\section{ Typical Wind Spectra}
We discuss here the properties of some representative spectra.
We assume a total event energy of $E=10^{51}$ erg and a geometry of a
spherical section, or jet, of opening angle $\theta = 0.1$ (the physics is
the same as in a spherical wind , provided $\theta > \Gamma^{-1}$).
We have investigated a range of dynamic parameters ($\eta, t_w$ and
$t_{var}$), and
used $q_f=2, q_r=0.5$, and $\kappa_f =\kappa_r$.
In figure {\ref{fig_long}} we present spectra for a long burst ($t_w =100 $
s) and
in figure {\ref{fig_short}} for a short one ($t_w= 1$ s), for different
$\eta, \lambda$ and $\kappa$
values.
The sharper spectral features present would be smoothed by inhomogeneities in a real flow.
The range of $\eta$ where a substantial fraction of the wind energy is
radiated with nonthermal spectra is considerably restricted
(eq. [\ref{eta_lim}]) and implies lower values than in impulsive models.
Larger $\eta$ may be difficult to produce in a natural way, while
lower values lead to shocks below the photosphere that make bursts too
dim to be observed. The allowed volume of parameter space is not very large
(\cite{ha96} [PM96]). Nonetheless, it allows for an appreciable variety of spectra,
since models differing only slightly in $\eta$ can produce significantly
different spectra. This is due to the strong
dependence of the photon number density on $\Gamma_s$
(i.e. $ n\acute{}_{\gamma} \propto \Gamma_s^{-7} $) which determines pair
production. Lower $\kappa$ spectra are less affected by pair production,
since they contain less energetic electrons and therefore fewer photons.
Generally, other parameters being equal, the higher $\eta$ models tend
to produce spectra that span over a wider range of frequencies.
For spectra like the majority of the ones observed by BATSE (\cite{Fi&Me_rev})
values of $\kappa < 10^2$ and $\lambda > 10^{-1}$ are excluded. Low
$\kappa$ spectra are either too dim or too soft; high $\lambda$ brings up the
synchrotron component and may be appropriate for a small percentage of
bursts
with low frequency excess (\cite{Preece96}).
In general, for a given $\kappa$ value a wide range of $\lambda$ values (3 - 6
orders of magnitude) is allowed, the trend being that higher $\kappa$ values
must combine with lower $\lambda$ values, fairly independently of $\eta$
(within the allowed range of values). This trend is due to the fact that
lower magnetic fields require higher electron energies in order for the break
to fall in the BATSE window (e.g. eq. [\ref{v_sy_min}], [\ref{v_abs}]
). High energy power laws (like those reported in {\cite{hanlon94}})
are common (see fig. \ref{fig_short} and right column of fig. \ref{fig_long}).
For a more complete discussion see PM96.
The effect of a longer $t_w$ is to push the pair cutoff to higher
frequencies, because, for fixed $E_0$, the flux and photon density are lower.
For the same $\eta$, $\kappa$ and $t_{var}$, longer bursts require
higher $\lambda$.
A greater $t_{var}$ with the rest of the parameters unchanged would
again require higher $\lambda$ in order to produce observed-like spectra.
The cases considered here were chosen
with $ t_w < t_{dec} \approx (E_0/n_{ex} m_p c^5 \eta^8)^{1/3}$ s.
\section{Conclusions}
\label{conclusions}
An unsteady relativistic wind provides an attractive scenario for the
generation of GRB, since it requires smaller bulk Lorentz factors than the
impulsive models, i.e. it can accommodate higher baryon loads (RM94).
In addition, the lack of kinematic restrictions (e.g. \cite{fe96}) allows it,
in principle, to describe events with arbitrarily complex light curves.
We have calculated spectra from optical through TeV frequencies for bursts
with a range of durations and variability timescales.
At cosmological distances the total energies and photon densities implied by
the model are likely to turn those spectra optically thick to
$\gamma\gamma$ pair production. Most of them may therefore
be missed by, or show a high energy cut-off in, the EGRET window, but
they would be prominent in the BATSE and HETE gamma-ray windows.
Low frequency (down to 5 keV) excess reported recently
({\cite{Preece96}}) may be attributed to a pronounced synchrotron
component due to a relatively high magnetic field, in a fraction of
the bursts.
The continued propagation of an unsteady wind flow should eventually lead to its
deceleration by the surrounding medium. If the latter is of any appreciable
density, it could lead to another burst with the ``external'' shock
characteristics (MRP94, \cite{mr94}), provided that not all of the wind energy
was radiated away by the ``internal'' ones ($\varepsilon_{sh} < 1$).
The spectra of the ``internal'' shocks are different from those coming
from the ``external'' ones, the main differences being that the former
cover a narrower range of frequencies and most have high energy cutoffs due to
pair production opacity (PM96). If GRB progenitors have escaped
their parent galaxies and are in a low density intergalactic medium
($n_{ex} <10^{-3}$ cm$^{-3}$), the ``external'' impulsive shocks would be of long
duration ($> 3\times10^3 /\eta_2^{8/3} $ s) and low
intensity, and most would be totally missed, hence the GRB could be entirely
due to unsteady wind ``internal'' shocks. If the GRB occur in a denser medium
(e.g. the galactic ISM), the last peak of multiple-peaked GRB would be smooth
and FRED-like but the earlier peaks, due to unsteady wind ``internal'' shocks,
may be arbitrarily complicated.
Those might also be responsible for delayed GeV emission ({\cite{mr94}}).
A number of unknown parameters enter into the calculation of GRB models, and
HETE may provide the information required to narrow the range of their allowed
values, as well as a test for the general scenario of unsteady relativistic
winds. For the HETE sensitivities indicated in figures \ref{fig_long}
and \ref{fig_short}, we expect that some bursts will show a simultaneous
X-ray counterpart. However, detection by the Ultraviolet Transient Camera
(UTC) on HETE should be rare for the majority of bursts (note though that
figures \ref{fig_long} and \ref{fig_short} refer to $z\sim$ 1
distances, and that at a few $\times 10^2$ Mpc a bright burst would be $30-100$
times brighter, thus increasing the likelihood of detection by the UTC). For
the majority of faint bursts, a detection in the ultraviolet would be
possible only for low values of $\kappa$, resulting in a very broad and flat
spectrum with upper cutoffs, if any, only at the highest energies.
\acknowledgements{This research has been supported through NASA NAG5-2362,
NAG5-2857 and NSF PHY94-07194. We are grateful to the Institute for Theoretical
Physics, UCSB, for its hospitality, and to participants in the Nonthermal
Gamma-Ray Source Workshop for discussions}.
\pagebreak
| proofpile-arXiv_065-419 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction and Sample Selection}
\subsection{Background}
The surface brightness fluctuation (SBF) method of distance
determination works by measuring the ratio of the second and
first moments of the stellar luminosity function in a galaxy.
This ratio, called $\overline L$, is then the luminosity-weighted,
average luminosity of a stellar population and is roughly equal
to the luminosity of a single giant star. In terms of magnitudes,
this quantity is represented as $\overline M$,
the absolute ``fluctuation magnitude.''
What we measure, of course, is the apparent fluctuation magnitude
in a particular photometric band, in our case the $I$~band,
\ifmmode\overline{m}_I\else$\overline{m}_I$\fi. In order to be useful as a distance estimator,
\ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ must be calibrated, either empirically, by tying
the measurements
to the Cepheid distance scale, or theoretically, according
to stellar population synthesis models.
Tonry and Schneider (1988) were the first to quantify the SBF phenomenon.
Their method was based on a measurement of the amount of power on the
scale of the point spread function in the power spectrum of a CCD image.
They applied this method to images of the galaxies M32 and NGC 3379.
Subsequent work by Tonry, Luppino, and Schneider (1988) and
Tonry, Ajhar, \& Luppino (1989, 1990) revised and refined the analysis
techniques and presented further observations in $V$, $R$,
and $I$ of early-type galaxies
in Virgo, Leo, and the Local Group.
Tonry et al.\ (1990) found that the $I$~band was most
suitable for measuring distances and attempted to calibrate
the SBF method theoretically using the Revised Yale Isochrones
(Green, Demarque, \& King 1987).
There were obvious problems with this calibration, however, so
a completely empirical calibration for \ifmmode\overline{M}_I\else$\overline{M}_I$\fi\ was presented
by Tonry (1991). The calibration was based on the variation of
\ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ with \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ color in the Fornax cluster and took
its zero point from the Cepheid distance to M31. Tonry used this
calibration to derive the Hubble constant.
A detailed review of the modern SBF technique can be found
in Jacoby et al.\ (1992), which also provides some historical
context for the method.
In recent years, the SBF technique has been used to measure distances
and study a variety of stellar populations in several different bands.
$K$-band SBF observations have been reported by Luppino \& Tonry (1993),
Pahre \& Mould (1994), and Jensen, Luppino, \& Tonry (1996). These
studies find that $\overline m_K$ is also a very good distance estimator.
Dressler (1993) has measured $I$-band SBF in Centaurus ellipticals,
finding evidence in support of the Great Attractor model.
Lorenz et al.\ (1993) have measured $I$-band SBF in Fornax, and
Simard \& Pritchet (1994) have reported
distances to two Coma~I galaxies using $V$-band SBF observations.
Ajhar \& Tonry (1994) reported measurements of \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ and
$\overline m_V$ for 19 Milky Way globular clusters and considered
the implications for both the distance scale and stellar populations.
Tiede, Frogel, \& Terndrup (1995) measured \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ and
$\overline m_V$ for the bulge of the Milky Way and derived the
distance to the Galactic center.
Sodemann \& Thomsen (1995, 1996) have used fluctuation colors and
radial gradients to investigate stellar populations in galaxies.
Finally, an enormous amount of progress has been achieved on the theoretical
SBF front through the stellar population models of Worthey (1993a,b, 1994),
Buzzoni (1993, 1995), and Yi (1996).
\subsection{Genesis of the SBF Survey}
When it became apparent that $I$-band SBF observations could indeed
provide accurate and reliable distances to galaxies, we undertook a
large survey of nearby galaxies. The sample selection is not
precisely defined because the measurement of SBF depends on a number
of criteria which are not ordinarily cataloged, such as dust content.
In addition, because the measurement of SBF is fairly expensive in
terms of telescope time and quality of seeing, it simply was not
possible to observe all early-type galaxies within some magnitude
limit out to a redshift which is large enough to make peculiar
velocities negligible. Nevertheless, we have tried to manage fairly
complete coverage of early-type galaxies within 2000~km/s and brighter
than $B = 13.5$, and we have significant coverage beyond those limits.
Comparison with
the Third Reference Catalog of Bright Galaxies
(de Vaucouleurs et al.\ 1991)
reveals that of the early-type galaxies ($T<0$) with $B \le 13.5$ in
the RC3, we have observed 76\% with heliocentric velocity
$v < 1000$~km/s, 73\% with $1000 < v < 1400$,
64\% with $1400 < v < 2000$, 49\% with $2000 < v < 2800$, and we have
data for more than 40 galaxies with $v > 2800$~km/s. Virtually all of the
galaxies closer than $v<2000$ where we lack data are S0s
for which measuring SBF is complicated by
dust and/or disk/bulge problems, and since many of them are in the cores of
clusters such as Virgo, we do not regard their distances as being
important enough to delay completion of our survey.
The survey is, however, an ongoing project, with some data still to be reduced.
About 50\% of our sample is listed as E galaxies ($T\le-4$), about
40\% as S0s, and 10\% as ``spirals'' ($T\ge0$).
Our sample of galaxies is drawn from the entire sky, and the
completeness was mainly driven by the vicissitudes of weather and
telescope time, so the sampling is fairly random. The survey
includes a large number of galaxies in the vicinity of the Virgo
supercluster, and the next paper in this series will present an analysis
of their peculiar motions.
The following section describes the SBF survey in more detail,
including the observations, photometric reductions, and consistency checks.
In Section 3 we use our observations of galaxies in groups to derive
the dependence of \ifmmode\overline{M}_I\else$\overline{M}_I$\fi\ on \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi. Seven of these groups
also have Cepheid distances, which we use to set the zero point of the
\ifmmode\overline{M}_I\else$\overline{M}_I$\fi--\ifmmode(V{-}I)\else$(V{-}I)$\fi\ relation. This new \ifmmode\overline{M}_I\else$\overline{M}_I$\fi\ calibration
agrees well with theory and supersedes the old calibration of Tonry
(1991). We then compare our distances
to those found using a number of other methods. In Section 4 we
discuss the tie to the large-scale Hubble flow and implications
for the value of $H_0$. The final section provides a summary of our
main conclusions.
\section{Observations and Reductions}
All in all, the SBF survey extends over some 40 observing runs at 6
telescopes. Table 1 lists these runs along with some salient
information. Note that the date of the run is coded in the name
as (Observatory)MMYY; the remaining columns are described below.
The normal observing procedure when the skies were clear was to
obtain sky flats each night and observe a number of Landolt standard
stars. We preferred observing the faint standard star fields of
Landolt (1992) in which there are several stars per CCD field and
where the observations are long enough that shutter timing is not a
problem. Table 2 gives our usual fields.
During a typical night
we would observe about 10 fields comprising perhaps 50 stars over a
range of airmasses from 1.1 to 2.0. We also strived to observe stars
over a wide
range of color ranging from $0 < \ifmmode(V{-}I)\else$(V{-}I)$\fi < 2$. Because there is substantial
fringing seen in the $I$ band with thin CCDs, at some point in a
run we would spend several hours looking at a blank field in order to
build up a ``fringe frame''. We have found the fringing pattern
for a given CCD and filter (although not the amplitude) to be
remarkably stable from night to night (even year to year). Hence, a
single fringe frame was used to correct an entire run's data, and we
usually collected a new one for each run.
The reductions of the photometry proceed by bias subtraction,
flattening, and following Landolt (1992),
summing the net flux from photometric standards within a
14\arcsec\ aperture. We also estimate a flux error from the sky brightness
and variability over the image and remove any stars whose expected
error is greater than 0.02 magnitude. Once all the photometric
observations from a run have been reduced,
we fit the results according to
\begin{equation}
m = m_1 - 2.5\log(f) - A\;\sec z + C\;\ifmmode(V{-}I)\else$(V{-}I)$\fi,
\end{equation}
where $f$ is the flux from the star in terms of electrons per second.
We have found that $m_1$ and $C$ are constant during a run
with a given CCD and filter, so we fit for a single value for these
parameters and extinction coefficients $A$ for each night. The rms
residual of the fit is typically about 0.01 magnitude which is
satisfactory accuracy for our purposes. Table 1 lists typical values
for $m_1$, $A$, and $C$ for each run in the two filters $V$ and $I$.
Note the havoc in the extinction caused by the eruption by Pinatubo in 1991.
Galaxy reductions proceed by first bias subtraction,
division by a flat field, and subtraction of
any fringing present in $I$ band data.
We always take multiple images of a galaxy with the telescope
moved by several arcseconds between images, and determine these offsets
to the nearest pixel. Any bad pixels or columns are
masked out, and the images are shifted into registration. We next
run a program called ``autoclean'' which identifies cosmic rays in
the stack of images and removes them by replacement with appropriately
scaled data from the rest of the stack. Autoclean also gives us
an estimate of how photometric the sequence of observations was by
producing accurate flux ratios between the exposures.
Finally we make a mask
of the obvious stars and companion galaxies in the cleaned image and
determine the sky background by fitting the outer parts of the galaxy
image with an $r^{1/4}$ profile plus sky level. This is usually done
simultaneously for $V$ and $I$ images, and when the sky levels are
determined, we also compute \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ colors as a function of radius
from the center of the galaxy.
In order to knit all of our observations into a consistent photometric
system, we attempted to make sure that there were overlap
observations between runs, and we developed a pair of programs called
``apphot'' and ``apcomp'' to compare observations. ``Apphot''
converts a galaxy image with its photometric calibration into a
table of circular aperture photometry. This only depends on
plate scale (which is well known) and therefore permits comparison of
different images regardless of their angular orientation. ``Apcomp''
then takes the aperture photometry from two observations and fits the
two profiles to one another using a photometry scale offset and a
relative sky level. These two programs can give us accurate offsets
between the photometry of two images, good at the 0.005 magnitude
level.
We learned, however, that good seeing is much more common than
photometric weather, and we realized that many of our
``photometric'' observations were not reliable at the 0.01 magnitude
level. As the survey progressed and the number of overlaps increased,
we also realized that although we only need 0.05 magnitude
photometry of \ifmmode\overline{m}_I\else$\overline{m}_I$\fi, \ifmmode\overline{M}_I\else$\overline{M}_I$\fi\ is sensitive enough to
\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ color that we needed better photometry. The existing
photoelectric (PE) photometry, although probably very good in the
mean, is neither extensive enough nor accurate enough to serve to
calibrate the survey.
We also became aware that there are many peculiar CCD and shutter
effects which make good photometry difficult. For example, we
have found photometry with Tektronix (SITe) CCDs particularly
challenging for reasons we do not fully understand. Because of their
high quantum efficiency and low noise they have been the detectors
of choice, but run to run comparisons with apphot and apcomp
show consistent zero point offsets at the 0.05 magnitude level. While
not a serious problem for \ifmmode\overline{m}_I\else$\overline{m}_I$\fi, we had to do much better in measuring
\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi.
Accordingly, we undertook an auxiliary survey in 1995 of a substantial
fraction of our SBF survey from the McGraw-Hill 1.3-m telescope at
the MDM Observatory. We
shared the time with another program and used only nights which were
photometric, as judged by the observer at the time and
as revealed later from the quality of the photometric standard
observations. We did not use Tek CCDs but primarily used the
front-illuminated, Loral
2048$^2$ CCD Wilbur (Metzger et al. 1993),
we used filters which match $V$ and $I_{KC}$ as
closely as possible, and the large field of view permitted us to make
very good estimates of sky levels. Over 5 runs this comprised about
600 observations in $V$ and $I$. We made certain to have a generous
overlap between these observations and all our other observing runs,
reaching well south to tie to the CTIO and LCO data.
We then performed a grand intercomparison of all the photometric
data in order to determine photometric offsets from run to run.
Using apphot and apcomp, we determined offsets between
observations, and we built up a large table of comparison pairs.
In addition, photoelectric (PE) photometry from deVaucouleurs
and Longo (1988), Poulain and Nieto (1994), and Buta and Williams (1995)
served as additional sources of comparison, and we
computed differences between PE and our photometry for every galaxy in
common. We have found that \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ colors often show somewhat better
agreement than the individual $V$ and $I$ measurements, presumably
because thin clouds are reasonably gray, so we also compared colors
directly in addition to photometric zero points.
The results are illustrated in Figure 1.
In each of three quantities $V$, $I$, and \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi, we fitted for
zero point offsets for each run (photoelectric sources were
considered to be a ``run''), minimizing the pairwise differences.
We set the overall zero point by insisting that the median run
offset be zero. Upon completion, we found that the rms
of the zero point offsets to be 0.029 mag, and the rms scatter
of individual comparisons between CCD data to be 0.030 mag in
$V$, 0.026 mag in $I$, and 0.024 mag in \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi. The scatter was
bigger for CCD-PE zero point comparisons, 0.047 mag in both $V$ and
$I$. The ``zero point offsets'' for the photoelectric
photometry were 0.003 mag in $V$, 0.017 mag in $I$, and 0.004 mag in
\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi, which we take to be close enough to zero that we did not choose to
modify our overall median zero point to force them to zero.
Finally, we chose zero point corrections for $V$ and $I$ for each run
according to these offsets. The difference of the corrections was
set to the \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ offset from the comparison, and the sum of the
corrections was the sum of the $V$ and $I$ offsets. We therefore
believe that (a) our photometry is very close to Landolt and photoelectric
in zero point, (b) the error in the $V$ or $I$ photometry for
a given observation is 0.02 mag, and (c) the error in a given \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\
color measurement is 0.017 mag (where we have divided by $\sqrt2$ to
get the error for single measurements). We also add in quadrature 0.25 of
the zero point offset which was applied. The offsets $\Delta V$ and
$\Delta I$ for each run are listed in Table 1.
The reductions of \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ are described elsewhere (e.g., Jacoby et al.
1992). Briefly, we fit a galaxy model to the summed, cleaned,
sky-subtracted galaxy image and subtract it. If there is dust present
(all too common in ellipticals and S0s as well as the large bulge
spiral galaxies we observe), we mask it out as well. Experiments with
masking different portions of M31 and M81 (where we used $B$ band
observations to show us clearly the location of the dust) indicate that
reasonable care in excising dust will produce a reliable \ifmmode\overline{m}_I\else$\overline{m}_I$\fi, both
because the extinction is less in the $I$ band and also because
the dust tends to cause structure at relatively large scales which
are avoided by our fit to the Fourier power spectrum. We run DoPhot
(Schechter et al. 1993)
on the resulting image to find stars, globular clusters, and background
galaxies; fit a luminosity function to the results; and derive a mask
of objects brighter than a completeness limit and an estimate of
residual variance from sources fainter than the limit. Applying the
mask to the model-subtracted image, we calculate the variance from the
fluctuations in a number of different regions. Finally, this variance
is converted to a value for \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ by dividing by the mean galaxy flux
and subtraction of the residual variance estimate from unexcised point
sources. Generally speaking, the various estimates of \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ are quite
consistent from region to region, and a weighted average and error
estimate are tabulated for each observation. If the observation was
photometric, we also record the \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ color found in the same region
in which we measure \ifmmode\overline{m}_I\else$\overline{m}_I$\fi.
There are many galaxies for which \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ and \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ have been measured
more than once, and intercomparison of the different observations
can be used to
evaluate whether our error estimates are reasonable. If we
consider all pairs and divide their difference by the
expected error, the distribution should be a Gaussian of unity variance.
Figure 2 illustrates these distributions for \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ and \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi.
Evidently, the error estimates are usually quite good, with discordant
observations occurring rarely. In most cases of discordances,
it is clear which of the observations is trustworthy, and
we simply remove the other observation from further consideration.
These excised observations occur 1.5\% of the time for \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ and
0.3\% of the time for \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi, and are an indication of how frequently
bad observations occur.
After observations are averaged together, they are subjected to some final
corrections. The mean \ifmmode(V{-}I)\else$(V{-}I)$\fi\ color of the
fluctuations is the mean of a galaxy's color \ifmmode(V{-}I)\else$(V{-}I)$\fi\ and the
``fluctuation color'' \ifmmode\overline{m}_V\else$\overline{m}_V$\fi--\ifmmode\overline{m}_I\else$\overline{m}_I$\fi, or $\ifmmode(V{-}I)\else$(V{-}I)$\fi\approx 1.85$ (since
the rms fluctuation is the square root of the flux from the galaxy
and the flux from \ifmmode\overline{m}_I\else$\overline{m}_I$\fi). The value of
\ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ is corrected according to this mean color and the color
term for the run's photometry. The values of \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ and \ifmmode(V{-}I)\else$(V{-}I)$\fi\ are
corrected for galactic extinction according to
\begin{equation}
A_V:A_{I_{\rm KC}}:E(B-V) = 3.04:1.88:1.00,
\end{equation}
where $E(B-V)$ comes from Burstein \& Heiles (1984), who give
$A_B = 4.0\, E(B-V)$,
the relative extinction ratio $A_{I_{\rm KC}} / A_V = 0.62$ is
taken from Cohen et al. (1981) for a star halfway between an A0 and an
M star,
and $A_V/E(B-V)$ is an adjustment of a value of 3.1 for
A0 stars common in the literature ({\it e.g.,} Cardelli, et al. 1989)
to a value of 3.04 more appropriate for early-type galaxies, following
the ratios given in Cohen et al.
The final modification is the application of K-corrections which brighten
magnitudes in $V$ and $I$ by 1.9 and $1.0 \times z$ respectively
(Schneider 1996), and brighten fluctuation magnitudes in $I$ by $7.0
\times z$ (Worthey 1996). Note that the very red color of SBF causes
flux to be shifted rapidly out of the $I$ band with redshift, but the
\ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ K-correction amounts to only 0.05 magnitude at a typical distance
of 2000~km/s.
\section{Calibrating \ifmmode\overline{M}_I\else$\overline{M}_I$\fi}
The next step we take in trying to establish how \ifmmode\overline{M}_I\else$\overline{M}_I$\fi\ varies according
to stellar population is to look at how \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ varies from galaxy to
galaxy within groups, where the distance to the galaxies is essentially
constant. We originally chose to observe SBF in the $I$ band because
stellar population models indicate that \ifmmode\overline{M}_I\else$\overline{M}_I$\fi\ is relatively constant
from population to population, and that the effects of age,
metallicity, and IMF are almost degenerate --- in other words, \ifmmode\overline{M}_I\else$\overline{M}_I$\fi\ is
nearly a one parameter family.
Guided by theoretical models we seek to establish whether three
statements are a fair description of our data:
\par\indent \hangindent2\parindent \textindent{(\S3.1)} \ifmmode\overline{M}_I\else$\overline{M}_I$\fi\ is a one-parameter family, with a
universal dependence on \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\break
(i.e., \ifmmode\overline{M}_I\else$\overline{M}_I$\fi\ is a function of \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ with small residual scatter).
\par\indent \hangindent2\parindent \textindent{(\S3.2)} The zero point of the \ifmmode\overline{M}_I\else$\overline{M}_I$\fi--\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ relation is universal.
\par\indent \hangindent2\parindent \textindent{(\S3.3)} The \ifmmode\overline{M}_I\else$\overline{M}_I$\fi--\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ relation is consistent with theoretical
models of stellar populations.
To this end we chose approximately 40 nearby groups where we currently have
(or will have) observed more than one galaxy. The groups are defined
by position on the sky and a redshift range and in most cases
correspond to one of the groups described by Faber et al. (1989).
Table 3 lists our groups.
Note that we are not trying to include all
groups, nor do we have to be complete in including all galaxies which
are members. We are simply trying to create samples of galaxies for
which we are reasonably confident that all galaxies are at the same
distance.
\subsection{Universality of the \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ dependence on \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi}
Figure 3 illustrates the \ifmmode\overline{m}_I\else$\overline{m}_I$\fi--\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ relationship in six groups where we
have measured SBF in a number of galaxies: NGC~1023, Leo, Ursa Major,
Coma~I\&II, Virgo, and Fornax. The lines are drawn with slope 4.5 and
zero point according to the fit to the data described below. We
see that
galaxies which meet the group criteria of position on the sky and
redshift are consistent with the same \ifmmode\overline{m}_I\else$\overline{m}_I$\fi--\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ relationship, where
the scatter reflects both the measurement error and the group depth
inferred from spread across the sky. In Virgo we find NGC~4600 much
brighter than the rest of the galaxies, NGC~4365 significantly
fainter, and NGC~4660 (the point at $\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi=1.21$ and $\ifmmode\overline{m}_I\else$\overline{m}_I$\fi = 28.9$) also
with an unusually bright \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ for its color. These three galaxies,
marked as smaller, square symbols, are discussed below.
Note that ellipticals and S0 galaxies are intermixed with spirals
(NGC~3368 in Leo, NGC~4548 in Virgo, NGC~891 in the NGC~1023 group,
and NGC~4565 and NGC~4725 in the Coma~I\&II group). The two galaxies
in Fornax marked as ``spiral'' (NGC~1373 and NGC~1386) might better
be classified as S0 on our deep CCD images. For this admittedly small
sample we see no offset between SBF measurements in spiral bulges and
early-type galaxies. We regard this as confirmation of our
assumption that SBF measurements are equally valid in spiral bulges
as in early-type galaxies.
In order to test the hypothesis that \ifmmode\overline{M}_I\else$\overline{M}_I$\fi\ has a universal dependence on
\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ in a more systematic way than fitting individual groups, we
simultaneously fit all our galaxies which match the group criteria with
\begin{equation}
\overline m_I = \langle\overline m_I^0\rangle_j + \beta \; [\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi-1.15],
\end{equation}
where we fit for values of $\langle\overline m_I^0\rangle_j$ for
each of j=1,N groups and a single value for $\beta$. The quantity
$\langle\overline m_I^0\rangle_j$ is the group mean value for \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ at a
fiducial galaxy color of $\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi=1.15$. The measurements of \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ and
\ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ carry errors which the pair-wise comparisons and the averaging
procedure of section 2 indicate are accurate.
We also anticipate that there will be an irreducable ``cosmic''
scatter in \ifmmode\overline{M}_I\else$\overline{M}_I$\fi. Accordingly, in fitting \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ as a function of \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi, we
include an error allowance for this cosmic scatter which is nominally
0.05 magnitudes (i.e. for this fit the error in \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ is enhanced by
0.05 magnitude in quadrature). In addition, we will also see scatter
because the galaxies within groups are not truly at the same distance.
We therefore calculate the rms angular position of the galaxies making
up each group, and divide this radius by $\sqrt2$ as an estimate of the
rms group depth. Converting this to a magnitude, we add it in
quadrature to the error in \ifmmode\overline{m}_I\else$\overline{m}_I$\fi. We then perform a linear fit of $N+1$
parameters which allows for errors in both the ordinate and abcissa,
according to the ``least distance'' method used by Tonry and Davis
(1981). (This also appears in a slightly different guise in the second
edition of {\it Numerical Recipes} by Press et al. 1992)
We remove the three Virgo galaxies which we believe are at
significantly different distances from the rest of the group (NGC~4365,
NGC~4660, and NGC~4600), mindful that what is considered to be part of
Virgo and what is not is somewhat arbitrary. We also choose to exclude
NGC~205 and NGC~5253 from the fit because recent starbursts make them
extremely blue --- we do not believe our modeling extends to such
young populations.
With 149 galaxies we have 117 degrees of freedom, and we find that
$\chi^2 = 129$, $\chi^2/N = 1.10$, and the slope of the \ifmmode\overline{M}_I\else$\overline{M}_I$\fi--\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\
relation is $4.5 \pm 0.25$. The galaxies contributing to the fit span
a color range of $1.0<\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi<1.3$. Because Virgo still contributes five
of the seven most discrepant points (the other two are in Cetus), the
rms depth used for Virgo ($2.35^\circ \rightarrow 0.08$~mag) may be too
small, making $\chi^2/N$ slightly bigger than one. If we replace the 3
Virgo galaxies we omitted earlier, we find that $\chi^2/N$ rises to
1.75 for 120 degrees of freedom and the slope changes to $4.7\pm0.25$,
showing that even though these galaxies are significantly outside of
Virgo, the slope is robust. When we experiment with adding and
removing different groups we find that the slope changes slightly, but
is always consistent with the error above.
These values for $\chi^2$ include an allowance for cosmic scatter of
0.05 magnitude and the nominal, rms group depth. These two,
ill-constrained sources of error can play off against each other: if we
double the group depth error allowance, we get $\chi^2/N = 1.0$ for
zero cosmic scatter; if we increase the cosmic scatter to 0.10
magnitude, we need to decrease the group depth to zero in order to make
$\chi^2/N = 1.0$. Therefore, even though we cannot unambiguously
determine how much cosmic scatter there is in the \ifmmode\overline{M}_I\else$\overline{M}_I$\fi--\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ relation, it
appears to be $\sim$0.05 mag.
The referee pointed out that even if we make no allowance for group
depth, the cosmic scatter of 0.10 mag makes SBF the most precise
tertiary distance estimator by far, and wanted to know how sensitive
this is to our estimates of observational error. There is not much
latitude for the cosmic scatter to be larger than 0.10 mag. The
distribution of measurement error in \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ and \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ (which also enter
$\chi^2$) starts at 0.06 mag, and has quartiles at 0.11, 0.16, and
0.20 mag. If we wanted to increase the cosmic scatter by $\sqrt2$ to
0.14 mag, we would have to have overestimated the observational errors
by 0.10 mag in quadrature, and apart from the fact that a quarter of
the measurements would then have imaginary errors, our pairwise
comparison of multiple observations from the previous section would
not allow such a gross reduction in observational error.
Figure 4 illustrates how \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ depends on \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ when all the group data
have been slid together by subtraction of the group mean at
$\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi=1.15$. Note again that spiral galaxies, in this case four
galaxies with both Cepheid and SBF distances, show no offset relative
to the other early-type galaxies making up the groups in which they
appear, other than the usual trend with \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi. The overall rms
scatter, 0.18 mag, arising from all the effects discussed above, is
a testament to the quality of SBF as a distance estimator.
The Local Group galaxies NGC~205, NGC~147, and NGC~185 have also
been plotted in Figure 4 (although they were not used in the fit), under
the assumption that they are at the same distance as M31 and M32.
This may or may not be a valid assumption for NGC~147 and NGC~185,
but they agree reasonably well with the mean relation. In contrast to
these two galaxies, which are blue because of extremely low
metallicity, NGC~205 has undergone a recent burst of star formation
and has a strong A star spectral signature. Because our models do
{\it not} extend to such young populations, the systematic deviation
from the mean relation is not unexpected.
The inset in Figure 4 extends the color range to show that this
deviation continues for two other galaxies where there has been recent
star formation: NGC~5253 and IC~4182. NGC~5253 is 0.5 mag fainter than
one would expect using a naive extrapolation of the relation to its
color of $\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi = 0.84$, and IC~4182 has an SBF magnitude which is 0.75
mag fainter than one would judge from its Cepheid distance and its
color of $\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi = 0.71$. Qualitatively this makes sense because the very
young stars change the overall color of the galaxy quite a bit but are
not very luminous in the $I$ band compared to the stars at the top of
the RGB which are the main contributors to the SBF \ifmmode\overline{m}_I\else$\overline{m}_I$\fi. It may be that
these very young populations can be understood well enough that one can
safely predict the SBF absolute magnitude from the mean color, but this
is beyond the scope of this paper.
Tammann (1992) expressed concern that there are residual stellar
population effects in SBF even after the correction for \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ color.
However, his critique was based on an early
attempt to correlate \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ as a function of \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ (Tonry et al. 1990).
Unfortunately, that work had the wrong sign for the slope (appropriate
for the $JHK$ bandpasses but not $I$), because it was based on the Revised
Yale Isochrones (Green et al. 1987), which did not properly model the
line blanketing in metal rich, high luminosity stars. The effect noted
by Tammann was a residual correlation of the corrected \ifmmode\overline{m}_I^0\else$\overline{m}_I^0$\fi\ with the
Mg$_2$ index among galaxies within a cluster. Figure 5 shows these trends
do not exist for the present data and the new
\ifmmode\overline{m}_I\else$\overline{m}_I$\fi--\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ relation: in both Fornax and Virgo there is no residual
correlation with either Mg$_2$ or galaxy magnitude.
We conclude that a one-parameter, linear relation between
\ifmmode\overline{M}_I\else$\overline{M}_I$\fi\ and \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ suffices to describe our data for $1.0 < \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi < 1.3$; the
slope of the \ifmmode\overline{M}_I\else$\overline{M}_I$\fi--\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ relation is universally $4.5 \pm 0.25$, and we
are indeed detecting cosmic scatter in \ifmmode\overline{M}_I\else$\overline{M}_I$\fi\ of order 0.05 mag. Very few
galaxies fail to follow the relation, and for every such galaxy
at least one of the following statements is true: (1) the measurement
of \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ or \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ is doubtful; (2) the galaxy may not be a member of the
group we assigned it to; (3) the stellar population is bluer than
$\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi=1.05$ due to recent star formation.
Note that this slope is steeper than the value of 3 tendered by Tonry
(1991) and used by Ciardullo et al. (1993) who suggested that it might
be as steep as 4. Basically, the reason for this is that the older
data were noiser and were fitted only to errors in the ordinate,
whereas in fact the errors in \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ are quite significant, particularly
for the better measured \ifmmode\overline{m}_I\else$\overline{m}_I$\fi, which count heavily in any weighted fit.
\subsection{Universality of the \ifmmode\overline{M}_I\else$\overline{M}_I$\fi\ zero point}
We have effectively tested the hypothesis that the zero point of SBF
is universal within groups, but in order to extend the test from group
to group we need independent distance estimates. Since the groups are
all nearby, the group's redshift is not an accurate
distance estimate --- there are likely to be substantial
non-Hubble velocities included in the group's recession velocity.
We therefore turn to other distance estimators: Cepheids,
planetary nebula luminosity function (PNLF), Tully-Fisher (TF),
\ifmmode D_n{-}\sigma\else$ D_n{-}\sigma$\fi, Type II supernovae (SNII) and Type I supernovae (SNIa).
Some of these estimators have zero points in terms of Mpc (such as
Cepheids and SNII), others have zero points in terms of km/s based on the
Hubble flow (such as \ifmmode D_n{-}\sigma\else$ D_n{-}\sigma$\fi), and a few have both (such as TF).
For our initial discussion we seek only to establish whether the
relative distances agree with SBF; for now we do not care about the
zero point, though it will soon be addressed.
Figures 6 and 7 show the comparison between the values of the SBF
parameters \ifmmode\langle\overline{m}_I^0\rangle\else$\langle\overline{m}_I^0\rangle$\fi\ derived previously for each of our groups and the
distances to the groups according to these 6 methods.
The results of fitting lines of unity slope
(allowing for errors in both coordinates) to the
data in each panel are given in Table 4. We use the published error
estimates for all of these other indicators so $\chi^2/N$ should be
viewed with some caution: outliers and non-Gaussian errors or
over-optimistic error estimates can inflate $\chi^2/N$ even though the
mean offset is still valuable.
Since each comparison is
very important, we briefly discuss them individually.
\subsubsection{Cepheids}
There is now a growing number of
Cepheid distances with which we compare, but we are faced with the
complication that Cepheids occur in young stellar populations, while
SBF is best measured where such populations are not present.
There are five galaxies which have
both Cepheid and SBF distances:
NGC~224 (Freedman \& Madore 1990),
NGC~3031 (Freedman et al 1994),
NGC~3368 (Tanvir et al. 1995),
NGC~5253 (Saha et al. 1995), and
NGC~7331 (Hughes 1996).
NGC~5253 is especially problematic for SBF, because its recent
starburst has produced a much younger and bluer stellar population than
we have calibrated. We can, of course, also compare distances
according to group membership. There are 7 groups where this is
currently possible:
Local Group,
M81,
CenA,
NGC~1023 (NGC~925 from Silbermann et al. 1996),
NGC~3379 (also including NGC~3351 from Graham et al. 1996),
NGC~7331, and
Virgo (including NGC~4321 from Ferrarese et al. 1996,
NGC~4536 from Saha et al. 1996a, and NGC~4496A from Saha et al. 1996b;
we exclude NGC~4639 from Sandage et al. 1996 because we are also
excluding NGC~4365 and the W cloud from the SBF mean). In the former
case we find that fitting a line to \ifmmode\langle\overline{m}_I^0\rangle\else$\langle\overline{m}_I^0\rangle$\fi\ as a function of
\ifmmode(m{-}M)\else$(m{-}M)$\fi\ yields a mean offset of $-1.75\pm0.05$ mag with $\chi^2/N$ of
3.4 for 4 degrees of freedom, and $-1.82\pm0.06$ mag with $\chi^2/N$ of
0.3 for 3 degrees of freedom when NGC~5253 is excluded. In the latter
case we get a mean offset of $-1.74\pm0.05$ mag with $\chi^2/N$ of 0.6
for 6 degrees of freedom. When NGC~5253 is excluded, the rms scatter
is remarkably small, only 0.12 magnitudes for the galaxy comparison and
0.16 magnitudes for the group comparison.
\subsubsection{PNLF}
Ciardullo et al. (1993) reported virtually perfect agreement between
SBF and PNLF, but recent publications (Jacoby et al. 1996) have raised some
discrepancies. Examination of Figure 7 reveals that our fit has two
outliers: Coma~I (e.g. NGC~4278) and Coma~II (e.g. NGC~4494). Because
we do not know how to resolve this issue at present, Table 4 gives the
result for $\ifmmode\langle\overline{m}_I^0\rangle\else$\langle\overline{m}_I^0\rangle$\fi - \ifmmode(m{-}M)\else$(m{-}M)$\fi_{\hbox{PNLF}}$ for the entire sample and
when these two outliers are removed. Since PNLF is fundamentally
calibrated on Cepheids, this is not independent of the previous
number, but it does confirm that PNLF and SBF are measuring the same
relative distances.
\subsubsection{SNII} The expanding photospheres method (EPM)
described most recently by
Eastman et al. (1996) offers distance estimates which are largely
independent of the Cepheid distance scale. There is only one galaxy
with both an EPM and an SBF distance (NGC~7331), but there have also been
two SNII in Dorado (NGC~1559 and NGC~2082), two in Virgo (NGC~4321 and
NGC~4579), and one in the NGC~1023 group (NGC~1058). The agreement
between EPM and SBF (Fig. 6) is good. The farthest outlier is
NGC~7331, for which SBF and Cepheid distances are discordant with the
SNII distance. Table 4 lists separately the zero point, scatter, and
$\chi^2/N$ when NGC~7331 is included and excluded.
\subsubsection{TF (Mpc calibration)} B. Tully (1996) was kind
enough to provide us with TF distances to the SBF groups in advance of
publication. The fit between TF and SBF gives $\ifmmode\langle\overline{m}_I^0\rangle\else$\langle\overline{m}_I^0\rangle$\fi - \ifmmode(m{-}M)\else$(m{-}M)$\fi =
-1.69\pm0.03$ mag. This is again not independent of the Cepheid
number, since the TF zero point comes from the same Cepheid
distances. Figure 7 demonstrates that the agreement is
generally good, despite the high $\chi^2/N$ which comes from a few
non-Gaussian outliers. We cannot tell whether these outliers reflect
non-Gaussian errors in the methods or simply the difficulties of
choosing spirals and early type galaxies in the same groups.
\subsubsection{TF (km/s calibration)} We applied the
SBF group criteria to the ``Mark II'' catalog of galaxy distances
distributed by D. Burstein. We selected all galaxies with ``good'' TF
distances (mostly from Aaronson et al. 1982) and computed an average
distance to the groups, applying the usual Malmquist bias correction
according to the precepts of Lynden-Bell et al. (1988) and the
error estimates from Burstein. Because these
distances have a zero point based on the distant Hubble flow, we derive
an average offset of $\ifmmode\langle\overline{m}_I^0\rangle\else$\langle\overline{m}_I^0\rangle$\fi\ - 5\log d = 13.55\pm0.08$ mag.
\subsubsection{\ifmmode D_n{-}\sigma\else$ D_n{-}\sigma$\fi} Most of the SBF groups are the same as
those defined by Faber et al. (1989). We compare their Malmquist bias
corrected distances to these groups (which are based on a zero point
from the distant Hubble flow) with SBF and find the same result as
Jacoby et al. (1992): the distribution of errors has a larger tail than
Gaussian, but the error estimates accurately describe the central core
of the distribution. $\chi^2/N$ is distinctly larger than 1, but the
difference histogram in Figure 7 reveals that this is because of the
tails of the distribution. The fit between \ifmmode D_n{-}\sigma\else$ D_n{-}\sigma$\fi\ and SBF gives
$\ifmmode\langle\overline{m}_I^0\rangle\else$\langle\overline{m}_I^0\rangle$\fi\ - 5\log d = 13.64\pm0.05$ mag.
\subsubsection{SNIa}
Extraordinary claims have been made recently about the quality of SNIa
as distance estimators. Some authors (e.g. Sandage and Tammann 1993)
claim that suitably selected (``Branch normal'') SNIa are standard
candles with a dispersion as little as 0.2 mag. Others (e.g. Phillips
1993) believe that they see a correlation between SNIa luminosity and
their rate of decline, parametrized by the amount of dimming 15
days after maximum, $\Delta m_{15}$. Still others (e.g. Riess et al.
1995) agree with Phillips (1993) but believe that they can categorize
SNIa better by using more information about the light curve shape than
just this rate of decline. Finally, there is the ``nebular SNIa
method'' of Ruiz-Lapuente (1996) which tries to determine the mass of
the exploding white dwarf by consideration of the emission lines from
the expanding ejecta. We therefore choose to compare SBF
distances with SNIa under two assumptions: that SNIa are standard
candles, and that $m_{max} - \alpha\Delta m_{15}$ is a better indicator
of distance. In both cases we restrict our fits to $0.8 < \Delta
m_{15} < 1.5$ as suggested by Hamuy et al. (1995) and use a distance
error of 0.225 mag for each SNIa.
SNIa have been carefully tied to a zero point according to the distant
Hubble flow (one of the main advantages of SNIa) by Hamuy et al.
(1995), under both assumptions. There have also been vigorous attempts
to tie the SNIa to the Cepheid distance scale which we have chosen not to
use because of the circularity with our direct comparison between SBF
and Cepheids.
The results are both encouraging and discouraging. We find that there
is indeed a good correlation between SNIa distance and SBF, with
average values of $\ifmmode\langle\overline{m}_I^0\rangle\else$\langle\overline{m}_I^0\rangle$\fi\ - 5\log d = 13.92\pm0.08$ mag and
$14.01\pm0.08$ mag for the group comparison under the two assumptions.
As illustrated in Figure 6, $m_{max} - \alpha\Delta m_{15}$ does
correlate better with distance than $m_{max}$, but as long as ``fast
declining'' SNIa are left out there is scant difference between the
zero point according to the two methods.
The panels of Figure 6 showing SBF and SNIa hint at a systematic
change between the nearest three and the farthest three groups, in the
sense that there appears to be a change in zero point by about 0.7 mag.
One might worry that this is evidence that SBF is ``bottoming out'',
but there is no hint of this in the comparisons with TF and \ifmmode D_n{-}\sigma\else$ D_n{-}\sigma$\fi\ in
Figure 7 which extend to much fainter \ifmmode\overline{m}_I\else$\overline{m}_I$\fi. One might also worry about
whether there are systematic differences in SNIa in spirals and
ellipticals, and biases from the lack of nearby ellipticals or S0
galaxies. However, it is probably premature to examine these points
in too much detail. For example, the point at $\ifmmode\langle\overline{m}_I^0\rangle\else$\langle\overline{m}_I^0\rangle$\fi \approx 28$ uses
the SBF distance to Leo~I, but the SNIa occurred in NGC~3627 which
lies $8^\circ$ away from the Leo~I group. This is a fundamental
difficulty in the SBF--SNIa comparison, which will improve as SBF
extends to greater distances and more nearby SNIa are observed.
There are seven galaxies bearing SNIa where SBF distances have been
measured: NGC~5253 (SN~1972E), NGC~5128 (SN~1986G), NGC~4526
(SN~1994D), NCG~2962 (SN~1995D), NGC~1380 (SN~1992A), NGC~4374
(SN~1991bg), NGC~1316 (SN~1980N). Inasmuch as two of these are slow
decliners (SN~1986G, SN~1991bg), we fit the remaining five using the
SBF distance to the galaxy instead of the group. We derive $\ifmmode\langle\overline{m}_I^0\rangle\else$\langle\overline{m}_I^0\rangle$\fi\ -
5\log d = 13.86\pm0.12$ mag and $14.01\pm0.12$ mag for the two
methods.
We regard the SBF distance to NGC~5253 as uncertain because we
have not calibrated \ifmmode\overline{M}_I\else$\overline{M}_I$\fi\ for such a young stellar population. We
thus also recompare SBF and SNIa with NGC~5253 removed from
consideration. $\chi^2/N$ becomes dramatically smaller in both cases
and $\ifmmode\langle\overline{m}_I^0\rangle\else$\langle\overline{m}_I^0\rangle$\fi\ - 5\log d$ become smaller by about 0.2 mag to
$13.64\pm0.13$ mag and $13.87\pm0.13$ mag.
\subsubsection{Zero point summary}
These comparisons demonstrate that the second hypothesis is
correct: the zero point of the \ifmmode\overline{M}_I\else$\overline{M}_I$\fi--\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ relationship is
universal. We use the SBF--Cepheid fit to derive a final,
empirical relationship between \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ and \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi:
\begin{equation}
\overline M_I = (-1.74\pm0.07)\, +\, (4.5\pm0.25)\, [\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi - 1.15].
\end{equation}
This zero point differs from that of Tonry (1991) by about 0.35
magnitude. The reason is simply that the 1991 zero point was based
entirely on M31 and M32, and the observational error in both \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ and
\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ worked in the same direction, as did the photometric zero point
errors (cf. Table 1 for K0990). The SBF distances which have been
published therefore increase by about 15 percent (for example Fornax
moves from 15~Mpc to 17~Mpc), except for Virgo,
where the earlier result included NGC~4365 which we now exclude
in calculating the average distance to the core of the cluster. This
new calibration is based on 10 Cepheid distances in 7 groups and 44
SBF distances. As seen in Figure 6 and Table 4, these are highly
consistent with one another with a scatter of about 0.15 mag.
Along with the extensive photometric recalibration, this zero point
should be accurate to $\pm0.07$ mag. This error estimate
makes an allowance of 0.05 mag for the uncertainty in the Cepheid zero
point in addition to the statistical error of 0.05 mag, and
the comparisons with theory and SNII give us confidence that this
truly is correct.
\subsection{Comparison with theory}
Finally we test our third hypothesis by comparing our \ifmmode\overline{M}_I\else$\overline{M}_I$\fi--\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\
relationship with theoretical models of stellar populations.
Figure 8 shows the model predictions of Worthey (1993a,b) along
with the empirical line.
When the theoretical models are fitted with the empirically determined
slope of 4.5, they yield a theoretical zero point of $-1.81$ mag with
an rms scatter of $0.11$ mag for the SBF relation. We enter this
value in Table 4, with the scatter offered as an ``error estimate'',
but it must be remembered that this is fundamentally different from
the other entries in the table.
There is good agreement here, although
the theoretical result for \ifmmode\overline{M}_I\else$\overline{M}_I$\fi\ may be slightly brighter (0.07 mag) or
slightly redder (0.015 mag) than the empirical result. Given the
difficulties that the theoretical models have in simultaneously
fitting the color and Mg$_2$ indices of real galaxies, we regard this
agreement as excellent confirmation of the empirical calibration.
\section{The Hubble Constant}
The scope of this paper does not extend to comparing SBF distances
with velocity; this will be the subject of the next paper in the
series. However, the comparison with other distance estimators
does provide us with a measurement of the Hubble constant.
The comparison with other estimators whose zero point is defined in
terms of Mpc tells us the absolute magnitude of SBF. At our fiducial
color of $\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi=1.15$, we find that Cepheids give us an absolute magnitude
$\ifmmode\overline{M}_I\else$\overline{M}_I$\fi = -1.74\pm0.05$. We prefer the group-based Cepheid comparison
because of the very few SBF measurements possible in spirals which
have Cepheids.
The other Mpc-based distance estimators are all
consistent with this zero point, as we would hope since they are
calibrated with the same Cepheid data.
The results from theoretical models of stellar populations and SNII are
also consistent with this zero point, and provide independent
confirmation of the validity of the Cepheid distance scale.
The comparison of SBF with estimators whose zero point is based
on the large scale Hubble flow is less consistent. The estimators
based on galaxy properties, TF and \ifmmode D_n{-}\sigma\else$ D_n{-}\sigma$\fi, are consistent with
one another and consistent with SBF in terms of relative distances.
They give a zero point for SBF at the fiducial color of $\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi=1.15$ of
$\ifmmode\langle\overline{m}_I^0\rangle\else$\langle\overline{m}_I^0\rangle$\fi = 5\log d\hbox{(km/s)} + 13.59\pm0.07$, where the error comes
from the rms divided by $\sqrt{N-1}$.
Supernovae and SBF are more interesting. The group membership of the
Cepheid galaxies was not difficult since they were specifically chosen
to be group members. In contrast, the SNIa are not easy to assign to
groups in many cases. Depending on (1) whether we fit galaxies
individually or groups, (2) whether we use the
``standard candle'' model for SNIa or the ``light curve decline''
relation, and (3) whether we include or exclude NGC~5253 for which we
regard our stellar population calibration as unknown, we get values
for the SBF zero point as low as 13.64 and as high as 14.01 (Table 4).
Averaging the two methods and again estimating uncertainties from rms
divided by $\sqrt{N-1}$, we find $13.96\pm0.17$ for groups and
$13.75\pm0.14$ for galaxies. Because these differ from the TF and
\ifmmode D_n{-}\sigma\else$ D_n{-}\sigma$\fi\ by $2.0\;\sigma$ and $1.0\;\sigma$ respectively, the discrepancy
may not be statistically significant.
It is possible that there are systematic errors in the tie to the
distant Hubble flow for TF and \ifmmode D_n{-}\sigma\else$ D_n{-}\sigma$\fi, whereas the SNIa appear to be
wonderfully consistent with the large scale Hubble flow. On the other
hand, the nearby SNIa do not agree with SBF or Cepheids as well as one
might hope from the scatter against the Hubble flow, which makes one
worry about the systematics with SN1a. For example, the SN1a
distances predicted for the Fornax clusters are significantly larger
than the very recent Cepheid measurement of the distance to the Fornax
cluster (Silbermann et al. 1996b). SNII appear to agree pretty well
with SBF and Cepheids, and there should eventually be enough of them
to tie very well to the large scale Hubble flow. In subsequent papers
we will present the direct tie between SBF and the Hubble flow, both
from ground-based observations as well as HST observations beyond
5000~km/s, but at present we depend on these other estimators to
tie to the Hubble flow. It is therefore
with some trepidation that we offer a value for $H_0$.
We have a calibration for \ifmmode\overline{M}_I\else$\overline{M}_I$\fi; we have several calibrations for \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\
in terms of $5\log d\hbox{(km/s)}$; and of course $\ifmmode(m{-}M)\else$(m{-}M)$\fi = 5\log
d\hbox{(km/s)} + 25 - 5\log H_0$. If we use the TF and \ifmmode D_n{-}\sigma\else$ D_n{-}\sigma$\fi\
calibration of SBF we get $H_0 = 86$~\hbox{km/s/Mpc}. Examining groups
and averaging the ``standard candle'' and the ``$\Delta m_{15}$''
assumptions about SNIa gives us $H_0 = 72$~\hbox{km/s/Mpc}. If we
compare galaxies directly without resorting to group membership, but
leave out NGC~5253, we get an average $H_0 = 80$~\hbox{km/s/Mpc}.
We suspect that there is more to the SNIa story than is currently
understood, so we therefore prefer not to use it to the exclusion of
all other distance estimators. The range we find for $H_0$ is
$$ H_0 = 72 - 86\;\hbox{km/s/Mpc,}$$
and our best guess at this point is derived by averaging the ties
to the Hubble flow from TF, \ifmmode D_n{-}\sigma\else$ D_n{-}\sigma$\fi, SNIa (both methods) in groups
and SNIa (both methods) galaxy by galaxy. This weights the SNIa
slightly more heavily than TF and \ifmmode D_n{-}\sigma\else$ D_n{-}\sigma$\fi\ and gives a zero point of 13.72
which translates to
$$ H_0 = 81\pm6\;\hbox{km/s/Mpc.}$$
The final error term includes a contribution of 0.07 magnitude from the
disagreement between the Cepheid and theory zero points (which we hope
is indicative of the true accuracy of our calibrations), and an
allowance of 0.13 magnitude for the uncertainty in the tie to the
distant Hubble flow (judged from the scatter among the various
methods).
In order to facilitate comparisons with SBF distances, we offer the
SBF distance to 12 nearby groups in Table 5. The relative distances
are completely independent of any other distance estimator, and the
zero point uses our Cepheid-based calibration. As we finish our
reductions and analysis, the remainder of the group and individual
galaxy distances will be published.
\section{Summary and Conclusions}
We have described the observational sample which comprises the SBF
Survey of Galaxy Distances. The survey was conducted over numerous
observing runs spanning a period of nearly seven years. The photometry
of the sample has been brought into internal consistency by applying
small systematic corrections to the photometric zero points of the
individual runs. Based on comparisons between overlapping galaxy
observations, we find that our error estimates for $(V-I)$ and
$\overline m_I$ are reliable, after correction for the
photometry offsets.
From our measurements of \ifmmode\overline{m}_I\else$\overline{m}_I$\fi\ within galaxy groups, we
conclude that \ifmmode\overline{M}_I\else$\overline{M}_I$\fi\ is
well described by a linear function of \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi.
Comparison of our relative distances with Cepheid distances to these
groups indicates that this linear relationship is universal and yields
the zero point calibration for the SBF method.
This calibration is applicable to galaxies that are in the color range
$1.0 < \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi < 1.3$ and which have not experienced
recent bursts of star formation. Any intrinsic, or ``cosmic,''
scatter about this relation is small, of order 0.05~mag.
Owing to many more data and improved photometry, this
new calibration differs in its zero point by 0.35 mag from the
earlier one of Tonry (1991), but is much closer to Worthey's (1993)
theoretical zero point, differing by just 0.07~mag.
We take this close agreement
to be an independent confirmation of the Cepheid distance scale.
An extensive set of comparisons between our SBF distances and those
estimated using other methods provides still further evidence for the
universality of the \ifmmode\overline{M}_I\else$\overline{M}_I$\fi--\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ relation.
We find that the various methods are all generally quite reliable,
apart from occasional outliers which serve to inflate the
$\chi^2$ values for the comparisons.
Coupled with our distance zero point, our comparisons with
methods tied to the distant Hubble flow yield values of $H_0$ in
the range 72--86 km/s/Mpc. The comparison with SNIa suggests values
between 72 and 80, and
\ifmmode D_n{-}\sigma\else$ D_n{-}\sigma$\fi\ and TF call for values around 86.
Thus, the controversy over $H_0$ continues, but the famous ``factor
of two'' is now a factor of 20 percent.
Although the SBF Survey is still a work in progress, it is near enough to
completion that the calibration presented in this paper
should not change in any significant way.
Future papers in this series will use the SBF survey distances to address
such issues as the velocity field of
the Local Supercluster and a direct determination of $H_0$, bulk flows,
the Great Attractor, and the specific details of our SBF analysis method,
including comprehensive listings of our \ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi\ and distance measurements
for individual galaxies.
\acknowledgments
We would like to thank many people for collecting SBF and photometry
observations for us: Bob Barr, Andre Fletcher, Xiaohui Hui, Gerry
Luppino, Mark Metzger, Chris Moore, and Paul Schechter.
This research was supported by NSF grant AST-9401519,
and AD acknowledges the support of the National Science Foundation
through grant AST-9220052.
\clearpage
\begin{deluxetable}{llllllllllll}
\small
\tablecaption{Observing Runs.\label{tbl1}}
\tablewidth{0pt}
\tablehead{
\colhead{Run} & \colhead{Telescope} & \colhead{CCD} &
\colhead{$^{\prime\prime}/p$} &
\colhead{$m_{1V}$} & \colhead{$A_V$} & \colhead{$C_V$} &
\colhead{$\Delta V$}&
\colhead{$m_{1I}$} & \colhead{$A_I$} & \colhead{$C_I$} &
\colhead{$\Delta I$}
}
\startdata
K0389 & KPNO4m & TI-2 & 0.299 & 26.15 & 0.150 & \llap{$-$}0.070 & 0.014 & 25.42 & 0.070 & 0.000 & 0.012 \nl
M1189 & MDM2.4m& ACIS & 0.465 & 23.47 & 0.179 & 0.013 & 0.000 & 22.44 & 0.065 & 0.000 & 0.000 \nl
C0990 & CTIO4m & TI & 0.299 & 26.23 & 0.160 & 0.0 & \llap{$-$}0.026 & 25.29 & 0.080 & 0.0 &\llap{$-$}0.003 \nl
K0990 & KPNO4m & TI-2 & 0.299 & 26.26 & 0.160 & 0.0 & 0.019 & 25.39 & 0.080 & 0.0 & 0.045 \nl
H0291 & CFH3.6m& SAIC & 0.131 & 24.86 & 0.089 & 0.0 & \llap{$-$}0.016 & 24.62 & 0.033 & 0.0 & 0.029 \nl
L0391 & LCO2.4m& Tek & 0.229 & 24.92 & 0.15 & 0.0 & 0.024 & 24.62 & 0.07 & 0.0 & 0.030 \nl
C0491 & CTIO4m & Tek1 & 0.472 & 26.06 & 0.16 & 0.0 & 0.069 & 26.02 & 0.11 & 0.0 & 0.061 \nl
K0691 & KPNO4m & TI-2 & 0.300 & 25.97 & 0.155 & 0.0 & \llap{$-$}0.002 & 25.36 & 0.06 & 0.0 & 0.019 \nl
C1091 & CTIO4m & Tek2 & 0.472 & 26.21 & 0.45 & \llap{$-$}0.007 & 0.019 & 26.08 & 0.3 & 0.025 & 0.040 \nl
H1091 & CFH3.6m& SAIC & 0.131 & & & & 0.000 & 24.62 & 0.07 & 0.0 & 0.000 \nl
L1191 & LCO2.4m& Tek & 0.229 & 24.92 & 0.15 & 0.0 & 0.034 & 24.62 & 0.07 & 0.0 &\llap{$-$}0.014 \nl
M1191 & MDM2.4m& ACIS & 0.257 & 25.11 & 0.205 & 0.0 & \llap{$-$}0.025 & 24.53 & 0.102 & 0.035 & 0.007 \nl
C0492 & CTIO4m & Tek2 & 0.472 & 26.05 & 0.220 & 0.005 & 0.014 & 25.93 & 0.145 & 0.030 &\llap{$-$}0.020 \nl
M0492 & MDM2.4m& Lorl & 0.343 & 24.69 & 0.33 & 0.000 & 0.010 & 24.84 & 0.20 & 0.045 & 0.007 \nl
H0592 & CFH3.6m& STIS & 0.152 & 25.91 & 0.210 & 0.0 & \llap{$-$}0.010 & 25.60 & 0.110 & 0.0 &\llap{$-$}0.038 \nl
M0892 & MDM2.4m& Lorl & 0.343 & 24.64 & 0.254 & 0.000 & 0.000 & 24.74 & 0.145 & 0.045 & 0.000 \nl
L1092 & LCO2.4m& Tek & 0.229 & 24.92 & 0.15 & 0.0 & 0.000 & 24.62 & 0.07 & 0.0 & 0.000 \nl
M1092 & MDM2.4m& Lorl & 0.343 & 24.68 & 0.32 & 0.000 & 0.010 & 24.82 & 0.22 & 0.046 & 0.029 \nl
L0493 & LCO2.4m& Tek & 0.229 & 24.92 & 0.15 & 0.0 & 0.027 & 24.62 & 0.07 & 0.0 &\llap{$-$}0.070 \nl
M0493 & MDM2.4m& Lorl & 0.343 & 24.67 & 0.21 & 0.022 & \llap{$-$}0.004 & 24.35 & 0.13 & 0.030 & 0.014 \nl
M0493 & MDM2.4m& Tek & 0.275 & 25.32 & 0.24 & 0.005 & \llap{$-$}0.004 & 24.60 & 0.11 & 0.012 & 0.014 \nl
M0593 & MDM2.4m& Lorl & 0.343 & 24.70 & 0.22 & 0.022 & \llap{$-$}0.033 & 24.35 & 0.138 & 0.030 &\llap{$-$}0.009 \nl
M0593 & MDM2.4m& Tek & 0.275 & 25.32 & 0.198 & 0.025 & \llap{$-$}0.033 & 24.74 & 0.134 & 0.030 &\llap{$-$}0.009 \nl
M0893 & MDM2.4m& Lorl & 0.343 & 24.82 & 0.19 & 0.012 & \llap{$-$}0.021 & 24.54 & 0.10 & 0.025 &\llap{$-$}0.006 \nl
M0294 & MDM2.4m& Tek & 0.275 & 25.11 & 0.150 & \llap{$-$}0.017 & \llap{$-$}0.040 & 24.84 & 0.058 & 0.015 &\llap{$-$}0.074 \nl
L0394 & LCO2.4m& Tek & 0.229 & 24.92 & 0.15 & 0.0 & 0.026 & 24.62 & 0.07 & 0.0 &\llap{$-$}0.030 \nl
L0994 & LCO2.4m& Tek & 0.229 & 24.92 & 0.15 & 0.0 & 0.016 & 24.62 & 0.07 & 0.0 & 0.022 \nl
G0395 & MDM1.3m& Lorl & 0.637 & 23.22 & 0.15 & 0.019 & \llap{$-$}0.029 & 22.84 & 0.064 & 0.026 &\llap{$-$}0.007 \nl
G0495 & MDM1.3m& Lorl & 0.637 & 23.15 & 0.15 & 0.015 & \llap{$-$}0.041 & 22.83 & 0.064 & 0.010 &\llap{$-$}0.012 \nl
G0495 & MDM1.3m& STIS & 0.445 & 23.11 & 0.15 & 0.015 & \llap{$-$}0.041 & 22.65 & 0.100 & 0.010 &\llap{$-$}0.012 \nl
G0695 & MDM1.3m& Lorl & 0.637 & 23.13 & 0.15 & 0.042 & \llap{$-$}0.015 & 22.79 & 0.061 & 0.026 & 0.000 \nl
G0995 & MDM1.3m& Lorl & 0.637 & 22.97 & 0.14 & 0.014 & \llap{$-$}0.009 & 22.81 & 0.05 & 0.026 &\llap{$-$}0.015 \nl
G1095 & MDM1.3m& STIS & 0.445 & 22.98 & 0.15 & 0.005 & \llap{$-$}0.015 & 22.70 & 0.06 & 0.008 &\llap{$-$}0.015 \nl
M0295 & MDM2.4m& Tek & 0.275 & 25.11 & 0.150 & \llap{$-$}0.014 & \llap{$-$}0.040 & 24.84 & 0.058 & 0.012 &\llap{$-$}0.074 \nl
M0395 & MDM2.4m& Tek & 0.275 & 25.11 & 0.150 & \llap{$-$}0.017 & \llap{$-$}0.040 & 24.84 & 0.058 & 0.015 &\llap{$-$}0.074 \nl
L0495 & LCO2.4m& Tek & 0.229 & 24.92 & 0.15 & 0.0 & 0.008 & 24.62 & 0.07 & 0.0 &\llap{$-$}0.027 \nl
L1095 & LCO2.4m& Tek & 0.229 & 24.92 & 0.15 & 0.0 & 0.000 & 24.62 & 0.07 & 0.0 & 0.000 \nl
M1295 & MDM2.4m& Tek & 0.275 & 25.11 & 0.150 & \llap{$-$}0.017 & \llap{$-$}0.040 & 24.84 & 0.058 & 0.015 &\llap{$-$}0.074 \nl
M0196 & MDM2.4m& Tek & 0.275 & 25.11 & 0.150 & \llap{$-$}0.017 & \llap{$-$}0.040 & 24.84 & 0.058 & 0.015 &\llap{$-$}0.074 \nl
M0396 & MDM2.4m& Tek & 0.275 & 25.11 & 0.150 & \llap{$-$}0.017 & \llap{$-$}0.040 & 24.84 & 0.058 & 0.015 &\llap{$-$}0.074 \nl
\enddata
\tablecomments{Columns:
Run name, telescope, detector, plate scale (\arcsec/pixel),
photometric zero point, extinction, color term, and run offset for the
$V$ band then $I$ band.}
\end{deluxetable}
\begin{deluxetable}{lrrrrrr}
\tablecaption{Landolt Fields.\label{tbl2}}
\tablewidth{0pt}
\tablehead{
\colhead{Field} & \colhead{RA} & \colhead{Dec} &
\colhead{$V_{min}$} & \colhead{$V_{max}$} &
\colhead{$\ifmmode(V{-}I)\else$(V{-}I)$\fi_{min}$} & \colhead{$\ifmmode(V{-}I)\else$(V{-}I)$\fi_{max}$}
}
\startdata
SA92-250 & 00 54 41 & $+$00 41 11 & 14.09 & 15.35 & 0.67 & 1.34 \nl
SA95-190 & 03 53 16 & $+$00 16 25 & 12.63 & 14.34 & 0.42 & 1.37 \nl
SA95-275 & 03 54 40 & $+$00 27 24 & 12.17 & 14.12 & 1.40 & 2.27 \nl
SA98-650 & 06 52 11 & $-$00 19 23 & 11.93 & 13.75 & 0.17 & 2.09 \nl
Rubin-149 & 07 24 13 & $-$00 31 58 & 11.48 & 13.87 & $-$0.11 & 1.13 \nl
PG0918+029 & 09 21 36 & $+$02 47 03 & 12.27 & 14.49 & $-$0.29 & 1.11 \nl
PG1323$-$085 & 13 25 44 & $-$08 49 16 & 12.08 & 14.00 & $-$0.13 & 0.83 \nl
PG1633+099 & 16 35 29 & $+$09 46 54 & 12.97 & 15.27 & $-$0.21 & 1.14 \nl
SA110-232 & 18 40 50 & $+$00 01 51 & 12.52 & 14.28 & 0.89 & 2.36 \nl
SA110-503 & 18 43 05 & $+$00 29 10 & 11.31 & 14.20 & 0.65 & 2.63 \nl
Markarian-A & 20 43 59 & $-$10 47 42 & 13.26 & 14.82 & $-$0.24 & 1.10 \nl
\enddata
\tablecomments{Columns:
Field name, J2000 coordinates, $V$ magnitude of the brightest and
faintest star, and the \ifmmode(V{-}I)\else$(V{-}I)$\fi\ colors of the bluest
and reddest star.}
\end{deluxetable}
\begin{deluxetable}{llrrrrrrr}
\tablecaption{Nearby SBF Groups.\label{tbl3}}
\tablewidth{0pt}
\tablehead{
\colhead{Group} & \colhead{Example} &
\colhead{RA} & \colhead{Dec} & \colhead{rad} &
\colhead{$v_{ave}$} & \colhead{$v_{min}$}& \colhead{$v_{max}$} &
\colhead{7S\#}
}
\startdata
LocalGroup & N0224 & 10.0 & 41.0 & 5 & \llap{$-$}300 & \llap{$-$}500 & \llap{$-$}100 & 282 \nl
Cetus & N0636 & 24.2 & \llap{$-$}7.8 & 10 & 1800 & 1500 & 2000 & 26 \nl
N1023 & N1023 & 37.0 & 35.0 & 9 & 650 & 500 & 1000 & \nl
N1199 & N1199 & 45.3 & \llap{$-$}15.8 & 2 & 2700 & 2500 & 3000 & 29 \nl
Eridanus & N1407 & 53.0 & \llap{$-$}21.0 & 6 & 1700 & 500 & 2300 & 32 \nl
Fornax & N1399 & 54.1 & \llap{$-$}35.6 & 6 & 1400 & 500 & 2100 & 31 \nl
Dorado & N1549 & 63.7 & \llap{$-$}55.7 & 5 & 1300 & 700 & 1700 & 211 \nl
N1700 & N1700 & 72.2 & \llap{$-$}3.5 & 3 & 4230 & 3600 & 4500 & 100 \nl
N2768 & N2768 & 136.9 & 60.2 & 4 & 1360 & 1100 & 1700 & 215 \nl
M81 & N3031 & 147.9 & 69.3 & 8 & \llap{$-$}40 & \llap{$-$}200 & 400 & \nl
N3115 & N3115 & 150.7 & \llap{$-$}7.5 & 8 & 700 & 100 & 900 & \nl
LeoIII & N3193 & 153.9 & 22.1 & 3 & 1400 & 1000 & 1700 & 45 \nl
LeoI & N3379 & 161.3 & 12.8 & 2 & 900 & 500 & 1200 & 57 \nl
LeoII & N3607 & 168.6 & 18.3 & 3 & 950 & 650 & 1500 & 48 \nl
N3640 & N3640 & 169.6 & 3.5 & 2 & 1300 & 1200 & 1800 & 50 \nl
UMa & N3928 & 180.0 & 47.0 & 8 & 900 & 700 & 1100 & \nl
N4125 & N4125 & 181.4 & 65.5 & 3 & 1300 & 1000 & 1700 & 54 \nl
VirgoW & N4261 & 184.2 & 6.1 & 2 & 2200 & 2000 & 2800 & \nl
ComaI & N4278 & 184.4 & 29.6 & 3 & 1000 & 200 & 1400 & 55 \nl
CVn & N4258 & 185.0 & 44.0 & 7 & 500 & 400 & 600 & \nl
N4386 & N4386 & 185.6 & 75.8 & 5 & 1650 & 1500 & 2100 & 98 \nl
N4373 & N4373 & 185.7 & \llap{$-$}39.5 & 2 & 3400 & 2500 & 3800 & 35 \nl
Virgo & N4486 & 187.1 & 12.7 & 10 & 1150 & \llap{$-$}300 & 2000 & 56 \nl
ComaII & N4494 & 187.2 & 26.1 & 5 & 1350 & 1200 & 1400 & 235 \nl
N4594 & N4594 & 189.4 & \llap{$-$}11.4 & 5 & 1100 & 900 & 1200 & \nl
M51 & N5194 & 200.0 & 45.0 & 4 & 480 & 380 & 580 & \nl
Centaurus & N4696 & 191.5 & \llap{$-$}41.0 & 3 & 3000 & 2000 & 5000 & 58 \nl
CenA & N5128 & 200.0 & \llap{$-$}39.0 & 15 & 550 & 200 & 600 & 226 \nl
N5322 & N5322 & 212.5 & 57.0 & 6 & 2000 & 1600 & 2400 & 245 \nl
N5638 & N5638 & 216.0 & 3.5 & 3 & 1650 & 1400 & 1900 & 68 \nl
N5846 & N5846 & 226.0 & 1.8 & 2 & 1700 & 1200 & 2200 & 70 \nl
N5898 & N5898 & 228.8 & \llap{$-$}23.9 & 2 & 2100 & 2000 & 2700 & 71 \nl
N6684 & N6684 & 281.0 & \llap{$-$}65.2 & 10 & 850 & 500 & 1200 & 78 \nl
N7144 & N7144 & 327.4 & \llap{$-$}48.5 & 6 & 1900 & 1500 & 2000 & 84 \nl
N7180 & N7180 & 329.9 & \llap{$-$}20.8 & 10 & 1500 & 1300 & 1900 & 265 \nl
N7331 & N7457 & 338.7 & 34.2 & 9 & 800 & 800 & 1100 & \nl
Grus & I1459 & 343.6 & \llap{$-$}36.7 & 5 & 1600 & 1400 & 2300 & 231 \nl
\enddata
\tablecomments{Columns:
Group name, sample member, RA and Dec (B1950), group radius (deg),
mean heliocentric velocity, minimum and maximum velocities for
inclusion in the group, and group number from Faber et al. (1989)}
\end{deluxetable}
\begin{deluxetable}{lccrrrrrl}
\tablecaption{Distance Comparisons.\label{tbl4}}
\tablewidth{0pt}
\tablehead{
\colhead{Estimator} & \colhead{Grp/gxy} &
\colhead{Distance} & \colhead{N} &
\colhead{$\langle\overline{m}_I^0\rangle-d$} & \colhead{$\pm$} &
\colhead{rms} & \colhead{$\chi^2/N$} & \colhead{Comments}
}
\startdata
Cepheid & Grp & (m-M) & 7 & \llap{$-$}1.74 & 0.05 & 0.16 & 0.6 & \nl
Cepheid & gxy & (m-M) & 5 & \llap{$-$}1.75 & 0.06 & 0.33 & 3.4 & \nl
Cepheid & gxy & (m-M) & 4 & \llap{$-$}1.82 & 0.07 & 0.12 & 0.3 & less N5253 \nl
PNLF & Grp & (m-M) & 12 & \llap{$-$}1.63 & 0.02 & 0.33 & 7.5 & \nl
PNLF & Grp & (m-M) & 10 & \llap{$-$}1.69 & 0.03 & 0.20 & 2.2 & less ComaI/II \nl
SNII & Grp & (m-M) & 5 & \llap{$-$}1.80 & 0.12 & 0.36 & 1.4 & \nl
SNII & Grp & (m-M) & 4 & \llap{$-$}1.76 & 0.12 & 0.22 & 1.1 & less N7331 \nl
TF & Grp & (m-M) & 26 & \llap{$-$}1.69 & 0.03 & 0.41 & 2.1 & \nl
TF (MkII) & Grp & 5logd & 29 & 13.55 & 0.08 & 0.59 & 2.1 & \nl
Dn-sigma & Grp & 5logd & 28 & 13.64 & 0.05 & 0.44 & 1.9 & \nl
SNIa ($M_{max}$)& Grp & 5logd & 6 & 13.92 & 0.08 & 0.38 & 3.6 & \nl
SNIa ($\Delta m_{15}$)& Grp & 5logd & 6 & 14.01 & 0.08 & 0.40 & 3.6 & \nl
SNIa ($M_{max}$)& gxy & 5logd & 5 & 13.86 & 0.12 & 0.54 & 4.9 & \nl
SNIa ($\Delta m_{15}$)& gxy & 5logd & 5 & 14.01 & 0.12 & 0.43 & 3.2 & \nl
SNIa ($M_{max}$)& gxy & 5logd & 4 & 13.64 & 0.13 & 0.22 & 1.0 & less N5253 \nl
SNIa ($\Delta m_{15}$)& gxy & 5logd & 4 & 13.87 & 0.13 & 0.30 & 1.8 & less N5253 \nl
Theory & & & & \llap{$-$}1.81 & & 0.11 & & \nl
\enddata
\tablecomments{Columns:
Name of the estimator, comparison
by group or by galaxy, estimator's zero point based on Mpc \ifmmode(m{-}M)\else$(m{-}M)$\fi\ or
Hubble flow (5logd~km/s), number of comparison points,
mean difference between SBF and the estimator,
expected error in this mean based on error estimates,
rms scatter in the comparison, $\chi^2/N$, and comments.}
\end{deluxetable}
\begin{deluxetable}{llrrrrrrrl}
\tablecaption{SBF Distances to Groups.\label{tbl5}}
\tablewidth{0pt}
\tablehead{
\colhead{Group} & \colhead{Example} &
\colhead{RA} & \colhead{Dec} &
\colhead{$v_{ave}$} & \colhead{$N$} & \colhead{\ifmmode(m{-}M)\else$(m{-}M)$\fi} &
\colhead{$\pm$} & \colhead{$d$} & $\pm$
}
\startdata
LocalGrp & N0224 & 10.0 & 41.0 & \llap{$-$}300 & 2 & 24.43 & 0.08
& \phn0.77 & 0.03 \nl
M81 & N3031 & 147.9 & 69.3 & \llap{$-$}40 & 2 & 27.78 & 0.08
& \phn3.6 & 0.2 \nl
CenA & N5128 & 200.0 & \llap{$-$}39.0 & 550 & 3 & 28.03 & 0.10
& \phn4.0 & 0.2 \nl
N1023 & N1023 & 37.0 & 35.0 & 650 & 4 & 29.91 & 0.09 & \phn9.6
& 0.4 \nl
LeoI & N3379 & 161.3 & 12.8 & 900 & 5 & 30.14 & 0.06 & 10.7 &
0.3 \nl
N7331 & N7331 & 338.7 & 34.2 & 800 & 2 & 30.39 & 0.10 & 12.0 &
0.6 \nl
UMa & N3928 & 180.0 & 47.0 & 900 & 5 & 30.76 & 0.09 & 14.2 &
0.6 \nl
ComaI & N4278 & 184.4 & 29.6 & 1000 & 3 & 30.95 & 0.08 & 15.5 &
0.6 \nl
ComaII & N4494 & 187.2 & 26.1 & 1350 & 3 & 31.01 & 0.08 & 15.9 &
0.6 \nl
Virgo & N4486 & 187.1 & 12.7 & 1150 & 27 & 31.03 & 0.05 & 16.1 &
0.4 \nl
Dorado & N1549 & 63.7 & \llap{$-$}55.7 & 1300 & 6 & 31.04 & 0.06 & 16.1 &
0.5 \nl
Fornax & N1399 & 54.1 & \llap{$-$}35.6 & 1400 & 26 & 31.23 & 0.06 & 17.6 &
0.5 \nl
\enddata
\tablecomments{Columns:
Group name, sample member, RA and Dec (B1950),
mean heliocentric velocity, number of SBF distances, SBF distance
modulus and error, and SBF distance (Mpc) and error.}
\end{deluxetable}
\clearpage
| proofpile-arXiv_065-420 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{INTRODUCTION}
This talk reviews the status of two distinct though related subjects:
our understanding of chiral extrapolations,
and results for weak matrix elements involving light (i.e. $u$, $d$
and $s$) quarks.
The major connection between these subjects
is that understanding chiral extrapolations allows
us to reduce, or at least estimate, the errors in matrix elements.
Indeed, in a number of matrix elements, the dominant errors are those
due to chiral extrapolation and to the use of the quenched approximation.
I will argue that an understanding of chiral extrapolations gives us
a handle on both of these errors.
While understanding errors in detail is a sign of a maturing field,
we are ultimately interested in the results for the matrix elements
themselves.
The phenomenological implications of these results were emphasized
in previous reviews \cite{martinelli94,soni95}.
Here I only note that it is important to calculate matrix elements
for which we know the answer, e.g. $f_\pi/M_\rho$ and $f_K/f_\pi$,
in order to convince ourselves, and others, that our predictions for
unknown matrix elements are reliable.
The reliability of a result for $f_D$, for example, will be gauged
in part by how well we can calculate $f_\pi$.
And it would be a real coup if we were able to show in detail that
QCD indeed explains the $\Delta I=1/2$ rule in $K\to\pi\pi$ decays.
But to me the most interesting part of the enterprise is the possibility
of using the lattice results to calculate quantities which allow us to
test the Standard Model. In this category, the light-quark matrix element
with which we have had the most success is $B_K$.
The lattice result is already used by phenomenological analyses which
attempt to determine the CP violation in the CKM matrix
from the experimental number for $\epsilon$.
I describe below the latest twists in the saga of the lattice
result for $B_K$.
What we would like to do is extend this success to the raft of
$B$-parameters which are needed to predict $\epsilon'/\epsilon$.
There has been some progress this year on the
contributions from electromagnetic penguins,
but we have made no headway towards calculating strong penguins.
I note parenthetically that another input into the prediction of
$\epsilon'/\epsilon$ is the strange quark mass.
The recent work of Gupta and Bhattacharya \cite{rajanmq96}
and the Fermilab group \cite{mackenzie96}
suggest a value of $m_s$ considerably smaller than the
accepted phenomenological estimates, which will substantially
increase the prediction for $\epsilon'/\epsilon$.
Much of the preceding could have been written in 1989, when I
gave the review talk on weak matrix elements \cite{sharpe89}.
How has the field progressed since then?
I see considerable advances in two areas.
First, the entire field of weak matrix elements involving heavy-light
hadrons has blossomed. 1989 was early days in our calculation of the
simplest such matrix elements, $f_D$ and $f_B$. In 1996 a plethora
of quantities are being calculated, and the subject deserves its own
plenary talk \cite{flynn96}.
Second, while the subject of my talks then and now is similar, there has
been enormous progress in understanding and reducing systematic errors.
Thus, whereas in 1989 I noted the possibility of using chiral loops
to estimate quenching errors, we now have a technology (quenched chiral
perturbation theory---QChPT) which allows us to make these estimates.
We have learned that the quenched approximation (QQCD) is most probably
singular in the chiral limit, and there is a growing body of numerical
evidence showing this, although the case is not closed.
The reduction in statistical errors has allowed us to
go beyond simple linear extrapolations in light quark masses,
and thus begin to test the predictions of QChPT.
The increase in computer power has allowed us to study systematically the
dependence of matrix elements on the lattice spacing, $a$.
We have learned how to get more reliable estimates
using lattice perturbation theory \cite{lepagemackenzie}.
And, finally, we have begun to use non-perturbative matching of lattice
and continuum operators, as discussed here by Rossi \cite{rossi96}.
The body of this talk is divided into two parts. In the first,
Secs. \ref{sec:whychi}-\ref{sec:querr}, I focus on chiral extrapolations:
why we need them, how we calculate their expected form,
the evidence for chiral loops, and how we can use them to estimate
quenching errors.
In the second part, Secs. \ref{sec:decayc}-\ref{sec:otherme},
I give an update on results for weak matrix elements.
I discuss $f_\pi/M_\rho$, $f_K/f_\pi$, $B_K$ and a few related $B$-parameters.
Results for structure functions have been reviewed here
by G\"ockeler \cite{gockeler96}.
There has been little progress on semi-leptonic form factors,
nor on flavor singlet matrix elements and scattering lengths,
since last years talks by Simone \cite{simone95} and Okawa \cite{okawa95}.
\section{WHY DO WE NEED QChPT?}
\label{sec:whychi}
Until recently linear chiral extrapolations
(i.e. $\alpha + \beta m_q$) have sufficed for most quantities.
This is no longer true.
This change has come about for two reasons. First, smaller statistical
errors (and to some extent the use of a larger range of quark masses)
have exposed the inadequacies of linear fits, as already stressed here
by Gottlieb \cite{gottlieb96}.
An example, taken from Ref. \cite{ourspect96}
is shown in Fig. \ref{fig:NDelta}.
The range of quark masses is $m_s/3- 2 m_s$,
typical of that in present calculations.
While $M_\Delta$ is adequately fit by a straight line,
there is definite, though small, curvature in $M_N$.
The curves are the result of a fit using QChPT \cite{sharpebary},
to which I return below.
\begin{figure}[tb]
\vspace{-0.1truein}
\centerline{\psfig{file=fig1.ps,height=3.0truein}}
\vspace{-0.6truein}
\caption{$M_N$ and $M_\Delta$ versus $M_\pi^2$,
with quenched Wilson fermions, at $\beta=6$, on $32^3\times64$
lattices. The vertical band is the range of estimates for $m_s$.}
\vspace{-0.2truein}
\label{fig:NDelta}
\end{figure}
The second demonstration of the failure of linear extrapolations
has come from studying the octet baryon mass splittings \cite{ourspect96}.
The new feature here is the consideration of baryons
composed of non-degenerate quarks.
I show the results for $(M_\Sigma-M_\Lambda)/(m_s-m_u)$ (with $m_u=m_d$)
in Fig. \ref{fig:sigdel}.
If baryons masses were linear functions of $m_s$ and $m_u$
then the data would lie on a horizontal line.
Instead the results vary by a factor of four.
This is a glaring deviation from linear behavior,
in contrast to the subtle effect shown in Fig. \ref{fig:NDelta}.
\begin{figure}[tb]
\vspace{-0.1truein}
\centerline{\psfig{file=fig2.ps,height=3.0truein}}
\vspace{-0.6truein}
\caption{Results for $(M_\Sigma-M_\Lambda)/(m_s-m_u)$ (diamonds).
The ``burst'' is the physical result using the linear fit shown.
Crosses are from a global chiral fit.}
\vspace{-0.2truein}
\label{fig:sigdel}
\end{figure}
Once there is evidence of non-linearity, we need to know
the appropriate functional form with which to extrapolate to
the chiral limit. This is where (Q)ChPT comes in.
In the second example, the prediction is \cite{LS,sharpebary}
\begin{eqnarray}
\lefteqn{{M_\Sigma - M_\Lambda \over m_s-m_u} \approx} \nonumber\\
&& - {8 D\over3}
+ d_1 {M_K^3-M_\pi^3 \over m_s-m_u}
+ d'_1 {M_{ss}^3-M_\pi^3 \over m_s-m_u} \nonumber \\
&& + e_1 (m_u + m_d + m_s) + e'_1 (m_u + m_s) \,,
\label{eq:chptsiglam}
\end{eqnarray}
where $M_{ss}$ is the mass of the quenched $\bar ss$ meson.\footnote{%
I have omitted terms which, though more singular in the chiral limit,
are expected to be numerically small for the range of $m_q$ under study.}
QChPT also implies that $|d'_1| < |d_1|$, which is why in
Fig. \ref{fig:sigdel} I have plotted the data versus
$(M_K^3-M_\pi^3)/(m_s-m_u)$. The good news is that the points do lie
approximately on a single curve---which would not be true for a poor
choice of y-axis. The bad news is that the fit to the
the $d_1$ term and a constant is not that good.
This example illustrates the benefits which can accrue if one knows
the chiral expansion of the quantity under study.
First, the data collapses onto a single curve, allowing
an extrapolation to the physical quark masses.
And, second, the theoretical input reduces the length of the
extrapolation.
In Fig. \ref{fig:sigdel}, for example, to reach the physical point requires an
extrapolation by less than a factor of 2.
This is much smaller than the ratio of the lightest quark mass to
the physical value of $(m_u+m_d)/2$---a factor of roughly 8.
In fact, in the present example,
the data are not yet accurate enough to distinguish between
the $d$ and $e$ terms in Eq. \ref{eq:chptsiglam}.
A global fit of the QChPT prediction to this and other mass differences
(the results of which are shown in the Figure)
implies that both types of term are present \cite{sharpebary}.
\section
CHIRAL PERTURBATION THEORY}
\label{sec:qchpt}
This brings me to a summary of QChPT.
In QCD, chiral perturbation theory predicts the form of the chiral expansion
for quantities involving one or more light quarks.
The expansion involves terms analytic in $m_q$ and the external momenta,
and non-analytic terms due to pion loops.\footnote{``Pion'' here refers to
any of the pseudo-Goldstone bosons.}
The analytic terms are restricted in form,
though not in magnitude, by chiral symmetry,
while the non-analytic terms are completely predicted given the
analytic terms.
The same type of expansions can be developed in quenched QCD using QChPT.
The method was worked out in Refs. \cite{morel,sharpestcoup,BGI,sharpechbk},
with the theoretically best motivated formulation
being that of Bernard and Golterman \cite{BGI}.
Their method
gives a precise meaning to the quark-flow diagrams which I use below.
They have extended it also to partially quenched theories
(those having both valence and sea quarks but with different masses)
\cite{BGPQ}.
Results are available for pion properties and the condensate \cite{BGI},
$B_K$ and related matrix elements \cite{sharpechbk},
$f_B$, $B_B$ and the Isgur-Wise function \cite{booth,zhang},
baryon masses \cite{sharpebary,LS},
and scattering lengths \cite{BGscat}.
I will not describe technical details, but rather focus on
the aims and major conclusions of the approach.
For a more technical review see Ref. \cite{maartench}.
As I see it, the major aims of QChPT are these. \\
$\bullet\ $
To predict the form of the chiral expansion, which can then be
used to fit and extrapolate the data. This was the approach taken above
for the baryon masses. \\
$\bullet\ $
To estimate the size of the quenching error by comparing the
contribution of the pion loops in QCD and QQCD,
using the coefficients of the chiral fits in QCD (from phenomenology)
and in QQCD (from a fit to the lattice data).
I return to such estimates in Sec. \ref{sec:querr}.
I begin by describing the form of the chiral expansions,
and in particular how they are affected by quenching.
The largest changes are to the non-analytic terms,
i.e. those due to pion loops.
There are two distinct effects.\\
(1) Quenching removes loops which, at the underlying quark level,
require an internal loop.
This is illustrated for baryon masses in Fig. \ref{fig:quarkloops}.
Diagrams of types (a) and (b) contribute in QCD,
but only type (b) occurs in QQCD.
These loops give rise to $M_\pi^3$ terms in the chiral expansions.
Thus for baryon masses, quenching only changes the coefficient of
these terms. In other quantities, e.g. $f_\pi$, they are
removed entirely. \\
(2) Quenching introduces artifacts due to $\eta'$ loops---as
in Fig. \ref{fig:quarkloops}(c).
These are chiral loops because the $\eta'$ remains light in QQCD.
Their strength is determined by the size of the ``hairpin'' vertex,
and is parameterized by $\delta$ and $\alpha_\Phi$
(defined in Sec. \ref{sec:chevidence} below).
\begin{figure}[tb]
\centerline{\psfig{file=fig3.ps,height=2.4truein}}
\vspace{-0.3truein}
\caption{Quark flow diagrams for $M_{\rm bary}$.}
\vspace{-0.2truein}
\label{fig:quarkloops}
\end{figure}
The first effect of quenching is of greater practical importance,
and dominates most estimates of quenching errors.
It does not change the form of the chiral expansion,
only the size of the terms.
The second effect leads to new terms in the chiral expansion,
some of which are singular in the chiral limit.
If these terms are large,
then one can be sure that quenching errors are large.
One wants to work at large enough quark mass so that these new terms are
numerically small. In practice this means keeping $m_q$ above
about $m_s/4-m_s/3$.
I illustrate these general comments using the results of
Labrenz and I for baryon masses \cite{LS}.
The form of the chiral expansion in QCD is\footnote{%
There are also $M_\pi^4 \log M_\pi$ terms in both QCD and QQCD,
the coefficients of which have not yet been calculated in QQCD.
For the limited range of quark masses used in simulations,
I expect that these terms can be adequately represented by the
$M_\pi^4$ terms, whose coefficients are unknown parameters.}
\begin{equation}
M_{\rm bary} = M_0 + c_2 M_\pi^2 + c_3 M_\pi^3 + c_4 M_\pi^4 + \dots
\end{equation}
with $c_3$ predicted in terms of $g_{\pi NN}$ and $f_\pi$.
In QQCD
\begin{eqnarray}
\lefteqn{
M_{\rm bary}^Q = M_0^Q + c_2^Q M_\pi^2 + c_3^Q M_\pi^3 + c_4^Q M_\pi^4 + }
\nonumber \\
&& \delta \left( c_1^Q M_\pi + c_2^Q M_\pi^2 \log M_\pi\right) + \alpha_\Phi
\tilde c_3^Q M_\pi^3 + \dots
\label{eq:baryqchpt}
\end{eqnarray}
The first line has the same form as in QCD,
although the constants multiplying the analytic terms in the two theories
are unrelated.
$c_3^Q$ is predicted to be non-vanishing, though different from $c_3$.
The second line is the contribution of $\eta'$ loops and is a quenched
artifact.
Note that it is the dominant correction in the chiral limit.
In order to test QChPT in more detail,
I have attempted to fit the expressions outlined above
to the octet and decuplet baryon masses from Ref. \cite{ourspect96}.
There are 48 masses, to be fit in terms of 19 underlying
parameters: the octet and decuplet masses in the chiral limit,
3 constants of the form $c_2^Q$, 6 of the form $c_4^Q$,
6 pion-nucleon couplings, $\delta$ and $\alpha_\Phi$.
I have found a reasonable description of the data with $\delta\approx0.1$,
but the errors are too large to pin down the values
of all the constants \cite{sharpebary}.
Examples of the fit are shown in Figs. \ref{fig:NDelta} and \ref{fig:sigdel}.
\begin{figure}[tb]
\vspace{-0.1truein}
\centerline{\psfig{file=fig4.ps,height=3.0truein}}
\vspace{-0.6truein}
\caption{Contributions to $M_N - M_0^Q$ in the global chiral fit.
All quantities in lattice units.
Vertical lines indicate range of estimates of $m_s$.}
\vspace{-0.2truein}
\label{fig:mnchfit}
\end{figure}
Although this is a first, rather crude, attempt at such a fit,
several important lessons emerge.\\
(1) For present quark masses, one needs several terms in the chiral
expansion to fit the data.
This in turn requires that one have high statistics results for a
number of light quark masses. \\
(2) One must check that the ``fit'' is not resulting from
large cancellations between different orders.
The situation for my fit is illustrated by Fig. \ref{fig:mnchfit},
where I show the different contributions to $M_N$.
Note that the relative size of the terms
is not determined by $M_N$, but rather by the fit as a whole.
The most important point is that the $M_\pi^4$ terms are considerably smaller
than those of $O(M_\pi^2)$ up to at least $m_q= m_s$.
The $M_\pi^3$ terms are part of a different series and need not be
smaller than those of $O(M_\pi^2)$.
Similarly the $M_\pi$ terms are the leading quenched artifact, and
should not be compared to the other terms.
Thus the convergence is acceptable for $m_q < m_s$, though it is dubious for
the highest mass point. \\
(3) The artifacts (in particular the $\delta M_\pi$ terms)
can lead to unusual behavior at small $M_\pi$,
as illustrated in the fit to $M_\Delta$ (Fig. \ref{fig:NDelta}). \\
(4) Since the ``$\delta$-terms'' are artifacts of quenching,
and their relative contribution increases as $M_\pi\to0$,
it makes more sense phenomenologically {\em to extrapolate
without including them}. In other words, a better estimate of the
unquenched value for $M_\Delta$ in the chiral limit
can probably be obtained simply using a linear extrapolation in $M_\pi^2$.
This is, however, a complicated issue which needs more thought.\\
(5) The output of the fit includes pion-nucleon couplings whose
values should be compared to more direct determinations. \\
(6) Finally, the fact that a fit can be found at all
gives me confidence to stick my neck out and
proceed with the estimates of quenching errors in baryon masses.
It should be noted, however, that a fit involving only analytic terms,
including up to $M_\pi^6$, can probably not be ruled out.
What of quantities other than baryon masses?
In Sec. \ref{subsec:bkchfit} I discuss fits to $B_K$,
another quantity in which chiral loops survive quenching.
The data is consistent with the non-analytic term predicted by QChPT.
Good data also exists for $M_\rho$.
It shows curvature,
but is consistent with either cubic or quartic terms\cite{sloan96}.
What do we expect from QChPT?
In QCD the chiral expansion for $M_\rho$
has the same form as for baryon masses \cite{rhochpt}.
The QChPT theory calculation has not been done,
but it is simple to see that form will be as for baryons,
Eq.~\ref{eq:baryqchpt}, {\em except that $c_3^Q=0$}.
Thus an $M_\pi^3$ term is entirely a quenched
artifact---and a potential window on $\alpha_\Phi$.
What of quantities involving pions, for which there is very good data?
For the most part, quenching simply
removes the non-analytic terms of QCD and replaces them with artifacts
proportional to $\delta$.
The search for these is the subject of the next section.
\section{EVIDENCE FOR $\eta'$ LOOPS}
\label{sec:chevidence}
The credibility of QChPT rests in part on the observation of
the singularities predicted in the chiral limit.
If such quenched artifacts are present,
then we need to study them if only to know at
what quark masses to work in order to avoid them!
What follows in this section is an update of the 1994 review of Gupta
\cite{gupta94}.
The most direct way of measuring $\delta$ is from the $\eta'$ correlator.
If the quarks are degenerate, then the part of the quenched chiral
Lagrangian bilinear in the $\eta'$ is \cite{BGI,sharpechbk}
\begin{eqnarray}
2 {\cal L}_{\eta'} &=& \partial_\mu \eta' \partial^\mu \eta' - M_\pi^2 \eta'^2
\\
&&+ {(N_f/3)} \left( \alpha_\Phi \partial_\mu \eta' \partial^\mu \eta' -
m_0^2 \eta'^2 \right) \,.
\end{eqnarray}
$\delta$ is defined by $\delta = m_0^2/(48 \pi^2 f_\pi^2)$.
In QQCD the terms in the second line must be treated as vertices, and
cannot be iterated. They contribute to the disconnected part of the
$\eta'$ correlator, whereas the first line determines the connected part.
Thus the $\eta'$ is degenerate with the pion, but has additional vertices.
To study this, various groups have looked at the ratio of the
disconnected to connected parts, at $\vec p=0$, whose predicted form
at long times is
\begin{equation}
R(t) = t (N_f/3) (m_0^2 - \alpha_\Phi M_\pi^2)/(2 M_\pi) \,.
\end{equation}
I collect the results in Table \ref{tab:etapres}, converting
$m_0^2$ into $\delta$ using $a$ determined from $m_\rho$.
\begin{table}[tb]
\caption{Results from the quenched $\eta'$ two point function.
$W$ and $S$ denote Wilson and staggered fermions.}
\label{tab:etapres}
\begin{tabular}{ccccc}
\hline
Ref. & Yr. &$\delta$&$\alpha_\Phi$ &$\beta$ W/S \\
\hline
JLQCD\cite{kuramashi94} &94&$0.14 (01)$ & $0.7$ & $5.7$ W \\
OSU\cite{kilcup95}&95& $0.27 (10)$ & $0.6$ & $6.0$ S \\
Rome\cite{masetti96}&96& $\approx 0.15$ & & $5.7$ W \\
OSU\cite{venkat96}& 96&$0.19(05)$ & $0.6$ & $6.0$ S \\
FNAL\cite{thacker96} &96& $< 0.02$ & $>0$ & $5.7$ W \\
\hline
\end{tabular}
\vspace{-0.2truein}
\end{table}
All groups except Ref. \cite{thacker96} report a non-zero value for $\delta$
in the range $0.1-0.3$.
What they actually measure, as illustrated in Fig. \ref{fig:osuR},
is the combination $m_0^2-\alpha_\Phi M_\pi^2$,
which they then extrapolate to $M_\pi=0$.
I have extracted the results for $\alpha_\Phi$ from such plots.
As the figure shows, there is a considerable cancellation between the
$m_0$ and $\alpha_\Phi$ terms at the largest quark masses, which
correspond to $M_\pi \approx 0.8\,$GeV.
This may explain why Ref. \cite{thacker96}
does not see a signal for $\delta$.
\begin{figure}[tb]
\vspace{-0.3truein}
\centerline{\psfig{file=fig5.ps,height=2.4truein}}
\vspace{-0.5truein}
\caption{$a^2(m_0^2-\alpha_\Phi M_\pi^2)$ from the OSU group.}
\vspace{-0.2truein}
\label{fig:osuR}
\end{figure}
Clearly, further work is needed to
sort out the differences between the various groups.
As emphasized by Thacker \cite{thacker96}, this is mainly an issue of
understanding systematic errors.
In particular, contamination from excited states leads to an
apparent linear rise of $R(t)$, and thus to an overestimate of $m_0^2$.
Indeed, the difference between the OSU results last year and this
is the use of smeared sources to reduce such contamination.
This leads to a smaller $\delta$, as shown in Fig. \ref{fig:osuR}.
Ref. \cite{thacker96} also find that $\delta$ decreases as the volume
is increased.
I want to mention also that the $\eta'$ correlator has also been studied
in partially quenched theories, with $N_f=-6, -4, -2$,
\cite{masetti96} and $N_f=2, 4$ \cite{venkat96}.
The former work is part of the ``bermion'' program which aims to
extrapolate from negative to positive $N_f$.
For any non-zero $N_f$ the analysis is different than in the quenched
theory, because the hairpin vertices do iterate, and
lead to a shift in the $\eta'$ mass. Indeed $m_{\eta'}$ is reduced (increased)
for $N_f<0$ ($>0$), and both changes are observed!
This gives me more confidence in the results of these groups at $N_f=0$.
The bottom line appears to be
that there is relatively little dependence of $m_0^2$ on $N_f$.
Other ways of obtaining $\delta$ rely on loop effects,
such as that in Fig. \ref{fig:quarkloops}(c).
For quenched pion masses $\eta'$ loops lead to terms which are
singular in the chiral limit \cite{BGI}\footnote{%
The $\alpha_\Phi$ vertex leads to terms proportional
to $M_\pi^2 \log M_\pi$ which are not singular in the chiral limit,
and can be represented approximately by analytic terms.}
\begin{eqnarray}
\lefteqn{{M_{12}^2 \over m_1 + m_2} = \mu^Q \left[
1 - \delta \left\{\log{\widetilde M_{11}^2\over \Lambda^2} \right. \right.}
\nonumber \\
&& \left.\left.
+ {\widetilde M_{22}^2 \over \widetilde M_{22}^2 -\widetilde M_{11}^2 }
\log{\widetilde M_{22}^2 \over \widetilde M_{11}^2} \right\}
+ c_2 (m_1 + m_2) \right]
\label{eq:mpichpt}
\end{eqnarray}
Here $M_{ij}$ is the mass of pion
composed of a quark of mass $m_i$ and antiquark of mass $m_j$,
$\Lambda$ is an unknown scale, and $c_2$ an unknown constant.
The tilde is relevant only to staggered fermions, and indicates that it
is the mass of the flavor singlet pion,
and not of the lattice pseudo-Goldstone pion, which appears.
This is important because, at finite lattice spacing,
$\tilde M_{ii}$ does not vanish in the chiral limit,
so there is no true singularity.
In his 1994 review, Gupta fit the world's data for
staggered fermions at $\beta=6$ having $m_1=m_2$.
I have updated his plot, including new JLQCD data \cite{yoshie96},
in Fig. \ref{fig:mpibymq}. To set the scale, note that $m_s a \approx 0.024$.
The dashed line is Gupta's fit to Eq. \ref{eq:mpichpt},
giving $\delta=0.085$,
while the solid line includes also an $m_q^2$ term, and gives $\delta=0.13$.
These non-zero values were driven by the results from Kim and Sinclair (KS),
who use quark masses as low as $0.1 m_s$ \cite{kimsinclair},
but they are now supported by the JLQCD results.
\begin{figure}[tb]
\vspace{-0.1truein}
\centerline{\psfig{file=fig6.ps,height=3.0truein}}
\vspace{-0.6truein}
\caption{Chiral fit to $\log(M_\pi^2/m_q)$ at $\beta=6$.
Some points have been offset for clarity.}
\vspace{-0.2truein}
\label{fig:mpibymq}
\end{figure}
Last year, Mawhinney proposed an alternative
explanation for the increase visible at small $m_q$ \cite{mawhinney95},
namely an offset in the intercept of $M_\pi^2$
\begin{equation}
M_{12}^2 = c_0 + \mu^Q (m_1 + m_2) + \dots
\label{eq:mpimawh}
\end{equation}
In his model, $c_0$ is proportional to the
minimum eigenvalue of the Dirac operator, and thus falls as $1/V$.
This model explains the detailed structure of his results for
$M_\pi^2$ and $\langle \bar\psi\psi \rangle$ at $\beta=5.7$.
It also describes the data of KS, {\em except for volume dependence of $c_0$}.
As the Fig. \ref{fig:mpibymq} shows,
the results of KS from $24^3$ and $32^3$ lattices are consistent,
whereas in Mawhinney's model the rise at small $m_q$
should be reduced in amplitude by $0.4$ on the larger lattice.
The fit in Fig. \ref{fig:mpibymq} is for pions composed of degenerate quarks.
One can further test QChPT by noting that Eq. \ref{eq:mpichpt} is
not simply a function of the average quark mass---there is a predicted
dependence on $m_1-m_2$. In Mawhinney's model, this dependence would
presumably enter only through a $(m_1-m_2)^2$ term, and thus would be
a weaker effect.
JLQCD have extensive data from the range $\beta=5.7-6.4$,
with both $m_1=m_2$ and $m_1\ne m_2$.
They have fit to Eq. \ref{eq:mpichpt}, and thus obtained
$\delta$ as a function of $\beta$.
They find reasonable fits, with $\delta\approx 0.06$ for most $\beta$.
I have several comments on their fits.
First, they have used $M_{ii}$ rather than $\widetilde M_{ii}$,
which leads to an underestimate of $\delta$, particularly at the
smaller values of $\beta$.
Second, their results for the constants, particularly $c_2$, vary rapidly with
$\beta$. One would expect that all dimensionless parameters in the fit
(which are no less physical than, say, $f_\pi/M_\rho$)
should vary smoothly and slowly with $\beta$.
This suggests to me that terms of $O(m_q^2)$ may be needed.
Finally, it would be interesting to attempt a fit to the JLQCD data
along the lines suggested by Mawhinney, but including a $(m_1-m_2)^2$ term.
Clearly more work is needed to establish convincingly that there
are chiral singularities in $M_\pi$.
One should keep in mind that the effects are small,
$\sim 5\%$ at the lightest $m_q$,
so it is impressive that we can study them at all.
Let me mention also some other complications.\\
(1) It will be hard to see the ``singularities'' with staggered fermions
for $\beta<6$. This is because $\widetilde M_{ii} - M_{ii}$ grows
like $a^2$ (at fixed physical quark mass).
Indeed, by $\beta=5.7$ the flavor singlet pion has a
mass comparable to $M_\rho$!
Thus the $\eta'$ is no longer light,
and its loop effects will be suppressed.
In fact, the rise in $M_\pi^2/m_q$ as $m_q\to0$ for $\beta=5.7$ is very
gradual\cite{gottlieb96}, and could be due to the $c_2$ term.\\
(2) It will be hard to see the singularities using Wilson fermions.
This is because we do not know, {\it a priori}, where $m_q$ vanishes,
and, as shown by Mawhinney, it is hard to distinguish
the log divergence of QChPT from an offset in $m_q$.\\
(3) A related log divergence is predicted for $\langle\bar\psi\psi\rangle$,
which has not been seen so far \cite{kimsinclair,mawhinney95}.
It is not clear to me that this is a problem for QChPT, however,
because it is difficult to extract the non-perturbative part of
$\langle\bar\psi\psi\rangle$ from the quadratically divergent perturbative
background.
Two other quantities give evidence concerning $\delta$.
The first uses the ratio of decay constants
\begin{equation}
R_{BG} = f_{12}^2/(f_{11} f_{22}) \,.
\end{equation}
This is designed to cancel the analytic terms proportional
to $m_q$ \cite{BGI}, leaving a non-analytic term proportional to $\delta$.
The latest analysis finds $\delta\approx 0.14$ \cite{guptafpi}.
It is noteworthy that a good fit once again requires
the inclusion of $O(m_q^2)$ terms.
The second quantity is the double difference
\begin{equation}
{\rm ES}2 = (M_\Omega - M_\Delta) - 3 (M_{\Xi^*} - M_{\Sigma^*}) \,,
\end{equation}
which is one measure of the breaking of the equal spacing rule for decuplets.
This is a good window on artifacts due to quenching because
its expansion begins at $O(M_\pi^5)$ in QCD, but contains terms
proportional to $\delta M_\pi^2 \log M_\pi$ in QQCD \cite{LS}.
The LANL group finds that ${\rm ES}2$ differs from zero
by 2-$\sigma$ \cite{ourspect96},
and I find that the data can be fit with $\delta\approx 0.1$
\cite{sharpebary}.
In my view, the preponderance of the evidence suggests a value of
$\delta$ in the range $0.1-0.2$. All extractions are complicated by
the fact that the effects proportional to $\delta$ are small with
present quark masses.
To avoid them, one should use quark masses above $m_s/4-m_s/3$.
This is true not only for the light quark quantities
discussed above, but also for heavy-light quantities such as $f_B$.
This, too, is predicted to be singular as the light quark mass vanishes
\cite{zhang}.
\section{QUENCHING ERRORS}
\label{sec:querr}
I close the first part of the talk by listing, in Table \ref{tab:querr},
a sampling of estimates of quenching errors, defined by
\begin{equation}
{\rm Error}({\rm Qty}) = { [ {\rm Qty}({\rm QCD})- {\rm Qty}({\rm QQCD})]
\over{\rm Qty}({\rm QCD})} \,.
\end{equation}
I make the estimates by taking
the numerator to be the difference between the pion loop contributions
in the full and quenched chiral expansions.
To obtain numerical values I set $\Lambda=m_\rho$ ($\Lambda$ is
the scale occurring in chiral logs), use $f=f^Q=f_K$, and assume
$\delta=0.1$ and $\alpha_\Phi=0$. For the estimates of heavy-light
quantities I set $g'=0$, where $g'$ is an $\eta'$-$B$-$B$ coupling defined
in Ref. \cite{zhang}. These estimates assume that the extrapolation
to the light quark mass is done linearly from $m_q\approx m_s/2$.
For example, $f_{B_d}$ in QQCD is {\em not} the quenched value with the
physical $d$-quark mass (which would contain a large artifact proportional
to $\delta$), but rather the value obtained by linear extrapolation from
$m_s/2$, where the $\delta$ terms are much smaller.
This is an attempt to mimic what is actually done in numerical simulations.
\begin{table}[tb]
\caption{Estimates of quenching errors.}
\label{tab:querr}
\begin{tabular}{ccl}
\hline
Qty. &Ref. & Error \\
\hline
$f_\pi/M_\rho$ &\cite{gassleut,sharpechbk} & $\,\sim 0.1$ \\
$f_K/f_\pi-1$ & \cite{BGI} & $\ 0.4$ \\
$f_{B_s}$ & \cite{zhang} & $\ 0.2$\\
$f_{B_s}/f_{B_d}$ & \cite{zhang} & $\ 0.16$ \\
$B_{B_s}/B_{B_d}$ & \cite{zhang} & $\,-0.04$ \\
$B_K$ ($m_d=m_s$) & \cite{sharpechbk} & $\ 0$ \\
$B_K$ ($m_d\ne m_s$) & \cite{sharpetasi} & $\ 0.05$ \\
$M_\Xi-M_\Sigma$& \cite{sharpebary} & $\ 0.4$ \\
$M_\Sigma-M_N$ & \cite{sharpebary} & $\ 0.3$ \\
$M_\Omega-M_\Delta$& \cite{sharpebary} & $\ 0.3$ \\
\hline
\end{tabular}
\vspace{-0.2truein}
\end{table}
For the first two estimates, I have used the facts that, in QCD,
\cite{gassleut}
\begin{eqnarray}
f_\pi &\approx& f\, [1 - 0.5 L(M_K)]
\,,\\
f_K/f_\pi &\approx& 1 - 0.25 L(M_K) - 0.375 L(M_\eta)
\,,
\end{eqnarray}
(where $L(M) = (M/4\pi f)^2 \log(M^2/\Lambda^2)$, and $f_\pi= 93\,$MeV),
while in QQCD \cite{BGI,sharpechbk}
\begin{eqnarray}
f_\pi &\approx& f^Q \,,\\
{f_K\over f_\pi} &\approx& 1 + {\delta \over 2} \left[
{M_K^2 \over M_{ss}^2 - M_\pi^2} \log {M_{ss}^2 \over M_\pi^2} - 1 \right]
\,.
\end{eqnarray}
I have not included the difference of pion loop contributions to $M_\rho$,
since the loop has not been evaluated in QChPT,
and a model calculation suggests that the difference is small
\cite{cohenleinweber}.
Details of the remaining estimates can be found in the references.
Let me stress that these are estimates and not calculations.
What they give is a sense of the effect of quenching on
the contributions of ``pion'' clouds surrounding hadrons---these clouds are
very different in QQCD and QCD!
But this difference in clouds could be cancelled numerically by differences
in the analytic terms in the chiral expansion.
As discussed in Ref. \cite{zhang}, a more conservative view is thus to
treat the estimates as rough upper bounds on the quenching error.
Those involving ratios (e.g. $f_K/f_\pi$)
are probably more reliable since some of the analytic terms do not contribute.
One can also form double ratios for which the error
estimates are yet more reliable
(e.g. $R_{BG}$ and ES2 from the previous section; see also Ref. \cite{zhang}),
but these quantities are of less phenomenological interest.
My aim in making these estimates is to obtain a sense of which
quenched quantities are likely to be more reliable and which less,
and to get an sense of the possible size of quenching errors.
My conclusion is that the errors could be significant
in a number of quantities, including those involving heavy-light mesons.
One might have hoped that the ratio $f_{B_s}/f_{B_d}$ would have
small quenching errors, but the chiral loops indicate otherwise.
For some other quantities, such as $B_{B_s}/B_{B_d}$ and $B_K$,
the quenching errors are likely to be smaller.
If these estimates work, then it will be worthwhile extending them
to other matrix elements of phenomenological interest, e.g.
$K\to\pi\pi$ amplitudes.
Then, when numerical results in QQCD are obtained,
we have at least a rough estimate of the quenching error in hand.
Do the estimates work? As we will see below,
those for $f_\pi/m_\rho$, $f_K/f_\pi$ and $B_K$ are consistent
with the numerical results obtained to date.
\section{RESULTS FOR DECAY CONSTANTS}
\label{sec:decayc}
For the remainder of the talk I will don the hat of a reviewer,
and discuss the status of results for weak matrix elements.
All results will be quenched, unless otherwise noted.
I begin with $f_\pi/M_\rho$, the results for which are shown
in Figs. \ref{fig:fpi_mrhoW} (Wilson fermions) and \ref{fig:fpi_mrhoCL}
(SW fermions, with tadpole improved $c_{SW}$).
The normalization here is $f_\pi^{\rm expt}=0.131\,$MeV,
whereas I use $93\,$MeV elsewhere in this talk.
\begin{figure}[tb]
\vspace{-0.6truein}
\centerline{\psfig{file=fig7.ps,height=2.5truein}}
\vspace{-0.6truein}
\caption{$f_\pi/M_\rho$ with quenched Wilson fermions.}
\vspace{-0.2truein}
\label{fig:fpi_mrhoW}
\end{figure}
Consider the Wilson data first. One expects a linear dependence on $a$,
and the two lines are linear extrapolations taken from Ref. \cite{guptafpi}.
The solid line is a fit to all the data,
while the dashed curve excludes the point with largest $a$
(which might lie outside the linear region).
It appears that the quenched result is lower than experiment,
but there is a $5-10\%$ uncertainty.
Improving the fermion action (Fig. \ref{fig:fpi_mrhoCL})
doesn't help much because of
uncertainties in the normalization of the axial current.
For the FNAL data, the upper (lower) points correspond to
using $\alpha_s(\pi/a)$ ($\alpha_s(1/a)$) in the matching factor.
The two sets of UKQCD95 points correspond to different normalization schemes.
Again the results appear to extrapolate to a point below experiment.
\begin{figure}[tb]
\vspace{-0.1truein}
\centerline{\psfig{file=fig8.ps,height=3.truein}}
\vspace{-0.6truein}
\caption{$f_\pi/M_\rho$ with quenched SW fermions.}
\vspace{-0.2truein}
\label{fig:fpi_mrhoCL}
\end{figure}
It is disappointing that we have not done better with such a basic quantity.
We need to reduce both statistical errors and normalization uncertainty.
The latter may require non-perturbative methods, or the use of
staggered fermions (where $Z_A=1$).
Note that chiral loops estimate that the quenched result
will undershoot by 12\%,
and this appears correct in sign, and not far off in magnitude.
Results for $(f_K-f_\pi)/f_\pi$ are shown in Fig. \ref{fig:fk_fpi}.
This ratio measures the mass dependence of decay constants.
Chiral loops suggest a 40\% underestimate in QQCD.
The line is a fit to all the Wilson data (including the largest $a$'s),
and indeed gives a result about half of the experimental value.
The new UKQCD results, using tadpole improved SW fermions,
are, by contrast, rising towards the experimental value.
It will take a substantial reduction in statistical errors to sort this out.
\begin{figure}[tb]
\vspace{-0.1truein}
\centerline{\psfig{file=fig9.ps,height=3.0truein}}
\vspace{-0.6truein}
\caption{$(f_K-f_\pi)/f_\pi$ in quenched QCD.}
\vspace{-0.2truein}
\label{fig:fk_fpi}
\end{figure}
\section{STATUS OF $B_K$: STAGGERED}
\label{sec:bks}
$B_K$ is defined by
\begin{equation}
B_K =
{\langle \bar K| \bar s \gamma_\mu^L d\, \bar s\gamma_\mu^L d | K \rangle
\over
(8/3) \langle \bar K| \bar s \gamma_\mu^L d|0 \rangle
\langle 0 |\bar s\gamma_\mu^L d | K \rangle } \,.
\label{eq:bkdef}
\end{equation}
It is a scale dependent quantity, and I will quote results in
the NDR (naive dimensional regularization) scheme at 2 GeV.
It can be calculated with very small statistical errors,
and has turned out to be a fount of knowledge about systematic errors.
This is true for both staggered and Wilson fermions,
though for different reasons.
There has been considerable progress with both types of fermions
in the last year. I begin with staggered fermions,
which hold the advantage for $B_K$ as they have a remnant chiral symmetry.
Back in 1989, I thought we knew what the
quenched answer was, based on calculations at $\beta=6$ on
$16^3$ and $24^3$ lattices:
$B_K = 0.70(2)$ \cite{sharpe89,ourbkprl}.
I also argued that quenching errors
were likely small (see Table \ref{tab:querr}).
I was wrong on the former, though maybe not on the latter.
By 1993, Gupta, Kilcup and I had found that $B_K$ had a
considerable $a$ dependence \cite{sharpe93}.
Applying Symanzik's improvement program, I argued that the discretization
errors in $B_K$ should be $O(a^2)$, and not $O(a)$.
Based on this, we extrapolated our data quadratically, and quoted
$B_K(NDR,2{\rm GeV}) = 0.616(20)(27)$ for the quenched result.
Our data alone, however, was not good enough to distinguish linear and
quadratic dependences.
Last year, JLQCD presented results from a more
extensive study (using $\beta=5.85$, $5.93$, $6$ and $6.2$) \cite{jlqcdbk95}.
Their data strongly favored a linear dependence on $a$.
If correct, this would lead to a value of $B_K$ close to $0.5$.
The only hope for someone convinced of an $a^2$ dependence was
competition between a number of terms.
Faced with this contradiction between numerical data and theory,
JLQCD have done further work on both fronts \cite{aoki96}.
They have added two additional lattice spacings, $\beta=5.7$ and $6.4$,
thus increasing the lever arm. They have also carried out finite volume
studies at $\beta=6$ and $6.4$, finding only a small effect.
Their data are shown in Fig. \ref{fig:jlqcdbk}.
``Invariant'' and ``Landau'' refer to two possible discretizations
of the operators---the staggered fermion operators are spread out over
a $2^4$ hypercube, and one can either make them gauge invariant
by including gauge links, or by fixing to Landau gauge and omitting the links.
The solid (dashed) lines show quadratic (linear) fits to the first five
points. The $\chi^2/{\rm d.o.f.}$ are
\begin{center}
\begin{tabular}{ccc}
\hline
Fit & Invariant & Landau \\
$a$ & 0.86 & 0.67 \\
$a^2$ & 1.80 & 2.21 \\
\hline
\end{tabular}
\end{center}
thus favoring the linear fit, but by a much smaller difference than
last year. If one uses only the first four points then linear
and quadratic fits are equally good.
What has changed since last year is that the new point at $\beta=6.4$
lies above the straight line passing through the next four points.
\begin{figure}[tb]
\vspace{-0.1truein}
\centerline{\psfig{file=fig10.ps,height=3.0truein}}
\vspace{-0.6truein}
\caption{JLQCD results for staggered $B_K$.}
\vspace{-0.2truein}
\label{fig:jlqcdbk}
\end{figure}
JLQCD have also checked the theoretical argument using a simpler method
of operator enumeration\cite{aoki96,ishizuka96}.\footnote{
A similar method has also been introduced by Luo \cite{luo96}.}
The conclusion is that there cannot be $O(a)$ corrections to $B_K$,
because there are no operators available with which one
could remove these corrections.
Thus JLQCD use quadratic extrapolation and quote
(for degenerate quarks)
\begin{equation}
B_K({\rm NDR}, 2\,{\rm GeV}) = 0.5977 \pm 0.0064
\pm 0.0166 \,,
\label{eq:jlqcdbk}
\end{equation}
where the first error is statistical, the second due to truncation of
perturbation theory.
This new result agrees with that from 1993 (indeed, the results are
consistent at each $\beta$), but has much smaller errors.
To give an indication of how far things have come, compare our
1993 result at $\beta=6$ with Landau-gauge operators, $0.723(87)$
\cite{sharpe93}, to the corresponding JLQCD result $0.714(12)$.
The perturbative error in $B_K$ arises from truncating the
matching of lattice and continuum operators to one-loop order.
The use of two different lattice operators allows one to estimate this
error without resort to guesswork about the higher order terms in the
perturbative expansion.
The difference between the results from the two operators is
of $O[\alpha(2\,{\rm GeV})^2]$, and thus should remain
finite in the continuum limit.
This is what is observed in Fig. \ref{fig:jlqcdbk}.
I will take Eq. \ref{eq:jlqcdbk} as the best estimate of $B_K$ in QQCD.
The errors are so much smaller than those in previous staggered results
and in the results with Wilson fermions discussed below,
that the global average is
not significantly different from the JLQCD number alone.
The saga is not quite over, however, since one should
confirm the $a^2$ dependence by running at even smaller lattice spacings.
JLQCD intend to run at $\beta=6.6$.
If the present extrapolation holds up, then it shows how one must
beware of keeping only a single term when extrapolating in $a$.
\subsection{Unquenching $B_K$}
To obtain a result for QCD proper, two steps remain:
the inclusion of dynamical quarks, and the use of $m_s\ne m_d$.
The OSU group has made important progress on the first step \cite{osubk96}.
Previous studies (summarized in Ref. \cite{soni95}) found that
$B_K$ was reduced slightly by sea quarks, although the effect was
not statistically significant.
The OSU study, by contrast, finds a statistically significant
increase in $B_K$
\begin{equation}
{ B_K({\rm NDR,2\,GeV},N_f=3) \over B_K({\rm NDR,2\,GeV},N_f=0)}
= 1.05 \pm 0.02 \,.
\label{eq:bkquerr}
\end{equation}
They have improved upon previous work by reducing statistical errors,
and by choosing their point at $N_f=0$ ($\beta=6.05$)
to better match the lattice spacing at $N_f=2$ ($\beta=5.7$, $m_qa=0.01$)
and $4$ ($\beta=5.4$, $m_qa=0.01$).
There are systematic errors in this result
which have yet to be estimated.
First, the dynamical lattices are chosen to have
$m_q^{\rm sea}=m_q^{\rm val} = m_s^{\rm phys}/2$,
and so they are truely unquenched simulations.
But $m_s^{\rm phys}$ is determined by
extrapolating in the valence quark mass alone,
and is thus a partially quenched result.
This introduces an uncertainty in $m_s$ which feeds into the estimate of the
$N_f$ dependence of $B_K$.
Similarly, $a$ is determined by a partially quenched extrapolation,
resulting in an uncertainty in the
matching factors between lattice and continuum operators.
But probably the most important error comes from the possibility
of significant $a$ dependence in the ratio in Eq. \ref{eq:bkquerr}.
The result quoted is for $a^{-1}=2\,$GeV, at which $a$ the
discretization error in the quenched $B_K$ is 15\%.
It is not inconceivable that, say, $B_K$ in QCD has very little dependence
on $a$, in which case the ratio would increase to $\sim 1.2$ in the
continuum limit.
Clearly it is very important to repeat the comparison at a different
lattice spacing.
Despite these uncertainties, I will take the OSU result and error
as the best estimate of the effect of quenching at $a=0$.
I am being less conservative than I might be because a small
quenching error in $B_K$ is consistent with the expectations of QChPT.
A more conservative estimate for the ratio would be $1.05\pm0.15$.
\subsection{$B_K$ for non-degenerate quarks}
\label{subsec:bknondegen}
What remains is to extrapolate from $m_s=m_d\approx m_s^{\rm phys}/2$
to $m_s=m_s^{\rm phys}$ and $m_d=m_d^{\rm phys}$.
This appears difficult because it requires
dynamical quarks with very small masses.
This may not be necessary, however, if one uses ChPT to guide
the extrapolation \cite{sharpetasi}.
The point is that the chiral expansion in QCD is \cite{bijnens,sharpechbk}
\begin{equation}
{B_K\over B} = 1 - \left(3+{\epsilon^2 \over 3}\right) y\ln{y}
+ b y + c y \epsilon^2 \,,
\label{eq:bkchqcd}
\end{equation}
where
\begin{equation}
\epsilon=(m_s-m_d)/(m_s+m_d)\,,\
y = M_K^2/(4 \pi f)^2 ,
\end{equation}
and $B$, $b$ and $c$ are unknown constants.
At this order $f$ can be equally well taken to be $f_\pi$ or $f_K$.
Equation \ref{eq:bkchqcd}
is an expansion in $y$, but is valid for all $\epsilon$.
The idea is to determine $c$ by working at small $\epsilon$,
and then use the formula to extrapolate to $\epsilon=1$.
This ignores corrections of $O(y^2)$, and so the errors
in the extrapolation are likely to be $\sim 25\%$.
Notice that $m_u$ does not enter into Eq. \ref{eq:bkchqcd}.
Thus one can get away with a simulation using only two
different dynamical quark masses, e.g. setting
$m_u=m_d < m_s^{\rm phys}/2$, while holding $m_s +m_d = m_s^{\rm phys}$.
To date, no such calculation has been done.
To make an estimate I use the chiral log alone, i.e. set $c=0$, yielding
\begin{equation}
B_K({\rm non-degen}) = (1.04-1.08) B_K({\rm degen}) \,.
\end{equation}
The range comes from using $f=f_\pi$ and $f_K$, and varying the
scale in the logarithm from $m_\rho-1\,$GeV.
Since the chiral log comes mainly from kaon
and $\eta$ loops \cite{sharpechbk}, I prefer $f=f_K$, which leads to
$1.04-1.05$ for the ratio.
To be conservative I take $1.05\pm0.05$, and assume that the generous
error is large enough to include also the error in the estimate
of the effect of unquenching. This leads to a final estimate of
\begin{equation}
B_K({\rm NDR},{\rm 2\,GeV,QCD}) = 0.66 \pm 0.02 \pm 0.03 \,,
\label{eq:finalqcdbk}
\end{equation}
where the first error is that in the quenched value, the second that
in the estimate of unquenching and using non-degenerate quarks.
Taking the more conservative estimate of the unquenching error (15\%),
and adding it in quadrature with the (5\%) estimate of the error
in accounting for non-degenerate quarks, increases the second error
in Eq. \ref{eq:finalqcdbk} to $0.11$.
It is customary to quote a result for the renormalization group
invariant quantity
\[
{\widehat{B}_K \over B_K(\mu)} = \alpha_s(\mu)^{-\gamma_0 \over2\beta_0}
\left(1 +
{\alpha_s(\mu) \over 4 \pi} \left[{\beta_1 \gamma_0 -\beta_0\gamma_1
\over 2 \beta_0^2} \right] \right)
\]
in the notation of Ref. \cite{crisafulli}.
Using $\alpha_s(2\,{\rm GeV})=0.3$ and $N_f=3$, I find
$\widehat{B}_K=0.90(3)(4)$, with the last error increasing to
$0.14$ with the more conservative error.
This differs from the result I quoted in Ref. \cite{sharpe93},
because I am here using the 2-loop formula
and a continuum choice of $\alpha_s$.
\subsection{Chiral behavior of $B_K$}
\label{subsec:bkchfit}
Since $B_K$ can be calculated very accurately, it provides a
potential testing ground for (partially) quenched ChPT.
This year, for the first time, such tests have been undertaken,
with results from OSU \cite{osubk96},
JLQCD \cite{aoki96}, and Lee and Klomfass \cite{lee96}.
I note only some highlights.
It turns out that, for $\epsilon=0$,
Eq. \ref{eq:bkchqcd} is valid for all $N_f$ \cite{sharpechbk}.
This is why my estimate of the quenching error for $B_K$ with degenerate
quarks in Table \ref{tab:querr} is zero.
Thus the first test of (P)QChPT is to see
whether the $-3 y\ln y$ term is present.
The OSU group has the most extensive data as a function of $y$,
and indeed observe curvature of the expected sign and magnitude for
$N_f=0,2,4$.
JLQCD also finds reasonable fits to the chiral form,
as long as they allow a substantial dependence of $f$ on lattice spacing.
They also study other $B$ parameters, with similar conclusions.
Not everything works. JLQCD finds that the volume dependence predicted
by the chiral log \cite{sharpechbk} is too small to fit their data.
Fitting to the expected form for $\epsilon\ne0$ in QQCD, they find
$\delta=-0.3(3)$, i.e. of the opposite sign to the other determinations
discussed in Sec. \ref{sec:chevidence}.
Lee and Klomfass have studied the $\epsilon$ dependence with $N_f=2$
(for which there is as yet no PQChPT prediction).
It will be interesting to see how things evolve.
My only comment is that one may need to include $O(y^2)$ terms in the
chiral fits.
\section{STATUS OF $B_K$: WILSON}
\label{sec:bkw}
There has also been considerable progress in the last year in the
calculation of $B_K$ using Wilson and SW fermions.
The challenge here is to account for the effects of the
explicit chiral symmetry breaking in the fermion action.
Success with $B_K$ would give one confidence to attempt
more complicated calculations.
The operator of interest,
\begin{equation}
{\cal O}_{V+A} = \bar s \gamma_\mu d\, \bar s \gamma_\mu d +
\bar s \gamma_\mu \gamma_5 d\, \bar s \gamma_\mu \gamma_5 d
\,,
\end{equation}
can ``mix'' with four other dimension 6 operators
\begin{equation}
{\cal O}_{V+A}^{\rm cont} = Z_{V+A} \left(
{\cal O}_{V+A} + \sum_{i=1}^{4} z_i {\cal O}_i \right) + O(a)
\end{equation}
where the ${\cal O}$ on the r.h.s. are lattice operators.
The ${\cal O}_i$ are listed in Refs. \cite{kuramashi96,talevi96}.
The meaning of this equation is that,
for the appropriate choices of $Z_{V+A}$ and the $z_i$,
the lattice and continuum operators will have the same
matrix elements, up to corrections of $O(a)$.
In particular, while the matrix elements of a general four fermion operator
has the chiral expansion
\begin{eqnarray}
\lefteqn{\langle \bar K | {\cal O} | K \rangle =
\alpha + \beta M_K^2 + \delta_1 M_K^4 +} \\
&& p_{\bar K}\cdot p_{K}
(\gamma + \delta_2 M_K^2 + \delta_3 p_{\bar K}\cdot p_{K} ) +\dots \,,
\end{eqnarray}
chiral symmetry implies that $\alpha=\beta=\delta_1=0$ for
the particular operator ${\cal O}={\cal O}_{V+A}^{\rm cont}$.
Thus, one can test that the $z_i$ are correct by checking that the
first three terms are absent.\footnote{I have ignored chiral logarithms,
which will complicate the analysis, but can probably be ignored given
present errors and ranges of $M_K$.}
Note that the $z_i$ must be known quite accurately because the
terms we are removing are higher order in the chiral
expansion than the terms we are keeping.
Five methods have been used to determine the $z_i$ and $Z_{V+A}$. \\
(1)
One-loop perturbation theory. This fails to give the correct
chiral behavior, even when tadpole improved. \\
(2)
Use (1) plus enforce chiral behavior by adjusting subsets of the
$z_i$ by hand \cite{bernardsoni89}. Different subsets give differing
results, introducing an additional systematic error.
Results for a variety of $a$ were presented by
Soni last year \cite{soni95}.\\
(3)
Use (1) and discard the $\alpha$, $\beta$ and $\gamma$ terms,
determined by doing the calculation at a variety of momenta.
New results using this method come from the LANL group \cite{gupta96}.
Since the $z_i$ are incorrect, there is, however,
an error of $O(g^4)$ in $B_K$. \\
(4)
Non-perturbative matching by imposing continuum normalization conditions
on Landau-gauge quark matrix elements. This approach has been pioneered
by the Rome group, and is reviewed here by Rossi \cite{rossi96}.
The original calculation omitted one operator \cite{donini},
but has now been corrected \cite{talevi96}. \\
(5)
Determine the $z_i$ non-perturbatively by
imposing chiral ward identities on quark matrix elements.
Determine $Z_{V+A}$ as in (4).
This method has been introduced by JLQCD \cite{kuramashi96}.
The methods of choice are clearly (4) and (5),
as long as they can determine the $z_i$ accurately enough.
In fact, both methods work well:
the errors in the non-perturbative results are much smaller
than their difference from the one-loop perturbative values.
And both methods find that the matrix element of ${\cal O}_{V+A}^{\rm cont}$
has the correct chiral behavior, within statistical errors.
What remains to be studied is the uncertainty introduced by the fact that
there are Gribov copies in Landau gauge. Prior experience suggests
that this will be a small effect.
It is not yet clear which, if either, of methods (4) and (5) is preferable
for determining the $z_i$.
As stressed in Ref. \cite{talevi96} the $z_i$ are unique,
up to corrections of $O(a)$.
In this sense, both methods must give the same results.
But they are quite different in detail,
and it may be that the errors are smaller with one method or the other.
It will be interesting to see a comprehensive comparison between them
and also with perturbation theory.
In Fig. \ref{fig:bkw} I collect the results for $B_K$.
All calculations use $m_s=m_d$ and the quenched approximation.
The fact that most of the results agree is a significant success,
given the variety of methods employed.
It is hard to judge which method gives the smallest errors,
because each group uses different ensembles and lattice sizes,
and estimates systematic errors differently.
The errors are larger than with staggered fermions mostly because
of the errors in the $z_i$.
\begin{figure}[tb]
\vspace{-0.1truein}
\centerline{\psfig{file=fig11.ps,height=3.0truein}}
\vspace{-0.6truein}
\caption{Quenched $B_K$ with Wilson fermions.}
\vspace{-0.2truein}
\label{fig:bkw}
\end{figure}
Extrapolating to $a=0$ using the data in Fig. \ref{fig:bkw} would
give a result with a large uncertainty.
Fortunately, JLQCD has found a more accurate approach.
Instead of $B_K$, they consider the ratio of the matrix element
of ${\cal O}_{V+A}^{\rm cont}$ to its vacuum saturation approximant.
The latter differs from the denominator of $B_K$ (Eq. \ref{eq:bkdef})
at finite lattice spacing. The advantage of this choice is
that the $z_i$ appear in both the numerator and denominator,
leading to smaller statistical errors.
The disadvantage is that the new ratio has the wrong chiral behavior at
finite $a$.
It turns out that there is an overall gain, and from their
calculations at $\beta=5.9$, $6.1$ and $6.3$ they find
$B_K({\rm NDR,2\,GeV})=0.63(8)$.
This is the result shown at $a=0$ in Fig. \ref{fig:bkw}.
It agrees with the staggered result, although it has much larger errors.
Nevertheless, it is an important consistency check,
and is close to ruling out the use of a linear extrapolation
in $a$ with staggered fermions.
\section{OTHER MATRIX ELEMENTS}
\label{sec:otherme}
The LANL group \cite{gupta96} has quenched results (at $\beta=6$)
for the matrix elements which determine the dominant part
of the electromagnetic penguin contribution to $\epsilon'/\epsilon$
\begin{eqnarray}
B_7^{I=3/2} &=& 0.58 \pm 0.02 {\rm (stat)} {+0.07 \atop -0.03} \,, \\
B_8^{I=3/2} &=& 0.81 \pm 0.03 {\rm (stat)} {+0.03 \atop -0.02} \,.
\end{eqnarray}
These are in the NDR scheme at 2 GeV.
The second error is from the truncation of perturbation matching factors.
These numbers lie at or below the lower end of the range used by
phenomenologists.
The LANL group also finds $B_D=0.78(1)$.
There are also new results for $f_\rho$ and $f_\phi$
\cite{guptafpi,yoshie96}, for the pion polarizability \cite{wilcox96},
and for strange quark contributions to magnetic moments \cite{dong96}.
\section{FUTURE DIRECTIONS}
\label{sec:future}
This year has seen the first detailed tests of the predicted
chiral behavior of quenched quantities.
Further work along these lines will help us make better extrapolations,
and improve our understanding of quenching errors.
It is also a warm-up exercise for the use of chiral perturbation theory
in unquenched theories. I have outlined one such application in
Sec. \ref{subsec:bknondegen}. I expect the technique to be
of wide utility given the difficulty in simulating light dynamical
fermions.
As for matrix elements, there has been substantial progress on $B_K$.
It appears that we finally know the quenched result,
thanks largely to the efforts of JLQCD.
At the same time, it is disturbing that the complicated $a$ dependence
has made it so difficult to remove the last 20\% of the errors.
One wonders whether similar complications lie lurking beneath
the relatively large errors in other matrix elements.
The improved results for $B_K$ with Wilson fermions show that
non-perturbative normalization of operators is viable.
My hope is that we can now return to an issue
set aside in 1989: the calculation of $K\to\pi\pi$ amplitudes.
The main stumbling block is the need
to subtract lower-dimension operators.
A method exists for staggered fermions, but the errors have so far
swamped the signal.
With Wilson fermions, one needs a non-perturbative method, and the hope
is that using quark matrix elements in Landau gauge
will do the job \cite{talevi96}.
Work is underway with both types of fermion.
Given the success of the Schr\"odinger functional method at calculating
current renormalizations \cite{wittig96},
it should be tried also for four fermion operators.
Back in 1989, I also described preliminary work
on non-leptonic $D$ decays, e.g. $D\to K\pi$.
Almost no progress has been made since then,
largely because we have been lacking
a good model of the decay amplitude for Euclidean momenta.
A recent proposal by Ciuchini {\em et al.} may fill this gap \cite{ciuchini}.
Enormous computational resources have been used to calculate
matrix elements in (P)QQCD.
To proceed to QCD at anything other than a snail's pace may well
require the use of improved actions.
Indeed, the large discretization errors in quenched staggered $B_K$
already cry out for improvement.
The fact that we know $B_K$ very accurately
will provide an excellent benchmark for such calculations.
Working at smaller values of the cut-off, $1/a$,
alleviates some problems while making others worse.
Subtraction of lower dimension operators becomes simpler,
but the evaluation of mixing with operators of the same dimension
becomes more difficult.
It will be very interesting to see how things develop.
\section*{Acknowledgements}
I am grateful to Peter Lepage for helpful conversations
and comments on the manuscript.
\def\PRL#1#2#3{{Phys. Rev. Lett.} {\bf #1}, #3 (#2)}
\def\PRD#1#2#3{{Phys. Rev.} {\bf D#1}, #3 (#2)}
\def\PLB#1#2#3{{Phys. Lett.} {\bf #1B} (#2) #3}
\def\NPB#1#2#3{{Nucl. Phys.} {\bf B#1} (#2) #3}
\def\NPBPS#1#2#3{{Nucl. Phys.} {\bf B({Proc.Suppl.}) {#1}} (#2) #3}
\def{\em et al.}{{\em et al.}}
| proofpile-arXiv_065-421 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
In papers \cite{1} was shown that the theory of
integrable systems (under the assumption of commutativety of all involved
functions) can be reformulated in a form, where the key role plays
the group of integrable mappings and its theory of representation.
It arose the question, what will happen with this construction,
if we will consider equations of motion for Heisenberg
operators, or in other words, when unknown functions of integrable
systems changed on non--commutative variables?
The goal of the present paper is to give the partial answer to this question.
Each quantum system with the same success can be described in many
different (in form) representations. The most known and used are
Schr\"odinger and Heisenberg pictures. The first deal with the wave
functions (the state vectors in a Hilbert space), the second with
the non--commutative Heisenberg operators and equations of motion under
appropriate initial conditions (commutation relations at the fixed moment of
time).
In this paper we will show how the group
of integrable mappings conception must be changed to include
the non--commutative variables case. The equations of evolution type
(after some modifications connected with the order of a multipliers)
remain invariant with respect to the corresponding quantum discrete
transformations without any assumption about commutation rules for unknown
functions (operators). Partially they can be $s\times s$ matrix functions or
some operators acting in the arbitrary representation space.The equations of
motion for quantum Heisenberg operators are containing within this
construction.
We use the discrete transformations method as the most
adequate to solution of such kind problems \cite{1}. We restrict ourselves
by some concrete examples of integrable mappings for non--commutative objects,
in usual and supersymmetrical two--dimensional spaces and by corresponding
hierarchies of $(1+2)$--dimensional integrable systems. We now have no idea
how to enumerate all possible integrable mappings for
non--commutative objects. In this connection we can only add that in
commutative case this problem yet is very far
{}From its final solution.
\section{Non--Commutative Darboux--Toda Substitution in Two--Dimensional
space}
\subsection{Definitions}
Let us denote $u,v$ the pair of operators defined in some representation
space and depending on $x,y$ coordinates of two--dimension space. We assume,
that partial derivatives up to some sufficient large order and inverse
operators $u^{-1}, v^{-1}$ are exist. Only associativety
is assumed for multiplication, nowhere we will assume any commutation
relations.
We will consider the following mapping:
\begin{equation}
\begin{array}{cc}
\lef{u}{}= v^{-1}&\lef{v}{}= [vu-(v_x v^{-1})_y]v\equiv v[uv- (v^{-1} v_y)_x],
\end{array}
\label{10}
\end{equation}
{\emergencystretch=5pt
where $\lef{u}{}; \lef{v}{}$
denotes final, transformed operators. In the case, when $u,v$ are some
$s\times s$ matrices (\ref{10}) was considered in \cite{10}. In classical
case, when $u,v$ are usual commutative functions (\ref{10}) is the well-known
Darboux--Toda substitution.
Substitution (\ref{10}) is invertible, i.e. the initial operators can be
expressed in the terms of final ones. Denoting $\rig{u}{}; \rig{v}{}$ the
result of inverse transformation we have:
\begin{equation} \begin{array}{cc}
\rig{v}{}= u^{-1}&\rig{u}{}= [uv-(u_y u^{-1})_x]u\equiv u[vu- (u^{-1}u_y)_x]
\end{array} \label{11}
\end{equation}
Operator $ f(u,v)$ after application of the s--times direct
transformation we will denote $ \lef{f}{s}\equiv f(\lef{u}{s},\lef{v}{s})$
and after s--times inverse transformation as
$\rig{f}{s}~\equiv~f~(\rig{u}{s}~,~\rig{v}{s}~)$,
with the agreement $\lef{f}{-m}\equiv\rig{f}{m}$, $m\ge0$.
}
\noindent If
\begin{equation}
\begin{array}{c}
u_t=F_1(u,v,u_x,v_x,u_y,v_y,...)\\
v_t=F_2(u,v,u_x,v_x,u_y,v_y,...)
\end{array}\label{611}
\end{equation}
is given evolution type system then
the condition of its invariance with respect to the transformation
(\ref{10}) (it means that in terms of $\lef{u}{},\lef{v}{}$
equations (\ref{611}) will be exactly the same as they are in terms of $u,v$)
can be derived by differentiation of (\ref{10}) by some parameter and has
the following form:
\begin{equation}
\begin{array}{rl} \lef{F}{}_1=
\lef{u}{}_t=&-v^{-1}v_t v^{-1}=-v^{-1}F_2 v^{-1}\\ \lef{F}{}_2=
\lef{v}{}_t=&([vu-(v_x v^{-1})_y]v)_t=[F_2 u+v F_1-(F_{2x}v^{-1})_y+\\
&+(v_xv^{-1}F_2v^{-1})_y]v+[vu-(v_xv^{-1})_y]F_2
\end{array} \label{13}
\end{equation}
This is the functional symmetry equation for substitution (\ref{10}).
Unknown operators here are $F_1, F_2$. If some operators $F_1(u,v), F_2(u,v)$
are solution of (\ref{13}) then the corresponding system (\ref{611}) will be
invariant with respect to (\ref{10}). (\ref{13}) is a
linear system, i.e. if ${F_1}', {F_2}'$ and ${F_1}'', {F_2}''$ are
solutions then $F_1= a{F_1}'+b{F_1}'', F_2=a{F_2}'+b{F_2}''$,
where $a, b$ are arbitrary numerical parameters, is also solution.
Every symmetry equation possesses trivial solution $F_1=au_x+bu_y,
F_2=av_x+bv_y$.
Substitution is called
integrable if its symmetry equation have at least one non--trivial
solution.
\subsection{Solution of the Symmetry Equation}
The method we will use here to find solutions of (\ref{13}) is analogues
to the method we used in \cite{14} in the case of commutative functions.
But it is not exactly the same as many
transformations can not be done because of non--commutativety of variables
under consideration.
First of all let us take $F_2=\alpha_0 v$, $F_1=u\beta_0$. We obtain:
\begin{equation}
\begin{array}{l}
\beta_0=-\rig{\alpha}{}_0\\ {\alpha_0}_{xy}=(\alpha_0-\lef{\alpha}{}_0)\lef{T_0}{\:}+
T_0(\alpha_0-\rig{\alpha_0}{\!})+\theta {\alpha_0}_y,
-{\alpha_0}_y\theta \label{2}
\end{array}
\end{equation}
where $T_0=vu$, $\theta=v_x v^{-1}$.
This system possesses obvious partial solution
${\alpha_0}^{(0)}=-{\beta_0}^{(0)}=1$, which gives the first term of hierarchy:
$F_1=-u, F_2=v$.
Two following equations for $ T_0 $ and $ \theta $, which are the direct
corollary of (\ref{10}), are important for father calculations:
\begin{equation}
{T_0}_x=\theta T_0-T_0 \rig{\theta}{}\qquad
\theta_y=T_0-\lef{T_0}{\:} \label{3}
\end{equation}
In fact, (\ref{3}) is the substitution (\ref{10}) rewritten in terms of
$T_0$ and $\theta$.
Now let us take ${\alpha_0}_y=\alpha_1\lef{T_0}{\:}+T_0\beta_1$. One can treat
this expression as analog to the decomposition of some vector by basic
vectors. {}From (\ref{2}), expressing $\lef{T_0}{\:}_x $ and $T_{0x}$ with the
help of (\ref{3}) and equating to zero coefficients before $\lef{T}{}_0$ and
after $T_0$ (which is some additional assumption), we have:
\begin{equation}
\begin{array}{l}
{\alpha_1}_x=\alpha_0-\lef{\alpha}{}_0+\theta\alpha_1-\alpha_1\lef{\theta}{}\\
{\beta_1}_x=\alpha_0-\rig{\alpha_0}{\!}+\rig{\theta}{}\beta_1-\beta_1\theta
\end{array}\label{1000}
\end{equation}
The second relation obviously can be rewritten as:
$$
{\lef{\beta}{}_1}_x=-(\alpha_0-\lef{\alpha}{}_0)+\theta\lef{\beta}{}_1-
\lef{\beta}{}_1\lef{\theta}{}
$$
{}From what it follows that system (\ref{1000}) possesses partial solution of
the form $\lef{\beta}{}_1=-\alpha_1$. After taking $y$--derivative of
equation for $\alpha_1$, this partial solution gives the following system:
\begin{equation}
\begin{array}{l}
\beta_1=-\rig{\alpha_1}{\!}\\
{\alpha_1}_{xy}=(\alpha_1-\lef{\alpha}{}_1)\lef{T_0}{2}+T_0(\alpha_1-\rig{\alpha_1}{\!})+
\theta{\alpha_1}_y -{\alpha_1}_y
\lef{\theta}{}\\
\end{array} \label{4}
\end{equation}
This system is analogues to the (\ref{2}). It also has
partial solution $\alpha_1=-\beta_1=1$, which leads to
${\alpha_0}^{(1)}=\int(\lef{T_0}{\:}-T_0)dy$, and gives the next solution of the
symmetry equation (\ref{13}). Taking now
${\alpha_1}_y=\alpha_2\lef{T_0}{2}+T_0\beta_2$ we are able to continue by the same
scheme. The system for $\alpha_2, \beta_2$ has the same structure as
previous systems. Its partial solution $\alpha_2=-\beta_2$ allows to find the
third term of hierarchy $$ {\alpha_0}^{(2)}=\int dy\,\Bigl[\Bigl(\int
dy\,\Bigl(\lef{T_0}{2}-T_0\Bigr)\Bigr) \lef{T_0}{\:}-T_0\int
dy\,\Bigl(\lef{T_0}{\:}-\rig{T_0}{\!}\Bigr)\Bigr] $$
By induction it can be proved that in general case equations for
$\alpha_n, \beta_n $ have the form:
\begin{equation}
\begin{array}{l}
\beta_n=-\rig{\alpha_n}{\!}\\
{\alpha_n}_{xy}=(\alpha_n-\lef{\alpha}{}_n)\lef{T_0}{(n+1)}+T_0(\alpha_n-\rig{\alpha_n}{\!})
+\theta{\alpha_n}_y-{\alpha_n}_{y}\lef{\theta}{n}\\
\end{array} \label{5}
\end{equation}
with partial solution $\alpha_n=-\beta_n=1$.
After this expression for ${\alpha_0}^{(n)}$ can be reconstructed in
the form of the sum of $2^n$ terms, which can be written in the following
symbolical form:
\begin{equation}
\begin{array}{rcl}
{\alpha_0}^{(n)}&=&(-1)^n \prod_{i=1}^n \left(1-L_i
exp\left[id_i+\sum_{i=k+1}^n d_k\right]\right)\times \\&&\\
&&\times\int dy (T_0 \int dy (\rig{T_0}{\!}\int dy (...\int dy
\rig{T_0}{(n-1)}...)))\label{20}
\end{array}
\end{equation}
where $exp\, d_p$ means shifts by the unity the argument of p\--repeated
integral.
$$
\dots\int dy\lef{T_0}{p}\to\dots\int dy\lef{T_0}{(p+1)}\dots
$$
and symbol $L_r$\---transposition of terms in the r\--th brackets
$$
(A_1(\dots(A_r[\dots])\dots)) \to (A_1(\dots([\dots]A_r)\dots))
$$
with the following multiplication rules:
$$
L_i exp[...]_1 L_j exp[...]_2=L_i L_j exp\left[ [...]_1+[...]_2\right]
$$
Comparing (\ref{20}) with \cite{14} we see that here for non-commutativety we
are forced to introduce the new operators $L_i$ which are discount the order
of the multipliers.
\subsection{Examples}
\subsubsection*{n=0}
$$
v_t=v\qquad u_t=-u
$$
\subsubsection*{n=1}
$$
v_t=v_x\qquad u_t=u_x
$$
\subsubsection*{n=2}
$$
v_t=v_{xx}-2\int (vu)_x dy\times v \quad u_t=-u_{xx}+2u\int (vu)_x dy
$$
This is the Davey--Stewartson system, described in \cite{12}
\subsubsection*{n=3}
$$
v_t=v_{xxx}-3\int (vu)_x dy\times v_x-3\int (v_x u)_x dy\times v-
$$
$$
-3\int \left[vu\int (vu)_x dy-\int (vu)_x dy\times vu\right]dy\times v
$$
\vspace{1em}
$$
u_t=-u_{xxx}-3u_x \int (vu)_x dy-3u\int (v_x u)_x dy-
$$
$$
-3u\int \left[vu\int dy (vu)_x-\left( \int dy (vu)_x \right) vu\right]dy
$$
In commutative case this is the Veselov--Novikov system.
\section{Non--Commutative Darboux--Toda Transformation in Two--Dimensional
Super Space}
\subsection{Definitions}
Here we will analyze the situation, when non-commutative operators under
consideration in addition to usual space and time coordinates $x, y, t$
are depend upon Grassman variables $\theta_+, \theta_-$. We will consider the
following mapping:
\begin{equation}
\begin{array}{cc}
\lef{u}{}= v^{-1}&\lef{v}{}=-[D_-(D_+v\times v^{-1})+vu]v\equiv
v[D_+(v^{-1}D_-v)-uv],
\end{array}
\label{31}
\end{equation}
where
$$
D_+=\frac{\partial}{\partial\theta_+}+\theta_+\frac{\partial}{\partial x}\quad
D_-=\frac{\partial}{\partial\theta_-}+\theta_-\frac{\partial}{\partial y}\quad
D^2_+=\frac{\partial}{\partial x}\quad D^2_-=\frac{\partial}{\partial y}
$$
Other notations are the same as in the
previous section. Substitution (\ref{31}) is invertible. Inverse
transformation has the form:
\begin{equation}
\begin{array}{cc}
\rig{v}{}= u^{-1}&\rig{u}{}=-[D_+(D_-u\times u^{-1})+uv]u\equiv
u[D_-(u^{-1}D_+u)-vu],
\end{array}
\label{33}
\end{equation}
The symmetry equation for (\ref{31}) is the following:
\begin{equation}
\begin{array}{rcl}
\lef{F_1}{\:}&=
& -v^{-1}F_2 v^{-1}\\ \lef{F_2}{\,}&=&
F_2[D_+(v^{-1}D_-v)-uv]+v[D_+(-v^{-1}F_2v^{-1}D_-v)+\\&&+D_+(v^{-1}D_-F_2)-F_1v-
uF_2]
\end{array} \label{35}
\end{equation}
\subsection{Solution of the Symmetry Equation}
Here we will get the hierarchy of solutions of the symmetry equation (\ref{35}).
For this we will use the same general method as in the previous section.
But there is an interesting and in some sense important difference. As we will
see bellow, partial solutions of (\ref{35}) can be found only at even steps,
when unknown operators are Bosonic--like variables, whereas at odd steps they
are Fermionic--like.
After substitution in (\ref{35}) $F_1=u\beta_0, F_2=\alpha_0 v$ we have:
\begin{equation}
\begin{array}{l}
\beta_0=-\rig{\alpha_0}{\!}\\ D_+D_-\alpha_0=(\lef{\alpha}{}_0-\alpha_0)\lef{T_0}{\:}+
T_0(\alpha_0-\rig{\alpha_0}{\!})+\theta D_-\alpha_0+D_-\alpha_0\theta, \label{62}
\end{array}
\end{equation}
where $T_0=vu, \theta=D_+v\times v^{-1}$. This system has partial solution
$\alpha_0=-\beta_0=1$, which correspond to: $F_1=-u,F_2=v$.
Transformation (\ref{31}) can be rewritten in terms of $T_0, \theta$ as:
\begin{equation}
D_+T_0=\theta T_0-T_0 \rig{\theta}{}\qquad
D_-\theta=-T_0-\lef{T_0}{\:} \label{43}
\end{equation}
Taking now
$D_-\alpha_0=\alpha_1\lef{T_0}{\:}+T_0\beta_1$
, for $\alpha_1, \beta_1$ we have:
$$
D_+\alpha_1=\lef{\alpha}{}_0-\alpha_0+\theta\alpha_1+\alpha_1\lef{\theta}{}
$$
$$
D_+\beta_1=\alpha_0-\rig{\alpha}{}_0+\rig{\theta}{}\beta_1+\beta_1\theta
$$
For $\lef{\beta}{}_1=\alpha_1$ the second relation directly follows from the first
one. For this case, acting on the equation for $\alpha_1$ with $D_-$ operator, we
have:
\begin{equation}
\begin{array}{l}
\beta_1=\rig{\alpha_1}{\!}\\
-D_+D_-\alpha_1=(\alpha_1+\lef{\alpha}{}_1)\lef{T_0}{2}-T_0(\alpha_1+\rig{\alpha_1}{\!})+
D_-\alpha_1\lef{\theta}{}-\theta D_-\alpha_1\\
\end{array} \label{44}
\end{equation}
This is the typical system for odd steps. Comparing it with (\ref{4}) we
notice that the difference between those systems is that (\ref{44}) have not
numerical partial solutions ($\alpha_1, \beta_1$ are Fermionic--like operators).
However it is possible to continue reduction using decomposition
$D_-\alpha_1=\alpha_2\lef{T_0}{2}+T_0\beta_2$. We have:
$$
-D_+\alpha_2=\lef{\alpha}{}_1+\alpha_1-\theta\alpha_2+\alpha_2\lef{\theta}{2}
$$
$$
-D_+\beta_2=-(\alpha_1+\rig{\alpha_1}{\!})-\rig{\theta}{}\beta_2+\beta_2\lef{\theta}{}
$$
Taking $\lef{\beta}{}_2=-\alpha_2$, after usual simple
calculations we will find:
\begin{equation}
\begin{array}{l}
\beta_2=-\rig{\alpha_2}{\!}\\
D_+D_-\alpha_2=(\lef{\alpha}{}_2-\alpha_2)\lef{T_0}{3}+T_0(\alpha_2-\rig{\alpha_2}{\!})+
D_-\alpha_1\lef{\theta}{2}+\theta D_-\alpha_2\\
\end{array} \label{54}
\end{equation}
The partial solution of this system is: $\alpha_2=-\beta_2=1$; it correspond
to the trivial system $F_1=au_x+bv_y, F_2=av_x+bv_y$.
All systems received on even steps will be similar to (\ref{54}). Partial
solution of each next system of that kind gives non--trivial, nonlinear
evolution type system invariant with respect to the transformation (\ref{31})
(see $k=2$ example).
By induction easily can be proved that for arbitrary $n=2k+1$ we will have:
$$
D_-\alpha_{n-1}=\alpha_n\lef{T_0}{\:n}+T_0\rig{\alpha}{}_n
$$
$$
D_-\alpha_n=\alpha_{n+1}\lef{T_0}{\:n+1}-T_0\rig{\alpha}{}_{n+1}
$$
\begin{equation}
\begin{array}{rr}
D_+D_-\alpha_{n-1}=(\lef{\alpha}{}_{n-1}-\alpha_{n-1})\lef{T_0}{n}-T_0(\alpha_{n-1}-
\rig{\alpha}{}_{n-1})+\\+D_-\alpha_{n-1}\lef{\theta}{n-1}+\theta D_-\alpha_{n-1}
\end{array}
\label{144}
\end{equation}
$$
-D_+D_-\alpha_n=(\lef{\alpha}{}_n+\alpha_n)\lef{T_0}{n+1}-
T_0(\alpha_n+\rig{\alpha}{}_{n})+
D_-\alpha_n\lef{\theta}{n}\\-\theta D_-\alpha_n \label{145}
$$
After this using $\alpha_{2k}=1$ partial solution of the system (\ref{144}) it is
possible to construct the $k$--th term of hierarchy. One can prove using
induction that the final result can be represented as:
\begin{equation}
\begin{array}{rcl}
{\alpha_0}^{(k)}&=&(-1)^k \prod_{i=1}^{2k} \left(1-(-1)^iL_i
exp\left[id_i+\sum_{i=k+1}^{2k} d_k\right]\right)\times \\&&\\
&&\times D_-^{-1}(T_0 D_-^{-1}(\rig{T_0}{\!}D_-^{-1} (...D_-{-1}
\rig{T_0}{(n-1)}...)))\label{320}
\end{array}
\end{equation}
The meaning of notations here is the same as in formula (\ref{20}).
\subsection{Examples}
\subsubsection*{k=0}
$$
v_t=v\qquad u_t=-u
$$
\subsubsection*{k=1}
$$
v_t=v_x\qquad u_t=u_x
$$
\subsubsection*{k=2}
$$
v_t=v_{xx}-2D_-^{-1}(vu)_x D_+v-2D_-^{-1}(vD_+ u)\times v+
$$
$$
+2D_-^{-1} \left[vuD_-^{-1}(vu)_x +D_-^{-1}(vu)_x \times vu\right]
$$
\vspace{1em}
$$
u_t=-u_{xx}+D_+uD_-^{-1}(vu)_x -2uD_-^{-1}(D_+vu)_x-
$$
$$
-2uD_-^{-1} \left[vuD_-^{-1}(vu)_x +D_-^{-1}(vu)_x \times vu\right]
$$
\section{Conclusion}
The main concrete result of the paper is the
explicit form of quantum integrable systems (\ref{20}), (\ref{320}) in the
mentioned above sense.
It is interesting that the scheme of our calculations is similar to the
computer program algorithm--there are many identical operations with
possibility to interrupt them at arbitrary step. Obviously, in this scheme is
coded the structure of
the group of integrable mappings, more exactly one of the possible
connections between the integrable system by itself and its
symmetry equation. If it will be possible to
translate it on the group--theoretical language, then we will be near to
understand the integrable substitutions role and near to the classification
theorem for them.
It is well known that quantum integrable systems are closely connected with
so--called "quantum" algebras \cite{16}. Moreover this object of mathematics in
essential part was discovered and developed under the investigations of
the integrable systems in quantum region.
So it arise more wide, deep and interesting problem--to find the
connection between the approach of this paper and sufficiently developed
formalism of quantum algebras. The equations for Heisenberg
operators, as it was mentioned in the introduction, are only one of the
possible representations of the quantum picture. We hope that further
investigations will find some bridge connecting quantum integrable mappings
of the present paper with quantum algebras of the traditional approach.
But now we are not ready and able to go so far and hope to return to this
problem in the future publications.
\section{Acknowledgments}
The authors wish to thank the Russian Foundation for Fundamental Researches
for partial support trough the Grant 95--01--00249
| proofpile-arXiv_065-422 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
The presence of ubiquitous X-ray absorption in cooling flow (CF) clusters
was first discovered by White et al. (1991) using the {\em Einstein}
Solid State Spectrometer (SSS), and was later verified using
{\em EXOSAT},
{\em ROSAT}, and {\em ASCA} (e.g. Allen et al. 1992; Fabian et al.
1994). The typical observed columns are a few $10^{20}$ to a few
$10^{21}$~cm$^{-2}$. Searches were conducted at other wavelengths in order
to verify the presence of the absorber and to understand its
nature. Emission line constraints rule out absorbing gas at
$T\sim 10^5-10^6$~K, and thus the absorbing gas must be mostly
H~I and/or H$_2$. Extensive searches for H~I absorption using the
21~cm hyperfine structure line (e.g. Jaffe 1990; Dwarakanath, van Gorkom,
\& Owen 1994) and searches for
CO associated with H$_2$ (e.g. O'dea et al. 1994; Antonucci \& Barvainis
1994) yielded upper limits typically well below
the X-ray columns. The 21~cm limits are linear with the
electron excitation temperature, and may thus be subject to significant
uncertainty.
For example, in the case of NGC~1275, the central galaxy in the
Perseus CF cluster, the X-ray column obtained by the {\em Einstein} SSS
is $1.3^{+0.3}_{-0.3}\times 10^{21}$~cm$^{-2}$ (White et al. 1991), by
{\em EXOSAT} $1.5^{+2.1}_{-0.9}\times 10^{21}$~cm$^{-2}$
(Allen et al. 1991), and by {\em ASCA}
$3-4\times 10^{21}$~cm$^{-2}$ (Fabian et al. 1994).
The 21~cm line, however, indicates an H~I
column of only $2\times 10^{18}T_s$~cm$^{-2}$ (Jaffe 1990).
In this paper we show that significantly tighter limits on
$N_{\rm H~I}$ can be obtained using Ly$\alpha$.
We first make a short comparison
of the absorption properties of Ly$\alpha$ versus the 21~cm line. We then
apply the results of curve of growth analysis for Ly$\alpha$ to
show that the H~I column $\sim 10-20$~kpc away from the center
of NGC~1275 is significantly
smaller than indicated by the 21~cm line, and demonstrate that improved
limits can be obtained with a higher quality UV spectrum.
We end with a short discussion of the implications of
the new upper limits on $N_{\rm H~I}$ on
the nature of the X-ray absorber.
\section{On Lyman $\alpha$ vs. 21 cm Absorption}
In this section we compare the absorption properties of Lyman $\alpha$
versus the 21~cm line. In a two level atom the absorption cross section
per atom is:
\[ \sigma_{\nu}=\frac{\pi e^2}{m_ec}f_{12}\phi(\nu)
f_{se}\frac{n_1}{n_1+n_2} \]
where $f_{12}$ is the oscillator strength, and
$\phi(\nu)$ is the line profile function (Voigt function for pure thermal
broadening). The parameter
$f_{se}$ is the correction for stimulated emission given by
\[ f_{se}\equiv1-\frac{n_2}{n_1}\frac{g_1}{g_2} \]
where $n_i, g_i$ are the population and degeneracy of level i.
The level population is accurately described by the Boltzmann ratio
since collisions dominate both excitations and deexcitaions, and thus
\[ f_{se}=1-e^{-\frac{\Delta E}{kT}}, \]
or $f_{se}\simeq \frac{\Delta E}{kT}$ for $\Delta E\ll kT$.
The value of the line profile function at line center is
\[ \phi(\nu_0)=\frac{1}{\sqrt{\pi}\Delta\nu_D}\]
assuming a Gaussian line shape (i.e. thermal broadening),
where $\Delta\nu_D=\nu_0\frac{b}{c}$ is the line width and
$b=\sqrt{\frac{2kT}{m_p}}$ is the Doppler parameter
(Rybicki \& Lightman 1979).
Table 1 compares the values of the parameters discussed above for
Ly$\alpha$ versus the 21~cm line.
\begin{table}
\caption{Ly$\alpha$ versus 21~cm absorption line parameters}
\begin{center}
\begin{tabular}{ccc}
Parameter & Ly$\alpha$ & 21~cm\\
\tableline
$f_{12}$ & 0.416 & $5.75\times 10^{-12}$ \\
$\phi(\nu_0)$ & $5.33\times 10^{-10}T^{-1/2}$ & $9.27\times 10^{-4}T^{-1/2}$\\
$f_{se}$ & $\simeq 1$ & $0.0682T^{-1}$\\
$n_2/n_1$ & $\ll 1$ & 3 \\
$ \sigma_{\nu_0}$ & $5.88\times10^{-12} T^{-1/2}$
& $2.41\times10^{-19} T^{-3/2}$ \\
\end{tabular}
\end{center}
\end{table}
The ratio of line center absorption cross sections is therefore
\[ \frac
{\sigma_{\nu_0}({\rm Ly}\alpha)}
{\sigma_{\nu_0}(21 {\rm cm})}
=2.44\times 10^6 T,
\]
and since $T\ge 2.73$~K, Ly$\alpha$ is $10^7$ times more
sensitive to H~I absorption than the 21~cm line, if both absorption
lines are optically thin.
Note that when $N_{\rm H~I}>2\times 10^{11}T^{1/2}$,
the Ly$\alpha$ line becomes optically thick, and the absorption
equivalent width $EW=\int (1-e^{-\tau_{\nu}})d\nu$ increases only as
$\sqrt{\ln \tau}$.
Figure 1 presents a comparison of the absorption profiles of the 21~cm
line vs. Ly$\alpha$ for $10^{20}\ge N_{\rm H~I}\ge 10^{17}$~cm$^{-2}$.
Note the large difference in absorption EW of the two lines.
\begin{figure}
\psfig{file=figcf1.ps,width=11.cm,angle=-90,silent=}
\caption{A comparison of the absorption profiles of the 21~cm
line (right) vs. Ly$\alpha$ (left)
for $10^{17}\ge N_{\rm H~I}\ge 10^{20}$~cm$^{-2}$.
Note the large difference in velocity scales in the two panels.}
\end{figure}
Jaffe (1990) measured in NGC~1275 21~cm absorption with
$N_{\rm H~I}=2\times 10^{18}T$
and FWHM=477~km~s$^{-1}$ (i.e. $b=286$~km~s$^{-1}$).
The expected Ly$\alpha$
absorption profile for $N_{\rm H~I}$ at various $T$
is displayed in Figure 2,
\begin{figure}
\psfig{file=figcf2.ps,width=11.cm,angle=-90,silent=}
\caption{
The predicted vs. observed Ly$\alpha$ absorption profile.
Left: The predicted absorption profile for different values of
T with $N_{\rm H~I}=2\times 10^{18}T$ and $b=286$~km~s$^{-1}$,
as measured by Jaffe (1990).
Right: The Ly$\alpha$ spectrum observed by Johnstone \& Fabian
(1995). Very
little, if any, absorption is present in Ly$\alpha$.}
\end{figure}
indicating that for all $T$ one expects a very broad absorption trough
with FWHM$\ge 2000$~km~s$^{-1}$. The Ly$\alpha$ region in NGC~1275
was observed by Johnstone \& Fabian (1995), and the observed spectrum
is displayed in Figure 2 (velocity scale is relative to
5260~km~s$^{-1}$). Clearly, the absorption predicted based on
the 21~cm $N_{\rm H~I}$ is not present. The small trough at the center
of Ly$\alpha$ suggests absorption with EW$\sim 0.5$~\AA, or
$120$~km~s$^{-1}$. Johnstone \& Fabian (1995) suggested that Ly$\alpha$
is double peaked, rather
than absorbed, in which case the absorption EW would be $\ll 0.5$~\AA.
Clearly, Ly$\alpha$ implies a much lower values for
$N_{\rm H~I}$ than the 21~cm line. To obtain the $N_{\rm H~I}$
implied by Ly$\alpha$ one
needs to calculate EW($N_{\rm H~I}$), i.e. use the standard ``curve of
growth'' analysis. Figure 3 displays on the left hand side the
Ly$\alpha$ EW versus $N_{\rm H~I}$ for various
values of the $b$ parameter (assuming a Gaussian velocity distribution).
The EW increases linearly with $N_{\rm H~I}$
when the line is optically thin, saturating to EW$\propto
\sqrt{\ln N_{\rm H~I}}$
when the line becomes optically thick, and recovering back to
EW$\propto \sqrt{N_{\rm H~I}}$ when the Lorenztian wings dominate the
absorption (`damped' absorption). The observed absorption EW of
120~km~s$^{-1}$ translates to
$10^{14}\ge N_{\rm H~I}\ge 4\times 10^{17}$~cm$^{-2}$, where the
upper limit is obtain if $b<10$~km~s$^{-1}$, and the lower limit is
obtained if $b>50$~km~s$^{-1}$. The right hand side
curves in Figure 3 represent
the curves of growth for 21~cm absorption by H~I at $T=10$~K. These curves
are identical to those for Ly$\alpha$ absorption, but shifted by a factor
of $2.4\times 10^7$ to the right hand side. The observed 21~cm absorption
EW translates to $N_{\rm H~I}=2\times 10^{19}$~cm$^{-2}$ for a reasonable
lower limit of $T=10$~K. The largest column allowed by Ly$\alpha$
is therefore $\sim 50$ times smaller than indicated by the 21~cm line.
\begin{figure}
\psfig{file=figcf3.ps,width=13.0cm,silent=}
\caption{The Ly$\alpha$ and the 21~cm curves of growth for different
velocity dispersions ($b$ parameters). The horizontal dashed lines indicate
the observed absorption EWs, and the vertical dashed lines indicate the
derived limits on $N_{\rm H~I}$. The largest column allowed by Ly$\alpha$
is $\sim 50$ times smaller than indicated by the 21~cm line.}
\end{figure}
\section{How can the 21~cm and Ly$\alpha$ columns be reconciled?}
The apparent contradiction between the 21~cm and the Ly$\alpha$ columns
can be understood if the H~I column is highly non-uniform, and the
spatial distributions of the background 21~cm and Ly$\alpha$ emission are
different. There is observational evidence that both these effects are
present in NGC~1275. According to the 21~cm continuum map presented by Jaffe
(1990) most of the continuum originates within $\pm 20''$ of the center.
Fabian, Nulsen, \& Arnaud (1984) discovered with IUE that Ly$\alpha$ is
also extended on $\sim 10''$ scale, and the recent HUT observations by
Van Dyke Dixon, Davidsen, \& Ferguson (1996) indicate that Ly$\alpha$
emission of comparable surface brightness extends out to $\sim 60''$
from the center. Evidence for spatially non-uniform absorption is seen
on much smaller scales in the VLBA observations of Walker et al. (1994) and
Vermeulen et al. (1994), who discovered a free-free absorbed counter jet to
the north of the nucleus.
The counter jet is most likely seen through a large column disk of
relatively cold gas close to the center of NGC~1275 (Levinson et al. 1995),
while the line of sight to the southren jet is clear.
The large $N_{\rm H~I}$ indicated by the 21~cm line most likely
resides on scales smaller than the $\sim 10-20$~kpc scale of the
Ly$\alpha$ emitting filaments. The low
$N_{\rm H~I}$
indicated by Ly$\alpha$ provides a constraint for the $\sim 100-200$~kpc
scale absorber indicated by the X-ray observations.
\section{How can the X-ray and Ly$\alpha$ columns be reconciled?}
The various X-ray telescopes mentioned above indicate an excess absorbing
column of $(1.5-4)\times 10^{21}$~cm$^{-2}$, and a covering factor close
to unity. Such a column becomes optically thick
at $E<0.6-1$~keV. At this energy range O is the dominant absorber
(e.g. Morrison
\& McCammon 1983). Thus, the X-ray spectra merely indicate the presence
of an O column of $(1.3-3.4)\times 10^{18}$~cm$^{-2}$. The O
X-ray absorption
is done by the inner K shell electrons, thus the ionization state of
the O can be anywhere from O~I to O~VII.
The X-ray and Ly$\alpha$ constraints imply that whatever is producing
the absorption on the $\sim 100$~kpc scale in the Perseus cooling flow
cluster
must have $3<N_{\rm O}/N_{\rm H~I}<3\times 10^4$, i.e. it is drastically
different from a neutral, solar abundance absorber, where
$N_{\rm O}/N_{\rm H~I}=8.5\times 10^{-4}$.
If the absorber has roughly solar abundance then H must be highly
ionized with $2.5\times 10^{-8}<N_{\rm H~I}/N_{\rm H}<2.5\times 10^{-4}$.
Can H be so highly ionized? The available ionizing flux is far too
low for significant photoionization. Collisional ionization requires
$5\times 10^6>T>5\times 10^4$~K, where the upper limit prevents O from
being too
highly ionized. However, this temperature range is excluded based on
the absence of significant line emission (e.g.
Voit \& Donahue 1995). It thus
appears that the required ionization state of H is ruled out.
The above constraints on $T$ assume equilibrium ionization states. It
remains to be studied whether plausible deviations from ionization equilibrium
can significantly affect the ionization state of H.
Another possibility is that the absorber has practically no H. This would be
the case if the absorption originates in O which resides in dust grains embeded
in hot gas. However, the dust sputtering time scales appear too short
to explain the absorption in the inner parts of clusters (Dwek et al. 1990;
Voit \& Donahue 1995).
Could most of the H be in molecular form?
CO emission was detected in the Perseus CF (e.g. Braine et al. 1995)
indicating that
some of the H is indeed in molecular form. However, the
H~I/H$_2$ fraction needs to be $<2.5\times 10^{-4}$, while theoretical
calculations (Ferland et al. 1994) indicate that most
of H ($>80$\%) would be in atomic form, even in extremely cold
clouds embeded in CFs.
It therefore appears that there is no satisfactory model which
explains the X-ray absorption together with the new tight limits on
$N_{\rm H~I}$ obtained from Ly$\alpha$.
\section{Future perspectives}
Given the difficulty in finding a plausible explanation for the X-ray
absorption, it is crucial to verify that this absorption is
indeed real. This can be achieved by the detection of an O bound-free
K edge at the CF cluster redshift (see Sarazin, these proceedings).
The edge energy will also indicate the ionization state of the absorber.
This can be achieved with next generation high resolution X-ray telescopes.
On a shorter time scale, significantly better constraints on
$N_{\rm H~I}$ in NGC~1275 can be obtained with HST. Currently,
the actual value of $N_{\rm H~I}$ in NGC~1275 is
uncertain by nearly 4 orders of magnitude, and a much more accurate
determination can be achieved through a higher resolution UV spectrum,
as demonstrated in Figure 4.
\begin{figure}
\psfig{file=figcf4.ps,width=11.cm,angle=-90,silent=}
\caption{The expected Ly$\alpha$ profile in NGC~1275 at a high
spectral resolution ($\sim 10$~km~s$^{-1}$). Left:
Emission + absorption profiles. Right: Blowout of the
absorption profile. Such a spectrum would allow a rather accurate
determination of $N_{\rm H~I}$. If the H~I gas has a very low
velocity dispersion the absorption will go black, and if the
velocity dispersion
is high the absorption profile will remain shallow.}
\end{figure}
The method described here can be extended to many more CF clusters. All
CF clusters have a large cD galaxy at their center, and these galaxies tend
to have a power-law continuum source at their center with significant UV
emission which can be used for the detection of
Ly$\alpha$ absorption. In addition, significant Ly$\alpha$ emission
most likely originates from the emission line filaments present in all CF
clusters. For example, Hu (1992) observed 10 CF cluster with
the IUE, detecting significant Ly$\alpha$ emission in 7 clusters, thus
demonstrating that the method described here can be applied to most CF
clusters.
The disadvantage of using the central cD galaxy is that if absorption
is detected it may be produced by gas local to the cD or the emission
line filaments, rather than the
large scale absorber. The lack of significant absorption can, however,
be used to place stringent limits on the nature of the X-ray absorber.
\acknowledgments
This research was partly supported by the E. and J. Bishop research fund,
and by the Milton and Lillian Edwards academic lectureship fund. The
UV spectrum of NGC~1275 was generously provided by R. M. Johnstone.
| proofpile-arXiv_065-423 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
The precision data collected to date have confirmed the
Standard Model to be a good description of physics below
the electroweak scale \cite{Schaile}.
Despite of its great success, there are many reasons to believe
that some kind of new physics must exist. On the other hand, the
non-abelian structure of the gauge
boson self-couplings is still poorly tested and one of the most sensitive
probes for new physics is provided by the trilinear gauge boson couplings
(TGC) \cite{TGC}.
Many studies have been devoted to the $WW\gamma$ and $WWZ$ couplings.
At hadron colliders and $e^+e^-$ colliders, the present bounds
(Tevatron \cite{Errede}) and prospects (LHC, LEP2 and
NLC \cite{TGC,LEP2}) are mostly based on diboson production ($WW$,
$W\gamma$ and $WZ$).
In $ep$ collisions, HERA could provide
further information
analyzing single $W$ production ($ep\to eWX$ \cite{ABZ})
and radiative charged current scattering
($ep\to\nu\gamma X$ \cite{hubert}). There is also some
literature on $WW\gamma$ couplings in $W$-pair production at future
very high energy photon colliders (bremsstrahlung photons in peripheral
heavy ion collisions \cite{HIC} and Compton backscattered laser
beams \cite{gg}).
Only recently, attention has been paid to the $Z\gamma Z$, $Z\gamma\g$ and
$ZZZ$ couplings. There is a detailed analysis of $Z\gamma V$
couplings ($V=\gamma,Z$) for hadron colliders in \cite{BB}.
CDF \cite{CDF} and D\O\ \cite{D0} have obtained bounds on the
$Z\gamma Z$ and $Z\gamma\g$ anomalous couplings, while L3 has studied
only the first ones \cite{L3}. Studies on the sensitivities to
these vertices in future $e^+e^-$ colliders,
LEP2 \cite{LEP2} and NLC \cite{Boudjema}, have been performed during
the last years.
Some proposals have been made to probe these neutral boson gauge
couplings at future photon colliders in $e\gamma\to Ze$ \cite{eg}.
In this work we study the prospects for measuring the
TGC in the process $ep\to
e\gamma X$. In particular, we will concentrate on the $Z\gamma\g$ couplings,
which can be more stringently bounded than the $Z\gamma Z$ ones
for this process.
In Section 2, we present the TGC. The next section deals with the
different contributions to the process $ep\to e\gamma X$ and the cuts
and methods we have employed
in our analysis. Section 4 contains our results
for the Standard Model total cross section and distributions and
the estimates of the sensitivity of these quantities to the
presence of anomalous couplings. Finally, in the last section we
present our conclusions.
\section{Phenomenological parametrization of the neutral TGC}
A convenient way to study deviations from the standard model predictions
consists of considering the most general lagrangian compatible with
Lorentz invariance, the electromagnetic U(1) gauge symmetry, and
other possible gauge symmetries.
For the trilinear $Z\gamma V$ couplings ($V=\gamma,Z)$ the most general vertex
function invariant under Lorentz and electromagnetic gauge transformations
can be described in terms of four independent dimensionless form
factors \cite{hagiwara}, denoted by $h^V_i$, i=1,2,3,4:
\begin{eqnarray}
\Gamma^{\a\b\mu}_{Z\gamma V} (q_1,q_2,p)=\frac{f(V)}{M^2_Z}
\{
h^V_1 (q^\mu_2 g^{\a\b} - q^\a_2 g^{\mu\b})
+\frac{h^V_2}{M^2_Z} p^\a (p\cdot q_2g^{\mu\b}-q^\mu_2 p^\b)
\nonumber \\
+h^V_3 \varepsilon^{\mu\a\b\r}q_{2_\r}
+\frac{h^V_4}{M^2_Z}p^\a\varepsilon^{\mu\b\r\sigma}p_\r q_{2_\sigma}
\}. \hspace{3cm}
\label{vertex}
\end{eqnarray}
Terms proportional to $p^\mu$, $q^\a_1$ and $q^\b_2$ are omitted as long as
the scalar components of all three vector bosons can be neglected
(whenever they couple to almost massless fermions) or they are zero
(on-shell condition for $Z$ or U(1) gauge boson character of the photon).
The overall factor, $f(V)$, is $p^2-q^2_1$ for $Z\gamma Z$ or $p^2$ for $Z\gamma\g$
and is a result of Bose symmetry and electromagnetic gauge invariance.
These latter constraints reduce the familiar seven form factors
of the most general $WWV$ vertex to only these four for the
$Z\gamma V$ vertex. There still remains a global factor that can be fixed,
without loss of generality, to $g_{Z\gamma Z}=g_{Z\gamma\g}=e$. Combinations
of $h^V_3 (h^V_1)$ and $h^V_4 (h^V_2)$ correspond to electric
(magnetic) dipole and magnetic (electric) quadrupole transition
moments in the static limit.
All the terms are $C$-odd. The terms proportional to $h^V_1$ and $h^V_2$
are $CP$-odd while the other two are $CP$-even. All the form factors
are zero at tree level in the Standard Model. At the one-loop level,
only the $CP$-conserving $h^V_3$ and $h^V_4$ are nonzero \cite{barroso}
but too small (${\cal O}(\a/\pi$)) to lead to any observable
effect at any present or planned experiment. However, larger effects
might appear in theories or models beyond the Standard Model,
for instance when the gauge bosons are composite objects
\cite{composite}.
This is a purely phenomenological, model independent parametrization.
Tree-level unitarity restricts the $Z\gamma V$ to the Standard Model
values at asympotically high energies \cite{unitarity}. This
implies that the couplings $h^V_i$ have to be described by form factors
$h^V_i(q^2_1,q^2_2,p^2)$ which vanish when $q^2_1$, $q^2_2$ or $p^2$
become large. In hadron colliders, large values of $p^2=\hat{s}$
come into play and the energy dependence has to be taken into
account, including unknown dumping factors \cite{BB}.
A scale dependence appears as an additional parameter (the scale
of new physics, $\L$). Alternatively,
one could introduce a set of operators invariant under SU(2)$\times$U(1)
involving the gauge bosons and/or additional would-be-Goldstone bosons
and the physical Higgs. Depending on the new physics dynamics,
operators with dimension $d$ could be generated at the scale $\L$,
with a strength which is generally suppressed by factors like
$(M_W/\L)^{d-4}$ or $(\sqrt{s}/\L)^{d-4}$ \cite{NPscale}.
It can be shown that $h^V_1$ and $h^V_3$ receive contributions from
operators of dimension $\ge 6$ and $h^V_2$ and $h^V_4$ from
operators of dimension $\ge 8$.
Unlike hadron colliders, in $ep\to e\gamma X$ at HERA energies, we can ignore
the dependence of the form factors on the scale. On the other
hand, the anomalous couplings are tested in a different kinematical region,
which makes their study in this process complementary to the ones
performed at hadron and lepton colliders.
\section{The process $ep\to e\gamma X$}
The process under study is $ep\to e\gamma X$, which is described in the
parton model by the radiative neutral current electron-quark and
electron-antiquark scattering,
\begin{equation}
\label{process}
e^- \ \stackrel{(-)}{q} \to e^- \ \stackrel{(-)}{q} \ \gamma .
\end{equation}
There are eight Feynman diagrams contributing to this process in
the Standard Model and three additional ones if one includes anomalous vertices:
one extra diagram for the $Z\gamma Z$ vertex and two for the $Z\gamma\g$
vertex (Fig. \ref{feyndiag}).
\bfi{htb}
\begin{center}
\bigphotons
\bpi{35000}{21000}
\put(4000,8000){(a)}
\put(200,17000){\vector(1,0){1300}}
\put(1500,17000){\vector(1,0){3900}}
\put(5400,17000){\line(1,0){2600}}
\drawline\photon[\S\REG](2800,17000)[5]
\put(200,\pbacky){\vector(1,0){1300}}
\put(1500,\pbacky){\vector(1,0){2600}}
\put(4100,\pbacky){\vector(1,0){2600}}
\put(6700,\pbacky){\line(1,0){1300}}
\put(0,13000){$q$}
\put(8200,13000){$q$}
\put(3300,\pmidy){$\gamma,Z$}
\drawline\photon[\SE\FLIPPED](4900,\pbacky)[4]
\put(0,18000){$e$}
\put(8200,18000){$e$}
\put(8200,\pbacky){$\gamma$}
\put(13000,8000){(b)}
\put(9500,17000){\vector(1,0){1300}}
\put(10800,17000){\vector(1,0){2600}}
\put(13400,17000){\vector(1,0){2600}}
\put(16000,17000){\line(1,0){1300}}
\drawline\photon[\S\REG](12100,17000)[5]
\put(9500,\pbacky){\vector(1,0){1300}}
\put(10800,\pbacky){\vector(1,0){3900}}
\put(14700,\pbacky){\line(1,0){2600}}
\drawline\photon[\NE\FLIPPED](14200,17000)[4]
\put(22000,8000){(c)}
\put(18500,17000){\vector(1,0){3250}}
\put(21750,17000){\vector(1,0){3250}}
\put(25000,17000){\line(1,0){1300}}
\drawline\photon[\S\REG](23700,17000)[5]
\put(18500,\pbacky){\vector(1,0){1300}}
\put(19800,\pbacky){\vector(1,0){2600}}
\put(22400,\pbacky){\vector(1,0){2600}}
\put(25000,\pbacky){\line(1,0){1300}}
\drawline\photon[\SE\FLIPPED](21100,\pbacky)[4]
\put(31000,8000){(d)}
\put(27500,17000){\vector(1,0){1300}}
\put(28800,17000){\vector(1,0){2600}}
\put(31400,17000){\vector(1,0){2600}}
\put(34000,17000){\line(1,0){1300}}
\drawline\photon[\S\REG](32700,17000)[5]
\put(27500,\pbacky){\vector(1,0){3250}}
\put(30750,\pbacky){\vector(1,0){3250}}
\put(33900,\pbacky){\line(1,0){1300}}
\drawline\photon[\NE\FLIPPED](30100,17000)[4]
\put(17800,0){(e)}
\put(17100,5500){$\gamma,Z$}
\put(17100,3000){$\gamma,Z$}
\put(14000,7000){\vector(1,0){1300}}
\put(15300,7000){\vector(1,0){3900}}
\put(19200,7000){\line(1,0){2600}}
\drawline\photon[\S\REG](16600,7000)[5]
\put(16750,\pmidy){\circle*{500}}
\put(14000,\pbacky){\vector(1,0){1300}}
\put(15300,\pbacky){\vector(1,0){3900}}
\put(19200,\pbacky){\line(1,0){2600}}
\drawline\photon[\E\REG](16750,\pmidy)[5]
\put(22300,\pbacky){$\gamma$}
\end{picture}
\end{center}
\caption{\it Feynman diagrams for the process $e^- q \to e^- q \gamma$.
\label{feyndiag}}
\end{figure}
Diagrams with $\gamma$ exchanged in the t-channel are dominant. Nevertheless,
we consider the whole set of diagrams in the calculation.
On the other side, u-channel fermion exchange poles appear, in the limit
of massless quarks and electrons (diagrams (c) and (d)).
Since the anomalous diagrams (e) do not present such infrared or
collinear singularities, it seems appropriate to avoid almost
on-shell photons exchanged and fermion poles by cutting the
transverse momenta of the final fermions (electron and jet) to
enhance the signal from anomalous vertices.
Due to the suppression factor coming from $Z$ propagator, the
anomalous diagrams are more sensitive to $Z\gamma\g$ than to $Z\gamma Z$ vertices.
In the following we will focus our attention on the former.
The basic variables of the parton level process are five. A
suitable choice is: $E_\gamma$ (energy of the final photon),
$\cos\th_\gamma$, $\cos\th_{q'}$ (cosines of the polar angles of the
photon and the scattered quark defined with respect to the proton direction),
$\phi$ (the angle between the transverse momenta of the photon and the
scattered quark in a plane perpendicular to the beam), and a
trivial azimuthal angle that is integrated out (unpolarized beams).
All the variables are referred to the laboratory frame. One needs
an extra variable, the Bjorken-x, to connect the partonic process
with the $ep$ process. The phase space integration over these six
variables is carried out by {\tt VEGAS} \cite{VEGAS} and has been
cross-checked with the {\tt RAMBO} subroutine \cite{RAMBO}.
We adopt two kinds of event cuts to constrain conveniently
the phase space:
\begin{itemize}
\item
{\em Acceptance and isolation} cuts. The former are to exclude
phase space regions
which are not accessible to the detector, because of angular or
efficiency limitations:\footnote{The threshold for the transverse
momentum of the scattered quark ensures that its kinematics can be
described in terms of a jet.}
\begin{eqnarray}
\label{cut1}
8^o < \theta_e,\ \theta_\gamma,\ \theta_{\rm jet} < 172^o; \nonumber\\
E_e, \ E_\gamma, \ p^{\rm q'}_{\rm T} > 10 \ {\rm GeV}.
\end{eqnarray}
The latter keep the final photon well separated
from both the final electron and the jet:
\begin{eqnarray}
\label{cut2}
\cos \langle \gamma,e \rangle < 0.9; \nonumber\\
R > 1.5,
\end{eqnarray}
where $R\equiv\sqrt{\Delta\eta^2+\phi^2}$ is the separation between
the photon and the jet in the rapidity-azimuthal plane, and $\langle \gamma,e \rangle$ is the angle between the photon and the scattered electron.
\item
Cuts for {\em intrinsic background suppression}. They consist of
strengthening some of the
previous cuts or adding new ones to enhance the signal of the anomalous
diagrams against the Standard Model background.
\end{itemize}
We have developed a Monte Carlo program for the simulation of the
process $ep\to e\gamma X$ where $X$ is the remnant of the proton plus one jet
formed by the scattered quark of the subprocess (\ref{process}). It
includes the Standard Model helicitity amplitudes computed using the {\tt HELAS} subroutines \cite{HELAS}. We added new code to account for the
anomalous diagrams. The squares of these anomalous amplitudes have been
cross-checked with their analytical expressions computed using {\tt FORM}
\cite{FORM}. For the parton distribution functions,
we employ both the set 1 of Duke-Owens' parametrizations \cite{DO}
and the modified MRS(A) parametrizations \cite{MRS}, with the scale chosen to
be the hadronic momentum transfer.
As inputs, we use the beam energies $E_e=30$ GeV and $E_p=820$ GeV,
the $Z$ mass $M_Z=91.187$ GeV, the weak angle $\sin^2_W=0.2315$
\cite{PDB} and the fine structure constant $\a=1/128$. A more correct choice
would be the running fine structure constant with $Q^2$ as the argument.
However, as we are interested in large $Q^2$ events, the value $\a(M^2_Z)$
is accurate enough for our purposes. We consider only
the first and second generations of quarks, assumed to be massless.
We start by applying the cuts (\ref{cut1}) and (\ref{cut2})
and examining the contribution to a set of observables of the
Standard Model and the anomalous diagrams, separately. Next, we
select one observable such that, when a cut on it is performed,
only Standard Model events are mostly eliminated. The procedure
is repeated with this new cut built in. After several runs, adding
new cuts, the ratio standard/anomalous cross sections is reduced
and hence the sensitivity to anomalous couplings is improved.
\section{Results}
\subsection{Observables}
The total cross section of $ep\to e\gamma X$ can be written as
\begin{equation}
\sigma=\sigma_{{\rm SM}} + \sum_{i} \t_i \cdot h^\gamma_i + \sum_{i}\sigma_i\cdot (h^\gamma_i)^2
+ \sigma_{12} \cdot h^\gamma_1 h^\gamma_2 + \sigma_{34} \cdot h^\gamma_3 h^\gamma_4.
\end{equation}
\bfi{htb}
\setlength{\unitlength}{1cm}
\bpi{8}{7}
\epsfxsize=11cm
\put(-1,-4){\epsfbox{eng_acciso.ps}}
\end{picture}
\bpi{8}{7}
\epsfxsize=11cm
\put(0.,-4){\epsfbox{ptg_acciso.ps}}
\end{picture}
\bpi{8}{6}
\epsfxsize=11cm
\put(-1,-5){\epsfbox{angge_acciso.ps}}
\end{picture}
\bpi{8}{6}
\epsfxsize=11cm
\put(0.,-5){\epsfbox{anggj_acciso.ps}}
\end{picture}
\bpi{8}{6}
\epsfxsize=11cm
\put(-1,-5){\epsfbox{angej_acciso.ps}}
\end{picture}
\bpi{8}{7}
\epsfxsize=11cm
\put(0.,-5){\epsfbox{q2e_acciso.ps}}
\end{picture}
\caption{\it Differential cross sections (pb) for the process $ep\to e\gamma X$ at
HERA, with only acceptance and isolation cuts.
The solid line is the Standard Model contribution and the dash (dot-dash) line
correspond to 10000 times the $\sigma_1$ ($\sigma_2$) anomalous contributions.\label{A}}
\end{figure}
The forthcoming results are obtained using the MRS'95
pa\-ra\-me\-tri\-za\-tion of the parton densities\footnote{The values
change $\sim 10$\% when using the (old) Duke-Owens' structure functions.}
\cite{MRS}.
The linear terms of the $P$-violating couplings $h^\gamma_3$
and $h^\gamma_4$ are negligible, as they mostly arise from the interference of
standard model diagrams with photon exchange ($P$-even) and anomalous
$P$-odd diagrams ($\t_3\simeq \t_4\simeq 0$). Moreover, anomalous diagrams with
different $P$ do not interfere either. On the other hand, the quadratic terms
proportional to $(h^\gamma_1)^2$ and $(h^\gamma_3)^2$ have identical expressions, and
the same for $h^\gamma_2$ and $h^\gamma_4$ ($\sigma_1=\sigma_3$, $\sigma_2=\sigma_4$). Only the
linear terms make their bounds different. The interference terms $\sigma_{12}$
and $\sigma_{34}$ are also identical.
\bfi{htb}
\setlength{\unitlength}{1cm}
\bpi{8}{7}
\epsfxsize=11cm
\put(-1,-4){\epsfbox{eng_bkgsup.ps}}
\end{picture}
\bpi{8}{7}
\epsfxsize=11cm
\put(0.,-4){\epsfbox{ptg_bkgsup.ps}}
\end{picture}
\bpi{8}{6}
\epsfxsize=11cm
\put(-1,-5){\epsfbox{angge_bkgsup.ps}}
\end{picture}
\bpi{8}{6}
\epsfxsize=11cm
\put(0.,-5){\epsfbox{anggj_bkgsup.ps}}
\end{picture}
\bpi{8}{6}
\epsfxsize=11cm
\put(-1,-5){\epsfbox{angej_bkgsup.ps}}
\end{picture}
\bpi{8}{7}
\epsfxsize=11cm
\put(0.,-5){\epsfbox{q2e_bkgsup.ps}}
\end{picture}
\caption{\it Differential cross sections (pb) for the process $ep\to e\gamma X$ at
HERA, after intrinsic background suppression.
The solid line is the Standard Model contribution and the dash (dot-dash) line correspond to 500 times the $\sigma_1$ ($\sigma_2$) anomalous contributions.\label{B}}
\end{figure}
We have analyzed the distributions of more than twenty
observables in the laboratory frame, including the energies, transverse
momenta and angular distributions of the jet, the photon and the final
electron, as well as their spatial, polar and azimuthal separations.
Also the bjorken-x, the leptonic and hadronic momenta transfer and other fractional energies are considered.
The process of intrinsic background suppression is illustrated
by comparing Figures \ref{A} and \ref{B}. For simplicity, only
the most interesting variables are shown: the energy $E(\gamma)$ and transverse
momentum $p_T(\gamma)$ of the photon; the angles between the photon and
the scattered electron $\langle \gamma,e \rangle$, the photon and the jet
$\langle \gamma,j \rangle$, and the scattered electron and the jet $\langle e,j
\rangle$; and the leptonic momentum transfer $Q^2(e)$.
In Fig.~\ref{A}, these variables
are plotted with only acceptance and isolation cuts
implemented.
All of them share the property of disposing of a range
where any anomalous effect is negligible, whereas the contribution
to the total SM cross section is large. The set of cuts
listed below were added to reach eventually the distributions of
Fig.~\ref{B}:
\begin{itemize}
\item
The main contribution to the Standard Model cross section comes from
soft photons with very low transverse momentum. The following cuts
suppress a 97$\%$ of these events, without hardly affecting the
anomalous diagrams which, conversely, enfavour high energy photons:
\begin{eqnarray}
E_\gamma > 30 \ {\rm GeV} \nonumber \\
p^\gamma_T > 20 \ {\rm GeV}
\label{cut3}
\end{eqnarray}
\item
Another remarkable feature of anomalous diagrams is the very different
typical momentum transfers. Let's concentrate on the leptonic momentum
transfer, $Q^2_e=-(p'_e-p_e)^2$. The phase space enhances high
$Q^2_e$, while the photon propagator of the Standard Model diagrams
prefer low values (above the threshold for electron detectability,
$Q^2_e>5.8$~GeV$^2$, with our required minimum energy and angle). On the
contrary, the anomalous diagrams have always a $Z$ propagator
which introduces a suppression factor of the order of $Q^2_e/M^2_Z$ and
makes irrelevant the $Q^2_e$ dependence, which is only determined by
the phase space. As a consequence, the following cut looks appropriate,
\begin{equation}
Q^2_e > 1000 \ {\rm GeV}^2
\label{cut4}
\end{equation}
\end{itemize}
It is important to notice at this point why usual form factors for the
anomalous couplings can be neglected at HERA. For our process, these
form factors should be proportional to $1/(1+Q^2/\L^2)^n$. With the scale of
new physics $\L=500$~GeV to 1~TeV, these factors can be taken to be one. This
is not the case in lepton or hadron high energy colliders where the diboson production in the s-channel needs dumping factors $1/(1+\hat{s}/\L^2)^n$.
The total cross section for the Standard Model with acceptance and isolation
cuts is $\sigma_{\rm SM}=21.38$~pb and is reduced to 0.37~pb when all the cuts are applied, while the quadratic contributions only change from
$\sigma_1=2\times10^{-3}$~pb, $\sigma_2=1.12\times10^{-3}$~pb to
$\sigma_1=1.58\times10^{-3}$~pb, $\sigma_2=1.05\times10^{-3}$~pb. The linear
terms are of importance and change from $\t_1=1.18\times10^{-2}$~pb, $\t_2=1.27\times10^{-3}$~pb to $\t_1=7.13\times10^{-3}$~pb, $\t_2=1.26\times10^{-3}$~pb. Finally, the interference term $\sigma_{12}=1.87\times10^{-3}$~pb changes to $\sigma_{12}=1.71\times10^{-3}$~pb.
The typical Standard Model events consist of soft and low-$p_T$ photons
mostly backwards, tending to go in the same direction of the scattered
electrons (part of them are emitted by the hadronic
current in the forward direction), close to the required angular separation ($\sim 30^o$). The low-$p_T$ jet goes opposite to both the photon and the scattered electron, also in the transverse plane.
On the contrary, the anomalous events have not so soft and high-$p_T$ photons,
concentrated in the forward region as it the case for the scattered electron
and the jet.
\subsection{Sensitivity to anomalous couplings}
In order to estimate the sensitivity to anomalous couplings, we
consider the $\chi^2$ function.
One can define the $\chi^2$, which is related to the likelihood
function ${\cal L}$, as
\begin{equation}
\label{chi2}
\chi^2\equiv-2\ln{\cal L}=
2 L \displaystyle\left(\sigma^{th}-\sigma^{o}+\sigma^{o}
\ln\displaystyle\frac{\sigma^{o}}{\sigma^{th}}\right)
\simeq L \displaystyle\dis\frac{(\sigma^{th}-\sigma^{o})^2}{\sigma^{o}},
\end{equation}
where $L=N^{th}/\sigma^{th}=N^o/\sigma^o$ is the integrated luminosity
and $N^{th}$ ($N^o$) is the number of theoretical (observed)
events. The last line of (\ref{chi2}) is a useful
and familiar approximation, only valid when $\mid \sigma^{th}-\sigma^o \mid/
\sigma^o \ll 1$.
This function is a measure of the probability that statistical
fluctuations can make undistinguisable the observed and the predicted
number of events, that is the Standard Model prediction. The well
known $\chi^2$-CL curve allows us to determine the corresponding
confidence level (CL).
We establish bounds on the anomalous couplings by fixing a
certain $\chi^2=\d^2$ and allowing for the $h^\gamma_i$
values to vary, $N^o=N^o(h^\gamma_i)$. The parameter $\d$ is often referred
as the number of
standard deviations or `sigmas'. A $95\%$ CL corresponds to almost
two sigmas ($\d=1.96$).
When $\sigma \simeq \sigma_{{\rm SM}} + (h^\gamma_i)^2 \sigma_i$ (case of the $CP$-odd
terms) and the anomalous contribution is small enough, the
upper limits present some useful, approximate scaling properties,
with the luminosity,
\begin{equation}
h^\gamma_i (L')\simeq\sqrt[4]{\frac{L}{L'}} \ h^\gamma_i (L).
\end{equation}
A brief comment on the interpretation of the results is in order.
As the cross section grows with $h^\gamma_i$, in the relevant range of
values, the $N^o$ upper limits can be regarded as the lowest number
of measured events that would discard the Standard Model, or the
largest values of $h^\gamma_i$ that could be bounded if no effect is
observed, with the given CL. This procedure approaches the
method of upper limits for Poisson processes when the
number of events is large ($\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$} 10$).
\bfi{htb}
\setlength{\unitlength}{1cm}
\bpi{8}{8}
\epsfxsize=12cm
\put(3.35,4.245){+}
\put(-2.5,-1.5){\epsfbox{conh1h2.nogrid.ps}}
\end{picture}
\bpi{8}{8}
\epsfxsize=12cm
\put(4.1,4.245){+}
\put(-1.75,-1.5){\epsfbox{conh3h4.nogrid.ps}}
\end{picture}
\caption{\it Limit contours for $Z\gamma\g$ couplings at HERA with an integrated luminosity of 10, 100, 250, 1000 pb$^{-1}$ and a 95\% CL.\label{contour}}
\end{figure}
In Fig. \ref{contour} the sensitivities for different luminosities are shown.
Unfortunately, HERA cannot compete with Tevatron, whose best
bounds, reported by the D\O\ collaboration \cite{D0}, are
\begin{eqnarray}
|h^\gamma_1|, \ |h^\gamma_3| &<& 1.9 \ (3.1), \nonumber
\\
|h^\gamma_2|, \ |h^\gamma_4| &<& 0.5 \ (0.8).
\end{eqnarray}
For the first value it was assumed that only one anomalous coupling contributes
(`axial limits') and for the second there are two couplings contributing (`correlated limits'). Our results are summarized in Table \ref{table}.
\begin{table}
\begin{center}
\begin{tabular}{|c|r|r|r|r|r|r|r|r|}
\hline
HERA & \multicolumn{2}{c|}{10 pb$^{-1}$} & \multicolumn{2}{c|}{100 pb$^{-1}$}
& \multicolumn{2}{c|}{250 pb$^{-1}$} & \multicolumn{2}{c|}{1 fb$^{-1}$} \\
\hline \hline
$h^\gamma_1$ & -19.0 & 14.5 & -11.5 & 7.0 & -9.5 & 5.5 & -8.0 & 3.5 \\
& -26.0 & 19.5 & -16.0 & 9.5 & -14.0 & 7.0 & -11.5 & 4.5 \\
\hline
$h^\gamma_2$ & -21.5 & 20.0 & -12.0 & 10.0 & - 9.5 & 8.0 & -7.0 & 6.0 \\
& -26.0 & 30.0 & -13.0 & 18.0 & -10.0 & 15.0 & - 7.5 & 12.0 \\
\hline
$h^\gamma_3$ & -17.0 & 17.0 & -9.0 & 9.0 & -7.5 & 7.5 & -5.5 & 5.5 \\
& -22.5 & 22.5 & -12.0 & 12.0 & -10.0 & 10.0 & -7.0 & 7.0 \\
\hline
$h^\gamma_4$ & -20.5 & 20.5 & -11.0 & 11.0 & -8.5 & 8.5 & -6.0 & 6.0 \\
& -27.5 & 27.5 & -14.5 & 14.5 & -12.0 & 12.0 & -8.5 & 8.5 \\
\hline
\end{tabular}
\end{center}
\caption{\it Axial and correlated limits for the $Z\gamma\g$ anomalous couplings
at HERA with different integrated luminosities and $95\%$ CL. \label{table}}
\end{table}
The origin of so poor results is the fact that, unlike diboson production
at hadron or $e^+e^-$ colliders, the anomalous diagrams of $ep\to e\gamma X$
have a $Z$ propagator decreasing their effect.
The process $ep\to eZX$ avoids this problem thanks to the absence
of these propagators: the Standard Model cross section is similar
to the anomalous one but, as a drawback, they are of the order of
femtobarns.
\section{Summary and conclusions}
The radiative neutral current process $ep\to e\gamma X$
at HERA has been studied. Realistic cuts have been applied in order to
observe a clean signal consisting of detectable and well separated
electron, photon and jet.
The possibility of testing the trilinear neutral gauge boson couplings
in this process has also been explored. The $Z\gamma Z$ couplings are
very suppressed by two $Z$ propagators. Only the $Z\gamma \gamma$ couplings
have been considered. A Monte Carlo program has been developed to
account for such anomalous vertex and further cuts have been implemented
to improve the sensitivity to this source of new physics.
Our estimates are based on total cross sections since the expected number
of events is so small that a distribution analysis is not possible.
The distributions just helped us to find the optimum cuts. Unfortunately,
competitive bounds on these anomalous couplings cannot be achieved at
HERA, even with the future luminosity upgrades.\footnote{We would like to
apologize for the optimistic but incorrect results that were presented
at the workshop due to a regrettable and unlucky mistake in our programs.}
As a counterpart, a different kinematical region is explored, in which
the form factors can be neglected.
\section*{Acknowledgements}
One of us (J.I.) would like to thank the Workshop organizers for financial
support and very especially the electroweak working group conveners
and the Group from Madrid at ZEUS for hospitality and useful conversations.
This work has been partially supported by the CICYT and the European Commission
under contract CHRX-CT-92-0004.
| proofpile-arXiv_065-424 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section*{References}
| proofpile-arXiv_065-425 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{INTRODUCTION}
\vskip 2mm
Processes of heavy quark and lepton pair
production on nucleon and nuclear targets at
high energies are very interesting from both theoretical and practical
reasons. These processes provide a method to study the internal
structure of hadrons in the very small $x$-region. Some limits on the possible
non-perturbative contributions to this region
can also be obtained. Realistic estimates of
the cross sections of heavy quark production are necessary in order to
plan experiments on the existing and future accelerators.
Predictions are usually obtained in the framework of perturbative QCD in the
leading and the next-to-leading order $\alpha_s$-expansion. In the case of a
nuclear target it is usually assumed that the cross section of heavy flavour
production (or any other hard process), $\sigma(Q\overline{Q})$, should be
proportional to $A^{\alpha}$, with $\alpha = 1$; that is, in agreement with the
most accurate experimental results of [1,2], where values of
$\alpha = 1.00 \pm 0.05$ for all $D$-meson production by pions at
$\sqrt{s} = 22$ GeV and $\alpha = 1.02 \pm 0.03 \pm 0.02$ for neutral
$D$-meson production by protons at $\sqrt{s} = 39$ GeV were obtained,
respectively.
However it seems important to consider the problem in more detail
because, for example, many experimental results are obtained by
extrapolation of results on a nuclear target to a nucleon target. It is
experimentally well-known [3-9] that parton distributions in nuclei (i.e., in
bound nucleons) are slightly different from the same distributions in free
nucleons. So the value of $\alpha$ can differ from unity and it is
interesting to estimate the size of this effect. Also the differences in the
distributions of gluons, sea quarks and valence quarks are not the same
\cite{Kari}. Moreover, in different hard processes different QCD
subprocesses give the main contributions, so the effects on the
$\alpha$-values can also be different.
In this paper we calculate the values of $\alpha$ as a function of
energy and Feynman-$x$ for the cases of open charm, beauty and
different mass Drell-Yan pair production in proton-nucleus and
nucleus-nucleus collisions.
In our previous paper \cite{APSS} we showed that the results for the
A-dependence of heavy flavour production depend very weakly
on the used set of parton distributions. Three different sets,
namely MRS-1 \cite{mrs}, MT S-DIS \cite{mt} and GRV HO \cite{grv}, which can
be found in CERN PDFLIB \cite{pdf}, gave practically the same results for the
$\alpha$ behaviour (see Table 1 of Ref. \cite{APSS}). Here we present
the results for two sets, GRV HO \cite{grv} and MRS-A \cite{mrsa} which are
in comparatively good agreement with the new HERA data and we see that
they give practically identical results for both open heavy flavour
and Drell-Yan pair production in proton-nucleus and nucleus-nucleus
collisions.
\vskip 9 mm
\section{CROSS SECTIONS OF HARD PROCESSES IN QCD}
\vskip 2mm
The standard QCD expression for the cross section of heavy quark production
in a hadron 1 - hadron 2 collision has the form
\begin{equation}
\sigma^{12\rightarrow Q\overline{Q}} = \int_{x_{a0}}^{1} \frac{dx_a}{x_a}
\int_{x_{b0}}^{1} \frac{dx_b}{x_b}\left [x_aG_{a/1}(x_a,\mu^{2})
\right ]\left [x_bG_{b/2}(x_b,\mu^{2})\right ]
\hat{\sigma}^{ab\rightarrow Q\overline{Q}}(\hat{s},m_Q,\mu^{2}) \;,
\label{eq:totalpf}
\end{equation}
where $x_{a0} = \textstyle 4m_Q^2/ \textstyle s$ and $x_{b0} = \textstyle
4m_Q^2/ \textstyle (sx_a)$. Here $G_{a/1}(x_a,\mu^{2})$ and
$G_{b/2}(x_b,\mu^{2})$ are the structure functions of partons $a$ and $b$
inside hadrons $1$ and $2$ respectively, and $\hat{\sigma}^{ab\rightarrow
Q\overline{Q}}(\hat{s},m_Q,\mu^{2})$ is the cross section of the subprocess
$ab\rightarrow Q\overline{Q}$ as given by standard QCD. The latter depends
on the parton center-of-mass energy \mbox{$\hat{s} = (p_a+p_b)^2 =
x_ax_bs$}, the mass of the produced heavy quark $m_Q$
and the QCD scale $\mu^{2}$. Eq. (\ref{eq:totalpf}) should
account for all possible subprocesses $ab \rightarrow Q\overline{Q}$. The
parton-parton cross section $\hat{\sigma}^{ab\rightarrow
Q\overline{Q}}$ can be written in the form \cite{NDE}
\begin{equation}
\hat{\sigma}^{ab\rightarrow Q\overline{Q}}(\hat{s},m_Q,\mu^{2}) =
\frac{\alpha^{2}_{s}(\mu^{2})}{m^{2}_{Q}}f_{ab}(\rho ,\mu^{2},m^{2}_{Q}) \;,
\end{equation}
with
\begin{equation}
\rho = 4m^{2}_{Q}/\hat{s}
\end{equation}
and
\begin{equation}
f_{ab}(\rho, \mu^2, m^2_Q) = f^{(0)}_{ab}(\rho) + 4\pi
\alpha_{s}(\mu^{2})[f^{(1)}_{ab}(\rho) +
\hat{f}^{(1)}_{ab}(\rho)\ln{(\mu^{2}/m^{2}_{Q})}]\;.
\end{equation}
The functions $f^{(0)}_{ab}(\rho)$, $f^{(1)}_{ab}(\rho)$ and
$\hat{f}^{(1)}_{ab}(\rho)$ can be found in \cite{NDE} for heavy flavour
production.
Formulae with the same structure as
(1)-(4) can be found in \cite{DY} for Drell-Yan pair
production. In this case $M_{l^+l^-}$ plays the role of $2m_Q$ in
$x_{a0}$, $x_{b0}$ and $\hat{\sigma}^{ab\rightarrow Q\overline{Q}}$.
For the numerical calculations we wrote the nuclear structure function
$G_{b/A}(x_b,\mu^{2})$ in the form
\begin{equation}
G_{b/A}(x_b,\mu^{2}) = A\cdot G_{b/N}(x_b,\mu^{2})\cdot R^A_b(x,\mu^{2}),
\end{equation}
similarly to Ref. \cite{ina} and we take the values of $R^A_b(x,\mu^{2})$
for gluons, valence and sea quarks from Ref. [10]. They are presented in
Fig. 1. The values of $R^A_b(x,\mu^{2})$ in [10] are given for
$x > 10^{-3}$ which is not small enough at high energies. So at $x < 10^{-3}$
we used two variants of the $R^A_b(x,\mu^{2})$ behaviour for gluon and sea
quark distributions: The first is the constant frozen at $x = 10^{-3}$
(dashed-dotted curves for gluons and solid curves for sea quarks in Fig. 1).
The second is the extrapolation as $x^{\beta}$ (dotted curves in Fig. 1)
of the distribution
which gives the main contribution to our cross section, with
$\beta =$ 0.096 and 0.040 for charm and beauty production respectively (only
for gluons), and $\beta$ = 0.109, 0.096 and 0.072 for Drell-Yan pair
invariant masses $M^2_{l^+l^-}=$ 5, 25 and 100 GeV$^2$ respectively (only for
sea quarks). Such behaviour is in qualitative agreement with the results of
\cite{lev}.
In the case of nucleus-nucleus collisions the parton distributions in both
incident nuclei should be written in the form (5). It is necessary to
note that in such a way the main part of the shadowing processes is taken
into account.
\vskip 9 mm
\section{A-DEPENDENCE OF HEAVY FLAVOUR PRODUCTION}
\vskip 2 mm
In the case of charm production we have used the values $m_c$ = 1.5 GeV and
$\mu^2$ = 4 GeV$^2$ ($\mu^2$ = 5 GeV$^2$ for the MRS-A set) and in the case of
beauty production $m_b$ = 5 GeV and $\mu^2$ = $m_b^2$.
The obtained results for $\alpha$ determined from the ratios of heavy quark
production cross section on a gold target and on the proton are presented in
Fig. 2 for the GRV HO and MRS-A sets\footnote{In
all the calculations in this paper
little difference, if any, has been found in the results for $\alpha$ between
these two sets of parton distributions.}
and the two variants of gluon distribution
ratios in the small $x$-region. Here $\sqrt{s_{NN}}$ is the c.m. energy for
the interaction of the incident proton with one target nucleon. One can see
that at fixed target energies the values of $\alpha$ are slightly higher
than unity, which is not in contradiction with the results of Refs. [1,2].
However $\alpha$ decreases with increasing energy and this effect is larger
in the case of charm production than in the case of beauty. One can see
also that the difference between the two variants for $R^A_b(x,\mu^{2})$ at
$x < 10^{-3}$ becomes important only at the highest energies.
We also calculate the values of $\alpha$ for different Feynman-$x$ ($x_F$)
regions using $x_F = x_a - x_b$ at energies $\sqrt{s_{NN}}$ = 39 GeV and
1800 GeV. The results are presented in Fig. 3. At negative and moderate $x_F$
(in the nucleus fragmentation region) the values of $\alpha$ are slightly
higher than unity. However in the case of charm production in the beam
fragmentation region (positive $x_F$) the values of $\alpha$ become
essentially smaller than unity. For beauty production the last effect is
expected only at very high energies.
The obtained results for A-dependences of charm and beauty production in the
symmetric case of gold-gold collisions are presented in Figs. 4 ($\alpha$
as a function of initial energy) and 5 ($\alpha$ as a function of $x_F$ at
two energies). Here $\sqrt{s_{NN}}$ is the c.m. energy for one
nucleon-nucleon interaction. One can see a qualitative agreement with the
case of proton-nucleus collisions (with the difference that the trivial value
of $\alpha$ is here equal to two). Again we can see that the values of
$\alpha$ are dependent on the initial energy and $x_F$ if the energy is high
enough.
Calculations for charm production in perturbative QCD (including
preequilibrium charm
production from secondary minijet gluons)
can
be found in \cite{wang} for central gold on gold collisions.
\vskip 9 mm
\section{ A-DEPENDENCE OF DRELL-YAN PAIR PRODUCTION}
\vskip 2mm
As said above, the QCD
expression for heavy lepton pair production in a
hadron 1 - hadron 2 collision has the same form as Eq. (1) but with another
matrix element which can be found in Ref. \cite{DY}. Now we have an
additional variable -- the
mass of the produced pair $M_{l^+l^-}$ which plays more
or less the same role as the mass of the heavy quark. However now its value can
be measured experimentally and do not lead to any uncertainty.
We have used the values of QCD scale $\mu^2 = M^2_{l^+l^-}$.
The obtained results for $\alpha$ determined in the same way as in heavy quark
production, i.e., from the ratios of Drell-Yan
pair production cross section in proton-gold and proton-proton
collisions, are presented in Fig. 6 for the GRV HO and MRS-A sets and the two
variants of sea quark distribution ratios in the small
$x$-region\footnote{As is well-known
gluons
are dominant for heavy flavour production, while sea quarks dominate
Drell-Yan production.}. Here again
$\sqrt{s_{NN}}$ is the c.m. energy for the interaction of the incident
proton with one target nucleon. Contrary to the case of heavy flavour
production the values of $\alpha$ are never higher than unity. They
decrease more or less monotonically with increasing energy and this effect
becomes smaller with increasing $l^+l^-$ mass. Again the two sets of parton
distributions give practically identical results for the energy dependence of
$\alpha$ and the difference between the two variants for $R^A_b(x,\mu^{2})$ at
$x < 10^{-3}$ becomes important only at the highest energies. The predicted
$M_{l^+l^-}$ dependence of $\alpha$ for Drell-Yan pair production in
proton-nucleus collisions at different energies is also shown in Fig. 6.
The results of our calculations of $\alpha$ as a function of $x_F$ for
heavy lepton pair production are presented in Fig. 7. The qualitative
picture is similar to the case of heavy flavour production; the
numerical difference of our predictions from the value $\alpha = 1$ is here
more significant for the case of a not very large mass of the lepton pair.
Predictions of A-dependence for Drell-Yan pair production cross section
in symmetric nucleus-nucleus collisions are shown in Figs. 8 and 9. As for
the case of proton-nucleus collisions, all the effects are
qualitatively the same but numerically larger than in the case of heavy
flavour production. Our results are in agreement with the
calculations of \cite{GGRV}.
\vskip 9 mm
\section{CONCLUSIONS}
\vskip 2 mm
In this paper
we calculate the A-dependence of charm and beauty as well as Drell-Yan pair
production using standard QCD formulas and accounting for the difference of
parton distributions in free and bound nucleons. If one parametrize these
cross sections as $\sigma \sim A^{\alpha}$, the value of $\alpha$ is slightly
different from unity at the available energies. For the case of heavy
flavour production at comparatively low energies the obtained values of
$\alpha$ are a little higher than unity. This should be connected with some
nucleon-nucleon correlations which change the large-$x$ parton
distributions. In the case of Drell-Yan pair production such effect is
absent because of the different relations in the contributions of valence
quarks, sea
quarks and gluons.
At higher energies the values of $\alpha$ decrease and become smaller than
unity. At $\sqrt{s_{NN}}$ = 1800 GeV we expect a value of $\alpha \sim 0.95$
for charm and low-mass Drell-Yan pair production.
The decrease of the ratio $R^A_b(x,\mu^{2})$, which results in a decrease of
$\alpha$, can be connected with the effects of parton density saturation
\cite{glr} which in heavy nuclei occur at $x$-values higher than in the
proton.
If we consider two small and different values of $x_a$ and $x_b$ in Eq. (1)
for the case of proton-nucleus interaction, it is clear that the contribution
to the inclusive cross section from the region $x_a < x_b$ should be larger
than the mirror contribution ($x_a > x_b$) because the value of the ratio
$R^A_b(x,\mu^{2})$ in the first case is larger. It means that heavy quark and
Drell-Yan pairs will be produced preferably in the nucleus fragmentation
hemisphere, i.e., asymmetrically, which is quite usual and has been confirmed
experimentally in the case of light quark production. From Figs. 3 and 7 it is
clear that charm and low-mass Drell-Yan pair production on nuclear targets at
LHC energies will give important information on the nuclear shadowing of the
structure functions at small $x$.
Besides,
the experimental measurements of the effects predicted for different hard
interactions should allow (in principle) to control the validity of the
conventional extraction of parton distributions from the experimental DIS
data.
Let us repeat again that almost all nuclear shadowing corrections are accounted
for in our calculations because they contribute to the $R^A_b(x,\mu^{2})$
ratios, which have been extracted from the experimental data. On the other hand
these shadowing corrections are not numerically large in the considered hard
processes. So we can assume that the corrections which are not accounted for
do not change significantly the obtained results.
In conclusion we express our gratitude to M.A.Braun for useful discussions and
both to K.J.Eskola and to I.Sarcevic for sending us their numerical results.
We thank the Direcci\'on General de Pol\'{\i}tica Cient\'{\i}fica and the
CICYT of Spain for financial support under contract AEN96-1673. C.A.S. also
thanks the Xunta de Galicia for financial support. The paper was supported in
part by INTAS grant 93-0079.
\newpage
\noindent{\Large \bf Figure captions}
\vspace{0.5cm}
\noindent{\bf Fig. 1.}
Functions $R^A_G(x,\mu^{2})$, $R^A_V(x,\mu^{2})$ and
$R^A_S(x,\mu^{2})$, which determine the ratios of the distributions for
protons in the nucleus versus free protons, for gluons (dashed-dotted and
dotted
curves, see text), valence quarks (dashed curves) and sea quarks (solid and
dotted
curves, see text)
respectively, for $\mu^2 =$ 5 GeV$^2$ (upper figure) and $\mu^2 = m_b^2$ (lower
figure).
\noindent{\bf Fig. 2.}
Energy dependence of $\alpha$ for charm and beauty
production in proton-gold collisions for GRV HO (solid and dashed curves) and
MRS-A (dotted and dashed-dotted curves) structure
functions and using extrapolated (dashed and dashed-dotted curves) and frozen at
$x= 10^{-3}$ (solid and dotted curves) ratios of gluon distributions.
\noindent{\bf Fig. 3.}
Feynman-$x$ dependence of $\alpha$ for charm
and beauty production in proton-gold collisions at $\sqrt{s_{NN}}$ = 39 GeV
(upper figure) and 1800 GeV (lower figure)
for GRV HO and MSR-A structure functions and using
extrapolated and frozen at $x = 10^{-3}$
ratios of gluon distributions
(with the same conventions as in Fig. 2).
\noindent{\bf Fig. 4.}
Energy dependence of $\alpha$ for charm and beauty
production in gold-gold collisions for GRV HO and MRS-A structure functions
and using extrapolated and frozen at $x = 10^{-3}$
ratios of gluon distributions
(with the same conventions as in Fig. 2).
\noindent{\bf Fig. 5.}
Feynman-$x$ dependence of $\alpha$ values for charm
and beauty production in gold-gold collisions at $\sqrt{s_{NN}}$ = 39 GeV
(upper figure) and 1800 GeV (lower figure)
for GRV HO and MSR-A structure functions and using
extrapolated and frozen at $x = 10^{-3}$
ratios of gluon distributions
(with the same conventions as in Fig. 2).
\noindent{\bf Fig. 6.}
Mass (upper figure)
and energy (lower figure)
dependence of $\alpha$ for
Drell-Yan pair production in proton-gold collisions for GRV HO and MRS-A
structure functions and using extrapolated and
frozen at $x = 10^{-3}$ ratios of sea quark distributions
(with the same conventions as in Fig. 2).
\noindent{\bf Fig. 7.}
Feynman-$x$ dependence of $\alpha$ values for heavy
lepton pair production in proton-gold collisions at $\sqrt{s_{NN}}$ = 39 GeV
(upper curves in each figure) and 1800 GeV (lower curves in each figure)
and different masses
for GRV HO and MSR-A structure functions and using
extrapolated and frozen at $x = 10^{-3}$
ratios of sea quark distributions
(with the same conventions as in Fig. 2). Note that at $\sqrt{s_{NN}}$ = 39 GeV
all curves coincide.
\noindent{\bf Fig. 8.}
Mass (upper figure)
and energy (lower figure)
dependence of $\alpha$ for Drell-Yan pair
production in gold-gold collisions for GRV HO and MRS-A structure functions
and using extrapolated and frozen at $x = 10^{-3}$
ratios of sea quark distributions
(with the same conventions as in Fig. 2).
\noindent{\bf Fig. 9.}
Feynman-$x$ dependence of $\alpha$ values for heavy
lepton pair production in gold-gold collisions at $\sqrt{s_{NN}}$ = 39 GeV
(upper curves in each figure) and 1800 GeV (lower curves in each figure)
and different masses
for GRV HO and MSR-A structure functions and using
extrapolated and frozen at $x = 10^{-3}$
ratios of sea quark distributions
(with the same conventions as in Fig. 2). Note that at $\sqrt{s_{NN}}$ = 39 GeV
all curves coincide.
\newpage
| proofpile-arXiv_065-426 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
In the last decades there has been an intensive activity in studying
(super)particles and (super) strings by use of
different approaches aimed at finding a formulation, which would be the most
appropriate for performing the covariant quantization of the models. Almost
all of the approaches use twistor variables in one form or another
\cite{penrose} -- \cite{bpstv}.
This allowed one to better understand the geometrical and group--theoretical
structure of the theory and to carry out a covariant
Hamiltonian analysis (and in some cases even the covariant quantization)
of (super)particle and (super)string dynamics in space--time
dimensions $D=3,4,6$ and 10, where conventional twistor relations take place.
It has been shown that twistor--like variables appear in a
natural way as superpartners of Grassmann spinor coordinates in a doubly
supersymmetric formulation \cite{spinsup} of Casalbuoni--Brink--Schwarz
superparticles and Green--Schwarz superstrings \cite{gsw}, the notorious
fermionic $\kappa$--symmetry \cite{ks}
of these models being replaced by
more fundamental local supersymmetry on the worldsheet supersurface swept
by the superparticles and superstrings in target superspace \cite{stvz}.
This has solved the problem of infinite reducibility of the fermionic
constraints associated with $\kappa$--symmetry \footnote{A
comprehensive
list of references on the subject the reader may find in \cite{bpstv}}.
As a result new formulation and methods of quantization of $D=4$
compactifications of
superstrings with manifest target--space supersymmetry have been
developed (see \cite{ber} for a review).
However, the complete
and simple solution of the problem of $SO(1,D-1)$ covariant quantization
of twistor--like superparticles and superstrings in $D~>~4$ is still lacking.
To advance in solving this problem one has to learn more on how to deal
with twistor--like variables
when performing the Hamiltonian analysis and the quantization of the
models. In this respect a bosonic relativistic particle in a twistor--like
formulation may serve as the simplest but rather nontrivial toy model.
The covariant quantization of the bosonic particle has been under intensive
study with both the operator and path--integral method
\cite{ferber,teit,polyakov,govaerts,sf,sg,bh,bfortschr}.
In the twistor--like approach the bosonic particle has been mainly quantized
by use of the operator formalism. For that different but classically
equivalent twistor--like particle actions have been considered
\cite{ferber,es,sg,bh,bfortschr}.
The aim of the present paper is to study some features of bosonic particle
path--integral quantization in the twistor--like approach
by use of the BRST--BFV quantization prescription \cite{bf} --
\cite{bff}.
In the course of the Hamiltonian analysis we shall observe links between
various
formulations of the twistor--like particle \cite{ferber,es,stvz}
by performing a
conversion of the Hamiltonian constraints of one formulation to another.
A particular feature of the conversion procedure \cite{fs}
applied to turn the
second--class constraints into the first--class constraints is that the
simplest Lorentz--covariant way to do this is to convert a
full mixed set of the initial first-- and second--class constraints rather
than explicitly extracting and converting only the second--class constraints.
Another novel feature of the conversion procedure applied below
(in comparison with the conventional
one \cite{bff,fs}) is that in the case of the $D=4$ and $D=6$ twistor--like
particle the number of new auxiliary Lorentz--covariant coordinates,
which one introduces to get a system of first--class constraints in an
extended phase space, exceeds the number
of independent second--class constraints of the original dynamical system,
(but because of an appropriate amount of the first--class constraints
we finally get, the number of physical degrees of freedom remains the
same).
In Section 2 the classical mechanics of a twistor--like bosonic
particle in D=3,4 and 6 is considered. The Hamiltonian analysis of the
constraints accompanied by the conversion procedure is carried out and a
classical BRST charge is constructed by introducing ghosts corresponding
to a set of the first--class constraints obtained as a result of
conversion.
In Section 3 the problem of admissible gauge choice for variables
describing the matter--ghost system of the model is discussed.
In Section 4 we perform the path--integral quantization of the model
in $D=3,4$ and $6$ space--time dimensions using
the extended BRST scheme \cite{mhenn}. We calculate the particle propagator
and show, that it coincides with that of the massless bosonic particle.
At the end of this Section we make a comment on problems of the D=10
case.
{\it Notation.} We use the following signature for the space-time metrics:
$(+,-,...,-)$.
\section{Classical Hamiltonian dynamics and the BRST-charge.}
\subsection{Preliminaries}
The dynamics of a massless bosonic particle in D=3,4,6 and 10 space--time
can be described by the action \cite{ferber}
\begin{equation} \label{201}
S={1\over 2}\int d\tau \dot x^m ({\bar \l}\gamma _m \l ), \qquad \end{equation}
where $x^m(\tau )$ is a particle space--time coordinate,
$\l ^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}(\tau )$ is an
auxiliary bosonic spinor variable, the dot stands for the time derivative
$\partial\over{\partial\tau}$ and $\gamma ^m $ are the Dirac matrices.
The derivation of the canonical momenta \footnote{In what
follows $P^{(..)}$ denotes the momentum conjugate to the
variable in the brackets}
$P^{(x)}_m={{\partial L}
\over{\partial{\dot x}^m}},~~
P^{(\l )}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}={{\partial L}\over
{\partial}{{\dot \l}^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}}$
results in a set of primary constraints $$
\Psi _m=P^{(x)}_m-{1\over 2}({\bar \l}\gamma _m \l )\approx 0, $$
\begin{equation} \label{202}
P^{(\l )}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}\approx 0.
\qquad \end{equation}
They form the following
algebra with respect to the Poisson brackets\footnote{The canonical
Poisson brackets are
$$
[P^{(x)}_m, x^n]_P=\delta ^n_m;
\qquad
[P^{(\l )}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}, \l^{\b}]_P=\delta ^{\b}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta} $$}
\begin{equation} \label{203}
[\Psi _m, \Psi _n]_P=0,~~ [P^{(\l )}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta},
P^{(\l )}_{\b}]_P=0,~~ [\Psi _m, P^{(\l )}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}]_P=({\gamma _m}\l )_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}.
\qquad \end{equation}
One can check that new independent secondary constraints do not appear
in the model.
In general,
Eqs. \p{202} are a mixture of first-- and second--class constraints.
The operator quantization of this
dynamical system in $D=4$ (considered previously in \cite{sg,bh}) was
based on the Lorentz--covariant
splitting of the first-- and second--class constraints and on the subsequent
reduction of the phase space (either by explicit solution of the
second--class constraints \cite{bh} or,
implicitly, by use of the Dirac brackets
\cite{sg}), while
in \cite{bz0,bfortschr} a conversion prescription \cite{bff,fs} was used.
The latter consists in the extension of the phase space of the particle
coordinates and momenta with auxiliary variables in such a way, that new
first--class constraints replace the original second--class ones. Then the
initial
system with the second--class constraints is treated as a gauge fixing of a
``virtual" \cite{bff} gauge symmetry generated by the additional first--class
constraints of the extended system \cite{bff,fs}.
This is achieved by taking the auxiliary conversion degrees of freedom to be
zero or expressed in terms of initial variables of the model.
The direct application of this procedure can encounter some technical
problems for systems, where the first-- and second--class constraints form a
complicated algebra
(see, for example, constraints of the $D=10$ superstring in a
Lorentz--harmonic formulation \cite{bzstr}).
Moreover, in order to perform the covariant
separation of the first-- and second--class constraints in the system
under consideration it is necessary either to introduce one more independent
auxiliary
bosonic spinor $\mu _{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ (the second component of a twistor
$Z^A=(\l^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta},\mu _{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta})$ \cite{penrose}) or to construct the second twistor
component from the variables at hand by use of a Penrose relation
\cite{penrose} ${\bar \mu}^{\dot \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}=ix^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta {\dot \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}\l _{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta} ~(D=4),~
\mu^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}=x^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}\l _{\b} ~(D=3)$. In the latter case the structure of the
algebra of the
first-- and second--class constraints separated this way
\cite{sfortschr,sg} makes the conversion procedure
rather cumbersome. To elude this one can try to simplify the procedure by
converting into the first class the whole set \p{202} of the mixed
constraints. The analogous trick was used to convert fermionic constraints
in superparticle models \cite{es,moshe}.
Upon carrying out the conversion procedure we get a system characterized by
the set of first--class constraints $T_i$ that form (at least on the mass
shell) a closed algebra with respect to the Poisson brackets
defined for all the variables of
the modified phase space. In order to perform the BRST--BFV quantization
procedure we associate with each constraint of Grassmann parity $\epsilon $
the pair of canonical conjugate auxiliary variables (ghosts) $\eta _i,~
P^{(\eta )}_i$ with Grassmann parity $\epsilon +1$ \footnote{If the
extended BRST--BFV method is used, with each constraint associated are also
a Lagrange multiplier, its conjugate momentum of Grassmann parity $\epsilon
$ and an antighost and its momentum of Grassmann parity $\epsilon +1$ (see
\cite{bf,mhenn} for details).}. The resulting system is required to be
invariant under gauge transformations generated by a nilpotent fermionic
BRST charge $\Omega $. This invariance substitutes the gauge symmetry,
generated by the first class constraints in the initial phase space.
The generator $\Omega $ is found as a series in
powers of ghosts
$$
\Omega ={\eta _i}T_i + higher~ order~ terms, $$
where the structure of higher--order terms reflects the noncommutative
algebraic structure of the constraint algebra \cite{mhenn}.
Being the generator of the BRST symmetry $\Omega$ must be a dynamical invariant:
$$
{\dot \Omega}=[\Omega ,H]_P=0,
$$
where $H$ is a total Hamiltonian of the system, which has the form
\begin{equation}\label{h}
H=H_0+[\chi ,\Omega]_P.
\end{equation}
In \p{h} $H_0$ is the initial Hamiltonian of the model and $\chi$ is a
gauge fixing fermionic function whose form is determined by admissible gauge
choices \cite{teit,govaerts,sf,west,bfortschr}
(see Section 3 for the discussion of this point).
Upon quantization $\Omega$ and $H$ become operators acting on quantum state
vectors. The physical sector of the model is singled out by the requirement
that the physical states are BRST invariant and vanish under the action of
$\Omega$. Another words, we
deal with a {\it quantum gauge theory}.
When the gauge is fixed, we remain only with physically nonequivalent
states, and the Hamiltonian $H$ is argued to reproduce the correct physical
spectrum of the quantum theory.
When the model is quantized by the path--integral method, we also
deal with a quantum gauge theory.
The
Hamiltonian \p{h} is used to construct an effective action and a corresponding
BRST-invariant generating functional
which allows one to get transition amplitudes between physical states of the
theory.
Below we consider the conversion procedure and construct the BRST
charge for the twistor--like particle model in dimensions $D=3,4$ and $6$.
\subsection{ D=3}
In $D=3$ the action \p{201} is rewritten as \begin{equation}\label{211}
S={1\over 2}\int d\tau \l ^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}{\dot x}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b} \l ^{\b}, \qquad
\end{equation}
where $\l^\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta$ is a real two-component commuting spinor (spinor indices are
risen and
lowered by the unit antisymmetric tensor $\epsilon _{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}$) and $x_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta
\b}=x_m\gamma ^m_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}$.
The system of
primary constraints \p{202}
\begin{eqnarray} \label{212}
\Psi_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}=P^{(x)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}-\l_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}\l_{\b}\approx 0, \nonumber \\ P^{(\l
)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}\approx 0, \qquad
\end{eqnarray}
is a mixture of a first--class constraint generating the
$\tau$--reparametrization transformations of $x$ $$
\phi = \l^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}P^{(x)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}\l^{\b}
$$
and four second--class constraints
\begin{equation} \label{2125}
(\l P^{(\l )}),~~ (\mu P^{(x)}\mu )-(\l \mu )^{2}, ~~(\mu P^{(\l )}),~~(\l
P^{(x)} \mu ), \qquad \end{equation} where $\mu ^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}=x^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}\l _{\b}$
(see \cite{sg} for details).
In order to perform a conversion of \p{212} into a system of first--class
constraints we introduce a pair of canonical conjugate bosonic spinors
$(\z^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta},~P^{(\z )}_{\b})$, $[P^{(\z )}_{\b}, \z^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}]_P=\delta ^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}
_{\b}$, and take the modified system of constraints, which is of the first
class, in the following form:
\begin{eqnarray} \label{213}
\Psi '_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}=P^{(x)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}-(\l_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}-\z_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta})(\l_{\b}-\z_{\b})\approx 0,
\nonumber \\
\Phi '_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}=P^{(\l )}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}+P^{(\z )}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}\approx 0.
\end{eqnarray}
Eqs. \p{213} reduce to \p{212}
by putting the auxiliary variables $\z^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ and $P^{(\z )}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ equal to
zero. This reflects the appearance in the model of a new gauge symmetry with
respect to which $\z^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ and $P^{(\z )}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ are pure gauge degrees of
freedom.
It is convenient to choose the following phase--space variables as
independent ones:
\begin{eqnarray} \label{214}
v^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}=\l^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}-\z^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}, \qquad
P^{(v)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}={1\over 2}(P^{(\l )}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}-P^{(\z )} _{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}), \nonumber \\
w^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}=\l^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}+\z^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta},
\qquad
P^{(w)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}={1\over 2}(P^{(\l )}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}+P^{(\z )}_ {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}), \qquad
\end{eqnarray}
Then Eqs. \p{213} take the following form \begin{eqnarray} \label{215}
\Psi '_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}=P^{(x)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}-v_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}v_{\b}\approx0, \nonumber \\ P^{(w)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}\approx
0. \qquad
\end{eqnarray}
These constraints form an Abelian algebra.
One can see that $w^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ variables do not enter the constraint relations,
and their conjugate momenta are zero. Hence, the quantum physical states of
the model will not depend on $w^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ .
Enlarging
the modified phase space with ghosts, antighosts and Lagrange multipliers in
accordance with the following table
\begin{tabular}{cccc}
&&&\\
Constraint & Ghost & Antighost & Lagrange~multiplier \\
${\Psi}'_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}$ & $c^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}$ & $\tilde c^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}$ & $e^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}$ \\
$P^{(w)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ & $b^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ & $\tilde b^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ & $f^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ \\
&&&
\end{tabular}
\noindent
we write the classical BRST charges \cite{bf,mhenn} of the model in the
minimal and extended BRST--BFV version as follows \begin{equation} \label{218}
\Omega_{min}=c^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}\Psi '_{\b \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}+b^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}P^{(w)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}, \qquad \end{equation}
\begin{equation} \label{217}
\Omega =P^{(\tilde c)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}P^{(e)\b \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}+P^{(\tilde b) \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}P^{(f)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}+
\Omega _{min}. \qquad
\end{equation}
\subsection{D=4}
In this dimension we use two--component $SL(2,C)$ spinors
$(\l^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}=\epsilon^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}\l_{\b}; ~{\bar \l}^{\dot
\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}=\epsilon^{{\dot \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta} {\dot \b}}{\bar \l}_{\dot \b}; ~\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta ,{\dot
\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}=1,2; ~ \epsilon^{12}=-\epsilon_{21}=1)$. Other notation
coincides with that of the $D=3$ case. Then in $D=4$ the action
\p{201} can be written as following
\begin{equation}\label{221}
S= {1\over 2}\int d \tau \l^\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta {\dot x}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta {\dot
\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}\bar{\l^{\dot \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}, \qquad
\end{equation}
where $ x_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta\dot \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}= x_{m}{\sigma}^{m}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta {\dot \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}} $, and
${\sigma}^m_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta {\dot
\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}$ are the relativistic Pauli matrices. The set of the primary
constraints \p{202} in this dimension
$$\Psi_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta {\dot {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}}= P^{(x)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta
{\dot {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}}- \bar{ \l}_{\dot \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta} \l_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta} \approx 0,$$
\begin{equation} \label{222} {
P^{(\l )}}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}\approx 0, \qquad
\end{equation}
$${\bar P}^{(\bar {\l} )}_{\dot
{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}\approx 0 $$ contains two first--class constraints and three pairs of
conjugate second--class constraints \cite{sg,sfortschr}. One of the first
class constraints generates the $\tau$-reparametrization transformations
of $x^{{\dot\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta} \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ $$ \phi= \l^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}P^{(x)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta {\dot {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}}{\bar
{\l}}^{\dot {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}} $$ and another one generates $U(1)$ rotations of the
complex spinor variables \begin{equation} \label{2221} U=i(\l^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta} P^{(\l )}_\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta- \bar
{\l}^{\dot{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}\bar P^{(\l )}_{\dot {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}). \end{equation}
The form of the second--class constraints is analogous to that in the D=3
case (see Eq. \p{2125} and \cite{sg}), and we do not present it explicitly
since it is not used below.
To convert the mixed system of the constraints \p{222} into first--class
constraints
one should introduce at least three pairs of canonical conjugate auxiliary
bosonic
variables, their number is to be equal to the number of the second--class
constraints in \p{222}. However, since we do not want to violate the
manifest Lorentz invariance, and the $D=4$ Lorentz group does not have
three--dimensional representations, we are to find a way round.
We introduce two
pairs of canonical conjugate conversion spinors $
(\zeta^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta},P^{(\zeta )}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}),~~
[\zeta^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta},P^{(\zeta)}_{\b}]_P=-\delta^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}_{\b}, ~~[\bar {\zeta}^{\dot
{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}},{\bar P}^{(\bar \zeta )}_{\dot {\b}}]_P=-\delta^ {\dot \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}_{\dot \b},
$
(i.e. four pairs of real auxiliary variables) and modify the constraints
\p{222} and the $U(1)$ generator, which becomes an independent first--class
constraint in the enlarged phase space.
Thus we get the following system of the first--class constraints:
$$\Psi^{'}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \dot {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}=P^{(x)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \dot {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}} -(\bar {\l}-\bar
{\zeta})_{\dot {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}} (\l-\zeta)_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}\approx 0,$$
\begin{equation} \label{223}
{ \Phi_\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}=P^{(\l )}_\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta+P^{(\zeta )}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}\approx 0 ,
\qquad \end{equation}
$$
\bar {\Phi}_{\dot {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}= \bar P^{(\bar \l )}_{\dot {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}+\bar P^{(\bar \zeta
)}_{\dot {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}\approx 0 ,
$$
$$
U=i(\l^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta} P^{(\l )}_\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta+\zeta^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}
P^{(\zeta )}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}- \bar {\l}^{\dot {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}\bar P^{(\l )}_{\dot {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}-\bar
{\zeta}^{\dot {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}\bar P^{(\zeta )}_{\dot {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}})\approx 0.
$$
One can see
(by direct counting), that the number of independent physical
degrees of freedom of the particle in the enlarged phase space is the
same as in the initial one. The latter is recovered by imposing gauge
fixing conditions on the new auxiliary variables \begin{equation} \label{2231}
\z^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}=0,\qquad{\bar \z}^{\dot \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}=0,\qquad P^{(\z )}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}=0, \qquad
P^{({\bar \z})}_{\dot \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}=0.
\end{equation}
By introducing a new set of the independent spinor variables analogous to
that in \p{214} one rewrites Eqs. \p{223} as follows
$$
\Psi'_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta {\dot{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}}=P^{(x)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta {\dot{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}}- v_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}{\bar
v}_{\dot{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}\approx 0, $$ $$ U=i(P^{(v)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}v^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}- P^{(\bar v)}_{\dot
{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}{\bar v}^{\dot {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}})\approx 0,
$$
\begin{equation} \label{224}
{ P^{(w)}}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}\approx 0, \qquad \end{equation} $$ P^{(\bar{w})}_{\dot{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}\approx 0.
$$
Again, as in the $D=3$ case, $w_\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta,~{\bar w}_{\dot\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ and their momenta
decouple from the first pair of the constraints \p{224}, and can be
completely excluded from the number of the dynamical degrees of freedom by
putting
\begin{equation}\label{es}
w_\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta=\l_\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta+\z_\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta=0, \qquad
{P^{(w)}}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}={1\over 2}({ P^{(\l )}}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}+{P^{(\z )}}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta})=0 \end{equation}
in the strong sense. This gauge choice, which differs from \p{2231},
reduces
the phase space of the model to that of a version of the twistor--like
particle dynamics, subject to the first pair of the first--class constraints
in \p{224}, considered by Eisenberg and Solomon \cite{es}.
The constraints \p{224} form an abelian algebra, as in the $D=3$ case.
In compliance with the BRST--BFV prescription we introduce ghosts,
antighosts and Lagrange
multipliers associated with the
constraints \p{224} as follows
\begin{tabular}{cccc}
Constraint & Ghost & Antighost & Lagrange~ multiplier \\ ${\Psi}
'_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta {\dot{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}}$ & $c^{\dot{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta} \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ & ${\tilde c}^{{\dot \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta} \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ &
$e^{\dot{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta} \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta} $ \\ $U$ & $a$ & ${\tilde a}$ & $g$ \\ $P^{(w)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ &
$b^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ & ${\tilde b}^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ & $f^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ \\ $P^{(\bar{w})}_{\dot{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}$ &
${\bar b}^{\dot{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}$ & ${\tilde{\bar b}}^{\dot{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}$ & ${\bar f}^{\dot
{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}$ \\ \qquad \end{tabular}
Then the BRST--charges of the $D=4$ model have the form
\begin{equation}
\label{227}
\Omega_{min}=c^{\dot {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta} \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}\Psi_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta {\dot {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}}+b^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}P^{(w)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}+
{\bar b}^{\dot{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}P^{({\bar w})}_{\dot{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}+aU,
\qquad \end{equation}
\begin{equation} \label{226}
\Omega=P^{({\tilde c})}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta {\dot{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}}P^{(e)
{\dot{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}+P^{({\tilde b})} _{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}P^{(f)\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}+P^{({\tilde{\bar
b}})}_{\dot{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}P^{({\bar f})\dot{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}+ P^{({\tilde
a})}P^{(g)}+\Omega_{min}.
\end{equation}
\subsection{D=6}
In $D=6$ a light--like vector $V^m$ can be represented in terms of commuting
spinors as follows
$$
V^m=\l^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}_i\gamma^m_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}\l^{\b i}, $$
where $\l^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}_i$ is an $SU(2)$--Majorana--Weyl
spinor which has the $SU^*(4)$ index
$\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta =1,2,3,4$ and the $SU(2)$ index $i=1,2$.
$\gamma^m_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}$ are $D=6$ analogs of the Pauli matrices (see
\cite{6spin,benght}). $SU(2)$ indices are risen and lowered by the unit
antisymmetric tensors $\epsilon_{ij},~~\epsilon^{ij}$. As to the $SU^*(4)$
indices, they can be risen and lowered only in pairs by the totally
antisymmetric tensors $\epsilon_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b \g \delta}\def\Om{\Omega}\def\s{\sigma}$, $\epsilon^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b \g \delta}\def\Om{\Omega}\def\s{\sigma}$
($\epsilon_{1234}=1)$.
Rewriting the action \p{201} in terms of $SU(2)$--Majorana--Weyl spinors,
one gets
\begin{equation} \label{231}
S={1\over 2}\int d \tau {\dot x}^m \l^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}_i(\gamma_m)_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}\l^{\b i}.
\end{equation}
The system of the primary constraints \p{202} takes the form
$$
\Psi_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}=P^{(x)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}-\epsilon_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b \g \delta}\def\Om{\Omega}\def\s{\sigma}\l^{\g}_i \l^{\delta}\def\Om{\Omega}\def\s{\sigma i}
\approx 0,
$$
\begin{equation} \label{232}
{ ~P^{(\l )i}}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}\approx 0, \qquad
\end{equation}
where $P^{(x)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}=P^{(x)}_m \gamma^m_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}$. $\Psi_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}$ is
antisymmetric in $\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta$ and $\b$ and contains six independent components.
(To get \p{232} we used the relation
$(\g_m)_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta\b}\g^m_{\g\delta}\def\Om{\Omega}\def\s{\sigma}\sim \epsilon_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b \g \delta}\def\Om{\Omega}\def\s{\sigma}$).
From Eqs. \p{232}
one can separate four first--class constraints by projecting \p{232}
onto $\l^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}_{i}$ \cite{sfortschr,benght}. One of the first--class
constraints generates the $\tau$--reparametrizations of $x^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta\b}$ $$ \phi
=\l^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}_{i}P^{(x)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}\l^{\b i}, $$ and another three ones form an
$SU(2)$ algebra $$ T_{ij}=\l^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}_{(i}P^{(\l )}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta j)}, $$ Braces denote
the symmetrization of $i$ and $j$. All other constraints in \p{232} are
of the second class.
The conversion of \p{232} into first--class constraints is carried out by
analogy with the $D=4$
case. According to the conventional conversion prescription we had to
introduce five pairs
of canonical conjugate bosonic variables. Instead, in order to preserve
Lorentz invariance, we introduce the canonical conjugate pair of bosonic
spinors $\z^{\b}_j$, $P^{(\z )i}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$
($[P^{(\z )i}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta},\z^{\b}_j]_P=\delta}\def\Om{\Omega}\def\s{\sigma^{\b}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}\delta}\def\Om{\Omega}\def\s{\sigma^i_j,$) modify the constraints
\p{232} and the $SU(2)$ generators.
This results in the set of independent first--class constraints
$$
\Psi '_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}=P^{(x)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}-\epsilon_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b \g \delta}\def\Om{\Omega}\def\s{\sigma}
(\l^{\g}_i-\z^{\g}_i)(\l^{\delta}\def\Om{\Omega}\def\s{\sigma i}-\z^{\delta}\def\Om{\Omega}\def\s{\sigma i})\approx 0, $$
\begin{equation} \label{233}
{ \Phi_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}^i=P^{(\l )i}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}+P^{(\z )i}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}\approx 0, \qquad \end{equation}
$$
T_{ij}=\l^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}_{(i}P^{(\l )}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta j)}-\z^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}_{(i}P^{(\z )}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta j)}\approx 0.
$$
In terms of spinors $v^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}_{i}$ and $w^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}_{i}$, and their momenta,
defined as in the $D=3$ case \p{214}, they take the following form
$$
\Psi '_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}=P^{(x)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}-\epsilon_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b \g \delta}\def\Om{\Omega}\def\s{\sigma}v^{\g}_iv^{\delta}\def\Om{\Omega}\def\s{\sigma i}\approx
0, $$
\begin{equation} \label{234}
{ T_{ij}}=v^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}_{(i}P^{(v)}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta j)}\approx 0, \qquad \end{equation}
$$ P^{(w)i}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}\approx 0. $$
These constraints form a closed algebra with respect to the Poisson
brackets.
The only nontrivial brackets in this algebra are \begin{equation} \label{235}
[T_{ij},T_{kl}]_{p}=\epsilon_{jk}T_{il}+\epsilon_{il}T_{jk}+
\epsilon_{ik}T_{jl} +\epsilon_{jl}T_{ik}, \qquad \end{equation}
which generate the $SU(2)$ algebra.
We introduce ghosts, antighosts and Lagrange multipliers related to the
constraints \p{235}
\begin{tabular}{cccc}
& & &\\
Constraint & Ghost & Antighost & Lagrange~ multiplier \\ ${\Psi} '_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta
\b}$ & $c^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}$ & ${\tilde c}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}$ & $e^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}$ \\ $T_{ij}$ &
$a^{ij}$ & ${\tilde a}_{ij}$ & $g^{ij}$ \\ $\Phi ^{i}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ & $b^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}_{i}$
& ${\tilde b}^{i}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ & $f^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}_{i}$ \\ &&& \end{tabular}
\noindent
and construct the BRST charges corresponding respectively, to the minimal
and extended BRST--BFV version, as follows \begin{equation}
\label{238}
\Omega_{min}=c^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}\Psi '_{\b \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}+b^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}_{i}P^{(w)i}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}+a^{ij}T_{ji}+
\qquad
\end{equation}
$$
(\epsilon_{jk}P^{(a)}_{il}+ \epsilon_{il}P^{(a)}_{jk}+
\epsilon_{ik}P^{(a)}_{jl}+\epsilon_{jl}P^{(a)}_{ik})a^{ij}a^{kl}.
$$
\begin{equation} \label{237}
\Omega=P^{({\tilde c})}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}P^{(e)\b \alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}+P^{({\tilde b})\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}
_{i}P^{(f)i}_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}+P^{({\tilde a})ij}P^{(g)}_{ji}+\Omega_{min}, \qquad \end{equation}
Higher order terms in ghost powers appear in \p{238} and \p{237} owing to the
noncommutative $SU(2)$ algebra of the $T_{ij}$ constraints \p{235}.
\section{Admissible gauge choice.}
One of the important problems in the quantization of gauge
systems is a correct gauge choice. In the frame of the BRST--BFV
quantization scheme gauge fixing is made by an appropriate choice of the
gauge fermion that determines the structure of the quantum Hamiltonian.
The Batalin and Vilkovisky theorem \cite{bf,mhenn} reads that the result of
path integration does not depend on the choice of the gauge fermions if they
belong
to the same equivalence class with respect to the BRST--transformations.
An analogous theorem
takes place in the operator BRST--BFV quantization scheme \cite{sf}.
Further
analysis of this problem for systems possessing the reparametrization
invariance showed that the result of path integration does not depend on
the choice of the
gauge fermion if only appropriate
gauge conditions are compatible with the boundary conditions for the
parameters of the corresponding gauge transformations
\cite{polyakov,govaerts,sf,west,bfortschr}. In particular, it was shown
that the so--called ``canonical gauge", when the worldline gauge field of the
reparametrization symmetry of the bosonic particle is fixed to be a
constant, is not admissible in this sense.
(see \cite{govaerts,bfortschr}
for details). Anyway one can use the canonical gauge as a consistent limit
of an admissible gauge \cite{sf}.
Making the analysis of the twistor--like model one can show that admissible
are the following gauge conditions on Lagrange multipliers from the
corresponding Tables of the previous section in the dimensions $D=3,$ 4 and
$6$ of space--time, respectively,
\begin{equation} \label{261}
D=3:\qquad {\dot e}^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta\b}=0;\qquad f^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}=0; \end{equation}
\begin{equation} \label{262}
D=4:\qquad {\dot e}^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta\dot\b}=0;\qquad f^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}=0; \qquad f^{\dot
\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}=0;\qquad g=0;
\end{equation}
\begin{equation} \label{263}
D=6:\qquad {\dot e}^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta\b}=0;\qquad f_i^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}=0;\qquad g^{ij}=0; \end{equation}
The canonical gauge
\begin{equation} \label{264}
e=constant,
\end{equation}
can be considered as a limit of
more general admissible gauge $e-\varepsilon {\dot e}=constant$ (at
$\varepsilon \rightarrow 0$) \cite{sf}.
Then the use of the gauge condition \p{264} does not lead to any problems
with the operator BRST--BFV quantization.
Below we shall use the ``relativistic" gauge conditions \p{261}, \p{262} and
\p{263} for the path--integral quantization. The use of the canonical
gauge \p{264} in this case would lead to a wrong form of the particle
propagator.
\section{Path--integral BRST quantization.}
In this section we shall use the
extended version of the BRST--BFV quantization procedure \cite{mhenn,bff}
and fix the gauge by applying the conditions \p{261}, \p{262}, \p{263}.
The gauge fermion, corresponding to this gauge choice, is
\begin{equation} \label{417}
\chi_D={1\over 2}P^{(c)}_me^m, ~~~D=3,4,6, \qquad
\end{equation}
The Hamiltonians constructed with \p{417} are \cite{bf,mhenn}
$$ {\it H}_D=[\Omega_D,\chi_D],\qquad D=3,4,6 $$ \begin{equation} \label{420}
{H}_3=e^{m}(P^{(x)}_{m}-{1\over 2}v^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}(\gamma_{m})_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}v^{\b})
-P^{(c)}_{m}P^{({\tilde c})m}, \qquad
\end{equation}
\begin{equation} \label{421}
{H}_4=e^{m}(P^{(x)}_{m}-{1\over 2}{\bar v}^{\dot {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}}(\sigma_{m})
_{\dot{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}v^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta})-P^{(c)}_{m}P^{({\tilde c})m}, \qquad \end{equation}
\begin{equation} \label{422}
{H}_6=e^{m}(P^{(x)}_{m}-{1\over 2}v^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}_i(\g_{m})_ {\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}v^{\b i})
-P^{(c)}_{m}P^{({\tilde c})m}, \qquad \end{equation}
We shall calculate the coordinate propagator
$Z=\langle x_{1}^{m}\mid U_0\mid x_{2}^{m}\rangle$ (where
$U_0=expiH(T_1-T_2)$ is the evolution operator),
therefore boundary conditions for the phase space
variables are fixed as follows: \begin{equation} \label{423} x^{m}(T_1)=x_{1}^{m},
\qquad x^{m}(T_2)=x_{2}^{m}, \qquad \end{equation} the boundary values of the ghosts,
antighosts and canonical momenta of the Lagrange multipliers are put equal
to zero (which is required by the BRST invariance of the boundary
conditions \cite{mhenn}), and we sum up over all possible values of the
particle momentum and the twistor variables.
The standard expression for the matrix element of the evolution operator is
\begin{equation} \label{424}
Z_D=\int[D\mu DP^{\mu}]_D
exp(i\int_{T_1}^{T_2} d \tau ([P^{\mu}{\dot {\mu}}]_D -{\it H}_D )),
\qquad D=3,4,6.
\end{equation}
$[D\mu DP^{\mu}]_D $ contains functional Liouville measures
of all the canonical variables of the BFV extended phase space \cite{bf}.
$[P^{\mu}{\dot{\mu}}]_D $ contains a sum of products
of the canonical momenta with the velocities.
For instance, an explicit expression for the path--integral measure
in the $D=3$ case is
$$
[D\mu DP^{\mu}]=DxDP^{(x)}DvDP^{(v)}DwDP^{(w)}DeDP^{(e)}DfDP^{(f)} $$
$$
DbDP^{(b)}DcDP^{(c)}D{\tilde b}DP^{({\tilde b})}D{\tilde c}DP^{({\tilde
c})}.
$$
We can perform straightforward integration over the all variables
that are not present in the Hamiltonians \p{420}, \p{421}, \p{422}
\footnote{ All calculations are done up to a multiplication constant,
which can always be absorbed by the integration measure.}.
Then \p{424} reduces to the product of two terms
\begin{equation} \label{425}
Z_D=I_DG_D, \qquad
\end{equation}
where
\begin{equation} \label{426}
G_D=\int DcDP^{(c)}D{\tilde c}DP^{({\tilde c})} exp(i\int_{T_1}^{T_2}
d \tau (P^{(c)}_m{\dot c}^m+P^{({\tilde c})}_m{\dot{\tilde c}}^m
-{1\over 2} P^{({\tilde c})}_mP^{(c)m})), \qquad
\end{equation}
and $I_D$ includes the integrals over bosonic variables entering \p{420},
\p{421}, \p{422} together with their conjugated momenta.
We use the method analogous to that in \cite{rivelles}
for computing these integrals.
The calculation of the ghost integral $G_D$ results in \begin{equation} \label{427}
G_D=(\Delta T)^D, \qquad
\Delta T=T_2-T_1, \qquad D=3,4,6.
\end{equation}
Let us demonstrate main steps of the $I_D$ calculation in the $D=3$
case
\begin{eqnarray}\label{801}
I_3&=&\int DxDP^{(x)}DeDP^{(e)}DvDP^{(v)}exp(i\int _{T_1}^{T_2}d\tau
(P_m^{(x)}{\dot x}^m+P_m^{(e)}{\dot e}^m+
P_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}^{(v)}{\dot v}^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta} \nonumber \\
& & -e^{m}(P^{(x)}_{m}-
{1\over 2}v^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}(\g _m)_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b} v^{\b}))
\end{eqnarray}
Integration over $P_m^{(e)}$ and $P_m^{(v)}$
results in the functional $\delta$-functions $\delta ({\dot e}),~ \delta
({\dot v})$ which reduce functional integrals over $e^m$ and $v^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ to
ordinary ones:
\begin{equation} \label{802}
I_3=\int DxDP^{(x)}d^{3}ed^{2}v~exp(ip_m\Delta x^m- \\
i\int _{T_1}^{T_2}d\tau (x^m{\dot P}^{(x)}_m+e^m(P^{(x)}_m-
{1\over 2}v^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}(\g _m)_{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta \b}v^{\b})),
\qquad \end{equation}
where $\Delta x^m=x_2^m-x_1^m$ \p{423}. Since the
integral over $v^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ is a usual Gauss integral after
integrating over $x^m$ and $v^{\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta}$ one obtains
\begin{equation} \label{803}
I_3=\int d^{3}pd^{3}e{1\over {\sqrt{e^me_m-i0}}}exp(i(p_m\Delta x^m-
e^mp_m\Delta T)).
\end{equation}
In general case of $D=3$, 4 and 6 dimensions, one obtains
\begin{equation} \label{428}
I_D=\int d^Dpd^De{{1\over{(e^me_m-i0)^{{D-2}\over 2}}}} exp(i(p_m\Delta
x^m- e^mp_m\Delta T)),
\end{equation}
that can be rewritten as
\begin{equation} \label{804}
I_D=\int
d^{D}pd^{D}e\int_0^{\infty}dc~exp(i(p_m\Delta x^m-e^mp_m\Delta T
+(e^me_m-i0)c^{2\over {D-2}})),
\qquad \end{equation}
where $c$ is an auxiliary variable.
Integrating over $p^m$ and $e^m$ one gets
\begin{equation} \label{436}
Z_D=\int_{0}^{\infty}dc{1\over{c^{D/2}}}
exp(i{{\Delta x^m\Delta x_m}\over{2c}} -c0), \qquad D=3,4,6,
\end{equation}
or
$$
Z_D={1\over {(\Delta x^m\Delta x_m-i0)^{{D-2}\over 2}}}, $$
which coincides with the coordinate propagator
for the massless bosonic particle in the
standard formulation \cite{govaerts}.
On the other hand integrating \p{428} only over $e^m$ we get
the massless bosonic particle causal propagator in the form $$
Z_D=\int d^Dp{1\over{p^mp_m+i0}}exp(ip_m\Delta x^m).
$$
\subsection{Comment on the $D=10$ case}
Above we have restricted our consideration to the space--time dimensions
3, 4 and 6. The case of a bosonic twistor--like particle in $D=10$ is
much more sophisticated. The Cartan--Penrose representation of a
$D=10$ light--like momentum vector is constructed out of a
Majorana--Weyl spinor $\lambda^\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta$ which has 16 independent components
\begin{equation}\label{pc}
P^m=\l\Gamma^m\l.
\end{equation}
Transformations of $\l^\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta$ which leave \p{pc} invariant take values on
an $S^7$-- sphere (see \cite{es,nispach,bfortschr}
and references therein). In contrast to
the $D=4$ and $D=6$ case, where such transformations belong to the
group $U(1)\sim S^1$ \p{224} and $SU(2)\sim S^3$ \p{233},
respectively, $S^7$
is not a Lie group and its corresponding algebra contains structure
functions instead of structure constants. Moreover, among the 10
constraints \p{pc} and 16 constraints $P^{(\l)}_\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta=0$ on the
momenta conjugate to $x^m$ and $\l^\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta$ $18=10+16-1-7$ (where 7 comes from
$S^7$ and 1 corresponds to local $\tau$--reparametrization) are of the
second class. They do not form a representation of the Lorentz group
and cause the problem for covariant Hamiltonian analysis.
One can overcome these problems in the framework of the
Lorentz--harmonic formalism (see \cite{bzstr,bpstv} and references
therein), where to construct a light--like vector one introduces eight
Majorana--Weyl spinors instead of one $\l^\alpha}\def\b{\beta}\def\l{\lambda}\def\g{\gamma}\def\z{\zeta$. Such a spinor matrix
takes values in a spinor representation of the double covering group
$Spin(1,9)$ of $SO(1,9)$ and satisfies second--class harmonic
conditions. The algebra of the constraints in this ``multi--twistor"
case is easier to analyze than that with only one commuting spinor
involved. The path--integral
BRST quantization of the $D=10$ twistor--like particle is in progress.
\section {Conclusion}
In the present paper the BRST--BFV quantization
of the dynamics of massless bosonic particle in $D=3,4,6$
was performed in the twistor--like formulation.
To this end the initially mixed system of the first-- and
second--class constraints was converted into the system of
first--class constraints by extending the initial phase space
of the model with auxiliary variables in a Lorentz--covariant way.
The conversion procedure (rather than having been a formal trick)
was shown to have a meaning of a symmetry transformation
which relates different twistor--like formulations of the bosonic particle,
corresponding to different gauge choices in the extended phase space.
We quantized the model by use of
the extended BRST--BFV scheme for the path--integral quantization.
As a result we have
presented one of the numerous proofs of the equivalence between
the twistor--like and conventional formulation of
the bosonic particle mechanics.
This example demonstrates peculiar features of treating the
twistor--like variables within the course of the covariant
Hamiltonian analysis and the BRST quantization,
which one should take into account when studying more
complicated twistor--like systems, such as superparticles and
superstrings.
\bigskip
\noindent
{\bf Acknowledgements.}
\noindent
The authors are grateful to P. Pasti, M. Tonin, D. V. Volkov and V. G. Zima
for useful discussion. I. Bandos and D. Sorokin acknowledge partial
support from the INTAS and Dutch Government
Grant N 94--2317 and the INTAS Grant N 93--493.
| proofpile-arXiv_065-427 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Guidelines}
It is well known that the effects on the physics of a field,
due to a much heavier field coupled to the former, are not detectable at
energies comparable to the lighter mass. More precisely the
Appelquist-Carazzone (AC) theorem~\cite{ac} states that for a Green's function
with only light external legs, the effects of the heavy loops
are either absorbable in a redefinition of the bare couplings
or suppressed by powers of $k/M$ where $k$ is the energy scale
characteristic of the Green's function
(presumably comparable to the light mass),
and $M$ is the heavy mass. However the AC theorem does not allow
to make any clear prediction when $k$ becomes close to $M$ and, in this
region one should expect some non-perturbative effect due to the onset of new
physics.
\par
In the following we shall make use of the Wilson's renormalization
group (RG) approach to discuss the physics of the light field from the
infrared region up to and beyond the mass of the heavy field.
Incidentally, the RG
technique has been already employed to proof the AC theorem~\cite{girar}.
The RG establishes the flow equations of the various coupling constants of
the theory for any change in the observational energy scale; moreover
the improved RG equations, originally derived by F.J. Wegner
and A. Houghton~\cite{weg},
where the mixing of all couplings (relevant and irrelevant) generated
by the blocking procedure is properly taken into account, should
allow to handle the non-perturbative features arising
when the heavy mass threshold is crossed.
\par
We shall discuss the simple case of two coupled scalar fields and since
we are interested in the modifications of the parameters
governing the light field, due to the heavy loops, we shall consider
the functional integration of the heavy field only. The action at a given
energy scale $k$ is
\begin{equation}
S_k(\phi,\psi)=\int d^4 x~\left ({1\over 2} \partial_\mu \phi
\partial^\mu \phi+
{1\over 2} W(\phi,\psi) \partial_\mu \psi \partial^\mu \psi+
U(\phi,\psi) \right ),
\label{eq:acteff}
\end{equation}
with polynomial $W$ and $U$
\begin{equation}
U(\phi,\psi)=\sum_{m,n}{{G_{2m,2n} \psi^{2m}\phi^{2n}}\over {(2n)!(2m)!}};
~~~~~~~~~~~~~~~~~~~~
W(\phi,\psi)=\sum_{m,n}{{H_{2m,2n}\psi^{2m}\phi^{2n}}
\over {(2n)!(2m)!}}.
\label{eq:svil}
\end{equation}
Since we want to focus on the light field, which we choose to be $\psi$,
we have simply set to 1 the wave function renormalization of $\phi$.
In the following we shall analyse the symmetric phase of the theory
with vanishing vacuum energy $G_{0,0}=0$.
\par
We do not discuss here the procedure employed~\cite{jan}
to deduce the RG coupled equations for the
couplings in Eq.~\ref{eq:svil}, because it is thoroughly explained in the
quoted reference. Since it is
impossible to handle an infinite set of equations and a truncation in the
sums in Eq.~\ref{eq:svil} is required, we keep in the action
only terms that do not exceed the sixth power in the fields and their
derivatives. Moreover we choose the initial condition for the RG equations
at a fixed ultraviolet scale $\Lambda$
where we set $H_{0,0}=1$, $G_{0,4}=
G_{2,2}=G_{4,0}=0.1$ and
$G_{0,6}=G_{2,4}=G_{4,2}=G_{6,0}=
H_{2,0}=H_{0,2}=0$,
and the flow of the various couplings is determined as a function of
$t=ln \left (k/ \Lambda\right )$, for negative $t$.
\begin{figure}
\psfig{figure=fig1.ps,height=4.3cm,width=12.cm,angle=90}
\caption{
(a): $G_{0,2}(t)/\Lambda^2$ (curve (1)) and $10^6\cdot
G_{2,0}(t)/\Lambda^2$ (curve (2)) vs $t=log\left ({{k}/{\Lambda}}\right )$.
\break
(b): $G_{2,2}(t)$ (1), $G_{0,4}(t)$ (2),
$G_{4,0}(t)$ (3) vs $t$.
\label{fig:funo}}
\end{figure}
\par
In Fig.~\ref{fig:funo}(a) $G_{0,2}(t)/\Lambda^2$ (curve (1) ) and
$10^6 \cdot G_{2,0}(t)/\Lambda^2$ (curve (2)) are plotted. Clearly the heavy
and the light masses become stable going toward the IR region and their value
at $\Lambda$ has been chosen in such a way that the stable IR values are,
$M\equiv\sqrt {G_{0,2}(t=-18)}\sim 10^{-4}\cdot \Lambda$
and $m\equiv\sqrt{G_{2,0}(t=-18)}\sim 2\cdot 10^{-7}\cdot\Lambda$.
So, in principle, there are three scales:
$\Lambda$, ($t=0$), the heavy mass $M$,
($t\sim -9.2$), the light mass $m$, ($t\sim -16.1$).
In Fig.~\ref{fig:funo}(b) the three renormalizable dimensionless
couplings are shown; the neat change around $t=-9.2$, that is $k \sim M$,
is evident and the curves become flat below this value.
The other four non-renormalizable couplings included in $U$ are
plotted in Fig.~\ref{fig:fdue}(a), in units of $\Lambda$.
Again everything is flat below $M$, and the values of the couplings
in the IR region coincide with their perturbative Feynman-diagram
estimate at the one loop level; it is easy to realize that
they are proportional to $1/M^2$, which, in units of $\Lambda$,
is a big number. Thus the large values in
Fig.~\ref{fig:fdue}(a) are just due to the scale employed and, since
these four couplings for any practical purpose, must
be compared to the energy scale at which they are calculated, it is
physically significant to plot them in units of
the running cutoff $k$:
the corresponding curves are displayed in Fig.~\ref{fig:fdue}(b);
in this case the couplings are strongly suppressed below $M$.
\begin{figure}
\psfig{figure=fig2.ps,height=4.3cm,width=12.cm,angle=90}
\caption{
(a): $G_{6,0}(t)\cdot \Lambda^2$ (1), $G_{0,6}(t)\cdot \Lambda^2$ (2),
$G_{4,2}(t)\cdot \Lambda^2$ (3) and
$G_{2,4}(t)\cdot \Lambda^2$ (4) vs $t$.\break
(b): $G_{6,0}(t)\cdot k^2$ (1), $G_{0,6}(t)\cdot k^2$ (2),
$G_{4,2}(t)\cdot k^2$ (3) and
$G_{2,4}(t)\cdot k^2$ (4) vs $t$.
\label{fig:fdue}}
\end{figure}
\par
It must be remarked that there is no change in the couplings when
the light mass threshold is crossed. This is a consequence of having
integrated the heavy field only: in this case one could check directly
from the equations ruling the coupling constants flow, that
a shift in the
initial value $G_{2,0}(t=0)$ has the only effect
(as long as one remains in the symmetric phase)
of modifying $G_{2,0}(t)$, leaving the other curves unchanged.
Therefore the results obtained are independent of $m$ and
do not change even if $m$ becomes much larger than $M$.
An example of the heavy mass dependence is shown in Fig.~\ref{fig:ftre}(a),
where $G_{6,0}(t)$ is plotted, in units of the running cutoff $k$, for three
different values of $G_{0,2}(t=0)$, which correspond respectively to
$M/\Lambda\sim 2\cdot 10^{-6}$, (1),
$M/\Lambda\sim 10^{-4}$, (2) and
$M/\Lambda\sim 0.33$, (3).
Note, in each curve, the change of slope when the $M$ scale is crossed.
$H_{0,0}=1,~~H_{0,2}=0$ is a constant solution of the corresponding equations
for these two couplings; on the other hand $H_{2,0}$ is not constant
and it is plotted in units of the running cutoff $k$ in Fig.~\ref{fig:ftre}(b),
for the three values of $M$ quoted above.
\begin{figure}
\psfig{figure=fig3.ps,height=4.3cm,width=12.cm,angle=90}
\caption{
(a): $G_{6,0}(t)\cdot k^2$ vs $t$
for $M/\Lambda \sim 2\cdot 10^{-6}$ (1),
$\sim 10^{-4}$ (2),
$\sim 0.33$ (3).\break
(b): $H_{2,0}(t)\cdot k^2$ for the three values of $M/\Lambda$
quoted in (a).
\label{fig:ftre}}
\end{figure}
\par
In conclusion, according to the AC theorem all couplings are constant at low
energies
and a change in the UV physics can only shift their values in the IR region.
Remarkably, for increasing $t$,
no trace of UV physics shows up until one reaches $M$,
that acts as a UV cut-off for the low energy physics.
Moreover, below $M$, no non-perturbative effect appears due to the
non-renormalizable couplings that vanish fastly in units of $k$.
Their behavior above $M$ is somehow constrained by the
renormalizability condition fixed at $t=0$, as clearly shown in
Fig.~\ref{fig:ftre}(a) (3). Finally, the peak of $H_{2,0}$ at $k\sim M$
in Fig.~\ref{fig:ftre}(b), whose width and
height are practically unchanged in the three examples,
is a signal of non-locality of the theory limited to the region
around $M$.
\section*{Acknowledgments}
A.B. gratefully acknowledges Fondazione A. Della Riccia and INFN
for financial support.
\section*{References}
| proofpile-arXiv_065-428 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
\label{sec:intro}
\subsection{The Lyman--alpha forest}
\label{sec:laf}
Spectroscopic observations towards quasars show a large number of
intervening absorption systems. This `forest' of lines is numerically
dominated by systems showing only the Lyman--alpha transition ---
these absorbers are called Lyman alpha clouds.
Earlier work suggests that the clouds are large, highly ionized
structures, either pressure confined (eg. Ostriker \& Ikeuchi 1983) or within
cold dark matter structures (eg. Miralda--Escud\'e \& Rees 1993; Cen, Miralda--Escud\'{e} \& Ostriker 1994;
Petitjean \& M\"{u}cket 1995; Zhang, Anninos \& Norman 1995). However, alternative models do
exist: cold, pressure confined clouds (eg. Barcons \& Fabian 1987, but see
Rauch et~al. 1993); various shock mechanisms (Vishniac \& Bust 1987,
Hogan 1987).
Low and medium resolution spectroscopic studies of the forest
generally measure the redshift and equivalent width of each cloud. At
higher resolutions it is possible to measure the redshift ($z$), H~I
column density ($N$, atoms per cm$^2$) and Doppler parameter ($b$,
\hbox{\rm km\thinspace s$^{-1}$}). These are obtained by fitting a Voigt profile to the data
(Rybicki \& Lightman 1979).
Using a list of $N$, $z$ and $b$ measurements, and their associated
error estimates, the number density of the population can be studied.
Most work has assumed that the density is a separable function of $z$,
$N$ and $b$ (Rauch et~al. 1993).
There is a local decrease in cloud numbers near the background quasar
which is normally attributed to the additional ionizing flux in that
region. While this may not be the only reason for the depletion (the
environment near quasars may be different in other respects; the cloud
redshifts may reflect systematic motions) it is expected for the
standard physical models wherever the ionising flux from the quasar
exceeds, or is comparable to, the background.
Since the generally accepted cloud models are both optically thin to
ionizing radiation and highly ionized, it is possible to correct
column densities from the observed values to those that would be seen
if the quasar were more remote. The simplest correction assumes that
the shape of the two incident spectra --- quasar and background ---
are similar. In this case the column density of the neutral fraction
is inversely proportional to the incident ionizing flux.
If the flux from the quasar is known, and the depletion of clouds is
measured from observations, the background flux can be determined. By
observing absorption towards quasars at different redshifts the
evolution of the flux can be measured. Bechtold (1993) summarises
earlier measurements of the ionising flux, both locally and at higher
redshifts.
Recently Loeb \& Eisenstein (1995) have suggested that enhanced clustering near
quasars causes this approach to overestimate the background flux. If
this is the case then an analysis which can also study the evolution
of the effect gives important information. In particular, a decrease
in the inferred flux might be expected after the redshift where the
quasar population appears to decrease. However, if the postulated
clustering enhancement is related to the turn--on of quasars at high
redshift, it may conspire to mask any change in the ionizing
background.
Section \ref{sec:model} describes the model of the population density
in more detail, including the corrections to flux and redshift that
are necessary for a reliable result. The data used are described in
section \ref{sec:data}. In section \ref{sec:errors} the quality of
the fit is assessed and the procedure used to calculate errors in the
derived parameters is explained. Results are given in section
\ref{sec:results} and their implications discussed in section
\ref{sec:discuss}. Section \ref{sec:conc} concludes the paper.
\section{The Model}
\label{sec:model}
\subsection{Population Density}
\label{sec:popden}
The Doppler parameter distribution is not included in the model since
it is not needed to determine the ionizing background from the proximity
effect. The model here assumes that $N$ and $z$ are uncorrelated.
While this is unlikely (Carswell 1995), it should be a good
approximation over the restricted range of column densities considered
here.
The model of the population without Doppler parameters or the
correction for the proximity effect is
\begin{equation} dn(N^\prime,z) = \, A^\prime
(1+z)^{\gamma^\prime}\,(N^\prime)^{-\beta}
\frac{c(1+z)}{H_0(1+2q_0z)^\frac{1}{2}} \ dN^\prime\,dz
\end{equation}
where $H_0$ is the Hubble parameter, $q_0$ is the cosmological
deceleration parameter and $c$ is the speed of light. Correcting for
the ionizing flux and changing from `original' ($N^\prime$) to
`observed' ($N$) column densities, gives
\begin{equation} dn(N,z) = \, A (1+z)^{\gamma^\prime}
\left(\frac{N}{\Delta_F}\right)^{-\beta}
\frac{c(1+z)}{H_0(1+2q_0z)^\frac{1}{2}} \ \frac{dN}{\Delta_F}\,dz
\end{equation}
where
\begin{equation}N = N^\prime \Delta_F \ , \end{equation}
\begin{equation}\Delta_F = \frac{ f_\nu^B }{ f_\nu^B + f_\nu^Q } \ ,
\end{equation}
and $f_\nu^B$ is the background flux, $f_\nu^Q$ is the flux from the
quasar ($4\pi J_\nu(z)$).
The background flux $J_\nu^B$ may vary with redshift. Here it is
parameterised as a constant, a power law, or two power laws with a
break which is fixed at $z_B=3.25$ (the mid--point of the available
data range). An attempt was made to fit models with $z_B$ as a free
parameter, but the models were too poorly constrained by the data to
be useful.
\begin{eqnarray}
J_\nu(z)=10^{J_{3.25}} & \hbox{model {\bf B}} \\
J_\nu(z)=10^{J_{3.25}}\left(\frac{1+z}{1+3.25}\right)^{\alpha_1} & \mbox{{\bf C}} \\
J_\nu(z)=10^{J_{z_B}}\times\left\{\begin{array}{ll}\left(\frac{1+z}{1+z_B}\right)^{\alpha_1}&\mbox{$z<z_B$}\\
\left(\frac{1+z}{1+z_B}\right)^{\alpha_2}&\mbox{$z>z_B$}\end{array}\right. & \mbox{{\bf D \& E}}
\end{eqnarray}
A large amount of information (figure \ref{fig:nz}) is used to
constrain the model parameters. The high--resolution line lists give
the column density and redshift, with associated errors, for each
line. To calculate the background ionising flux the quasar luminosity
and redshift must be known (table \ref{tab:objects}). Finally, each
set of lines must have observational completeness limits (table
\ref{tab:compl}).
\subsection{Malmquist Bias and Line Blending}
\label{sec:malm}
Malmquist bias is a common problem when fitting models to a population
which increases rapidly at some point (often near an observational
limit). Errors during the observations scatter lines away from the
more populated regions of parameter space and into less populated
areas. Line blending occurs when, especially at high redshifts,
nearby, overlapping lines cannot be individually resolved. This is a
consequence of the natural line width of the clouds and cannot be
corrected with improved spectrographic resolution. The end result is
that weaker lines are not detected in the resultant `blend'. Both
these effects mean that the observed population is not identical to
the `underlying' or `real' distribution.
\subsubsection{The Idea of Data Quality}
To calculate a correction for Malmquist bias we need to understand the
significance of the error estimate since any correction involves
understanding what would happen if the `same' error occurs for
different column density clouds. The same physical cloud cannot be
observed with completely different parameters, but the same
combination of all the complex factors which influence the errors
might affect a line with different parameters in a predictable way.
If this idea of the `quality' of an observation could be quantified it
would be possible to correct for Malmquist bias: rather than the
`underlying' population, one reflecting the quality of the observation
(ie. including the bias due to observational errors) could be fitted
to the data.
For example, if the `quality' of an observation was such that,
whatever the actual column density measured, the error in column
density was the same, then it would be trivial to convolve the
`underlying' model with a Gaussian of the correct width to arrive at
an `observed' model. Fitting the latter to the data would give
parameters unaffected by Malmquist bias. Another example is the case
of galaxy magnitudes. The error in a measured magnitude is a fairly
simple function of source brightness, exposure time, etc., and so it
is possible to correct a flux--limited galaxy survey for Malmquist
bias.
\subsubsection{Using Errors as a Measure of Quality}
It may be possible to describe the quality of a spectrum by the signal
to noise level in each bin. From this one could, for a given line,
calculate the expected error in the equivalent width. The error in
the equivalent width might translate, depending on whether the
absorption line was in the linear or logarithmic portion of the `curve
of growth', to a normal error in either $N$ or $\log(N)$. But in this
idealised case it has been assumed that the spectrum has not been
re--binned, leaving the errors uncorrelated; that the effect of
overlapping, blended lines is unimportant; that there is a sudden
transition from a linear to logarithmic curve of growth; that the
resulting error is well described by a normal distribution. None of
this is likely to be correct and, in any case, the resulting analysis,
with different `observed' populations for every line, would be too
unwieldy to implement, given current computing facilities.
A more pragmatic approach might be possible. A plot of the
distribution of errors with column density (figure~\ref{fig:ndn})
suggests that the errors in $\log(N)$ are of a similar magnitude for a
wide range of lines (although there is a significant correlation
between the two parameters). Could the error in $\log(N)$ be a
sufficiently good indicator of the `quality' of an observation?
\begin{figure}
\epsfxsize=8.5cm
\epsfbox{nn.ps}
\epsfverbosetrue
\caption{The distribution of errors in column density.}
\label{fig:ndn}
\end{figure}
If the number density of the underlying population is $n^\prime(N)\
\hbox{d}\log(N)$ then the observed population density for a line with
error in $\log(N)$ of $\sigma_N$ is:
\begin{equation} n(N)\ \hbox{d}\log(N)
\propto\int_{-\infty}^{\infty}n^\prime\left(N10^x\right)\,\exp\left(\frac{-x^2}{2\sigma_N^2}\right)\,\hbox{d}x\
. \end{equation}
For a power law distribution this can be calculated analytically
and gives an increased probability of seeing lines with larger errors,
as expected. For an underlying population density $N^{-\beta}\
\hbox{d}N$ the increase is $\exp\left((1-\beta)^2(\sigma_N\log
10)^2/2\right)$.
This gives a lower statistical weight to lines with larger errors when
fitting. For this case --- a power law and log--normal errors --- the
weighting is not a function of $N$ directly, which might imply that
any correction would be uniform, with little bias expected for
estimated parameters.
In practice this correction does not work. This is probably because
the exponential dependence of the correction on $\sigma_N$ makes it
extremely sensitive to the assumptions made in the derivation above.
These assumptions are not correct. For example, it seems that the
correlation between \hbox{$\log( N )$}\ and the associated error is important.
\subsubsection{An Estimation of the Malmquist Bias}
It is possible to do a simple numerical simulation to gauge the
magnitude of the effect of Malmquist bias. A population of ten
million column densities were selected at random from a power law
distribution ($\beta=1.5, \log(N_{\hbox{min}})=10.9,
\log(N_{\hbox{max}})=22.5$) as an `unbiased' sample. Each line was
given an `observed' column density by adding a random error
distributed with a normal or log--normal (for lines where $13.8 <
\log(N) < 17.8$) distribution, with a mean of zero and a standard
deviation in $\log(N)$ of $0.5$. This procedure is a simple
approximation to the type of errors coming from the curve of growth
analysis discussed above, assuming that errors are approximately
constant in $\log(N)$ (figure~\ref{fig:ndn}). The size of the error
is larger than typical, so any inferred change in $\beta$ should be an
upper limit.
Since a power--law distribution diverges as $N\rightarrow0$ a normal
distribution of errors in $N$ would give an infinite number of
observed lines at every column density. This is clearly unphysical
(presumably the errors are not as extended as in a normal distribution
and the population has some low column density limit). Because of
this the `normal' errors above were actually constrained to lie within
3 standard deviations of zero.
\begin{figure}
\epsfxsize=8.5cm
\epsfbox{malm.ps}
\epsfverbosetrue
\caption{A model including Malmquist bias. The bold, solid line is the original sample, the bold, dashed line is the distribution after processing as described in the text. Each curve is shown twice, but the upper right plot has both axes scaled by a factor of 3 and only shows data for $12.5<\log(N)<16$. Reference lines showing the evolution expected for $\beta=1.5$ and $1.45$ (dashed) are also shown (centre).}
\label{fig:malm}
\end{figure}
The results (figure~\ref{fig:malm}) show that Malmquist bias has only
a small effect, at least for the model used here. The main solid line
is the original sample, the dashed line is the observed population.
Note that the results in this paper come from fitting to a sample of
lines with $12.5<\log(N)<16$ (section~\ref{sec:data}) ---
corresponding to the data shown expanded to the upper right of the
figure. Lines with smaller column densities are not shown since that
fraction of the population is affected by the lower density cut--off
in the synthetic data.
A comparison with the two reference lines, showing the slopes for a
population with $\beta=1.5$ or $1.45$ (dashed), indicates that the
expected change in $\beta$ is $\sim0.05$. The population of lines
within the logarithmic region of the curve of growth appears to be
slightly enhanced, but otherwise the two curves are remarkably
similar. The variations at large column densities are due to the
small number of strong lines in the sample.
\subsubsection{Other Approaches}
What other approaches can be used to measure or correct the effects of
Malmquist bias and line blending? Press \& Rybicki (1993) used a completely
different analysis of the Lyman--$\alpha$ forest. Generating and reducing synthetic
data, with a known background and cloud population, would allow us to
assess the effect of blending. Changing (sub--setting) the sample of
lines that is analysed will alter the relative (and, possibly,
absolute) importance of the two effects.
The procedure used by Press \& Rybicki (1993) is not affected by Malmquist bias
or line blending, but it is difficult to adapt to measure the ionizing
background.
Profile fitting to high--resolution data is a slow process, involving
significant manual intervention (we have tried to automate
profile--fitting with little success). An accurate measurement of the
systematic error in the ionizing background would need an order of
magnitude more data than is used here to get sufficiently low error
limits. Even if this is possible --- the analysis would need a
prohibitive amount of CPU time --- it would be sufficient work for a
separate, major paper (we would be glad to supply our software to
anyone willing to try this).
Taking a sub--set of the data is not helpful unless it is less likely
to be affected by the biases described above. One approach might be
to reject points with large errors, or large relative errors, in
column density since these are more affected by Malmquist bias.
However, this would make the observations incomplete in a very poorly
understood way. For example, relative errors are correlated with
column density (as noted above) and so rejecting lines with larger
relative errors would preferentially reject higher column density
lines. There is no sense in trying to measure one bias if doing so
introduces others.
Unlike rejecting lines throughout the sample, changing the completeness
limit does not alter the coverage of the observations (or rather, it
does so in a way that is understood and corrected for within the
analysis). Raising the completeness limits should make line blending
less important since weaker lines, which are most likely to be
blended, are excluded from the fit. Whether it affects the Malmquist
bias depends on the distribution of errors.
For blended lines, which tend to be weak, raising the completeness
limit should increase the absolute value of $\beta$ since the more
populous region of the (hypothetical) power--law population of column
densities will no longer be artificially depleted. The effect on
$\gamma$ is more difficult to assess since it is uncertain whether the
completeness limits are correct at each redshift. If the limits
increase too rapidly with redshift, for example, then raising them
further will reduce blending most at lower redshifts, lowering
$\gamma$. But if they are increasing too slowly then the converse
will be true.
\subsubsection{Conclusions}
Until either profile--fitting is automated, or the method of
Press \& Rybicki (1993) can be modified to include the proximity effect, these
two sources of uncertainty --- Malmquist bias and line blending ---
will continue be a problem for any analysis of the Lyman--$\alpha$ forest. However,
from the arguments above, it is likely that the effect of Malmquist
bias is small and that, by increasing the completeness limit, we can
assess the magnitude of the effect of line blending.
\subsection{Flux Calculations}
\label{sec:fluxcal}
\subsubsection{Galactic Extinction}
Extinction within our Galaxy reduces the apparent luminosity of the
quasars and so lowers the estimate of the background. Since the
absorption varies with frequency this also alters the observed
spectral slope.
Observed fluxes were corrected using extinction estimates derived from
the H~I measurements of Heiles \& Cleary (1979) for Q2204--573 and
Stark et~al. (1992) for all the other objects. H I column densities were
converted to $E(B-V)$ using the relationships: \begin{eqnarray}
E(B-V)&=&\frac{N_{\hbox{\footnotesize
H~I}}}{5.27\,10^{21}}\quad\hbox{if\ } \frac{N_{\hbox{\footnotesize
H~I}}}{5.27\,10^{21}}<0.1\\ E(B-V)&=&\frac{N_{\hbox{\footnotesize
H~I}}}{4.37\,10^{21}}\quad\hbox{otherwise} \end{eqnarray} where the
first value comes from Diplas \& Savage (1994) and the second value, which
compensates for the presence of H$_2$, is the first scaled by the
ratio of the conversions given in Bohlin, Savage \& Drake (1978). A ratio
$R=A(V)/E(B-V)$ of 3.0 (Lockman \& Savage 1995) was used and variations of
extinction with frequency, $A(\lambda)/A(V)$ were taken from
Cardelli, Clayton \& Mathis (1989).
The correction to the observed index, $\alpha_o$, of the power--law
continuum,
\begin{equation} f_\nu\propto\nu^{-\alpha}\ , \end{equation}
was calculated using
\begin{equation}
\alpha_o=\alpha+\frac{A(V)}{2.5}\frac{\partial}{\partial\ln\nu}\frac{A(\nu)}{A(V)}
\end{equation}
which, using the notation of Cardelli, Clayton \& Mathis (1989), becomes
\begin{equation}
\alpha_o=\alpha+\frac{A(V)}{2.5\,10^6c}\,\nu\ln(10)\,\frac{\partial}{\partial
y}\left(a(x)+\frac{b(x)}{R}\right)\ . \end{equation}
\subsubsection{Extinction in Damped Systems}
\label{sec:dampcor}
Two quasars are known to have damped absorption systems along the line
of sight (Wolfe et~al. 1995). The extinction due to these systems is not
certain, but model {\bf E} includes the corrections listed in
table~\ref{tab:damp}. These have been calculated using the SMC
extinction curve in Pei (1992), with a correction for the
evolution of heavy element abundances taken from Pei \& Fall (1995). The
SMC extinction curve is most suitable for these systems since they do
not appear to have structure at 2220~\AA\ (Boiss\'{e} \& Bergeron 1988), unlike LMC
and Galactic curves.
\begin{table}
\begin{tabular}{lllll}
Object&$\log(N_{\hbox{\footnotesize H I}})$&$z_{\hbox{\footnotesize abs}}$&$A(V)$&$\Delta_\alpha$\\
Q0000--263&21.3&3.39&0.10 &0.078\\
Q2206--199&20.7&1.92&0.14 &0.10\\
$ $&20.4&2.08&0.019 &0.049\\
\end{tabular}
\caption{
The damped absorption systems and associated corrections (at 1450~\AA\ in the quasar's rest--frame) for model {\bf E}.}
\label{tab:damp}
\end{table}
\subsubsection{Absorption by Clouds near the Quasar}
\label{sec:internal}
The amount of ionizing flux from the background quasar incident on a
cloud is attenuated by all the other clouds towards the source. If
one of the intervening clouds has a large column density this can
significantly reduce the extent of the effect of the quasar.
To correct for this the fraction of ionizing photons from the quasar
not attenuated by the intervening H~I and He~II absorption is
estimated before fitting the model. A power--law spectrum is assumed
and the attenuation is calculated for each cloud using the
cross--sections given in Osterbrock (1989). The ratio
$n(\hbox{He~II})/n(\hbox{H~I})$ within the clouds will depend on
several unknown factors (the true energy distribution of the ionizing
flux, cloud density, etc.), but was assumed to be 10 (Sargent et~al. 1980,
Miralda--Escud\'e 1993).
The attenuation is calculated using all the observed intervening
clouds. This includes clouds which are not included in the main fit
because they lie outside the column density limits, or are too close
to the quasar ($\Delta z \leq 0.003$).
For most clouds ($\log(N)\sim13.5$) near enough to the quasar to
influence the calculation of the background this correction is
unimportant (less than 1\%). However, large ($\log(N)\sim18$ or
larger) clouds attenuate the flux to near zero. This explains why
clouds with $\Delta_f\sim1$ are apparent close to the QSO in
figure~\ref{fig:prox}.
This relatively sudden change in optical depth at $\log(N)\sim18$ is
convenient since it makes the correction insensitive to any
uncertainties in the calculation (eg. $n(\hbox{He~II})/n(\hbox{H~I})$,
the shape of the incident spectrum, absorption by heavier elements)
--- for most column densities any reasonable model is either
insignificant ($\log(N)<17$) or blocks practically all ionizing
radiation ($\log(N)>19$).
In fact, the simple correction described above is in reasonable
agreement with CLOUDY models, for even the most critical column
densities. A model cloud with a column density of $\log(N)=13.5$ and
constant density was irradiated by an ionizing spectrum based on that
of Haardt \& Madau (1996). Between the cloud and quasar the model included an
additional absorber (constant density, $\log(N)=18$) which modified
the quasar's spectrum. The effect of the absorber (for a range of
heavy element abundances from pure H to primordial to 0.1 solar) on
the ionized fraction of H~I was consistent with an inferred decrease
in the quasar flux of about 80\%. In comparison, the correction
above, using a power--law spectrum with $\alpha=1$, gave a reduction
of 60\% in the quasar flux. These two values are in good agreement,
considering the exponential dependence on column densities and the
uncertainty in spectral shape. At higher and lower absorber column
densities the agreement was even better, as expected.
\subsection{Redshift Corrections}
\label{sec:redcor}
Gaskell (1982) first pointed out a discrepancy between the redshifts
measured from Lyman $\alpha$ and C~IV emission, and those from lower
ionization lines (eg. Mg~II, the Balmer series). Lower ionization
lines have a larger redshift. If the systemic redshift of the quasar
is assumed to be that of the extended emission (Heckman et~al. 1991),
molecular emission (Barvainis et~al. 1994), or forbidden line emission
(Carswell et~al. 1991), then the low ionization lines give a better measure
of the rest--frame redshift.
Using high ionization lines gives a reduced redshift for the quasar,
implies a higher incident flux on the clouds from the quasar, and, for
the same local depletion of lines, a higher estimate of the
background.
Espey (1993) re--analysed the data in Lu, Wolfe \& Turnshek (1991), correcting
systematic errors in the quasar redshifts. The analysis also
considered corrections for optically thick and thin universes and the
differences between the background and quasar spectra, but the
dominant effect in reducing the estimate from 174 to 50~J$_{23}$\ was the
change in the quasar redshifts.
To derive a more accurate estimate of the systemic velocity of the
quasars in our sample we made use of published redshift measurements
of low ionization lines, or measured these where spectra were
available to us. The lines used depended on the redshift and line
strengths in the data, but typically were one or more of
Mg~II$\,2798\,$\AA, O~I$\,1304\,$\AA, and C~II$\,1335\,$\AA.
When no low ionization line observations were available (Q0420--388,
Q1158--187, Q2204--573) we applied a mean correction to the high
ionization line redshifts. These corrections are based on
measurements of the relative velocity shifts between high and low
ionization lines in a large sample of quasars (Espey \& Junkkarinen 1996). They
find a correlation between quasar luminosity and mean velocity
difference ($\Delta_v$) with an empirical relationship given by:
\begin{equation} \Delta_v=\exp(0.66\log L_{1450}-13.72)\
\hbox{\rm km\thinspace s$^{-1}$}\end{equation} where $L_{1450}$ is the rest--frame luminosity
(ergs Hz$^{-1}$ s$^{-1}$) of the quasar at 1450~\AA\ for $q_0=0.5$ and H$_0=100\
\hbox{\rm km\thinspace s$^{-1}$}/\hbox{Mpc}$.
\section{The Data}
\label{sec:data}
\begin{table*}
\begin{tabular}{lrrrrrrrr}
&&&\multicolumn{2}{c}{$L_\nu(1450)$}&\multicolumn{4}{c}{Typical change in $\log(\hbox{J$_{23}$})$}\\
\hfill Object \hfill&\hfill $z$ \hfill&\hfill $\alpha$ \hfill&\hfill $q_0=0$ \hfill&$\hfill q_0=0.5$ \hfill&\hfill $z$ \hfill&\hfill $f_\nu$ \hfill&\hfill $\alpha$ \hfill&\hfill Total \hfill\\
Q0000--263 & 4.124 & 1.02 & 13.5\ten{31} & 2.8\ten{31} & $-$0.09 & $+$0.02 & $+$0.01 & $-$0.05\\
Q0014+813 & 3.398 & 0.55 & 34.0\ten{31} & 8.6\ten{31} & $-$0.19 & $+$0.33 & $+$0.21 & $+$0.36\\
Q0207--398 & 2.821 & 0.41 & 5.6\ten{31} & 1.7\ten{31} & $-$0.16 & $+$0.03 & $+$0.02 & $-$0.11\\
Q0420--388 & 3.124 & 0.38 & 10.9\ten{31} & 3.0\ten{31} & $-$0.16 & $+$0.04 & $+$0.02 & $-$0.10\\
Q1033--033 & 4.509 & 0.46 & 5.5\ten{31} & 1.0\ten{31} & $-$0.05 & $+$0.12 & $+$0.00 & $+$0.06\\
Q1100--264 & 2.152 & 0.34 & 13.8\ten{31} & 5.3\ten{31} & $-$0.42 & $+$0.19 & $+$0.11 & $-$0.13\\
Q1158--187 & 2.454 & 0.50 & 42.2\ten{31} & 14.4\ten{31} & $-$0.46 & $+$0.09 & $+$0.06 & $-$0.31\\
Q1448--232 & 2.223 & 0.61 & 9.6\ten{31} & 3.5\ten{31} & $-$0.34 & $+$0.28 & $+$0.17 & $+$0.11\\
Q2000--330 & 3.783 & 0.85 & 12.7\ten{31} & 2.9\ten{31} & $-$0.10 & $+$0.16 & $+$0.10 & $+$0.15\\
Q2204--573 & 2.731 & 0.50 & 42.8\ten{31} & 13.3\ten{31} & $-$0.35 & $+$0.06 & $+$0.03 & $-$0.25\\
Q2206--199 & 2.574 & 0.50 & 19.4\ten{31} & 6.3\ten{31} & $-$0.31 & $+$0.05 & $+$0.03 & $-$0.22\\
Mean & 3.081 & 0.56 & 19.1\ten{31} & 5.7\ten{31} & $-$0.24 & $+$0.12 & $+$0.07 & $-$0.04\\
\end{tabular}
\caption{
The systemic redshifts, power law continuum exponents. and rest frame luminosities (ergs Hz$^{-1}$ s$^{-1}$\ at 1450~\AA) for the quasars used. $H_0=100$ \hbox{\rm km\thinspace s$^{-1}$}/Mpc and luminosity scales as $H_0^{-2}$. The final four columns are an estimate of the relative effect of the various corrections in the paper (systemic redshift, correction for reddening to flux and spectral slope).}
\label{tab:objects}
\end{table*}
Objects, redshifts and fluxes are listed in table \ref{tab:objects}.
A total of 1675 lines from 11 quasar spectra were taken from the
literature. Of these, 844 lie within the range of redshifts and
column densities listed in table \ref{tab:compl}, although the full
sample is used to correct for absorption between the quasar and
individual clouds (section~\ref{sec:internal}). The lower column
density limits are taken from the references; upper column densities
are fixed at $\hbox{$\log( N )$}=16$ to avoid the double power law distribution
discussed by Petitjean et~al. (1993). Fluxes are calculated using standard
formulae, assuming a power law spectrum ($f_\nu \propto
\nu^{-\alpha}$), with corrections for reddening. Low ionization line
redshifts were used where possible, otherwise high ionization lines
were corrected using the relation given in section~\ref{sec:redcor}.
Values of $\alpha$ uncorrected for absorption are used where possible,
corrected using the relation above. If no $\alpha$ was available, a
value of 0.5 was assumed (Francis 1993).
References and notes on the calculations for each object follow:
\begin{description}
\item[Q0000--263] Line list from Cooke (1994). There is some
uncertainty in the wavelength calibration for these data, but the
error ($\sim30 \hbox{\rm km\thinspace s$^{-1}$}$) is much less than the uncertainty in the quasar
redshift ($\sim900 \hbox{\rm km\thinspace s$^{-1}$}$) which is taken into account in the error
estimate (section~\ref{sec:erress}). Redshift this paper
(section~\ref{sec:redcor}). Flux and $\alpha$ measurements from
Sargent, Steidel \& Boksenberg (1989).
\item[Q0014+813] Line list from Rauch et~al. (1993). Redshift this paper
(section~\ref{sec:redcor}). Flux and $\alpha$ measurements from
Sargent, Steidel \& Boksenberg (1989).
\item[Q0207--398] Line list from Webb (1987). Redshift (O I line)
from Wilkes (1984). Flux and $\alpha$ measurements from
Baldwin et~al. (1995).
\item[Q0420--388] Line list from Atwood, Baldwin \& Carswell (1985). Redshift, flux and
$\alpha$ from Osmer (1979) (flux measured from plot). The redshifts
quoted in the literature vary significantly, so a larger error (0.01)
was used in section~\ref{sec:erress}.
\item[Q1033--033] Line list and flux from Williger et~al. (1994). From their
data, $\alpha=0.78$, without a reddening correction. Redshift this
paper (section~\ref{sec:redcor}).
\item[Q1100--264] Line list from Cooke (1994). Redshift from
Espey et~al. (1989) and $\alpha$ from Tytler \& Fan (1992). Flux measured from
Osmer \& Smith (1977).
\item[Q1158--187] Line list from Webb (1987). Redshift from
Kunth, Sargent \& Kowal (1981). Flux from Adam (1985).
\item[Q1448--232] Line list from Webb (1987). Redshift from
Espey et~al. (1989). Flux and $\alpha$ measured from Wilkes et~al. (1983),
although a wide range of values exist in the literature and so a
larger error (0.6 magnitudes in the flux) was used in
section~\ref{sec:erress}.
\item[Q2000--330] Line list from Carswell et~al. (1987). Redshift this paper
(section~\ref{sec:redcor}). Flux and $\alpha$ measurements from
Sargent, Steidel \& Boksenberg (1989).
\item[Q2204--573] Line list from Webb (1987). Redshift from
Wilkes et~al. (1983). V magnitude from Adam (1985).
\item[Q2206--199] Line list from Rauch et~al. (1993). Redshift this paper
(section~\ref{sec:redcor}). V magnitude from Hewitt \& Burbidge (1989).
\end{description}
\begin{table}
\begin{tabular}{cccccc}
Object&\multispan2{\hfil$N$\hfil}&\multispan2{\hfil$z$\hfil}&Number\\
name&Low&High&Low&High&of lines\\
Q0000--263& 14.00 & 16.00 & 3.1130 & 3.3104 & 62 \\
& & & 3.4914 & 4.1210 & 101 \\
Q0014+813& 13.30 & 16.00 & 2.7000 & 3.3800 & 191 \\
Q0207--398& 13.75 & 16.00 & 2.0765 & 2.1752 & 11 \\
& & & 2.4055 & 2.4878 & 7 \\
& & & 2.6441 & 2.7346 & 6 \\
& & & 2.6852 & 2.7757 & 9 \\
& & & 2.7346 & 2.8180 & 8 \\
Q0420--388& 13.75 & 16.00 & 2.7200 & 3.0800 & 73 \\
Q1033--033& 14.00 & 16.00 & 3.7000 & 3.7710 & 16 \\
& & & 3.7916 & 3.8944 & 21 \\
& & & 3.9191 & 4.0301 & 24 \\
& & & 4.0548 & 4.1412 & 25 \\
& & & 4.1988 & 4.3139 & 30 \\
& & & 4.3525 & 4.4490 & 23 \\
& & & 4.4517 & 4.4780 & 2 \\
Q1100--264& 12.85 & 16.00 & 1.7886 & 1.8281 & 2 \\
& & & 1.8330 & 1.8733 & 8 \\
& & & 1.8774 & 1.9194 & 13 \\
& & & 1.9235 & 1.9646 & 9 \\
& & & 1.9696 & 2.0123 & 10 \\
& & & 2.0189 & 2.0617 & 6 \\
& & & 2.0683 & 2.1119 & 18 \\
Q1158--187& 13.75 & 16.00 & 2.3397 & 2.4510 & 9 \\
Q1448--232& 13.75 & 16.00 & 2.0847 & 2.1752 & 9 \\
Q2000--330& 13.75 & 16.00 & 3.3000 & 3.4255 & 23 \\
& & & 3.4580 & 3.5390 & 15 \\
& & & 3.5690 & 3.6440 & 18 \\
& & & 3.6810 & 3.7450 & 11 \\
Q2204--573& 13.75 & 16.00 & 2.4467 & 2.5371 & 10 \\
& & & 2.5454 & 2.6276 & 12 \\
& & & 2.6441 & 2.7280 & 8 \\
Q2206--199& 13.30 & 16.00 & 2.0864 & 2.1094 & 2 \\
& & & 2.1226 & 2.1637 & 8 \\
& & & 2.1760 & 2.2188 & 5 \\
& & & 2.2320 & 2.2739 & 7 \\
& & & 2.2887 & 2.3331 & 7 \\
& & & 2.3471 & 2.3940 & 10 \\
& & & 2.4105 & 2.4574 & 4 \\
& & & 2.4754 & 2.5215 & 11 \\
\multicolumn{2}{l}{Total: 11 quasars }& & & & 844 \\
\end{tabular}
\caption{Completeness limits.}
\label{tab:compl}
\end{table}
Table~\ref{tab:objects} also gives an estimate of the relative effect
of the different corrections made here. Each row gives the typical
change in $\log(\hbox{J$_{23}$})$ that would be estimated using that
quasar alone, with a typical absorption cloud 2~Mpc from the quasar
($q_0=0.5, H_0=100\,\hbox{\hbox{\rm km\thinspace s$^{-1}$}}$). The correction to obtain the
systemic redshift is not necessary for any quasar whose redshift has
been determined using low ionization lines. In such cases the value
given is the expected change if the redshift measurement had not been
available.
Using the systematic redshift always reduces the background estimate,
while correcting for reddening always acts in the opposite sense. The
net result, in the final column of table~\ref{tab:objects}, depends on
the relative strength of these two factors. For most objects the
redshift correction dominates, lowering $\log(\hbox{J$_{23}$})$ by $\sim
0.15$ (a decrease of 30\%), but for four objects the reddening is more
important (Q0014+813, the most reddened, has $B-V = 0.33$; the average
for all other objects is $0.09$).
\begin{figure*}
\hbox{
\epsfxsize=8.5cm
\epsfbox{nz0.ps}
\hfill
\epsfxsize=8.5cm
\epsfbox{nz1.ps}
}
\epsfverbosetrue
\caption{
The lines in the full sample (left) used to calculate the attenuation
of the quasar flux by intervening clouds and the restricted sample
(right) to which the model was fitted.}
\label{fig:nz}
\end{figure*}
Figure \ref{fig:nz} shows the distribution of column density, $N$, and
redshift, $z$, for the lines in the sample. The completeness limit
was taken from the literature and depends on the quality of the
spectra. There is also a clear trend with redshift as the number
density increases and weak lines become less and less easy to separate
in complex blends, whatever the data quality (see
section~\ref{sec:malm} for a more detailed discussion of line
blending).
\section{Fit Quality and Error Estimates}
\label{sec:errors}
\subsection{The Quality of the Fit}
\label{sec:finalq}
\begin{table}
\begin{tabular}{c@{\hspace{3em}}cc@{\hspace{3em}}cc}
&\multicolumn{2}{c}{Without Inv. Eff.\hfill}&\multicolumn{2}{c}{With Inv. Eff.\hfill}\\
Variable&Statistic&Prob.&Statistic&Prob.\\
$N$ & 1.11 & 0.17 & 1.05 & 0.22 \\
$z$ & 1.12 & 0.16 & 1.05 & 0.22 \\
\end{tabular}
\caption{
The K--S statistics measuring the quality of the fit.}
\label{tab:ks}
\end{table}
Figures \ref{fig:cum1} and \ref{fig:cum2} show the cumulative data and
model for each variable using two models: one includes the proximity
effect (model {\bf B}), one does not (model {\bf A}). The
probabilities of the associated K--S statistics are given in table
\ref{tab:ks}. For the column density plots the worst discrepancy
between model and data occurs at $\log(N)=14.79$. The model with the
proximity effect (to the right) has slightly more high column density
clouds, as would be expected, although this is difficult to see in the
figures (note that the dashed line --- the model --- is the curve that
has changed). In the redshift plots the difference between the two
models is more apparent because the changes are confined to a few
redshifts, near the quasars, rather than, as in the previous figures,
spread across a wide range of column densities. The apparent
difference between model and data is larger for the model that
includes the proximity effect (on the right of figure~\ref{fig:cum2}).
However, this is an optical illusion as the eye tends to measure the
vertical difference between horizontal, rather than diagonal, lines.
In fact the largest discrepancy in the left figure is at $z=3.323$,
shifting to $z=3.330$ when the proximity effect is included. It is
difficult to assess the importance of individual objects in cumulative
plots, but the main difference in the redshift figure occurs near the
redshift of Q0014+813. However, since this is also the case without
the proximity effect (the left--hand figure) it does not seem to be
connected to the unusually large flux correction for this object
(section~\ref{sec:data}).
In both cases --- with and without the proximity effect --- the model
fits the data reasonably well. It is not surprising that including
the proximity effect only increases the acceptability of the fit
slightly, as the test is dominated by the majority of lines which are
not influenced by the quasar. The likelihood ratio test that we use
in section \ref{sec:disevid} is a more powerful method for comparing
two models, but can only be used if the models are already a
reasonable fit (as shown here).
\begin{figure*}
\hbox{
\epsfxsize=8.5cm
\epsfbox{ks0.n.ps}
\hfill
\epsfxsize=8.5cm
\epsfbox{ks1.n.ps}
}
\epsfverbosetrue
\caption{
The cumulative data (solid line) and model (dashed), integrating over
$z$, for the lines in the sample, plotted against column
density (\hbox{$\log( N )$}). The model on the right includes the proximity effect.}
\label{fig:cum1}
\end{figure*}
\begin{figure*}
\hbox{
\epsfxsize=8.5cm
\epsfbox{ks0.z.ps}
\hfill
\epsfxsize=8.5cm
\epsfbox{ks1.z.ps}
}
\epsfverbosetrue
\caption{
The cumulative data (solid line) and model (dashed), integrating over
$N$, for the lines in the sample, plotted against redshift.
The model on the right includes the proximity effect.}
\label{fig:cum2}
\end{figure*}
\subsection{Sources of Error}
\label{sec:erress}
There are two sources of stochastic uncertainty in the values of
estimated parameters: the finite number of observations and the error
associated with each observation (column densities, redshifts, quasar
fluxes, etc.).
The first source of variation --- the limited information available
from a finite number of observations --- can be assessed by examining
the distribution of the posterior probabilities for each parameter.
This is described in the following section.
The second source of variation --- the errors associated with each
measurement --- can be assessed by repeating the analysis for
simulated sets of data. In theory these errors could have been
included in the model and their contribution would have been apparent
in the posterior distribution. In practice there was insufficient
information or computer time to make a detailed model of the error
distribution.
Instead, ten different sets of line--lists were created. Each was
based on the original, with each new value, $X$, calculated from the
observed value $x$ and error estimate $\sigma_X$: \begin{equation} X =
x + a \sigma_X\ ,\end{equation} where $a$ was selected at random from
a (approximate) normal distribution with zero mean and unit variance.
The redshift (standard error 0.003) and luminosity (standard error 0.2
magnitudes) of each background quasar were also changed. For
Q0420--388 the redshift error was increased to 0.1 and, for
Q1448--232, the magnitude error was increased to 0.6 magnitudes. The
model was fitted to each data set and the most likely values of the
parameters recorded. A Gaussian was fitted to the distribution of
values. In some cases (eg.\ figure~\ref{fig:alphas_d}) a Gaussian
curve may not be the best way to describe the distribution of
measurements. However, since the error in the parameters is dominated
by the small number of data points, rather than the observational
errors, using a different curve will make little difference to the
final results.
Since the two sources of stochastic error are not expected to be
correlated they can be combined to give the final distribution of the
parameters. The Gaussian fitted to the variation from measurement
errors is convolved with the posterior distribution of the variable.
The final, normalized distribution is then a good approximation to the
actual distribution of values expected.
This procedure is shown in figures \ref{fig:beta_gamma_d}\ to
\ref{fig:alphas_d}. For each parameter in the model the `raw'
posterior distribution is plotted (thin line and points). The
distribution of values from the synthetic data is shown as a dashed
histogram and the fitted Gaussian is a thin line. The final
distribution, after convolution, is the heavy line. In general the
uncertainties due to a finite data set are the main source of error.
\subsection{Error Estimates from Posterior Probabilities\label{sec:postprob}}
\newcommand{{\bf y}}{{\bf y}}
\newcommand{\tb}{{\bf\theta}}
\newcommand{\rb}{{\bf R_\nu}}
\newcommand{\bn}{{b_\nu}}
If $p({\bf y}|\tb)$ is the likelihood of the observations (${\bf y}$), given
the model (with parameters $\tb$), then we need an expression for the
posterior probability of a `parameter of interest', $\eta$. This
might be one of the model parameters, or some function of the
parameters (such as the background flux at a certain redshift):
\begin{equation}\eta = g(\tb)\ .\end{equation}
For example, the value of J$_{23}$\ at a particular redshift for models
{\bf C} to {\bf E} in section~\ref{sec:popden} is a linear function of
several parameters (two or more of $J_{3.25}, J_{z_B}, \alpha_1$, and
$\alpha_2$). To calculate how likely a particular flux is the
probabilities of all the possible combinations of parameter values
consistent with that value must be considered: it is necessary to
integrate over all possible values of $\beta$ and $\gamma^\prime$, and
all values of $J_{z_B}, \alpha_1$, etc. which are consistent with
J$_{23}$(z) having that value.
In other words, to find the posterior distribution of $\eta$,
$\pi(\eta|\tb)$, we must marginalise the remaining model parameters:
\begin{equation}\pi(\eta|\tb)=\lim_{\gamma \rightarrow 0}
\frac{1}{\gamma} \int_D \pi(\tb|{\bf y})\,d\tb\ ,\end{equation} where D is
the region of parameter space for which $\eta \leq g(\tb) \leq \eta +
\gamma$ and $\pi(\tb|{\bf y}) \propto \pi(\tb) p({\bf y}|\tb)$, the posterior
density of $\tb$ with prior $\pi(\tb)$.
A uniform prior is used here for all parameters (equivalent to normal
maximum likelihood analysis). Explicitly, power law exponents and the
logarithm of the flux have prior distributions which are uniform over
$[-\infty,+\infty]$.
Doing the multi--dimensional integral described above would require a
large (prohibitive) amount of computer time. However, the
log--likelihood can be approximated by a second order series expansion
in $\tb$. This is equivalent to assuming that the other parameters
are distributed as a multivariate normal distribution, and the result
can then be calculated analytically. Such a procedure is shown, by
Leonard, Hsu \& Tsui (1989), to give the following procedure when $g(\tb)$ is a
linear function of $\tb$: \begin{equation}\bar{\pi}(\eta|{\bf y}) \propto
\frac{\pi_M(\eta|{\bf y})}{|\rb|^{1/2}(\bn^T\rb^{-1}\bn)^{1/2}}\
,\end{equation} where \begin{eqnarray} \pi_M(\eta|{\bf y}) & = &
\sup_{\tb:g(\tb)=\eta} \pi(\tb|{\bf y})\\&=&\pi(\tb_\eta|{\bf y})\ ,\\ \bn & =
& \left.\frac{\partial g(\tb)}{\partial \tb}\right|_{\tb=\tb_\eta}\
,\\ \rb & = & \left.\frac{\partial^2 \ln
\pi(\tb|{\bf y})}{\partial(\tb\tb^T)} \right|_{\tb=\tb_\eta}\
. \end{eqnarray} The likelihood is maximised with the constraint that
$g(\tb)$ has a particular value. $\rb$ is the Hessian matrix used in
the fitting routine (Press et~al. 1992) and $\bn$ is known (when $\eta$
is the average of the first two of three parameters, for example, $\bn
= 0.5,0.5,0$).
This quickens the calculation enormously. To estimate the posterior
distribution for, say, J$_{23}$, it is only necessary to choose a series
of values and, at each point, find the best fit consistent with that
value. The Hessian matrix, which is returned by many fitting
routines, can then be used --- following the formulae above --- to
calculate an approximation to the integral, giving a value
proportional to the probability at that point. Once this has been
repeated for a range of different values of J$_{23}$\ the resulting
probability distribution can be normalised to give an integral of one.
Note that this procedure is only suitable when $g(\tb)$ is a linear
function of $\tb$ --- Leonard, Hsu \& Tsui (1989) give the expressions needed for
more complex parameters.
\section{Results}
\label{sec:results}
A summary of the results for the different models is given in table
\ref{tab:fpars}. The models are:
\begin{description}
\item[{\bf A}] --- No Proximity Effect. The population model
described in section \ref{sec:model}, but without the proximity
effect.
\item[{\bf B}] --- Constant Background. The population model
described in section \ref{sec:model} with a constant ionising
background.
\item[{\bf C}] --- Power Law Background. The population model
described in section \ref{sec:model} with an ionising background which
varies as a power law with redshift
\item[{\bf D}] --- Broken Power Law Background. The population model
described in section \ref{sec:model} with an ionising background whose
power law exponent changes at $z_B=3.25$.
\item[{\bf E}] --- Correction for Extinction in Damped Systems. As
{\bf D}, but with a correction for absorption in known damped
absorption systems (section \ref{sec:dampcor}).
\end{description}
In this paper we assume $q_0$ = 0.5 and $H_0$ = 100~\hbox{\rm km\thinspace s$^{-1}$}/Mpc.
\subsection{Population Distribution}
\label{sec:resmpars}
\begin{table*}
\begin{tabular}{crlrlrlrlrlrc}
Model & \multispan2{\hfill$\beta$\hfill} & \multispan2{\hfill$\gamma$\hfill} & \multispan2{\hfill $J_{z_B}$\hfill} & \multispan2{\hfill $\alpha_1$\hfill} & \multispan2{\hfill $\alpha_2$\hfill} & \multispan1{\hfill $z_B$\hfill} & -2 log--likelihood\\
{\bf A} & 1.66 & $\pm0.03$ & 2.7 & $\pm0.3$ & \multicolumn{7}{c}{No background} & 60086.2 \\
{\bf B} & 1.67 & $\pm0.03$ & 2.9 & $\pm0.3$ & $-21.0$ & $\pm0.2$ & & & & & & 60052.8 \\
{\bf C} & 1.67 & $\pm0.03$ & 3.0 & $\pm0.3$ & $-21.0$ & $\pm0.2$ & $-1$ & $\pm3$ & & & & 60052.6 \\
{\bf D} & 1.67 & $\pm0.04$ & 3.0 & $\pm0.3$ & $-20.9$ & $\pm0.3$ & 0 & $+5,-6$ & $-2$ & $\pm4$ & 3.25 & 60052.4 \\
{\bf E} & 1.67 & $\pm0.03$ & 3.0 & $\pm0.3$ & $-20.9$ & $+0.3,-0.2$ & 0 & $+5,-6$ & $-2$ & $+7,-4$ & 3.25 & 60051.4 \\
\end{tabular}
\caption{
The best--fit parameters and expected errors for the models.}
\label{tab:fpars}
\end{table*}
The maximum likelihood `best--fit' values for the parameters are given
in table \ref{tab:fpars}. The quoted errors are the differences (a
single value if the distribution is symmetric) at which the
probability falls by the factor $1/\sqrt{e}$. This is equivalent to a
`$1\sigma$ error' for parameters with normal error distributions.
The observed evolution of the number of clouds per unit redshift is
described in the standard notation found in the literature
\begin{equation}dN/dz = A_0 (1+z)^{\gamma}\ .\end{equation} The
variable used in the maximum likelihood fits here, $\gamma^\prime$,
excludes variations expected from purely cosmological variations and
is related to $\gamma$ by: \begin{equation} \gamma = \left\{
\begin{array}{ll}\gamma^\prime + 1 & \mbox{ if $q_0 = 0$} \\
\gamma^\prime + \frac{1}{2} & \mbox{ if $q_0 = 0.5$ \ .}\end{array}
\right.\end{equation}
Figure \ref{fig:var_d}\ shows the variation in population parameters
for model {\bf D} as the completeness limits are increased in steps of
$\Delta\hbox{$\log( N )$}=0.1$. The number of clouds decreases from 844 to 425
(when the completeness levels have been increased by
$\Delta\hbox{$\log( N )$}=0.5$).
\begin{figure*}
\hbox{
\epsfxsize=8.5cm
\epsfbox{beta.ps}
\hfill
\epsfxsize=8.5cm
\epsfbox{gamma.ps}
}
\epsfverbosetrue
\caption{The expected probability distribution of the model parameters $\beta$ and $\gamma$ (heavy line) for model {\bf D}. The dashed histogram and Gaussian (thin line) show how the measured value varies for different sets of data. The dash-dot line shows the uncertainty in the parameter because the data are limited. These are combined to give the final distribution (bold). See section \ref{sec:erress} for mode details.}
\label{fig:beta_gamma_d}
\end{figure*}
\begin{figure*}
\hbox{
\epsfxsize=8.5cm
\epsfbox{nu1_b.ps}
\hfill
\epsfxsize=8.5cm
\epsfbox{nu1_d.ps}
}
\epsfverbosetrue
\caption{The expected probability distribution of the log background flux at $z=3.25$ (heavy line) for models {\bf B} (left) and {\bf D}. The uncertainty from the small number of lines near the quasar (line with points) is significantly larger than that from uncertainties in column densities or quasar properties (thin curve). See section \ref{sec:erress} for a full description of the plot.}
\label{fig:nu1_d}
\end{figure*}
\begin{figure*}
\hbox{
\epsfxsize=8.5cm
\epsfbox{nu2_d.ps}
\hfill
\epsfxsize=8.5cm
\epsfbox{nu3_d.ps}
}
\epsfverbosetrue
\caption{The expected probability distribution of the model parameters $\alpha_1$ and $\alpha_2$ (heavy line) for model {\bf D}. See section \ref{sec:erress} for a full description of the plot.}
\label{fig:alphas_d}
\end{figure*}
\begin{figure*}
\hbox{
\epsfxsize=8.5cm
\epsfbox{beta_all.ps}
\hfill
\epsfxsize=8.5cm
\epsfbox{gamma_all.ps}
}
\epsfverbosetrue
\caption{The expected probability distribution of the population parameters for model {\bf D}. The top curve is for all data, each lower curve is for data remaining when the column density completeness limits are progressively increased by $\Delta\hbox{$\log( N )$}=0.1$.}
\label{fig:var_d}
\end{figure*}
\subsection{Ionising Background}
\label{sec:resbackg}
Values of the ionising flux parameters are show in table
\ref{tab:fpars}. The expected probability distributions for models
{\bf B} and {\bf D} are shown in figures \ref{fig:nu1_d} and
\ref{fig:alphas_d}. The background flux relation is described in
section \ref{sec:popden}.
The variables used to describe the variation of the flux with redshift
are strongly correlated. To illustrate the constraints more clearly
the marginalised posterior distribution (section \ref{sec:postprob})
of J$_{23}$\ was calculated at a series of redshifts. These are shown
(after convolution with the combination of Gaussians appropriate for
the uncertainties in the parameters from observational errors) for
model {\bf D} in figure \ref{fig:flux_both}. The distribution at each
redshift is calculated independently. This gives a conservative
representation since the marginalisation procedure assumes that
parameters can take all possible values consistent with the background
at that redshift (the probability that the flux can be low at a
certain redshift, for example, includes the possibility that it is
higher at other redshifts). Figure \ref{fig:flux_both} also compares
the results from the full data set (solid lines and smaller boxes)
with those from the data set with column density completeness limits
raised by $\Delta\log(N)=0.5$ (the same data as the final curves in
figure \ref{fig:var_d}).
Table~\ref{tab:modeld} gives the most likely flux (at probability
$p_m$), an estimate of the `1$\sigma$ error' (where the probability
drops to $p_m/\sqrt{e}$), the median flux, the upper and lower
quartiles, and the 5\% and 95\% limits for model {\bf D} at the
redshifts shown in figure~\ref{fig:flux_both}. It is difficult to
assess the uncertainty in these values. In general the central
measurements are more reliable than the extreme limits. The latter
are more uncertain for two reasons. First, the distribution of
unlikely models is more likely to be affected by assumptions in
section~\ref{sec:postprob} on the normal distribution of secondary
parameters. Second, the tails of the probability distribution are
very flat, making the flux value sensitive to numerical noise.
Extreme limits, therefore, should only be taken as a measure of the
relevant flux magnitude. Most likely and median values are given to
the nearest integer to help others plot our results --- the actual
accuracy is probably lower.
\begin{table}
\begin{tabular}{crrrrrr}
&$z=2$&$z=2.5$&$z=3$&$z=3.5$&$z=4$&$z=4.5$\\
$p_m/\sqrt{e}$ & 30 & 50 & 60 & 60 & 40 & 30 \\
$p_m$ & 137 & 129 & 118 & 103 & 80 & 63 \\
$p_m/\sqrt{e}$ &1000 & 400 & 220 & 180 & 160 & 170 \\
5\% & 10 & 30 & 50 & 40 & 30 & 20 \\
25\% & 70 & 80 & 80 & 70 & 60 & 40 \\
50\% & 232 & 172 & 124 & 108 & 87 & 75 \\
75\% &1000 & 400 & 200 & 160 & 100 & 200 \\
95\% &30000&3000 & 400 & 300 & 400 & 600 \\
\end{tabular}
\caption{
The fluxes (J$_{23}$) corresponding to various posterior probabilities for model {\bf D}. See the text for details on the expected errors in these values.}
\label{tab:modeld}
\end{table}
\begin{figure*}
\hbox{
\epsfxsize=8.5cm
\epsfbox{flux_bothdy2.ps}
\hfill
\epsfxsize=8.5cm
\epsfbox{box_bothy2.ps}
}
\epsfverbosetrue
\caption{The expected probability distribution of the log background flux for model {\bf D}, comparing the results from the full data set with those obtained when the column density completeness limit is raised by $\Delta\log(N)=0.5$ (dashed line, left; larger boxes, right). The box plots show median, quartiles, and 95\% limits.}
\label{fig:flux_both}
\end{figure*}
\section{Discussion}
\label{sec:discuss}
\subsection{Population Parameters}
\label{sec:dispop}
Parameter values for the different models are given in table
\ref{tab:fpars}. They are generally consistent with other estimates
(Lu, Wolfe \& Turnshek 1991; Rauch et~al. 1993). Including the proximity effect
increases $\gamma$ by $\sim0.2$. Although not statistically
significant, the change is in the sense expected, since local
depletions at the higher redshift end of each data set are removed.
Figure \ref{fig:var_d}\ shows the change in population parameters as
the completeness limits for the observations are increased. The most
likely values (curve peaks) of both $\beta$ and $\gamma$ increase as
weaker lines are excluded, although $\gamma$ decreases again for the
last sample.
The value of $\beta$ found here ($\sim 1.7$) is significantly
different from that found by Press \& Rybicki (1993) ($\beta \sim 1.4$) using a
different technique (which is insensitive to Malmquist bias and line
blending). The value of $\beta$ moves still further away as the
column density completeness limits are increased. This is not
consistent with Malmquist bias, which would give a smaller change in
$\beta$ (section~\ref{sec:malm}), but could be a result of either line
blending or a population in which $\beta$ increases with column
density. The latter explanation is also consistent with
Cristiani et~al. (1995) who found a break in the column density distribution
with $\beta$ = 1.10 for log($N$)$ < 14.0$, and $\beta$ = 1.80 above
this value. Later work (Giallongo et~al. 1996, see section~\ref{sec:prevhi})
confirmed this.
Recent work by Hu et~al. (1995), however, using data with better
signal--to--noise and resolution, finds that the distribution of
column densities is described by a single power law ($\beta\sim 1.46$)
until $\hbox{$\log( N )$}\sim 12.3$, when line blending in in their sample becomes
significant. It might be possible that their sample is not
sufficiently large (66 lines with $\log(N)>14.5$, compared with 192
here) to detect a steeper distribution of high column density lines.
The change in $\gamma$ as completeness limits are raised may reflect
the decrease in line blending at higher column densities. This
suggests that the value here is an over--estimate, although the shift
is within the 95\% confidence interval. No estimate is significantly
different from the value of 2.46 found by Press \& Rybicki (1993) (again, using
a method less susceptible to blending problems).
\subsection{The Proximity Effect}
\subsubsection{Is the Proximity Effect Real?}
\label{sec:disevid}
The likelihood ratio statistic (equivalent to the `F test'), comparing
model {\bf A} with any other, indicates that the null hypothesis (that
the proximity effect, as described by the model here, should be
disregarded) can be rejected with a confidence exceeding 99.9\%. Note
that this confirmation is based on the likelihood values in
table~\ref{tab:fpars}. This test is much more powerful than the K--S
test (section~\ref{sec:finalq}) which was only used to see whether the
models were sufficiently good for the likelihood ratio test to be
used.
To reiterate: if model {\bf A} and model {\bf B} are taken as
competing descriptions of the population of Lyman--$\alpha$\ clouds, then the
likelihood ratio test, which allows for the extra degree of freedom
introduced, strongly favours a description which includes the
proximity effect. The model without the proximity effect is firmly
rejected. This does not imply that the interpretation of the effect
(ie.\ additional ionization by background radiation) is correct, but
it does indicate that the proximity effect, in the restricted,
statistical sense above, is `real' (cf. R\"{o}ser 1995).
If the assumptions behind this analysis are correct, in particular
that the proximity effect is due to the additional ionising flux from
the quasar, then the average value of the background is
100\elim{50}{30}~J$_{23}$\ (model {\bf B}).
If a more flexible model for the background (two power laws) is used
the flux is consistent with a value of 120\elim{110}{50}~J$_{23}$\
(model {\bf D} at $z=3.25$).
\subsubsection{Systematic Errors\label{sec:syserr}}
Five sources of systematic error are discussed here: Malmquist bias
and line blending; reddening by damped absorption systems; increased
clustering of clouds near quasars; the effect of gravitational
lensing.
The constraints on the background given here may be affected by
Malmquist bias and line blending (sections \ref{sec:malm} and
\ref{sec:dispop}). The effects of line blending will be discussed
further in section~\ref{sec:gerry}, where a comparison with a
different procedure suggests that it may cause us to over--estimate
the flux (by perhaps $0.1$ dex). Malmquist bias is more likely to
affect parameters sensitive to absolute column densities than those
which rely only on relative changes in the observed population. So
while this may have an effect on $\beta$, it should have much less
influence on the inferred background value.
Attenuation by intervening damped absorption systems will lower the
apparent quasar flux and so give an estimate for the background which
is too low. This is corrected in model {\bf E}, which includes
adjustments for the known damped systems (section \ref{sec:dampcor},
table~\ref{tab:damp}). The change in the inferred background flux is
insignificant (figure \ref{fig:evoln}, table~\ref{tab:fpars}),
implying that the magnitude of the bias is less than 0.1 dex.
If quasars lie in regions of increased absorption line clustering
(Loeb \& Eisenstein 1995; Yurchenko, Lanzetta \& Webb 1995) then the background flux may be
overestimated by up to 0.5, or even 1, dex.
Gravitational lensing may change the apparent brightness of a quasar
--- in general the change can make the quasar appear either brighter
or fainter. Absorption line observations are made towards the
brightest quasars known (to get good quality spectra). Since there
are more faint quasars than bright ones this will preferentially
select objects which have been brightened by lensing (see the comments
on Malmquist bias in section~\ref{sec:malm}). An artificially high
estimate of the quasar flux will cause us to over--estimate the
background.
Unfortunately, models which assess the magnitude of the increase in
quasar brightness are very sensitive to the model population of
lensing objects. From Pei (1995) an upper limit consistent with
observations is an increase in flux of about 0.5 magnitudes,
corresponding to a background estimate which is too high by a factor
of 1.6 (0.2 dex). The probable effect, however, could be much smaller
(Blandford \& Narayan 1992).
If bright quasars are more likely to be lensed we can make a
rudimentary measurement of the effect by splitting the data into two
separate samples. When fitted with a constant background (model {\bf
B}) the result for the five brightest objects is indeed brighter than
that for the remaining six, by 0.1 dex. The errors, however, are
larger (0.3 dex), making it impossible to draw any useful conclusions.
The effects of Malmquist bias, line blending and damped absorption
systems are unlikely to change the results here significantly. Cloud
clustering and gravitational lensing could be more important --- in
each case the background would be over--estimated. The magnitude of
these last two biases is not certain, but cloud clustering seems more
likely to be significant.
\subsubsection{Is there any Evidence for Evolution?}
\label{sec:noevoln}
More complex models allow the background flux to vary with redshift.
If the flux does evolve then these models should fit the data better.
However, there is no significant change in the fit when comparing the
likelihood of models {\bf C} to {\bf E} with that of {\bf B}. Nor are
$\alpha_1$ or $\alpha_2$ significantly different from zero. So there
is no significant evidence for a background which changes with
redshift.
The asymmetries in the wings of the posterior distributions of
$\alpha_1$ or $\alpha_2$ for model {\bf D} (figure \ref{fig:alphas_d})
are a result of the weak constraints on upper limits (see next
section). The box plots in figure \ref{fig:flux_both} illustrate the
range of evolutions that are possible.
\subsubsection{Upper and Lower Limits\label{sec:lims}}
\begin{figure*}
\hbox{
\epsfxsize=8.5cm
\epsfbox{delta_z.ps}
\hfill
\epsfxsize=8.5cm
\epsfbox{delta_d.ps}
}
\epsfverbosetrue
\caption{On the left, absorber redshifts are plotted against $\Delta_F$. On the right $\Delta_F$ is plotted against the distance between cloud and quasar. Note that the correction for the quasar's flux, and hence the upper limit to the estimate of the background, is significant for only a small fraction of the clouds.}
\label{fig:prox}
\end{figure*}
While there is little evidence here for evolution of the background,
the upper limits to the background flux diverge more strongly than the
lower limits at the lowest and highest redshifts. Also, the posterior
probability of the background is extended more towards higher values.
The background was measured by comparing its effect with that of the
quasar. If the background were larger the quasar would have less
effect and the clouds with $\Delta_F < 1$ would not need as large a
correction to the observed column density for them to agree with the
population as a whole. If the background was less strong then the
quasars would have a stronger influence and more clouds would be
affected.
The upper limit to the flux depends on clouds influenced by the
quasar. Figure \ref{fig:prox} shows how $\Delta_F$ changes with
redshift and proximity to the background quasar. From this figure it
is clear that the upper limit is dominated by only a few clouds.
However, the lower limit also depends on clouds near to, but not
influenced by, the quasar. This involves many more clouds. The lower
limit is therefore stronger, more uniform, and less sensitive to the
amount of data, than the upper limit.
Other procedures for calculating the errors in the flux have assumed
that the error is symmetrical (the only apparent exception is
Fern\'{a}ndez--Soto et~al. (1995) who unfortunately had insufficient data to normalize
the distribution). While this is acceptable for $\beta$ and $\gamma$,
whose posterior probability distributions
(figure~\ref{fig:beta_gamma_d}) can be well--approximated by Gaussian
curves, it is clearly wrong for the background
(eg. figure~\ref{fig:flux_both}), especially where there are less data
(at the lowest and highest redshifts).
An estimate based on the assumption that the error is normally
distributed will be biased in two ways. First, since the extended
upper bound to the background has been ignored, it will underestimate
the `best' value. Second, since the error bounds are calculated from
the curvature of the posterior distribution at its peak (ie. from the
Hessian matrix) they will not take account of the extended `tails' and
so will underestimate the true range of values. In addition, most
earlier work has calculated errors assuming that the other population
parameters are fixed at their best--fit values. This will also
under--estimate the true error limits. All these biases become more
significant as the amount of data decreases.
The first of these biases also makes the interpretation of the
box--plots (eg. figures \ref{fig:flux_both} and \ref{fig:evoln}) more
difficult. For example, the curves in the left--hand plot in
figure~\ref{fig:flux_both} and the data in table~\ref{tab:modeld} show
that the value of the flux with highest probability at $z=2$ is
$140$~J$_{23}$\ (for model {\bf D}). In contrast the box--plot on the
right shows that the median probability is almost twice as large
($230$~J$_{23}$). Neither plot is `wrong': this is the consequence of
asymmetric error distributions.
\begin{figure*}
\epsfxsize=15.cm
\epsfbox{box_bothe3.ps}
\epsfverbosetrue
\caption{The expected probability distribution of the log background
flux for models {\bf D} (left) and {\bf E} (right, including a
correction for the known damped absorption systems). The box plots
show median, quartiles, and 95\% limits. The shaded area covers the
range of backgrounds described in Fall \& Pei (1995). The lower boundary
is the expected background if all quasars are visible, the higher
fluxes are possible if an increasing fraction of the quasar population
is obscured at higher redshifts. The crosses and arrows mark the
extent of previous measurements from high resolution spectra --- see
the text for more details.}
\label{fig:evoln}
\end{figure*}
\subsection{Comparison with Previous Estimates}
\subsubsection{Earlier High--Resolution Work}
\label{sec:prevhi}
Fern\'{a}ndez--Soto et~al. (1995) fitted high signal--to--noise data towards three
quasars. For $2 < z < 2.7$\ they estimate an ionizing background
intensity of 32~J$_{23}$, with an absolute lower limit (95\% confidence)
of 16~J$_{23}$\ (figure~\ref{fig:evoln}, the leftmost cross). They were
unable to put any upper limit on their results.
Cristiani et~al. (1995) determined a value of 50~J$_{23}$\ using a sample of five
quasars with a lower column density cut--off of log($N$) = 13.3. This
sample was recently extended (Giallongo 1995). They find that the
ionizing background is roughly constant over the range $1.7 < z <
4.0$\ with a value of 50~J$_{23}$\ which they considered a possible
lower limit (figure~\ref{fig:evoln}, the middle lower limit).
While this paper was being refereed Giallongo et~al. (1996) became available,
extending the work above. Using a maximum likelihood analysis with an
unspecified procedure for calculating errors they give an estimate for
the background of $50\pm10$~J$_{23}$. They found no evidence for
evolution with redshift when using a single power law exponent.
Williger et~al. (1994) used a single object, Q1033--033, which is included in
this sample, to give an estimate of $10-30$~J$_{23}$\
(figure~\ref{fig:evoln}, the rightmost cross). The error limits are
smaller than those found here, even though they only use a subset of
this data, which suggests that they have been significantly
underestimated.
If the errors in Williger et~al. (1994) are indeed underestimates then these
measurements are consistent with the results here. However, the
best--fit values are all lower than those found here. This may be, at
least partly, because of the biases discussed in
section~\ref{sec:lims}.
Williger et~al. (1994) used a more direct method than usual to estimate the
background. This gives a useful constraint on the effect of
line blending in the procedures used, and is explored in more detail
below.
\subsubsection{Q1033--033 and Line Blending\label{sec:gerry}}
The measured value of the background, 80\elim{80}{40}~J$_{23}$\ (model
{\bf D} at $z=4$), is larger than an earlier estimate using a subset
of this data (Williger et~al. 1994, Q1033--033, $10-30$~J$_{23}$).
As has already been argued, it is difficult to understand how a
procedure using much less data could have smaller error limits than
the results here, so it is likely that the error was an underestimate
and that the two results are consistent. However, it is interesting
to see if there is also a systematic bias in the analyses used.
The correction for galactic absorption is not very large for this
object (about 20\%). More importantly, the procedures used differ
significantly in how they are affected by blended lines. These are a
problem at the highest redshifts, where the increased Lyman--$\alpha$\ cloud
population density means that it is not always possible to resolve
individual clouds. Williger et~al. (1994) added additional lines ($\hbox{$\log( N )$} =
13.7$, $b$ = 30~\hbox{\rm km\thinspace s$^{-1}$}) to their $z$ = 4.26 spectra of Q1033--033 and
found that between 40\% and 75\% would be missed in the line list.
As the lower column density limit is raised Williger et~al. (1994) find that
the observed value of $\gamma$ also increases. The resulting stronger
redshift evolution would make the deficit of clouds near the quasar
more significant and so give a lower estimate of the background.
Although not significant at the 95\% level, there is an indication
that $\gamma$ also increases with higher column density in this
analysis (section \ref{sec:dispop}, figure \ref{fig:var_d}). While it
is possible that $\gamma$ varies with column density the same
dependence would be expected if line blending is reducing the number
of smaller clouds. To understand how line blending can affect the
estimates, we will now examine the two analyses in more detail.
Line blending makes the detection of lines less likely. Near the
quasar lines are easier to detect because the forest is more sparse.
In the analysis used in this paper the appearance of these `extra'
lines reduces the apparent effect of the quasar. Alternatively, one
can say that away from the quasar line blending lowers $\gamma$.
Both arguments describe the same process and imply that the estimated
background flux is too large.
In contrast, Williger et~al. (1994) take a line--list from a crowded region,
which has too few weak lines and correspondingly more saturated lines,
and reduce the column densities until they agree with a region closer
to the quasar. Since a few saturated lines are less sensitive to the
quasar's flux than a larger number of weaker lines, the effect of this
flux is over--estimated (and poorly determined), making the background
seem less significant and giving a final value for the background flux
which is too small. This method is therefore biased in the opposite
sense to ours and so the true value of the background probably lies
between their estimate and ours.
The comparison with Williger et~al. (1994) gives one estimate of the bias from
line blending. Another can be made by raising the completeness limits
of the data (section~\ref{sec:malm}). This should decrease the number
of weak, blended lines, but also excludes approximately half the data.
In figure \ref{fig:flux_both} the flux estimates from the full data
set are shown together with those from one in which the limits have
been raised by $\Delta\log(N)=0.5$. There is little change in the
lowest reasonable flux, an increase in the upper limits, and in
increase in the `best--fit' values. The flux for $z<3$ is almost
unconstrained by the restricted sample (section~\ref{sec:lims}
explains the asymmetry).
An increase of 0.5 in $\log(N)$ is a substantial change in the
completeness limits. That the lower limits remain constant (to within
$\sim 0.1$ dex) suggests that line blending is not causing the flux to
be significantly over--estimated. The increase in the upper limits is
expected when the number of clouds in the sample decreases
(section~\ref{sec:lims}).
In summary, the total difference between our measurement and that in
Williger et~al. (1994) is 0.7 dex which can be taken as an upper limit on the
effect of line blending. However, a more typical value, from the
constancy of the lower limits when completeness limits are raised, is
probably $\sim0.1$ dex.
\subsubsection{Results from Lower Resolution Spectra}
Bechtold (1994) analysed lower resolution spectra towards 34 quasars
using equivalent widths rather than individual column density
measurements. She derived a background flux of 300~J$_{23}$\
($1.6<z<4.1$), decreasing to 100~J$_{23}$\ when a uniform correction was
applied to correct for non--systemic quasar redshifts. With
low--resolution data a value of $\beta$ is used to change from a
distribution of equivalent widths to column densities. If $\beta$ is
decreased from 1.7 to a value closer to that found for narrower lines
(see section~\ref{sec:dispop}) then the inferred background estimate
could decrease further.
The evolution was not well--constrained ($-7<\alpha<4$). No
distinction was made between the lower and upper constraints on the
flux estimate, and it is likely that the wide range of values reflects
the lack of strong upper constraints which we see in our analysis.
It is not clear to what extent this analysis is affected by line
blending. Certainly the comments above --- that relatively more
clouds will be detected near the quasar --- also apply.
\subsubsection{Lower Redshift Measurements}
The background intensity presented in this paper is much larger than
the 8~J$_{23}$\ upper limit at $z=0$ found by
Vogel et~al. (1995). Kulkarni \& Fall (1993) obtain an even lower value of
0.6\elim{1.2}{0.4}~J$_{23}$\ at $z=0.5$ by analysing the proximity
effect in HST observations. However, even an unevolving flux will
decrease by a factor of $\sim 50$ between $z=2$ and $0$, so such a
decline is not inconsistent with the results given here.
\subsection{What is the Source of the Background?}
\label{sec:source}
\subsubsection{Quasars}
Quasars are likely to provide a significant, if not dominant,
contribution to the extragalactic background. An estimate of the
ionizing background can be calculated from models of the quasar
population. Figure \ref{fig:evoln} shows the constraints from models
{\bf D} and {\bf E} and compares them with the expected evolution of
the background calculated by Fall \& Pei (1995). The background can take
a range of values (the shaded region), with the lower boundary
indicating the expected trend for a dust--free universe and larger
values taking into account those quasars that may be hidden from our
view, but which still contribute to the intergalactic ionizing flux.
The hypothesis that the flux is only from visible quasars (the
unobscured model in Fall \& Pei 1995) is formally rejected at over the
95\% significance level since the predicted evolution is outside the
95\% bar in the box plots at higher redshift.
Although our background estimate excludes a simple quasar--dominated
model based on the observed number of such objects, the analysis here
may give a background flux which is biased (too large) from a
combination of line blending (section~\ref{sec:gerry}) and clustering
around the background quasars. From the comparison with
Williger et~al. (1994), above, there is an upper limit on the correction for
line blending, at the higher redshifts, of 0.7 dex. However, an
analysis of the data when column density completeness limits were
increased by $\Delta\log(N)=0.5$ suggests that a change in the lower
limits here of $\sim 0.1$ dex is more likely. A further change of up
to between 0.5 and 1 dex is possible if quasars lie in regions of
increased clustering (section~\ref{sec:syserr}). These two effects
imply that at the highest redshifts the flux measured here could
reasonably overestimate the real value by $\sim 0.5$ dex. This could
make the measurements marginally consistent with the expected flux
from the observed population of quasars.
There is also some uncertainty in the expected background from quasars
since observations could be incomplete even at the better understood
lower redshifts (eg.~Goldschmidt et~al. 1992) and while absorption in damped
systems is understood in theory (Fall \& Pei 1993) its effect is
uncertain (particularly because the distribution of high column
density systems is poorly constrained).
The highest flux model (largest population of obscured quasars) from
Fall \& Pei (1995) is consistent with the measurements here (assuming that
the objects used in this paper are not significantly obscured).
\subsubsection{Stars}
The background appears to be stronger than the integrated flux from
the known quasar population. Can star formation at high redshifts
explain the discrepancy?
Recent results from observations of low redshift starbursts
(Leitherer et~al. 1995) suggest that very few ionizing photons ($\leq 3\,$\%)
escape from these systems. If high redshift starbursts are similar in
their properties, then the presence of cool gas in these objects would
similarly limit their contribution to the ionizing background.
However, Madau \& Shull (1996) estimate that if star formation occurs in
Lyman--$\alpha$\ clouds, and a significant fraction of the ionizing photons
($\sim 25\,\%$) escape, then these photons may contribute a
substantial fraction of the ionizing background photons in their
immediate vicinity. As an example, at $z \sim 3$\ they estimate that
$J_\nu \leq 50~\hbox{J$_{23}$}$\ if star formation sets in at
$z\sim3.2$. This flux would dominate the lowest (no correction for
obscuration) quasar background shown in figure \ref{fig:evoln} and
could be consistent with the intensity we estimate for the background
at this redshift, given the possible systematic biases discussed above
and in section~\ref{sec:syserr}.
\section{Conclusions}
\label{sec:conc}
A model has been fitted to the population of Lyman--$\alpha$\ clouds. The model
includes the relative effect of the ionising flux from the background
and nearby quasars (section~\ref{sec:model}).
The derived model parameters for the population of absorbers are
generally consistent with earlier estimates. There is some evidence
that $\beta$, the column density power law population exponent,
increases with column density, but could also be due to line blending
(section~\ref{sec:dispop}).
The ionising background is estimated to be 100\elim{50}{30}~J$_{23}$\
(model {\bf B}, section~\ref{sec:resbackg}) over the range of
redshifts ($2<z<4.5$) covered by the data. No strong evidence for
evolution in the ionizing background is seen over this redshift range.
In particular, there is no significant evidence for a decline for
$z>3$ (section~\ref{sec:noevoln}). Previous results may have been
biased (too low, with optimistic error limits ---
section~\ref{sec:lims}).
Constraints on the evolution of the background are shown in figure
\ref{fig:evoln}. The estimates are not consistent with the background
flux expected from the observed population of quasars
(section~\ref{sec:source}). However, two effects are likely to be
important. First, both line blending and increased clustering of
clouds near quasars lead to the measured background being
overestimated. Second, a significant fraction of the quasar
population at high redshifts may be obscured. Since their
contribution to the background would then be underestimated this would
imply that current models of the ionizing background are too low.
Both of these would bring the expected and measured fluxes into closer
agreement. It is also possible that gravitational lensing makes the
measurement here an overestimate of the true background.
The dominant source of errors in our work is the limited number of
lines near the background quasar (eg. figures \ref{fig:nu1_d}\ and
\ref{fig:prox}). Systematic errors are smaller and become important
only if it is necessary to make standard (unobscured quasar) models
for the background consistent with the lower limits presented here.
Further data will therefore make the estimate here more accurate,
although observational data are limited by confusion of the most
numerous lower column density systems ($\hbox{$\log( N )$}< 13.0$) so it will
remain difficult to remove the bias from line blending. An
improvement in the errors for the highest redshift data points, or a
determination of the shape of the ionizing spectrum (e.g. from
He~II/H~I estimates in Lyman--$\alpha$\ clouds) would help in discriminating
between current competing models for the ionizing background.
Finally, a determination of the background strength in the redshift
range $0.5 < z < 2.0$ is still needed.
\section{Acknowledgements}
We would like to thank Yichuan Pei for stimulating discussions and for
making data available to us. Tom Leonard (Dept. of Statistics,
Edinburgh) gave useful comments and guidance on the statistics used in
this paper. We would also like to thank an anonymous referee for
helpful and constructive comments.
| proofpile-arXiv_065-429 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
The study of the cosmic velocity field is a very promising and
crucial area for the understanding of large--scale structure
formation. Since the early work of Rubin et al. (1976) and Burstein et
al. (1987) a lot of effort has been invested in the measurement of the
large--scale velocity flows (see Dekel 1994 and Strauss \& Willick
1995 for recent reviews of the subject). Particularly important
for the analysis of these measured velocity fields has been the
development of the parameter-free POTENT method by Bertschinger
\& Dekel (1989). Based on the plausible assumption of potential flow
it enabled the construction and study of the full three-dimensional
velocity field in a fair fraction of the local Universe out of the
measurements of galaxy line-of-sight peculiar velocities. This
opened up and triggered a host of studies addressing various issues
and aspects of cosmic velocity flows and provided a versatile
ground for testing the scenario of large--scale structure formation.
The cosmic velocity field is particularly interesting because of its
close and direct relation to the underlying field of mass
fluctuations. Indeed, on these large scales the acceleration, and
therefore the velocity, of any object is expected to have an
exclusively gravitational origin so that it should be independent of its
nature, whether it concerns a dark matter particle or a bright
galaxy. Moreover, the linear theory of the generic gravitational
instability scenario predicts that at every location in the Universe
the local velocity is related to the local acceleration, and hence the
local mass density fluctuation field, through the same universal
function $f(\Omega)\approx\Omega^{0.6}$ (Peebles 1980). As the linear
theory provides a good description on those large scales the
use of this straightforward relation implies the possibility of a
simple inversion of the measured velocity field into a field that is
directly proportional to the field of local mass density
fluctuations $\delta = \rho/\mg\rho\md -1$.
Such a procedure can be used to infer the value of
$\Omega$, through a comparison of the resulting field with the field of mass
density fluctuations in the same region. However, the determination of this
mass density fluctuation field through the measurement of the
local galaxy density fluctuation field, $\delta_g$, may
be contrived. The galaxy distribution may be representing a biased
view of the underlying mass density fluctuation field. A common and
rather simplistic assumption is that $\delta_g$ and $\delta$ are
related via a linear bias factor $b$,
\begin{equation}
\delta_g(\vr)=b\ \delta(\vr).
\label{eq:def_bias}
\end{equation}
However, although several physical mechanisms have been invoked to
explain such a {\it linear bias model} (see e.g. Dekel \& Rees
1987), by lack of a complete and self-consistent theory of galaxy
formation it should as yet only be considered as a numerical
factor roughly describing the contrast of galaxy density fluctuations
with respect to the mass density fluctuations.
The comparison between the observed local galaxy density fluctuation
field and the local cosmic velocity field, invoking
equation~(\bibitem{eq:def_bias}), will therefore provide an estimate of
the ratio,
\begin{equation}
\beta=\frac{f(\Omega)}{b}\approx \frac{\Omega^{0.6}}{b}.
\end{equation}
Various studies, most notably the ones based on a comparison of the
galaxy density field inferred from the IRAS redshift survey of Strauss
et al. (1990) and the local velocity field reconstructed by the POTENT
algorithm (Bertschinger et al. 1990; Dekel, Bertschinger \& Faber 1990;
see Dekel 1994), have yielded estimates of $\beta$ in the range
$\beta \approx 0.5-1.2$ (see Dekel 1994, Strauss \& Willick 1995,
for compilations of results).
\begin{figure}
\vskip 7.5 cm
\special{hscale=50 vscale=50 hoffset=0 voffset=-80 psfile=sketch.ps}
\caption{
The PDF of the velocity divergence as given by Eq. (\bibitem{eq:PDF_theor_exact})
showing its dependence with $\Omega$ and $\sigma$.}
\label{fig:PDF_theor}
\end{figure}
It is then of crucial interest to find ways to disentangle the
contribution of $\Omega$ and $b$ to $\beta$. A variety of methods using {\it
intrinsic} properties of the large-scale velocity field have
been proposed to achieve this. One such attempt is based on the
reconstruction of the initial density field from the observed
distribution of matter through the use of the Zel'dovich
approximation. The further assumption of Gaussianity of the
initial density probability distribution leads to a constraint on
$\Omega$ (Nusser \& Dekel 1993). Although promising, the quantitative
results of this approach may be questionable, as the Zel'dovich
approximation provides non-exact results for the induced
non-Gaussian properties of the density and velocity field (Bouchet et
al 1992, Bernardeau 1994a). Another interesting attempt is the one
by Dekel \& Rees (1994), who exploit the simple observation that
voids have a maximal ``emptiness'' of $\delta=-1$. On the basis
of the corresponding analysis of the measured velocity field in and
around a void region these authors inferred a lower limit on $\Omega$
of about 0.3.
In this paper we focus on a method to determine the value of
$\Omega$ that finds its origin in a statistical analysis of the
velocity field. The foundation for this method is formed by analytical
work within the context of the perturbation theory for the evolution
of density and velocity fluctuations. In particular, it focuses on
the statistical properties of the divergence of the
locally smoothed velocity field $\theta$, which is defined as
\begin{equation}
\theta \equiv \frac{\nabla \cdot {\bf v}}{H},
\label{eq:def_theta}
\end{equation}
where $\bf v$ is the time derivative of the comoving coordinate
$\bf x$, and $\nabla = {\partial}/{\partial \bf x}$. The method,
proposed by Bernardeau (1994a) and Bernardeau et al. (1995),
exploits the relations between the lower order moments of the
probability distribution function (PDF) of $\theta$, and the
explicit dependence on $\Omega$ of these relations. The specific
form of these relations were derived under the assumption of
Gaussian initial conditions.
The viability of the method was demonstrated by Bernardeau et
al. (1995), who used the strong $\Omega$-dependence of the skewness
factor of $\theta$, or rather the third normalized moment $T_3$,
\begin{equation}
T_3 \equiv \mg\theta^3\md/\mg\theta^2\md^2 \propto
\Omega^{-0.6}
\end{equation}
to estimate successfully the density parameter in N-body simulations
of structure formation. A tentative application of this method to the
observed velocity field as processed by the POTENT method yielded results
consistent with $\Omega=1$. Moreover, subsequent work by Bernardeau \& van de
Weygaert (1996) showed that the theoretical predictions concerning the
complete overall shape of the PDF of $\theta$ were valid. They
demonstrated this by comparison of the analytically predicted PDF with
the PDF determined from a large CDM N-body simulation with
$\Omega=1$. In order to deal with the major complication of obtaining
clean and straightforward numerical estimates of the statistical
properties of the velocity divergence field from the discretely
sampled velocity field, they developed two new techniques. These
techniques exploit the minimum triangulation properties of Voronoi and
Delaunay tessellations (Voronoi 1908, Delaunay 1934, see Van de
Weygaert 1991, 1994 for references and applications).
A Voronoi tessellation of a set of particles is a space filling
network of polyhedral cells, with each cell being defined by
one of the particles as its nucleus and delimiting the part of space
closer to this nucleus than to any other of the particles. Closely
related to the Voronoi tessellation is the Delaunay tessellation,
a space filling lattice of tetrahedra (in three dimensions). Each of
the tetrahedra in the Delaunay tessellation has four particles of
the set as its vertices, such that the corresponding circumscribing
sphere does not have any other particle inside. Through the duality
relation of the Voronoi and the Delaunay tessellation it is possible
to obtain one from the other.
In the method based on the Voronoi tessellation -- the ``Voronoi method''
-- the velocity field is defined to be uniform within each Voronoi
polyhedron, with the velocity at every location within each cell being
equal to that of its nucleus. The obvious implication of such an
interpolation scheme is that the only regions of space where the
velocity divergence, as well as the shear and vorticity, acquires a
non-zero value is in the polygonal walls that separate the cells.
While the Voronoi method can be regarded as a zeroth order
interpolation scheme, yielding a discontinuous velocity field, the
``Delaunay method'' can be seen as the corresponding first order
scheme. Basically it constructs the velocity field within each
Delaunay tetrahedron through linear interpolation between the
velocities of the four defining particles. Evidently, the velocity
field constructed by the Delaunay method is a field of uniform
velocity field gradients within each Delaunay cell.
In the following we consider the statistical properties of the local
velocity divergence when it is filtered by a top-hat window function
following a {\it volume weighting} prescription.
In practice the local filtered divergence is computed on $50^3$
grid points in all cases. Depending on the method used to define the
velocity field, the filtered divergence is given by
a sum of intersections of a sphere with either planar polygonal walls or
tetrahedra (each of them being multiplied by the local divergence)
divided by the volume of the sphere. In such a volume weighting
scheme the filtered quantities do not depend
on the local number density of tracers. But of course the more
numerous they are, the more accurately determined are the local divergences.
For an extensive description and preliminary tests of the two techniques
we refer to Bernardeau \& van de Weygaert (1996).
With the intention of demonstrating the potential and practical
applicability of our method, this paper presents the results of
a systematic study of the $\Omega$ dependence of the moments and
the PDF of the velocity divergence $\theta$ in N-body simulations
of structure formation, using the numerical schemes of the Voronoi
and the Delaunay method. To this end, we will first recall the
relevant theoretical results on the statistical properties of
$\theta$ in Section \bibitem{sec:PT}, thereby underlining the main
features of importance to our study. This theoretical groundwork
is followed by Section \bibitem{sec:num}, containing the presentation of
the numerical results of the statistical analysis of PM N-body
simulations of structure formation in $\Omega=1$ and $\Omega<1$
universes. While the first subsection, \S~\bibitem{sec:large_sample},
concerns the statistical quantities that were determined in a very large
sample and thus with maximum attainable accuracy, the second
subsection \S~\bibitem{sec:dilut} is devoted to the issue to what extent
these results get affected when the sample is strongly diluted. The
latter is particularly important as we are interested in the
reliability of our method for samples with a number density comparable
to that of available galaxy catalogues. Also of immediate
relevance for a practical application is the question of how far
the results of the statistical analysis get influenced when the
velocity of the particles in the sample are known along only one
direction. This issue, of crucial importance within the context
of the statistical analysis of observational catalogues, is treated
in a third subsection, in \S~\bibitem{sec:1D}. Finally, following the
successful application of our method under the circumstances
described above, we conclude with a summary and a discussion of
possible complications and prospects for our statistical method
to infer a bias-independent value of $\Omega$.
\section{Perturbation Theory of Structure Formation and the
velocity field Probability Distribution Function}
\label{sec:PT}
Perturbation Theory (PT) is extremely useful for the study and
analytical description of the mildly non linear evolution of density
and velocity fields. In particular within the context of structure
forming out of Gaussian initial conditions perturbation theory has
been extensively developed in a large body of work (see e.g.
Bernardeau 1994a,b). In the case of these Gaussian initial conditions
the complete set of moments of the smoothed velocity and density fields
can be computed analytically, in particular if these fields are
top-hat filtered. The corresponding PDF can be computed through
re-summation of the series of moments.
One of the most straightforward and useful results in the context of
perturbation theory is the relation between the third moment $\mg\theta^3\md$
and the second moment $\mg\theta^2\md$ of the probability distribution
function of $\theta$,
\begin{equation}
\mg\theta^3\md=T_3\,\mg\theta^2\md^2 = T_3\,\sigma_{\theta}^4.
\end{equation}
The coefficient $T_3$ depends on the cosmological parameter $\Omega$,
on the shape of the power spectrum, on the geometry of the window
function that has been used to filter the velocity field and even
on the value of the cosmological constant $\Lambda$, although the
latter is an almost negligible weak dependence. In fact, the
dependence of $T_3$ on $\Omega$ is substantially stronger than the
one of the
equivalent coefficient for density field. For
instance, for a top-hat window function and a power law initial power
spectrum of index $n$, i.e.
\begin{equation}
P(k) \equiv \mg\delta({\bf k})^2\md
\propto \mg\theta({\bf k})^2\md
\propto k^{n},
\end{equation}
one obtains the following expression for $T_3$,
\begin{equation}
T_3={-1\over \Omega^{0.6}}\left[{26\over7}-(3+n)\right].
\end{equation}
As $T_3$ can be directly determined from observations, one can use its
strong dependence on $\Omega$ to obtain an estimate of $\Omega$, as
has been done by Bernardeau et al. (1995).
More generally, Perturbation Theory enables one to infer the whole set of
the cumulants $\mg\theta^p\md$ to their leading order. All of
them are related to the second moment via the relation
\begin{equation}
\mg\theta^p\md=T_p\,\mg\theta^2\md^{p-1},
\label{eq:def_Tp}
\end{equation}
and as in the case of $T_3$ all the coefficients $T_p$ possess a
strong dependence on the value of $\Omega$ (Bernardeau 1994b). To a good
approximation, this dependence on cosmological parameters can be
written as
\begin{equation}
T_p(\Omega,\Lambda)\approx{1\over\Omega^{(p-2)\,0.6}}T_p(\Omega=1,\Lambda=0).
\end{equation}
This property, given here for the moments, naturally
extends itself to the shape of the complete velocity divergence
PDF $p(\theta)$. This can be directly appreciated from the work
by Bernardeau (1994b), who showed that the PDF can be calculated
from its moments $T_p$ and the value of $\sigma_{\theta}^2$
through a Laplace transform of its generating function $\varphi_{\theta}$
\begin{equation}
p(\Omega,\theta){\rm d}\theta=\int_{-{\rm i}\infty}^{+{\rm i}\infty}
{{\rm d} y\over
2\pi{\rm i}\sigma_{\theta}^2}\exp\left[-{ \varphi_{\theta}(\Omega,y)\over
\sigma_{\theta}^2}
+{ y\theta\over \displaystyle \sigma_{\theta}^2}\right]
{\rm d}\theta\,.
\label{eq:PDF_theor_exact}
\end{equation}
The moment generating function $\varphi_{\theta}(\Omega,y)$, given by
\begin{equation}
\varphi_{\theta}(\Omega,y)=\sum_{p=2}^{\infty}\ -T_p(\Omega)
\frac{(-y)^p}{p!},
\label{eq:gener_f}
\end{equation}
can be related to the spherical collapse dynamics in the cosmology
under consideration (see Appendix A). Although this calculation is
almost intractable in the general case, it has been shown that at
least in one particular case one can evaluate this expression
analytically. In the specific case of a power law density perturbation
spectrum with an index $n=-1$ in combination with a top-hat smoothing,
it is possible to invoke some approximations that enable the
derivation of a simple analytic fit for the PDF
$p(\theta)$\footnote{This fit is obtained
through approximations which tend to lower the values of the
moments (appendix A).
The PDF (\bibitem{eq:PDF_theor}) presented here is actually more
accurate if $n\approx -1.3$.} (see Appendix A). This fit is given by
\begin{eqnarray}
p(\theta){\rm d}\theta &=& {([2\kappa-1]/\kappa^{1/2}+
[\lambda-1]/\lambda^{1/2})^{-3/2} \over \kappa^{3/4} (2\pi)^{1/2}
\sigma_{\theta}}\, \nonumber \\
& \times &
\exp\left[-{ \theta^2\over 2\lambda
\sigma_{\theta}^2}\right]\,{\rm d}\theta,
\label{eq:PDF_theor}
\end{eqnarray}
with
$\kappa=1+\theta^2/\left(9\lambda \Omega^{1.2}\right),$
and $\lambda=1-2\theta/\left(3\Omega^{0.6}\right).$
\begin{figure*}
\vskip 16.7 cm
\special{hscale=90 vscale=90 voffset=-150 hoffset=-10 psfile=pdf_large_sample.ps}
\caption{The PDF of the velocity divergence for various values
of $\Omega$. The dotted lines correspond to the approximate analytic
fit (Eq.\bibitem{eq:PDF_theor}) and the solid lines to the theoretical predictions
using (A.15) and (A.14) with $n=-0.7$ obtained for the measured values
of $\sigma$ and $\Omega$.
The dashed lines are the predictions for $\Omega=1$ and the
same variance. The numerical estimations have been obtained using the
Delaunay method.}
\label{fig:PDF_large_sample}
\end{figure*}
\begin{figure*}
\vskip 7.8 cm
\special{hscale=90 vscale=90 voffset=-390 hoffset=-10 psfile=pdf_small_sample.ps}
\caption{Effects of dilution for $\Omega=0.33$ (left panels)
and $\Omega=1$ (right panels). The PDF-s have been obtained
with only 10,000 tracers in total which gives an average of
about 10 particles per cell.
The squares show the results of the Voronoi
method and the triangles the results of the Delaunay method.
The solid lines correspond to the theoretical predictions
for the variance obtained with each method and the ``right''
assumption for $\Omega$ and the dashed line with the ``wrong''
assumption for each case and method.}
\label{fig:PDF_small_sample}
\end{figure*}
The behaviour of this function $p(\theta)$ has been illustrated in
figure \bibitem{fig:PDF_theor} for various values of $\Omega$ and
$\sigma_{\theta}$. Qualitatively, one can see that the dependence of
the shape on $\Omega$ reveals itself in two ways: (1) the location of
its cut-off at high positive values of $\theta$ and (2) the location
of its peak.
As for (1), the maximum value that $\theta$ can obtain is known exactly
and is not dependent on the approximations that have been invoked to
derive expression (\bibitem{eq:PDF_theor})
\begin{equation}
\theta_{\rm max}=1.5\ \Omega^{0.6}.
\end{equation}
The value of $\theta_{\rm max}$ determines the location of the cut-off,
and therefore the maximum expansion rate in voids. The value of 1.5 is
the difference in value of the Hubble parameter in an empty,
$\Omega=0$, Universe and that of an Einstein-de Sitter Universe,
$\Omega=1$. Evidently, this is reflecting the fact that the interior of
the deepest voids locally mimic the behaviour of an $\Omega=0$
Universe.
Recall that the suggestion by Dekel \& Rees (1994) of
using the maximum emptiness of voids to constrain $\Omega$ is also
based on a similar feature.
Also quite sensitive to the value of $\Omega$ are the position of the
peak of the distribution function $p(\theta)$, i.e. the most likely
value of $\theta$, and the overall shape of $p(\theta)$. Using the Edgeworth
expansion (Juszkiewicz et al. 1995, Bernardeau \& Kofman 1995)
one can show that the value of $\theta$ for which the distribution
reaches its maximum is given by
\begin{equation}
\theta_{\rm peak}\approx
-{T_3\over 2}\ \sigma_{\theta}={1\over \Omega^{0.6}} \sigma_{\theta}.
\end{equation}
In fact, a procedure exploiting this dependence of shape and peak
location of $p(\theta)$ will probably yield a more robust measure of
$\Omega$ than the maximum value of $\theta_{\rm max}$ as it will be
less bothered by the noise in the tails.
\section{The $\Omega$ dependence in numerical simulations}
\label{sec:num}
By means of numerical simulations we have investigated the discussed
dependence of the PDF of $\theta$. These N-body simulations use a
Particle-Mesh (PM) code (Moutarde et al. 1991) with a $256^3$ grid to
follow the evolution of a system of $256^3$ particles. For our
project we used two simulations, one with $\Omega$ having a value of
$\Omega=1$ and the second one of $\Omega < 1$. By analyzing the latter
at different time-steps we explore situations for different
values of $\Omega$. The particle distribution in the two simulations
corresponds to a density and velocity fluctuation field with a
$P(k)\propto k^{-1}$ spectrum.
As can be seen in Table~\bibitem{tab:cumulant}, the variances
$\sigma_{\theta}$ do not differ significantly for the different values
of $\Omega$ for a given filtering radius. The fact that the values of
the variance are comparable simplifies a comparison of the PDF
substantially, which makes the interpretation
in terms of the intrinsic $\Omega$ dependence more straightforward.
\subsection{Measurements with a large number of tracers}
\label{sec:large_sample}
The first step of our analysis concerns an exploration of the velocity
field using a large number of tracers. For this
study the number of selected particles in each simulation is about
70,000, which for a cell radius of about 6\% of the box size leads to
a mean number of 67 particles per cell. The selection procedure used
here is deliberately biased towards low-density regions by inducing it
to retain a uniform density of particles all over the simulation
box. Except for its goal of achieving a better velocity
field coverage of low-density regions such a selection bias is not
expected to influence the velocity field analysis.
The methods that we use to analyze the simulations are exactly the
same ones as described by Bernardeau \& van de Weygaert
(1996). In fact, at this stage we only used the Delaunay method
to calculate numerically the shape of the PDF of the velocity
divergence. The results for the case of a large number of
velocity field tracers are shown in Fig.~\bibitem{fig:PDF_large_sample}.
The results are in good agreement with the theoretical predictions for
the values of $T_3$ and $T_4$ (see Table~\bibitem{tab:cumulant}), as well
as with the theoretical shape of the PDF (Fig.~\bibitem{fig:PDF_theor}).
It is in particular worth noting that the specific features expected
from equation~(\bibitem{eq:PDF_theor}) are indeed confirmed by the
numerical results. Notably, the locations of the cut-off, which are
very sensitive to rare event discrepancies, are
well reproduced (solid and dotted lines).
Moreover, as can be observed from the insets,
also the position and shape of the peak have been reproduced
very well (solid lines),
providing a strong discriminatory tool between different values of $\Omega$.
Within the context of these observations, we should issue a few
side remarks. Although the shape (\bibitem{eq:PDF_theor})
is very attractive because
it is a close analytic form, one should have in mind that it is only
approximate. Indeed it is derived from an approximate expression
for the cumulant generating function. The differences do not reveal
for the overall shape (the logarithmic plots) but are significant
for the shape of the peaks. For calculating the theoretical
predictions we are then forced to use a more accurate description
of the cumulants. To achieve this we use the relations (A.13, A.14)
with $n=-0.7$ (the expression \bibitem{eq:PDF_theor}
corresponds to $n=-1$) in the integral (A.15) which is then
computed numerically.
It is still an approximate expression, but
it yields the correct value for $T_3$ and a very good
approximation for the higher order cumulants. We should emphasize
that this slight
modification is only instrumental in obtaining the correct shape of the PDF
around its maximum, which is indeed almost entirely
determined by the values of the low order moments. This may be
understood for example from the properties of the Edgeworth expansion, for
which we refer to Juszkiewicz et al. 1995 and Bernardeau \& Kofman 1995.
\subsection{The effects of dilution}
\label{sec:dilut}
In order to check the robustness of the results when only
a limited number of tracers for the velocity field is available,
we performed numerical experiments where only 10,000 particles are
used to trace the velocity field. The selection of the sample
points in this diluted sample is completely random and does not
invoke the specific biased selection procedure that was used in
the case described in the former subsection. For this case of diluted
samples, we used both the Delaunay and the Voronoi methods for
analysis.
Figure~\bibitem{fig:PDF_small_sample} shows the PDFs obtained with both
methods, for an $\Omega=1$ simulation and for an $\Omega=0.33$
simulation. Both the Voronoi and the Delaunay methods appear to
yield numerical results that are in reasonably good agreement with the
predicted PDF. Particularly encouraging is
the result born out by the insets, namely the fact that the shape of
the peak can still be used as a strong discriminatory tool between different
values of $\Omega$.
Investigating Fig.~\bibitem{fig:PDF_small_sample} in somewhat more detail,
we can observe that the results obtained by the two methods are affected
somewhat differently by the dilution procedure. The PDFs obtained with the
Voronoi method generally possess a less sharply defined tail at the
side of the high $\theta$ values. This effect appears to be stronger for the
low $\Omega$ case. This behaviour may originate in the fact
that the divergence is localized to a limited part of space, non-zero
values being confined to the walls of the tessellation. In such a
situation Poisson like errors in the measurements are expected to
become particularly prominent in the heavily diluted areas of the void
regions. An additional difference is a slight underestimation of the
values of the $T_p$ coefficients by the Voronoi method (see
Table~\bibitem{tab:cumulant}). Unfortunately, we do not see a possibility
to correct for such effects. On the other hand, the Delaunay method
seems to be more robust against such effects. However, it tends to
underestimate the value of the variance and the higher order moments.
The latter is probably a consequence of the fact that the effective
filtering radius tends to be larger.
\subsection{The effects of reducing the information to one velocity component}
\label{sec:1D}
A major complication in the analysis of velocity fields under practical
circumstances is the fact that only the velocity component along the
line of sight can be measured. This therefore forms the second issue
that we address in this paper.
As yet we restrict ourselves to an artificial situation with ideal
measurements, not yet to investigate the determination of the statistical
quantities associated with the velocity field under realistic
circumstances. To simplify furthermore our investigations we assume
that we can use
the approximation of an infinitely remote observer so that the radial
velocity can be identified with one velocity component, namely
the $x$-direction in the following.
More specifically, we address the effects of the
reduction of information concerning the velocity field traced by
a diluted sample of points. In principle, the fact that the velocity
field is only known along one direction should not pose any problem.
In the usual structure formation scenarios based on gravitational
instability the large scale velocity field is expected to be
non-rotational, implying it to be a potential flow and therefore
the gradient of a potential that can be inferred from the
measurement of only one component of the velocity (Bertschinger et
al. 1990, Dekel et al. 1990).
\begin{figure*}
\vskip 7.8 cm
\special{hscale=90 vscale=90 voffset=-390 hoffset=-10 psfile=pdf_1D_sample.ps}
\caption{Effects of reduction of the information on the velocity
field to only one component for $\Omega=0.33$
(left panel) and $\Omega=1$ (right panel). We used the estimator
(\bibitem{eq:theta_w_estim}) with $\epsilon=0.1$ to determine
numerically the local divergences.
The solid lines correspond to the model (\bibitem{eq:PDF_1D_estim})
obtained from (\bibitem{eq:theta_estim_model})
with the parameters $\mu$ and $\sigma_e$ given in Table~\bibitem{tab:par1D}.
The dashed lines are the resulting shapes of the PDF-s when one
uses the same parameters but the ``wrong'' value for $\Omega$
($1$ instead of $0.33$ in the left panel, $0.33$ instead of $1$ in the
right one).}
\label{fig:PDF_1D}
\end{figure*}
\begin{figure*}
\vskip 11.5 cm
\special{hscale=90 vscale=90 voffset=-300 hoffset=-30 psfile=scatter.ps}
\caption{Scatter plots that show the differences of the various
estimators of the local divergence measured at 500 different
random locations. }
\label{fig:scatt_div}
\end{figure*}
In the Voronoi method non-zero values of the divergence $\theta$ are
restricted to the walls of the Voronoi tessellation, where the local
divergence is given by
\begin{equation}
\theta_{\rm wall}=\vn\,.\,\Delta \vv,
\label{eq:theta_w_exact}
\end{equation}
with $\vn$ being the normed vector orthogonal to the wall and
$\Delta \vv$ the difference of the velocities on the opposite sides
of the Voronoi wall. The expression for the local vorticity is given by
\begin{equation}
{\bf \omega}} %\def\vomega{{\vec \omega}_{\rm wall}=\vn\,\times\,\Delta\vv.
\label{eq:vortic_w_exact}
\end{equation}
Assuming potential flow, and hence a zero value of ${\bf \omega}} %\def\vomega{{\vec \omega}$, this
implies relations between the various components of $\Delta \vv$ and
the following expressions for $\theta_{\rm wall}$:
\begin{equation}
\theta_{\rm wall}={\Delta v_x\over n_x}={\Delta v_y\over n_y}=
{\Delta v_z\over n_z}.
\end{equation}
In practice, this introduces the numerically unstable operation of
dividing by one component $\vn$ as it can be arbitrarily close to
zero. It may therefore be more reasonable to try to estimate the
value of $\theta_{\rm wall}$ using the stable, but ad-hoc,
prescription of
\begin{equation}
\theta_{\rm wall}^{\rm estim.}=
\Delta v_x { n_x\over n_x^2+\epsilon},
\label{eq:theta_w_estim}
\end{equation}
where $\epsilon$ is a small parameter of the order of $\epsilon
\approx 0.1$.
Note that such a prescription is not self-consistent in reproducing
the full 3D velocity field.\footnote{This can
be readily appreciated from the fact that the normal $\vn$ to a wall
in the Voronoi tessellation is proportional to the vector $\Delta\vr$
between the two points on each side of the wall.
Thus, if it were possible to build a consistent velocity filed from the
constraints (17) it would imply $\Delta\vv\propto
\Delta\vr$ yielding not only a vanishing vorticity
but also a vanishing shear.}Indeed, depending on the way one goes from
cell A to cell B -- and there are infinitely many ways to go from A to
B, even while they are direct neighbours -- one will not necessarily
find the same value for $\theta_{\rm wall}^{\rm estim.}$ using the
equivalent prescriptions for $\Delta v_y$ or $\Delta v_z$.
The method that we adopt here will
therefore certainly not be the most accurate method. However, we may
expect it to be a reasonable approximation, and will therefore use
it here to illustrate the properties in which we are interested.
Figure~\bibitem{fig:PDF_1D} displays the results obtained with
(\bibitem{eq:theta_w_estim}) for $\epsilon=0.1$. Evidently, the results
get affected to quite some extent by the transition from
(\bibitem{eq:theta_w_exact}) to (\bibitem{eq:theta_w_estim}). This leads to
the key question whether it is still feasible to reliably
recover the statistical information on $\theta$ or not. As can be
observed from the scatter plots in Fig.~\bibitem{fig:scatt_div}, the
change from a situation in which one has knowledge of the full
velocity to one where this has been limited to only one
component thereof introduces a large scatter. However, we found that
it is possible to define a meaningful representation of
the scatter plots in terms of the following empirical description:
\begin{equation}
\theta_{\rm estim.}=\mu\ (\theta+e)\,.
\label{eq:theta_estim_model}
\end{equation}
Within this expression the coefficient $\mu$ is a constant with some
fixed value. The scatter is represented by the quantity $e$, a
Gaussian random variable whose value is {\em independent} of $\theta$
and which has a vanishing mean. Using
the measured values of the variance and the skewness of the
distribution it is possible to estimate the value of $\mu$ and the
value of the rms fluctuation $\sigma_e$ of $e$. From
(\bibitem{eq:theta_estim_model}) one can readily infer that
\begin{eqnarray}
\sigma_{\rm estim.}&=&\mu\ \sqrt{\sigma_{\theta}^2+\sigma_e^2} \nonumber \\
T_{3\rm\ estim.}&=&{\sigma_{\theta}^4\over \mu\,(\sigma_{\theta}^2+\sigma_e^2)^2}\ T_3
\end{eqnarray}
where $\sigma_{\theta}$ is the exact rms fluctuations of $\theta$ and $T_3$
its third cumulant, while $\sigma_{\rm estim.}$ and $T_{3\rm\ estim.}$
are the corresponding estimated values. By solving this set of
equations one can find the values of $\mu$ and $\sigma_e$. Their
values for various $\Omega$ and $\epsilon$ are listed in
Table~\bibitem{tab:par1D}.
Moreover, significant within the context of the ultimate goal of
developing an unbiased estimator of $\Omega$, is that these
parameters were found to be almost independent of the value of
$\Omega$. More specifically, it turns out that the value of $\mu$
only depends on the adopted value of $\epsilon$, whereas $\sigma_e$
is independent of $\Omega$ and only marginally dependent on
$\epsilon$. This can clearly be appreciated from the bottom right-hand
panel of Fig.~\bibitem{fig:scatt_div}, which demonstrates that the two
estimations of $\theta$, one based on $\epsilon=0.1$ and the other
on $\epsilon=0.2$, are basically proportional to each other.
It may therefore be argued that it is quite natural to expect that
the noise $e$ introduced by this method is somehow intrinsic to
the distribution.
In Appendix B we describe an extremely simple model based on the
assumption that the relative velocity of two particles is proportional
to their relative position. This allows us to compute analytically the
parameters $\mu$ and $\sigma_e$ entailed by the use of the numerical scheme
(\bibitem{eq:theta_w_estim}). One can show that they are both independent on
$\Omega$, and only depend on $\epsilon$ and $\sigma_{\theta}$, with $\sigma_e
< \sigma_{\theta}$. These analytical predictions are listed in
Table~\bibitem{tab:par1D_anal}. They appear to be fairly close to their
numerical measurements. The discrepancy
between the analytical and numerical estimations of these parameters is
due to the loss of information associated with the projection of the
velocity from three to one dimensions. Although this is not fairly
represented by our model, the discrepancy seems to be quite small.
\begin{table*}
\begin{minipage}{140mm}
\caption{Cumulants from the Perturbation Theory
and as estimated by the various numerical methods. \hfill}
\label{tab:cumulant}
\halign{\quad\hfil#\hfil\quad&
\quad\hfil$#$\hfil\quad&\quad\hfil$#$\hfil\quad&\quad\hfil$#$\hfil\quad&\quad\hfil$#$\hfil\quad&\quad\hfil$#$\hfil\quad\cr
\noalign{\hrule}
\noalign{\medskip}
\# tracers per cell & {\rm cumulants} & \Omega=0.2 & \Omega=0.33 & \Omega=0.4 & \Omega=1 \cr
\noalign{\medskip}
\noalign{\hrule}
\noalign{\medskip}
$\infty$ & T_3^{\rm PT} & -4.5 & -3.33 & -2.97 & -1.71 \cr
Perturbation Theory & T_4^{\rm PT} & 31.41 & 17.22 & 13.67 & 4.55 \cr
\noalign{\medskip}
\noalign{\hrule}
\noalign{\medskip}
$\approx 65$& \sigma^{\rm Del} & 0.37\pm0.01& 0.41\pm0.01& 0.42\pm0.01&0.38\pm0.005\cr
Delaunay Method & T_3^{\rm Del} & -4.52\pm0.2& -3.3\pm0.2 & -2.9\pm0.2 &-1.54\pm0.1\cr
& T_4^{\rm Del} & 31.5\pm2.5 & 17.3\pm3 &12.8\pm2.5 &3.1\pm0.7\cr
\noalign{\medskip}
\noalign{\hrule}
\noalign{\medskip}
$\approx 10$& \sigma^{\rm Del} & - & 0.36\pm0.01& - & 0.35\pm0.005\cr
Delaunay Method & T_3^{\rm Del} & - & -3.8\pm0.2 & - &-1.84\pm0.1\cr
& T_4^{\rm Del} & - & 24.7\pm3 & - &5.4\pm1\cr
\noalign{\medskip}
\noalign{\hrule}
\noalign{\medskip}
$\approx 10$& \sigma^{\rm Vor} & - & 0.40\pm0.01& - & 0.38\pm0.003\cr
Voronoi Method & T_3^{\rm Vor} & - & -3.2\pm0.2 & - & -1.51\pm0.04\cr
& T_4^{\rm Vor} & - & 19.4\pm5.5 & - & 3.62\pm1.5\cr
\noalign{\medskip}
\noalign{\hrule}
\noalign{\medskip}
$\approx 10$& \sigma^{\rm Vor} & - & 0.35\pm0.006& - & 0.34\pm0.009\cr
one velocity & T_3^{\rm Vor} & - & -2.40\pm0.3& - & -1.14\pm0.07\cr
component, $\epsilon=0.1$ & T_4^{\rm Vor} & - & 11.9\pm4 & - & 2.9\pm2.3\cr
\noalign{\medskip}
\noalign{\hrule}}
\end{minipage}
\end{table*}
\begin{table}
\caption{Numerical values of the
parameters $\mu$ (bias) and $\sigma_e$ (noise) introduced by the
calculation (\bibitem{eq:theta_w_estim}) of the velocity divergence.}
\label{tab:par1D}
\centering
\begin{tabular}{lcc}
\hline
\ $\Omega=0.33$\ &\ $\epsilon=0.1$ &\ $\epsilon=0.2$\ \\
\hline
\ $\mu$ & 0.73 & 0.56 \\
\ $\sigma_e$ & 0.25 & 0.27 \\
\hline
\ $\Omega=1$\ &\ $\epsilon=0.1$ &\ $\epsilon=0.2$\ \\
\hline
\ $\mu$ & 0.75 & 0.57 \\
\ $\sigma_e$ & 0.24 & 0.26 \\
\hline
\end{tabular}
\end{table}
On the basis of the simple model described in
Eq.~(\bibitem{eq:theta_estim_model}) it is possible to reconstruct the
corresponding shape of the distribution of $\theta_{\rm estim}$,
\begin{equation}
p_{\rm estim.}(\theta_{\rm estim.})=
\int_{-\infty}^{+\infty}
p({\theta_{\rm estim.} \over \mu}-e)\ {\exp(-e^2/2/\sigma_e^2)\over
(2\,\pi)^{1/2}}\,{{\rm d} e\over\sigma_e}
\label{eq:PDF_1D_estim}
\end{equation}
where $p(\theta)$ is given by Eq.~(\bibitem{eq:PDF_theor}). A comparison
of the resulting distribution (\bibitem{eq:PDF_1D_estim}) with the
measured histograms is shown in Fig.~\bibitem{fig:PDF_1D}. The agreement
appears to be quite good, rendering this phenomenological description
a quite valuable one.
\begin{table}
\caption{Analytical estimations of the
parameters $\mu$ and $\sigma_e$ (see Appendix B).}
\label{tab:par1D_anal}
\centering
\begin{tabular}{lcc}
\hline
\ $\Omega=0.33$\ &\ $\epsilon=0.1$ &\ $\epsilon=0.2$\ \\
\hline
\ $\mu(\epsilon)$ & 0.60 & 0.49 \\
\ $\sigma_e(\epsilon,\sigma_{\theta})\ $ & 0.20 & 0.23 \\
\hline
\ $\Omega=1$\ &\ $\epsilon=0.1$ &\ $\epsilon=0.2$\ \\
\hline
\ $\mu(\epsilon)$ & 0.60 & 0.49 \\
\ $\sigma_e(\epsilon,\sigma_{\theta})\ $ & 0.19 & 0.22 \\
\hline
\end{tabular}
\end{table}
Evidently, we are still able to clearly distinguish between
the scenario with a high value of $\Omega$ and the ones with a low
value, although the distinction is not as clear as in the previous
cases based on ideal sampling circumstances. Looking into some detail,
we come to the conclusion that the signal has been diluted somehow by
the noise $e$, and that
there is a competition between the true rms of the divergence
$\sigma_{\theta}$ and the value of $\sigma_e$. In this respect it is also
important to note that it is crucial for a successful determination
of $\Omega$ to have good estimates of both $\mu$ and $\sigma_e$.
Once these parameters have been determined, and as long as
$\sigma_{\theta} > \sigma_e$, we can see no further problem in distinguishing
between the different cosmological scenarios. Fig.~\bibitem{fig:PDF_1D}
gives an idea of the magnitude
of the discrepancy that one gets when one assumes a wrong value
of $\Omega$. In the left panel this concerns the case wherein one
take a value of $\Omega=1$ instead of the actual value of
$\Omega=0.33$, while in the right-hand panel it concerns the reverse
situation.
At this point it is worthwhile to stress once more that so far we only
tested the method for {\em ideal} measurements of the line-of-sight
velocities. Noise in the measured values of these velocities was
not yet taken into account. Moreover, besides the fact that this
noise has quite a large value, an evaluation of its influence is
substantially complicated as it not only concerns random measurement
errors but also contains contributions from a plethora of, partially
un-understood, systematic effects. In a work in preparation
we will attempt to develop specific techniques for estimating the
velocity divergence PDF from such noisy line-of-sight measurements.
\section{Discussion and Conclusions}
\label{sec:conclusion}
We have tested and confirmed the validity of the strong $\Omega$
dependence of the Probability Density Function (PDF) of the velocity
divergence that had been predicted on the basis of analytical
Perturbation Theory calculations. These tests are based on a numerical
analysis of N-body simulations of structure formation in a Universe
with a Gaussian initial density and velocity fluctuation field.
On the basis of this verification we may conclude that the analytical
predictions of Perturbation Theory yield very accurate results for
a wide range of cosmological models.
The main practical implication of our work is the basis it offers
for a potentially very valuable and promising estimator of the
value of $\Omega$, an estimator independent of a possible bias
between the distribution of galaxies and the underlying matter
distribution. The successful tests presented in this work demonstrate
the validity of the equations of Perturbation Theory that form
a basis for the estimates of $\Omega$ which are based on the statistical
properties of the velocity divergence field. Their unbiased nature
finds its origin in the fact that the relations between the various
statistical moments do not contain any explicit dependence on a
bias between the galaxy and the matter distribution.
Moreover, the estimated values of $\Omega$ are even more
direct and straightforward to interpret as the relevant statistical
relations only involve a very weak dependence on the cosmological constant
$\Lambda$ is expected to be very weak (Bernardeau 1994a, b).
Not only relations between the statistical moments of the $\theta$
distribution, also the shape and general functional behaviour provide
a useful indicator for the value of $\Omega$. When we focus on the
details of this functional behaviour of the velocity divergence
distribution function -- illustrated in Fig.~\bibitem{fig:PDF_large_sample} and
\bibitem{fig:PDF_small_sample}) -- we can draw a few conclusions with
regards the practical feasibility of obtaining reliable estimates
from the shape of $p(\theta)$. Both location and
shape of the peak of this distribution appear to be robust indicators
of the value of $\Omega$. On the other hand, the location of the
maximum of the divergence $\theta$ -- i.e. the cutoff value of
$p(\theta)$ -- appears to be much more sensitive to a poor sampling.
This may make it harder for it to provide reliable estimates
of $\Omega$ from currently available observational catalogues
(see Fig.~\bibitem{fig:PDF_small_sample}). However, it is all the more
encouraging that even on the basis of the cutoff value we obtained
a reasonable agreement between theoretical predictions and numerical
measurements.
Finally, we addressed one further crucial issue towards an application
of our estimation procedures to real data sets. This concerns the
problem of not being able to obtain directly the full
three-dimensional velocity field. Instead, the velocity of a galaxy
can only be measured along the line-of-sight. In a preliminary attempt
to study the consequences of this fact for the feasibility of
our method, we introduced a partially empirically defined extension
of our method. Despite of the extreme crudeness and rather ad-hoc
nature of this algorithm to reconstruct the full velocity field, it
is quite encouraging that we are able to distinguish between
the velocity PDF obtained in a flat Universe and that obtained in
an open Universe. The major obstacle towards a successful application
of our methods therefore appears to be the one of noisy data sets
and systematic sampling errors.
We have not yet dealt with these
problems, deferring them to a forthcoming paper.
An additional and useful application of our numerical work involves
a test for structure indeed having emerged through the process of
gravitational growth of an initially Gaussian
random density and velocity field. Having shown Perturbation Theory
to be valid, we can exploit its prediction that the PDF of $\theta$
is only dependent on a few parameters, in particular $\sigma_{\theta}$ and
$\Omega$. If no values of $\sigma_{\theta}$ and $\Omega$ can be found to
produce an acceptable fit to the observed velocity field, this will
force us to conclude that it is unlikely that the structure
developed as described within the standard framework of gravitational
instability and Gaussian initial conditions. In this context it
is interesting to point out that a negative skewness has been observed
in the currently available datasets (Bernardeau et al. 1995), which is
an indication in favour of standard scenarios.
Summarizing, we may conclude that the combined machinery of the
analytical perturbation theory results and the developed numerical
methods and their application on the intrinsic statistical properties of the
velocity field provides us with a reliable new estimator of the
cosmological density parameter $\Omega$. This estimator is all the
more useful as it is one of the very few which will yield values
of $\Omega$ completely independent of galaxy-density field biases
and almost independent of the value of $\Lambda$.
\section*{Acknowledgments}
F. Bernardeau would like to thank IAP, where a large part of the work
has been completed, for its warm hospitality. We would like to thank
A. Dekel for encouraging comments and discussions. FB and RvdW are
grateful for the hospitality of the Hebrew University of Jerusalem,
where the last part of this contribution was finished.
R. van de Weygaert is supported by a fellowship of the Royal
Netherlands Academy of Arts and Sciences. Part of this work was done
while EH was at the Institut d'Astrophysique de Paris (CNRS),
supported by the Minist\`ere de la Recherche et de la Technologie.
Additional partial support to EH was provided by the Danish National Research
Foundation through the establishment of the Theoretical Astrophysics
Center. The computational means were made available to us thanks to the
scientific council of the Institut du D\'eveloppement et des Resources
en Informatique Scientifique (IDRIS).
| proofpile-arXiv_065-430 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
Elimination of fermionic degrees of freedom is very desirable in
many problems of quantum field theory and elementary particle
physics. First, in lattice gauge theories integration with respect
to Grassmannian variables leads to serious complications of
numerical simulations. The second, appearance of fermionic variables
in functional integrals hampers the application of stationary phase
method. As a consequence, one cannot also apply quasiclassical
expansions to evaluation of functional integrals in theories with
fermions except some especially simple cases.
Indeed, typical Green function can be written as
\begin{equation}
G(x_1, ..., x_n) = \int DA D\Psi D\bar{\Psi} e^{\frac{i}{\hbar}
(S_{YM}(A) + S_{ferm}(\bar{\Psi}, \Psi, A))}
{\cal O}_1(x_1) ...{\cal O}_n(x_n)
\label{1}
\end{equation}
\noindent where ${\cal O}_i$ are some operators whereas $S_{YM}$ and
$S_{ferm}$ are Yang-Mills and fermionic actions respectively.
Quasiclassical approximation is defined, up to some subtleties, by
stationary point equations
\begin{equation}
\frac{\delta S_{YM}}{\delta A}=[\mbox{the source of YM field}]
\label{2}
\end{equation}
But what must be written in R.H.S. of eq. (\ref 2) ? Of course, one
cannot put
\begin{equation}
[\mbox{the source of YM field}]=\frac{\delta S_{ferm}}{\delta A}
\label 3
\end{equation}
\noindent because $S_{ferm}$ depends on Grassmanian variables.
Moreover, without preliminary exception of fermionic variables one
cannot write in R.H.S. of (\ref 2) nothing except zero. But this
means that in zero approximation YM field can be considered as free.
It seems inappropriate in all cases in which interaction is strong.
So for application of quasiclassical methods as well as for
facilitation of numerical simulations on the lattice fermionic
variables in functional integrals of the type (\ref 1)
must be integrated out and result must be represented as functional
integral with respect to only bosonic variables. In other words, the
theory must be bosonized.
The problem of bosonization of fermionic theories has a long
history. Most likely, the first example of bosonisation of fermionic
theory was given by Schwinger in his famous paper \cite{1}
concerning full solution of massless $\mbox{QED}_2$. Then the problem
of bosonisation was investigated by many authors, but more or less
exhaustive solution was obtained only in two dimensional case and
in some three dimensional models (see, for instance, papers \cite{2}
and references therein). In realistic four dimensional case only
partial success was achieved (see, for instance, \cite 3). In fact in
all proposed bosonization schemes in four dimensions it is necessary
to evaluate (exactly or in some approximation) fermionic determinant
-- but it is just the main problem that must be solved by means of
bosonization.
To author's knowledge, the only exceptions are recent papers by
Lusher \cite 4 and Slavnov \cite 5 (see also \cite 6). In Ref. \cite
4 fermionic determinant on the finite lattice is represented as
infinite some of bosonic determinants. In Ref. \cite 5 fermionic
determinant in $D$ dimensions is expressed via bosonic one in $D+1$
dimensions. These approaches seem useful in lattice theories
but they cannot be applied to investigation of
quasiclassical approximation.
So hitherto no quite satisfactory representation for fermionic
determinant in terms of bosonic fields is known, and at present
paper we will develop another approach to bosonization. Namely, we
will derive pure bosonic worldline path integral representation for
fermionic determinants, Green functions and Wilson loops.
Worldline approach to quantum field theory also has very long
history. It was originated many years ago in classical works by
Feynman \cite 7 and Schwinger \cite 8 . The main idea of this
approach is to represent fermionic determinants and fermionic Green
functions as functional integral over trajectory of a single
relativistic particle.
Let us consider, for instance, fermionic determinant for $SU(N)$
Yang-Mills theory in Euclidean space:
\begin{equation}
D \equiv \det (i\hat{\nabla} +im)
=\det (i\hat{\nabla} +im)\gamma^5
\label 4
\end{equation}
\noindent where $\hat{\nabla}=\gamma^{\mu} \nabla_{\mu} =\gamma^{\mu}
(\partial_{\mu} - iA_{\mu}), \ \ (\gamma^{\mu})^{\dag}=\gamma^{\mu},
\ \ \{\gamma^{\mu},\gamma^{\nu}\}=2\delta^{\mu \nu}, \ \
iA_{\mu}(x)\in su(N)$.
One can write:
\begin{equation}
\ln D =\frac{1}{2} \ln \det[(i\hat{\nabla} +im)\gamma^5]^2=
\frac{1}{2} \ln {\det} (-\nabla_{\mu}\nabla_{\mu} +
\sigma^{\mu\nu}F_{\mu\nu} +m^2)
\label 5
\end{equation}
\noindent where
\begin{equation}
F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-
i[A_{\mu},A_{\nu}],
\label 6
\end{equation}
\begin{equation}
\sigma^{\mu\nu}=\frac{i}{4}[\gamma^{\mu},\gamma^{\nu}]
\label 7
\end{equation}
Further, eq. (\ref 5) can be written, up to inessential constant, as
\begin{eqnarray}
\ln D&=&
\frac{1}{2} \mbox{tr}\: \ln (-\nabla_{\mu}\nabla_{\mu} +
\sigma^{\mu\nu}F_{\mu\nu} +m^2)\nonumber\\
&=&-\frac{1}{2} \int_0^{\infty}\frac{dT}{T} e^{-m^2T}
\mbox{tr}\:
e^{-T(
-\nabla_{\mu}\nabla_{\mu} +
\sigma^{\mu\nu}F_{\mu\nu})}
\label 8
\end{eqnarray}
The integral (\ref 8) is divergent at $T=0$ and must be regularized.
One can use, for instance, $\zeta$-function regularization:
\begin{equation}
\ln D =-\frac{1}{2} \frac{d}{ds} \frac{1}{\Gamma(s)} \int_0^{\infty}
dT \ T^{s-1} e^{-m^2 T}
\mbox{tr}\:
e^{-T(
-\nabla_{\mu}\nabla_{\mu} +
\sigma^{\mu\nu}F_{\mu\nu})}
\raisebox{-9pt}{\rule{0.4pt}{20pt}${\scriptstyle s=0}$}
\label 9
\end{equation}
\noindent However, in what follows we shall write, for short, formal
expression (\ref 8) for $\ln D$ bearing in mind any suitable
regularization.
Further, trace in eq.(\ref 8) can be represented as functional
integral:
\begin{eqnarray}
&& \mbox{Tr}
e^{-T(
-\nabla_{\mu}\nabla_{\mu} +
\sigma^{\mu\nu}F_{\mu\nu})}\nonumber\\
&=& \int _{PBC} Dq\: \mbox{tr P}\exp \left\{ -\int_0^1 dt
\left(\frac{\dot{q}^2}{4T} -i\dot{q}^{\mu}A_{\mu}(q) +T
\sigma^{\mu\nu}F_{\mu\nu}(q) \right) \right\}
\label{10}
\end{eqnarray}
\noindent Here PBC means 'periodic boundary conditions'. Path
ordering in (\ref{10}) corresponds to both spin and colour matrix
structures.
Let us suppose for a moment that spin and colour degrees of
freedom are absent in (\ref{10}). (This is just the case of scalar
electrodynamics). Then, integrating in (\ref 1) over fermionic
fields and using for arising fermionic determinants and fermionic
Green functions formulae of the type (\ref 8), (\ref{10}), one
obtains formulation of quantum field theory in terms of particles
interacting with gauge field $A$. This is just result of classical
work bu Feynman \cite 7.
In realistic models, however, one must take into account spin and
colour degrees of freedom. So, to develop worldline formulation of
quantum field theory, it is necessary to represent the path ordered
exponent in (\ref{10}) as functional integral. In the case of QED
this was done by Fradkin \cite 9 in terms of fermionic path
integral. This representation and their modifications were
successfully used, in particular, for construction of derivative
expansion in QED \cite{10} and for investigation of complicated
Feynman diagrams \cite{11}. It is also intensively used for
investigation of hidden supersymmetry in fermionic theories (see,
for example, \cite{12} and references therein). Recently, D'Hoker
and Gagne derived fermionic path integral representation for
fermionic determinants for particles coupled with arbitrary tensor
field \cite{13}.
In papers \cite{8}-\cite{13} fermionic path integral representations
were derived and used only for spin degrees of freedom. For colour
P-exponent in (\ref{10}) fermionic path integral representation also
can be derived (see, for example, \cite{14}) though it seems not so
elegant as one for spin P-exponent. Apropos, works \cite{14} were, to
author's knowledge, the first attempts to obtain non-perturbative
information about QCD in worldline formalism.
But, as we already stated at the beginning of the present paper,
there exist important problems for which pure bosonic worldline path
integral representation for fermionic determinants and Green
functions is very desirable. We see that for solution of the latter
problem it is sufficient to derive bosonic worldline path integral
representation for the trace of path ordered exponent
\begin{equation}
Z=\mbox{tr P} e^{i\int_0^1 dt B(t)}
\label{11}
\end{equation}
\noindent with matrix $B(t) \in GL(N),\; B(0)=B(1)$. Indeed,
substituting such representation with
\begin{equation}
B(t)=\dot{q}^{\mu}A_{\mu}(q(t)) - \sigma^{\mu\nu}F_{\mu\nu}(q(t))
\label{12}
\end{equation}
\noindent in (\ref{10}), one obtains desired representation for
fermionic determinant. As we will show later, bosonic path integral
representation for $Z$ allows also to obtain bosonic worldline path
integral representation for fermionic Green functions.
Different bosonic path integral representations for $Z$ in the case
$B(t) \in SU(2)$ were proposed by several authors (see
\cite{15}-\cite{18}). For more general case $B(t) \in su(N)$ the an
analogous representation was pointed out in \cite{19}.
The typical result for $B \in SU(2)$ is
\begin{equation}
Z=\int DS(t) \exp \left\{ \frac{i}{2} \int _0^1 dt \left[ \mbox{tr}
\sigma^3 \left( S(t)B(t)S^{\dag}(t) +iS(t)\dot{S}^{\dag}(t) \right)
\right] \right\}
\label{13}
\end{equation}
The integration in (\ref{13}) is carried out over all trajectories
in the group $SU(2), \; S \in SU(2), \; \sigma^3$ is a Pauli matrix.
The last result has two disadvantages. First, in any parametrization
of $SU(2)$ the "action" in (\ref{13}) is rather complicated
non-polinomial function. This hampers the usage of (\ref{13}) in
practice. The second, it appears that integral (\ref{13}) is
ill-defined and needs insertion of some
regulators in the "action" (see
the discussion of this point in \cite{16,19} and in more recent
paper \cite{20}).
Recently Dyakonov and Petrov found much more elegant formula for
path ordered exponent. Namely, for the case when $Z$ is Wilson loop
and gauge group is $SU(2)$, they derived from (\ref{13}) the
following expression for $Z$:
\begin{eqnarray}
Z&=&\mathop{\mbox{tr P}}
e^{ i\oint_{\gamma} \mathop{dx^{\mu}} A_{\mu}(x) }\nonumber\\
&=&\int \mathop{Dn(x)} \prod_{x\in\Sigma} \d(n^a(x) n^a(x) -1)
\nonumber\\
&&\exp\left\{
\frac{i}{4}\int_{\Sigma}\mathop{dx^{\mu}\wedge dx^{\nu}} \left[ -F^a_{\mu\nu}n^a +\varepsilon^{abc}n^a
D_{\mu}n^b D_{\nu} n^c \right] \right\}
\label{14}
\end{eqnarray}
\noindent where $\Sigma$ is two dimensional surface spanned on contour
$\gamma$, $a=1,2,3$, $D^{\mu}$ -- covariant derivative. This formula can
be considered as non-Abelian variant of Stokes theorem. We shall
continue the discussion of this result in section~4.
In the present paper we will derive alternative bosonic path
integral representation for $Z$ in the general case $B\in su(N)$.
The "action" in this representation is quadratic and so it is much
more simpler then one in (\ref{13}) and (\ref{14}). This derivation
is presented in the section 2. In the section 3 we check the
representation obtained by direct evaluation of the functional
integral that defines $Z$. In fact, we give alternative proof of
results obtained in the section 2. In the section 4 we derive
an non-Abelian analog of Stokes theorem and compare our results with
those due to Dyakonov and Petrov. In the section 5 we derive bosonic
worldline path integral representation for fermionic determinant and
Green functions in Euclidean QCD. In the section 6 we get analogous
results for Minkowski space and then, applying stationary phase
method, derive quasiclassical equations of motion in QCD. In the
last section we summarize our results and discuss perspectives of
future investigations. In two appendixes we derive some auxiliary
formulae that are used in the main text of the paper.
\section{Bosonic path integral representation for the trace of path
ordered exponent}
Let $N\times N$ matrix $U(t)$, $0\le t\le 1$, be defined by equations
\begin{eqnarray}
\frac{dU}{dt}&=&iB(t)U(t) \nonumber\\
U(0)&=&I_N
\label{15}
\end{eqnarray}
\noindent where $I_N$ is $N\times N$ unit matrix. Then, obviously,
\begin{equation}
Z=\mathop{{\rm tr}} U(1)
\label{16}
\end{equation}
First, we consider the case $B \in su(N)$, that is
\begin{equation}
B^{\dag }=B, \ \ \mathop{{\rm tr}} B=0
\label{16.1}
\end{equation}
Let us consider an operator
\begin{equation}
\hat B = a^{\dag}_r B^r_s a^s
\label{17}
\end{equation}
\noindent that acts in some Fock space ${\cal F}$. Operators $a^{\dag}_r$
and $a^s$ in (\ref{17}) are usual {\it bosonic} creation and
annihilation ones, $[a^r,a^{\dag}_s]=\delta^r_s$.
Let ${\cal H}_n$ be $n$-particle subspace of ${\cal F}$, $\Pi_n$ is
orthogonal projector on ${\cal H}_n$. Then
\begin{equation}
Z= \mathop{{\rm tr}} \Pi_1 \mathop{{\rm P}} e^{i\int_0^1 \mathop{dt} \hat B (t)}
\label{18}
\end{equation}
Indeed, if $\hat N=a^{\dag}_r a^r$, then
\begin{equation}
\left[ \hat B, \hat N \right] =0
\label{19}
\end{equation}
So $\hat B {\cal H}_n \subset {\cal H}_n$ and in one dimensional
subspace ${\cal H}_1$ the operator $\hat B$ can be identified with
the matrix $B$ via relation:
\begin{equation}
\hat B a^{\dag}_r \phi^r |0> = a^{\dag}_r (B^r_s\phi^s) |0>
\label{20}
\end{equation}
Projector $\Pi_1$ can be represented as
\begin{eqnarray}
\Pi_1&=&\int_{1-\varepsilon}^{1+\varepsilon \delta(\hat N-\lambda)\nonumber \\
&=&\int_{1-\varepsilon}^{1+\varepsilon\int\limits_{-\infty}^{\infty e^{-i\lambda\eta + i\hat N \eta}
\label{21}
\end{eqnarray}
\noindent where $0<\varepsilon<1$.
So
\begin{eqnarray}
Z&=&\lim_{\delta \to +0} \mathop{{\rm tr}} \Pi_1 e^{-\delta\hat N} \mathop{{\rm P}}
e^{i\int_0^1 \mathop{dt} \hat B (t)} \nonumber \\
&=&\lim_{\delta \to +0} \int_{1-\varepsilon}^{1+\varepsilon \int\limits_{-\infty}^{\infty \mathop{{\rm tr}} e^{-i\lambda\eta +(i\eta-\delta)\hat N}
e^{i\int_0^1 \mathop{dt} \hat B (t)}\nonumber\\
&=&\lim_{\delta \to +0} \int_{1-\varepsilon}^{1+\varepsilon \int\limits_{-\infty}^{\infty \mathop{{\rm tr}} e^{-i\lambda\eta} \mathop{\mbox{tr P}}
e^{i\int_0^1 \mathop{dt} (\hat B (t) +(\eta +i\delta)\hat N)}
\label{22}
\end{eqnarray}
\noindent The latter equality is valid due to (\ref{19}).
The trace of ordered exponent in (\ref{22}) can be represented as
functional integral. In our case its explicit form essentially
depends on some subtleties in its definition. So let me remind in a
few words the general construction of functional integral. For more
detailed discussion see, for instance, \cite{22}.
Let
\begin{equation}
\check Z=\mathop{\mbox{tr P}} e^{\int^1_0 \mathop{dt} \hat H}
\label{23}
\end{equation}
\noindent where $\hat H=\hat H(a^{\dag},a;t)$ is an operator in Fock
space ${\cal F}$.
In our case
\begin{equation}
\hat H=(i\eta-\delta)\hat N +i\hat B
\label{24.1}
\end{equation}
Equivalently, we can write
\begin{equation}
\hat H=\hat H(\hat p,\hat q;t)
\label{24.2}
\end{equation}
\noindent
\begin{equation}
\hat p=\frac{a+a^{\dag}}{\sqrt 2}, \qquad \hat q=
\frac{a-a^{\dag}}{i\sqrt 2}
\label{24}
\end{equation}
Let $H=H(p,q;t)\equiv H(\bar z,z;t)$ be Weyl, or normal, or any
other symbol of the operator $\hat H$,
\begin{equation}
z=\frac{p-iq}{\sqrt 2},\qquad \bar z=\frac{p+iq}{\sqrt 2}
\label{25}
\end{equation}
Further, let $*$ be the operation that represent the multiplication
of operators in the language of symbols. This means that if $\hat
K=\hat K(\hat p,\hat q)$ and $\hat L=\hat L(\hat p,\hat q)$ are
some operators with symbols $ K=K( p, q)$ and $ L= L( p, q)$ and
$\hat M=\hat K \hat L$ then
\begin{equation}
M(p,q)=(K*L)(p,q)
\label{26}
\end{equation}
In this terms
\begin{equation}
\check Z=\lim_{n\to \infty}\int\mathop{dpdq}\left(
e^{\frac{1}{n}H(\cdot,\cdot;0)}*
e^{\frac{1}{n}H(\cdot,\cdot;\frac{1}{n})}*\ldots*
e^{\frac{1}{n}H(\cdot,\cdot;\frac{n-1}{n})}\right)(p,q)
\label{27}
\end{equation}
The last formula can be rewritten as functional integral. Its
concrete form depends on the choice of a kind of symbols used
in (\ref{27}). In particular, for Weyl symbols
\begin{equation}
\check Z={\cal N}\int_{PBC}DpDq\: e^{\int_0^1 dt\:(ip(t)\dot q(t)+
H_W(p(t),q(t);t))}
\label{28}
\end{equation}
\noindent whereas for normal symbols
\begin{equation}
\check Z=\lim_{\epsilon \to +0}{\cal N}' \int_{PBC} DzD\bar z \: e^{
\int_0^1 \mathop{dt}(\bar z(t)\dot z(t)+H_{norm}(\bar z(t),z(t+\epsilon))}
\label{29}
\end{equation}
\noindent Here $PBC$ means 'periodic boundary conditions', $z$ and $\bar
z$ are {\rm independent} complex variables, and ${\cal N,N'}$ are
normalization constants (that we will usually omit in what
follows).
In our case
\begin{eqnarray}
H_W(p,q;t)&=&H_W(z^{\dag},z;t)\nonumber\\
&=&(i\eta-\delta)z^{\dag}z+iz^{\dag} Bz-\frac{N}{2}(i\eta-\delta)
\label{30}
\end{eqnarray}
\noindent where $p,q$ are connected with $z$ by formulae (\ref{25}),
and
\begin{equation}
H_{norm}(\hat z,z)=(i\eta-\delta)z^{\dag} z +iz^{\dag} B z
\label{31}
\end{equation}
Formulae (\ref{28}), (\ref{29}) correspond to standard sign
conventions. However, for our purposes it is convenient to change
variables
$$ z(t)\to z(1-t)$$
\noindent Then formulae (\ref{28}), (\ref{29}) with symbols (\ref{30}),
(\ref{31}) can be rewritten as
\begin{eqnarray}
\check Z&=&\int_{PBC} D^2 z\:\exp\left\{ i\int_0^1\mathop{dt} \left[ iz^{\dag} (t)
\dot z(t) +z^{\dag} (t) B(t) z(t)\right.\right.\nonumber\\
& +&\left.\left. (\eta
+i\delta)z^{\dag}(t)z(t)-\frac{N}{2}(\eta+i\delta)\right]\right\}
\label{32}
\end{eqnarray}
\noindent where
$D^2z\equiv D({\rm Re} z) D({\rm Im} z)$ and
\begin{eqnarray}
\check Z& =&\lim_{\epsilon \to +0} \int DzD\bar z \: \exp\left\{ i\int^1_0 \mathop{dt} \left[
i\bar z(t) \dot z(t) +\bar z(t)B(t)z(t-\epsilon)\right.\right.\nonumber\\
&{}&\left.\left. +(\eta+i\delta)\bar z(t)z(t-\epsilon) \right] \right\}
\label{33}
\end{eqnarray}
\noindent respectively.
There are several essential differences between formulae (\ref{32})
and (\ref{33}). First, in (\ref{32}) $z$ and $z^{\dag}$ are complex
conjugated variables whereas in (\ref{33}) $\bar z$ and $z$ are
independent. The second, the last term in the "action" in (\ref{32})
is absent in (\ref{33}). The third, there is the shift of "time"
variable in the last two terms in the "action" in (\ref{33}) that is
absent in (\ref{32}). In the next section we will show by explicit
calculations that this shift just compensates the absence of the
term
$$ -\frac{N}{2}(\eta +i\delta)$$
\noindent in (\ref{33}). Finally, it is worth to note that the limit
in (\ref{33}) must be evaluated {\it after} functional integration in
(\ref{33}) because this two operations don't commute. It will be
confirmed by explicit calculations in the section 3.
Substituting (\ref{32}) and (\ref{33}) in (\ref{22}) and evaluating
limits $\varepsilon \to 0, \ \ \delta \to 0$ (bat not the limit $\epsilon \to +0$!),
ones obtains, respectively,
\begin{eqnarray}
Z&=&\int_{PBC} D^2z \: \int\limits_{-\infty}^{\infty e^{i\int^1_0 \mathop{dt} (iz^{\dag}\dot z+z^{\dag} Bz+
\eta(z^{\dag} z-1-\frac{N}{2})}
\label{34}
\end{eqnarray}
\noindent and
\begin{eqnarray}
Z&=&\lim_{\epsilon \to +0} \int_{PBC} DzD\bar z \int\limits_{-\infty}^{\infty \exp
\left\{ i\int_0^1\mathop{dt} i\bar z
\dot z +\bar z B e^{-\epsilon \frac{d}{dt}}z \right.
\nonumber\\
&+&\left.\eta(\bar ze^{-\epsilon \frac{d}{dt}}z -1)\right\}
\label{35}
\end{eqnarray}
We won't try to justify here the validity of limiting procedure $\varepsilon
\to 0$, $\mbox{$\delta \to 0$}$ because in the next section we will check
formulae (\ref{34}), (\ref{35}) by direct calculation.
In (\ref{34}) one can integrate with respect to $\eta$:
\begin{equation}
Z=\Dzi
e^{i\int_0^1 \mathop{dt}(iz^{\dag}\dot z +z^{\dag} Bz)}
\label{36}
\end{equation}
But we cannot obtain $\delta$-function by integration with respect to
$\eta$ in (\ref{35}) because $\bar z$ and $z$ in (\ref{35}) are
independent complex variables.
One can also get another useful form of the representation
( \ref{36}), namely
\begin{equation}
Z=\int_{PBC}D^2z \: \prod \limits _t \delta(z^{\dag} (t)z(t)-1-\frac{N}{2})
e^{i\int_0^1 \mathop{dt}(iz^{\dag}\dot z +z^{\dag} Bz)}
\label{37}
\end{equation}
In what follows, we will sometimes omit symbol $\prod$ in formulae
of the type (\ref{37}).
To get formula (\ref{37}), it is sufficient to insert projectors
$\Pi_1$ represented in the form (\ref{21}) between each pair of
adjacent factors in (\ref{27}) and to repeat all calculations that
have led us to representation (\ref{36}).
The same arguments allow also to obtain an analogous variant of
(\ref{35}):
\begin{eqnarray}
Z&=&\lim_{\epsilon \to +0} \int_{PBC} Dz(t)D\bar z(t)D\eta(t) \exp
\left\{ \right. i\int_0^1\mathop{dt} i\bar z (t)
\dot z(t) +\bar z(t) B(t) e^{-\epsilon \frac{d}{dt}}z(t)
\nonumber\\
&+&\eta(t)(\bar z(t)e^{-\epsilon \frac{d}{dt}}z(t) -1)\left.\right\}
\label{37.1}
\end{eqnarray}
Thus we have derived four different representations
(\ref{35})-(\ref{37.1}) for one quantity $Z$. They all appear to be
useful in quantum field theory.
We have got (\ref{35})-(\ref{37.1}) assuming that $B\in su(N)$.
But these representation remains valid for every trace free matrix
by virtue of analytical continuation. Representation for $B\in
GL(N)$ can be obtained by extracting of the trace part of the matrix
$B$ at the beginning of calculations. Indeed, if
$$B=B'+ \frac{1}{N}I_N\mathop{{\rm tr}} B$$
\noindent then $\mathop{{\rm tr}} B'=0$ and
\begin{equation}
\mathop{\mbox{tr P}} e^{i\int_0^1 \mathop{dt} B}=
e^{\frac{i}{N}\int_0^1 \mathop{dt} \mathop{{\rm tr}} B}
\mathop{\mbox{tr P}} e^{i\int_0^1 \mathop{dt} B'}
\label{39}
\end{equation}
Another representation for matrices with non-zero trace will be
given in the next section (see eq. (\ref{58})).
Finally, let us derive a kind of representation (\ref{37}) for the
case
\begin{equation}
B(t)=\sum_{j=1}^{M} C_j(t) \otimes D_j(t)
\label{39.1}
\end{equation}
\noindent where $C_j$ and $D_j$ are $N_1\times N_1$ and $N_2\times N_2$
matrices. Let $a_{(i)}^{r_i},\ \ a^{\dag}_{(i)r_i},\ \ i=1,2
\ \ r_i=1,2,\ldots ,N_i$ be two sets of annihilation and creation
operators that act in some Fock space, and $\Pi_{1\otimes 1}$ is a
projector on the space
\begin{equation}
{\cal H}_{1\otimes 1}=\left\{\phi^{r_1 r_2}a^{\dag}_{(1)r_1}
a^{\dag}_{(2)r_2}|0>\right\}
\label{39.2}
\end{equation}
One can write
\begin{eqnarray}
\lefteqn{
\mathop{\mbox{tr P}} e^{i\int_0^1 \mathop{dt} B}}\nonumber\\
&&=\mathop{{\rm tr}} \Pi_{1\otimes 1}\: \mbox{P}\: \exp\left\{ i\int^1_0 \mathop{dt}
\sum\limits_{j=1}^M(a^{\dag}_{(1)}C_j a_{(1)})
(a^{\dag}_{(2)}D_j a_{(2)})\right\}
\label{39.3}
\end{eqnarray}
\noindent The last formula ia an analog of (\ref{18}). Then, repeating the
calculations that have been done for derivation of eq. (\ref{36})
from eq. (\ref{18}), one obtains:
\begin{eqnarray}
\mathop{\mbox{tr P}} e^{i\int_0^1 \mathop{dt} B}
&=&\int D^2z_1 D^2 z_2 \prod\limits_{i=1}^2
\delta (z^{\dag}_{(i)}z_{(i)}-1-\frac{N}{2})
\nonumber\\
&&\exp \left\{ i\int_0^1 \mathop{dt} \left[ i\sum\limits_{i=1}^2
z^{\dag}_{(i)}\dot z_{(i)}\right.\right.\nonumber\\
&&\left.\left. +\sum\limits_{j=1}^M
(z^{\dag}_{(1)}C_j z_{(1)} -\frac{1}{2}\mathop{{\rm tr}} C_j)
(z^{\dag}_{(2)}D_j z_{(2)}-\frac{1}{2}\mathop{{\rm tr}} D_j)
\right] \right\}
\label{39.4}
\end{eqnarray}
Similar arguments allow to obtain the analog of eqs. (\ref{35}),
(\ref{37}), and (\ref{37.1}) for an matrix $B$ defined by eq.
(\ref{39.1}).
\section{Alternative proof of bosonic path integral representation
for the trace of path ordered exponent}
At first, we will evaluate the integral (\ref{36}) assuming that
$B\in su(N)$ . One can write:
\begin{eqnarray}
Z&=&\lim_{\epsilon \to +0} \Dzi e^{-\epsilon\int^1_0 \mathop{dt} z^{\dag} z}e^{i\int^1_0 \mathop{dt}(iz^{\dag} \dot z
+z^{\dag} B z)}\nonumber\\
&=&\lim_{\epsilon \to +0} \int\limits_{-\infty}^{\infty e^{-i\eta\left( 1+\frac{N}{2}\right)}\int_{PBC} D^2z \:
e^{\int^1_0 \mathop{dt} z^{\dag}\left(-\frac{d}{dt} +iB+i\eta -\epsilon\right)z}
\label{40}
\end{eqnarray}
Performing functional integration, one gets:
\begin{equation}
Z=\lim_{\epsilon \to +0} \int\limits_{-\infty}^{\infty \frac{e^{-i\eta\left( 1+\frac{N}{2}\right)}}{
{\det}\left(-\frac{d}{dt} +i(B+\eta)-\epsilon\right)_{PBC}}
\label{41}
\end{equation}
To evaluate integral in (\ref{41}), let us consider eigenvalue
problem
\begin{equation}
\left(-\frac{d}{dt} +i(B+\eta)-\epsilon\right)\phi (t)=\lambda \phi(t)
\label{42}
\end{equation}
\begin{equation}
\phi(0)=\phi(1)
\label{43}
\end{equation}
The general solution of eq. (\ref{42}) is
\begin{equation}
\phi=e^{(-\lambda +i\eta -\epsilon)t} U(t)\chi
\label{44}
\end{equation}
\noindent where $U(t)$ is a solution of (\ref{15}) and vector $\chi$
doesn't depend on $t$.
Let $e^{i\alpha_r}$ and $\xi_r,\ \ r=1,\ldots,N$ be eigenvalues and
eigenvectors of the matrix $U(1)$:
\begin{equation}
U(1)\xi_r=e^{i\alpha_r}\xi_r
\label{45}
\end{equation}
One notes that $U(1)\in SU(N)$ and so $\alpha_r$ are real and
\begin{equation}
\sum_{r=1}^N \alpha_r=0
\label{46}
\end{equation}
Further, to satisfy (\ref{43}) we must put
\begin{equation}
\lambda\equiv \lambda_{rn}=-\epsilon +i\eta +i\alpha_r +2\pi i n,\ \ n=0,\pm 1,\ldots
\label{47}
\end{equation}
\noindent and $\chi=\xi_r$. So
\begin{eqnarray}
\lefteqn{{\det}\left(-\frac{d}{dt} +i(B+\eta)-\epsilon\right)_{PBC}=
\prod\limits_{r=1}^{N}
\prod\limits_{n=-\infty}^{\infty}(2\pi i n+i\alpha_r +i\eta-\epsilon)}
\nonumber\\
&=&\left[-i\prod_{n=-\infty}^{\infty}(2\pi in)\right]^r
\prod\limits_{r=1}^{N}
(\alpha_r+\eta +i\epsilon)\prod\limits_{n\ne 0}
\left(1+\frac{\alpha_r +\eta +i\epsilon}{2\pi n}\right)\nonumber\\
&=&\left[-i\prod_{n=-\infty}^{\infty}(2\pi in)\right]^r
\prod\limits_{r=1}^{N}(\alpha_r+\eta +i\epsilon)
\prod\limits_{n=1}^{\infty}
\left( 1-\frac{(\alpha_r+\eta +i\epsilon)^2}{4\pi^2n^2}\right)
\label{48}
\end{eqnarray}
\noindent Then, omitting irrelevant infinite constant and using well-known
formula
\begin{equation}
\prod\limits_{n=1}^{\infty}\left(
1-\frac{a^2}{n^2}\right)=\frac{1}{\pi a} \sin \pi a \label{49} \end{equation}
\noindent one gets
\begin{equation}
{\det}\left(-\frac{d}{dt}
+i(B+\eta)-\epsilon\right)_{PBC}=
\prod_{r=1}^{N}\sin \left(
\frac{\alpha_r+\eta+i\epsilon}{2}\right)
\label{50}
\end{equation}
Substituting (\ref{50}) in (\ref{41}), one obtains contour integral
\begin{equation}
Z=\lim_{\epsilon \to +0} \int\limits_{-\infty}^{\infty \frac{e^{-i\eta\left(1+\frac{N}{2}\right)}}{
\prod_{r=1}^{N}\sin \left(
\frac{\alpha_r+\eta+i\epsilon}{2}\right)
}
\label{51}
\end{equation}
One can close the contour of integration in (\ref{51}) in the lower
half plane and represent $Z$ as the sum of residues in the poles
\begin{equation}
\eta_{rn}=-\alpha_r -i\epsilon +2\pi n
\label{52}
\end{equation}
The contribution of the pole with given $r$ and $n$ is
\begin{equation}
2\pi i(-1)^{n+1}\frac{e^{i(\alpha_r+i\epsilon)}}{
\prod\limits_{{1\le s\le N \atop s\ne r}}\sin \left(
\frac{\alpha_s-\alpha_r}{2}\right)
}
\label{53}
\end{equation}
\noindent So, omitting again inessential constant, one gets:
\begin{eqnarray}
Z&=&\lim_{\epsilon \to +0} \sum\limits_{r,n}
\left(
\begin{array}{c}
\mbox{contribution of the residue in}\\
\eta_{rn}=-\alpha_r -i\epsilon +2\pi n
\end{array}
\right)
\nonumber\\
&=&
\sum\limits_{r=1}^N\frac{e^{i\alpha_r(1+\frac{N}{2})}}{
\prod\limits_{
{\scriptstyle {s=1 \atop s\ne r}}
}^N
\sin \left( \frac{\alpha_s-\alpha_r}{2} \right)}
\label{54}
\end{eqnarray}
\noindent Finally, using elementary but rather non-obvious identity
\begin{equation}
\sum\limits_{r=1}^N\frac{e^{i\alpha_r(1+\frac{N}{2})}}{
\prod\limits_{
{\scriptstyle
{s=1\atop s\ne r}}}
^N
\sin \left( \frac{\alpha_s-\alpha_r}{2} \right)}
=
(-2i)^{N-1}
\left(e^{\frac i 2 \sum\limits_{r=1}^N \alpha_r}\right)
\sum\limits_{r=1}^N e^{i\alpha_r}
\label{55}
\end{equation}
\noindent (that is valid for any complex numbers $\alpha_r$; see Appendix A
for proof), one obtains:
\begin{equation}
Z=
\sum\limits_{r=1}^N e^{i\alpha_r}=\mathop{{\rm tr}} U(1) =\mathop{\mbox{tr P}} e^{i \int^1_0 \mathop{dt} B(t)}
\label{56}
\end{equation}
\noindent This proves the representation (\ref{36}).
Throughout the proof we didn't control inessential numerical
normalization factors arising in front of functional integrals,
determinants, etc. They can be easily reconstructed from
normalization condition
\begin{equation}
Z\raisebox{-10pt}{\rule{0.4pt}{20pt}${\scriptstyle B=0}$}
=N
\label{57}
\end{equation}
The representation (\ref{36}) is valid for traceless matrix $B$. In
general case the following representation is valid
\begin{equation}
Z=e^{-\frac i 2 \int^1_0 \mathop{dt} \mathop{{\rm tr}} B(t)}
\int D^2z \: \delta \left( \int^1_0 \mathop{dt} z^{\dag} z -1- \frac N 2\right)
e^{i\int_0^1 \mathop{dt}(iz^{\dag}\dot z +z^{\dag} Bz)}
\label{58}
\end{equation}
\noindent This representation is alternative to one given by eq.
(\ref{39}).
To prove (\ref{58}), it is sufficient to repeat the proof given for
representation (\ref{36}) but without using the condition (\ref{46})
. One can trace the cancellation of the pre-integral factor
in (\ref{58}) and contribution of the factor
$$
e^{\frac i 2 \sum\limits_{r=1}^N \alpha_r}
$$
\noindent in (\ref{55}).
Now let us check the representation (\ref{37}).
Representing $\delta$-function in (\ref{37}) as
\begin{equation}
\prod_t \delta \left( z^{\dag} z -1 -\frac N 2 \right)=\int
D\eta \: e^{i\int^1_0 \mathop{dt}
\left( z^{\dag} z -1 -\frac N 2 \right)\eta
}
\label{60}
\end{equation}
\noindent and integrating over $z^{\dag}, z$, one obtains the analog of
(\ref{41}):
\begin{equation}
Z=\lim_{\epsilon \to +0} \int D\eta(t)\: \frac{e^{-i\left( 1+\frac{N}{2}\right)
\int^1_0 \mathop{dt} \eta(t)}}{
{\det}\left(-\frac{d}{dt} +i(B(t)+\eta(t))-\epsilon\right)_{PBC}}
\label{61}
\end{equation}
The eigenvalue problem
\begin{equation}
\left(-\frac{d}{dt} +i(B(t)+\eta(t))-\epsilon\right)\phi (t)=\lambda \phi(t)
\label{62}
\end{equation}
\begin{equation}
\phi(0)=\phi(1)
\label{62.1}
\end{equation}
\noindent has the solution
\begin{equation}
\lambda\equiv \lambda_{rn}=-\epsilon +i\alpha_r
+i\int^1_0 \mathop{dt} \eta(t)
+2\pi i n,\ \ n=0,\pm 1,\ldots
\label{63}
\end{equation}
\noindent where numbers $\alpha_r$ are defined by equation (\ref{45}). So
determinant in (\ref{61}) depends on $\eta(t)$ only via
\begin{equation}
\eta=\int^1_0 \mathop{dt} \eta(t)
\label{64}
\end{equation}
\noindent Therefore, if one introduce new variables $\eta$ and
\begin{equation}
\eta_n=\int^1_0 \mathop{dt} e^{2\pi int} \eta(t),\ \ n\ne 0
\label{65}
\end{equation}
\noindent instead of $\eta(t)$ in (\ref{61}), one finds that integration
with respect to $\eta_n$ gives only inessential constant and so the
integral (\ref{61}) transforms just in the integral (\ref{41}). But
this means that we reduced the proof of representation (\ref{37}) to
one of the representation (\ref{36}) which we have already proved.
Finally, let us prove the validity of the representations (\ref{35})
and (\ref{37.1}). The proof can be reduced again to one of the
representation (\ref{36}) by means of the formula
\begin{equation}
\lim_{\epsilon \to +0} {\det} \oee =e^{\frac i 2 \int^1_0 \mathop{dt} \mathop{{\rm tr}} B(t)}
{\det} \left( -\frac{d}{dt} +iB(t) \right)
\label{66}
\end{equation}
\noindent that is valid for any $N\times N$ matrix $B(t)$ (the proof of
(\ref{66}) is given in Appendix~B). Indeed, evaluating the integral
with respect to $\bar z, \, z$ in (\ref{35}), one obtains, after
inserting of the factor $\exp\left( -\delta \int^1_0 \mathop{dt} z^{\dag}
(t)z(t+\epsilon)\right)$,
\begin{equation}
Z=\lim_{\delta \to +0}\lim_{\epsilon \to +0} \int\limits_{-\infty}^{\infty \frac{e^{-i\eta
}}{ {\det}\left(-\frac{d}{dt}
+i(B+\eta+i\delta)e^{-\epsilon\frac{d}{dt}}\right)_{PBC}}
\label{67}
\end{equation}
\noindent and further application of (\ref{66}) reduce (\ref{67}) to
(\ref{41}). (Remind, that $\mathop{{\rm tr}} B=0$ by assumption.)
The same arguments allow to check the representation(\ref{37.1}).
Thus we have checked all four representations for the trace of path
ordered exponent derived in the section 2. We see the proofs given
in the present section are very unlike ones given in the section 2.
This can be considered as strong confirmation of the validity of
the representations
obtained.
\section{A variant of non-Abelian Stokes theorem}
Analogs of Stokes theorem for Wilson loop
\begin{equation}
Z=\wl
\label{68}
\end{equation}
\noindent were proposed by several authors \cite{23}. In this works
Wilson loop is expressed via some area integral over surface spanned
on $\gamma$ with rather complicated path ordered prescriptions. Recently
Dyakonov and Petrov derived another, very smart variant of
non-Abelian Stokes theorem. Their result we already
cited (see (\ref{14})).
In this section we will prove a new variant of non-Abelian Stokes
theorem for $A_{\mu}\in su(N)$ that is similar to one given by
Dyakonov and Petrov. For the case $N=2$ we will be able to derive
from our results the Dyakonov-Petrov's formula (\ref{14}) but in
slightly corrected form. The little discrepancy between our results
and those by Dyakonov and Petrov arises, most likely, because of some
subtleties in the definition of the corresponding functional
integrals.
Let the path $\gamma$ in (\ref{68}) be parametrized as
\begin{equation}
\gamma=\{ x^{\mu}=q^{\mu}(t),\:0\le t\le 1,\: q^{\mu}(0)=q^{\mu}(t)\}
\label{69}
\end{equation}
\noindent Then
\begin{equation}
Z=\mathop{\mbox{tr P}} e^{i\int^1_0 \mathop{dt} \dot q ^{\mu}(t) A_{\mu}(q(t))}
\label{70}
\end{equation}
\noindent and we can apply the representation (\ref{37}):
\begin{equation}
Z=
\int D^2\psi \delta (z^{\dag} z -1- \frac N 2)
e^{\int^1_0 \mathop{dt} z^{\dag}\left(-\frac{d}{dt} +i\dot q A\right)z}
\label{71}
\end{equation}
Let $\xi^r=\xi^r(x), \: r=1,\ldots,N$ be any field such that
\begin{equation}
\xi^r(q(t))=z^r(t)
\label{72}
\end{equation}
\noindent The formula (\ref{71}) can be rewritten as
\begin{equation}
Z=
\int \mathop{D^2\xi} \prod\limits_{x\in\gamma}\delta\left( \xi^{\dag}(x)
\xi(x)-1-\frac N 2 \right)
e^{-\oint_{\gamma} dx^{\mu} \: \xi^{\dag}D_{\mu}\xi(x)}
\label{73}
\end{equation}
\noindent where $D_{\mu}=\partial_{\mu}-iA_{\mu}$ is the usual covariant
derivative. Further,
applying the classical Stokes theorem, one can easy to prove that
\begin{equation}
\oint_{\gamma} dx^{\mu} \: \xi^{\dag}D_{\mu}\xi(x)
=
\int_{\Sigma} \mathop{dx^{\mu}\wedge dx^{\nu}} \left(D_{\mu}\xi^{\dag}D_{\nu}\xi -\frac i 2
\xi^{\dag}F_{\mu\nu}\xi \right)
\label{74}
\end{equation}
\noindent where $\Sigma$ is any surface for which $\partial \Sigma=\gamma$.
Substituting (\ref{74}) in (\ref{73}), we get our variant of
non-Abelian Stokes theorem:
{\samepage
\begin{eqnarray}
Z
\equiv&&\wl\nonumber\\
=&&
\int \mathop{D^2\xi} \prod\limits_{x\in\gamma}\delta\left( \xi^{\dag}(x)
\xi(x)-1-\frac N 2 \right)
e^{
\int_{\Sigma} \mathop{dx^{\mu}\wedge dx^{\nu}} \left(-D_{\mu}\xi^{\dag}D_{\nu}\xi +\frac i 2
\xi^{\dag}F_{\mu\nu}\xi \right)
}\nonumber\\
\label{75}
\end{eqnarray}
}
Now let us transform eq. (\ref{75}) into the form that would be
similar to eq. (\ref{14}).
Let $\lambda^a,\: a=1,\ldots,N^2-1$ be Hermitian generators of $SU(N)$
in the fundamental representation. They can be normed as
\begin{equation}
\mathop{{\rm tr}} \lambda^a\lambda^b=2\delta^{ab}
\label{76}
\end{equation}
\noindent and satisfy equations
\begin{equation}
[\lambda^a, \lambda^b] =2 i f^{abc}\lambda^c
\label{77}
\end{equation}
\begin{equation}
[\lambda^a, \lambda^b]_{+} = \frac 4 N \delta^{ab}I_N +2d^{abc}\lambda^c
\label{78}
\end{equation}
One notes that
\begin{equation}
A_{\mu}=\frac{\lambda^a}{2}A_{\mu}^a,\: \ F_{\mu\nu}=
\frac{\lambda^a}{2}F_{\mu\nu}^a=
\frac{\lambda^a}{2}(\partial_{\mu}A^a_{\nu}-
\partial_{\nu}A^a_{\mu} +f^{abc}A^b_{\mu}A^c_{\nu})
\label{78'}
\end{equation}
Matrices $\lambda^a$ also obey "Fierz" identities:
\begin{eqnarray}
\frac 1 2 \lambda^a_{ij}\lambda^a_{kl}+
\frac 1 N \delta_{ij}\delta_{kl}&=&
\delta_{il}\delta_{kj}
\label{79}\\
if^{abc}\lambda^a_{ij}\lambda^b_{kl}\lambda^c_{mn}&=&
2(\delta_{il}\delta_{mj}\delta_{km}-
\delta_{in}\delta_{jk}\delta_{ml})
\label{80}
\end{eqnarray}
One introduces new variables
\begin{equation}
I^a(x)=\xi^{\dag}(x)\lambda^a\xi(x)
\label{81}
\end{equation}
Variables $I^a$ are not independent. Using "Fierz" identity
(\ref{79}), formula (\ref{81}) can be represented as
\begin{eqnarray}
\frac{1}{\xi^{\dag} \xi}(I^a\lambda^a)_{ij}+\frac 1 N \delta_{ij}=
\xi_i\xi^{\dag}_j\frac{1}{(\xi^{\dag}\xi)}
\label{82}
\end{eqnarray}
Let $I\equiv I^a\lambda^a, \: \xi^{\dag}\xi\equiv c$. One notes that
$c=1+\frac N 2$ on the surface of integration in (\ref{75}).
The matrix $I$ can be represented in the form
\begin{equation}
I=U diag(a_1,\ldots,a_N)U^{\dag}
\label{83}
\end{equation}
\noindent where
\begin{equation}
\sum_{i=1}^{N}a_i=0,\ \ U\in SU(N)
\label{84}
\end{equation}
The matrix in R.H.S. of eq. (\ref{82}) is a projector on the one
dimensional subspace. This means that eigenvalues of the matrix in
the L.H.S. of eq. (\ref{82}) are all equal to zero except only one
that is equal to 1. So, up to renumbering of eigenvalues,
\begin{eqnarray}
a_i&=&-\frac{2c}{N},\ \ i=1,\ldots,N-1\nonumber\\
a_N&=&2c\frac{(N-1)}{N}
\label{85}
\end{eqnarray}
One notes that
\begin{eqnarray}
\mathop{{\rm tr}} I^k&=&(N-1)\left( -\frac{2c}{N}\right)^k +\left[
2c\frac{(N-1)}{N}\right]^k\nonumber\\
&=&\frac{(2c)^k (N-1)}{N^k}
\left[ (N-1)^{k-1} +(-1)^k
\right]
\label{94}
\end{eqnarray}
Thus the matrix $I$ has $N-1$ coincident eigenvalues. But this means
that the solutions of eq. (\ref{82}) are in one-to-one
correspondence with the points of a coset $SU(N)/U(1)\times SU(N-1)${} $\approx CP^{N-1}$.
Indeed, the matrix $U$ in formula (\ref{83}) can be presented in the
form
\begin{equation}
U=U_1V_1V_2
\label{85.1}
\end{equation}
\noindent where
\begin{eqnarray}
V_1=
\left(
\begin{array}{cc}
\tilde{V}_1&
\begin{array}{c}
0\\
\vdots\\ 0
\end{array}\\
0\ldots 0&1
\end{array} \right), \ \ \tilde V_1 \in SU(N-1)
\label{85.2} \\
V_2=\exp \{ diag(t,\ldots,t,-(N-1)t \}\nonumber
\end{eqnarray}
\noindent and a matrix $U_1$ represents an element of the coset $SU(N)/U(1)\times SU(N-1)${} . So, due to
coincidence of the first $N-1$ eigenvalues of the matrix $I$, one
finds
\begin{equation}
I=U_1 diag\left(-\frac{2c}{N},\ldots,-\frac{2c}{N},
2c\frac{(N-1)}{N}\right) U^{\dag}_1 \label{86} \end{equation}
\noindent Therefore the set of solutions of eq. (\ref{82}) is isomorfic to
coset $SU(N)/U(1)\times SU(N-1)${} .
Further, using identities (\ref{79}), (\ref{80}), one finds that on
the surface $\xi^{\dag} \xi$$=c=const$
\begin{eqnarray}
D_{\mu}\xi^{\dag} D_{\nu}\xi&=&\frac{1}{8c^2}\mathop{{\rm tr}} ID_{[\mu}ID_{\nu]}I
\label{87}\\
\xi^{\dag} F_{\mu\nu}\xi&=&\frac 1 2 \mathop{{\rm tr}} IF_{\mu\nu}
\label{88}
\end{eqnarray}
\noindent where
\begin{equation}
D_{\mu}I=\partial_{\mu}I-i[A_{\mu},I]
\label{89}
\end{equation}
Now let us insert in (\ref{75}) an identity
\begin{equation}
1=\prod_{x\in\Sigma}\int\prod_a dI^a(x)\delta(\xi^{\dag}(x) \lambda^a \xi(x) -I^a(x))
\label{90}
\end{equation}
\noindent By virtue of (\ref{87}), (\ref{88}), the eq. (\ref{75}) can be
rewritten in the form
\begin{equation}
Z=\int D\mu(I)\exp\left\{
-i\int\limits_{\Sigma}\mathop{dx^{\mu}\wedge dx^{\nu}}\left[
\frac{1}{8c^2}\mathop{{\rm tr}} ID_{[\mu}ID_{\nu]}I-\frac 1 4
\mathop{{\rm tr}} IF_{\mu\nu}\right]\right\}
\label{91}
\end{equation}
\noindent where
\begin{equation}
D\mu(I)=\left(\prod_a DI^a\right)
\int D^2\xi \delta(\xi^{\dag} \xi- 1-\frac N 2)\prod_a\delta(\xi^{\dag} \lambda^a \xi-I^a)
\label{92}
\end{equation}
The consideration given above shows that
$D\mu(I)$ is nothing but $SU(N)$-invariant measure on the
coset $SU(N)/U(1)\times SU(N-1)${}{}
The explicit form of the measure $D\mu(I)$ is rather cumbersome but,
fortunately, it appears that measure $D\mu(I)$ in (\ref{91}) can be
replaced by the measure
\begin{equation}
D\mu'(I)\equiv \left(DI^a\right) \prod_{k=2}^N \delta(\mathop{{\rm tr}} I^k- c_{k,N})
\label{93}
\end{equation}
\noindent where numbers $c_{k,N}$ are defined by formula (\ref{94}) in
which one must put $2c=N+2$ by virtue of the first $\delta$-function in
eq.(\ref{92}).
Indeed, numbers $\mathop{{\rm tr}} I^k$ define uniquely
characteristic equation for the matrix $I$ and, consequently, its
eigenvalues. This leads to representation (\ref{86}). So on the
surface $\mathop{{\rm tr}} I^k=c_{k,N}$ any functional $\Phi(I)$ is invariant under
transformation
\begin{equation}
\Phi(I)\to \Phi(V_1 V_2 IV^{\dag}_2V^{\dag}_1)
\end{equation}
\noindent where a matrix $V_1 V_2$ is defined by eqs. (\ref{85.2}).
Therefore in any integral
\begin{equation}
\int D\mu'(I)\Phi(I)
\end{equation}
\noindent one can perform all integrations except those corresponding to
integration over coset $SU(N)/U(1)\times SU(N-1)${} :
\begin{equation}
\int D\mu'(I)\Phi(I)
=const \int D\mu(I)\Phi (U_1diag(-\frac{2c}{N},\ldots,-\frac{2c}{N},
2c\frac{N-1}{N})U^{\dag}_1)
\end{equation}
\noindent (see (\ref{86})). But this just means, in particular, that one
can replace the measure $D\mu(I)$ in (\ref{91}) by the measure
$D\mu'(I)$ defined by eq. (\ref{93}).
Now we can formulate our final result:
\begin{eqnarray}
\lefteqn{
\mathop{\mbox{tr P}} \exp\left\{i\oint_{\gamma} \mathop{dx^{\mu}} A_{\mu} \right\}}\nonumber\\
&=&\int DI\prod_{k=2}^N \delta(\mathop{{\rm tr}} I^k-c_{k,N})\nonumber\\
&&\exp \left\{
i\int_{\Sigma}\mathop{dx^{\mu}\wedge dx^{\nu}}\left[
-\frac{1}{2(N+2)^2}\mathop{{\rm tr}} (ID_{\mu}ID_{\nu}I) +\frac 1 4 \mathop{{\rm tr}} IF_{\mu\nu}
\right]\right\}\nonumber\\
&&\label{94'}
\end{eqnarray}
\noindent where
\begin{equation}
c_{k,N}=\frac{(N-1)(N+2)^k}{N^k}\left[ (N-1)^{k-1} +(-1)^k
\right]
\label{95}
\end{equation}
Let us compare our results with those due to Dyakonov and Petrov
\cite{21}. Putting in (\ref{94'}) $N=2,\ \ I=I^a\sigma^a$ and the
changing $I^a\to 2I^a$ , we get:
\begin{eqnarray}
\lefteqn{
\mathop{\mbox{tr P}} \exp\left\{i\oint_{\gamma} \mathop{dx^{\mu}} A_{\mu} \right\}}\nonumber\\
&=&
\int DI \delta(I^2-1)\exp\left\{-\frac i 2 \int \mathop{dx^{\mu}\wedge dx^{\nu}}
\left( \varepsilon^{abc}I^aD_{\mu}I^b D_{\nu}I^c-I^a F^a_{\mu\nu}
\right)\right\}\nonumber\\
&&\label{97}
\end{eqnarray}
Comparing formulae (\ref{97}) and (\ref{14}), we see that they
differ by the factor 1/2 in front of the "action". Most likely this
discrepancy arises because of some subtleties in the definition of
the functional integrals that play the important role in our
discussion. In particular, if one ignores the difference
between functional integrals constructed by means of Weyl and normal
symbols, one obtains the following representation instead of
(\ref{73}):
\begin{equation}
Z=
\int \mathop{D^2\xi} \prod\limits_{x\in\gamma}\delta\left( \xi^{\dag}(x)
\xi(x)-1 \right)
e^{-\oint_{\gamma} dx^{\mu} \: \xi^{\dag}D_{\mu}\xi(x)}
\label{98}
\end{equation}
\noindent Further, using formulae (\ref{87}) and (\ref{94}) with $c=1$
instead of $c=1+\frac N 2$, one can easy trace that representation
(\ref{98}) leads exactly to Dyakonov-Petrov formula (\ref{14}). But
representation (\ref{98}) is wrong. So just formula (\ref{97}) must
be considered as correct version of non-Abelian Stokes theorem for
$N=2$.
This doesn't mean, however, that Dyakonov-Petrov formula (\ref{14})
is incorrect. But it means that the definition of functional
integral in (\ref{14}) must be clarified.
Another discrepancy between our results and those due to Dyakonov
and Petrov arises in the case $N\ge 3$. Dyakonov and Petrov pointed
out in their work \cite {21} that integration in the functional
integral representation for Wilson loop must be performed over the
coset $SU(N)/[U(N)]^{N-1}$
whereas in our representation (\ref{94'}) integration is carried out,
in fact, over the coset $SU(N)/U(1)\times SU(N-1)${} . But nowadays it is hard to discuss this
discrepancy because no explicit formulae for the case $N\ge 3$ were
given in \cite{21}.
{\hsize=13.26cm
{\section{Bosonic worldline path integral representation for
fermionic determinants and
Green functions in Euclidean space}}
Bosonic worldline path integral representation for fermionic
determinants can be obtained directly from formulae (\ref{8}),
(\ref{10}) and results of the section 2:
\begin{eqnarray}
\lefteqn{
\ln{\det}(i\hat{\nabla}+im)}\nonumber\\
&=&\dT\dQ\nonumber\\
&&
{\cal N}_T\exp\left\{
\int^1_0 \mathop{dt} \left[ -\frac{\dot q^2}{4T}-z^{\dag} \dot z
-\psi^{\dag}\dot{\psi}+i\dot q z^{\dag} A_{\mu} z\right.\right.\nonumber\\
&&\left.\left.
-T(\psi^{\dag}\sigma^{\mu\nu}\psi)(z^{\dag} F_{\mu\nu} z)
\vphantom{
\int^1_0 \mathop{dt} \left[ -\frac{\dot q^2}{4T}-z^{\dag} \dot z\right.
}
\right] \right\}
\label{99}
\end{eqnarray}
Here variables $z=\{z^r, \ \ r=1,\ldots,N\}$ and $\psi=\{\psi^i,\:
i=1,,2,3,4\}$ describe colour and spin degrees of freedom
respectively, $z^{\dag}$ and $\psi^{\dag}$ are complex conjugated to $z$
and $\psi$,
\begin{eqnarray}
D^2 z&\equiv&\prod\limits_t \prod\limits_{r=1}^N
\mathop{d(\mbox{Re}\, z^r(t))}\mathop{d(\mbox{Im}\, z^r(t))}\label{100}\\
D^2 \psi&\equiv&\prod\limits_t \prod\limits_{r=1}^N
\mathop{d(\mbox{Re}\, \psi^i(t))}\mathop{d(\mbox{Im}\, \psi^i(t))}
\label{100.1}
\end{eqnarray}
\noindent and ${\cal N}_T$ is a normalization constant. The latter can be
evaluated from the condition
\begin{equation}
<x|\mathop{{\rm tr}}
e^{-T\wH}|x>\raisebox{-10pt}{\rule{0.4pt}{20pt}${\scriptstyle A=0}$}=
4N<x|e^{-T\partial_{\mu}\partial_{\mu}}|x>=\frac{N}{2\pi^2T^2}
\label{101}
\end{equation}
Indeed, putting $A=0$ in (\ref{99}) and comparing the result with
(\ref{101}), one obtains:
\begin{eqnarray}
{\cal N}_T^{-1}
&=&\left( \frac{N}{2\pi^2T^2}\right)^{-1}
\int\limits_{q(0)=q(1)=x} Dq\:\int_{PBC}D^2\psi D^2 z\delta\left(
z^{\dag}z -1-\frac N 2 \right)\nonumber\\
&&\delta\left( \psi^{\dag}\psi
-3\right)
\exp\left\{
\int^1_0 \mathop{dt} \left[ -\frac{\dot q^2}{4T}-z^{\dag} \dot z
-\psi^{\dag}\dot{\psi}\right]\right\}
\label{102}
\end{eqnarray}
\noindent Obviously, ${\cal N}_T$ doesn't depend on $x$.
Remind, that in Euclidean space $\psi^{\dag}\psi$ is $SO(4)$ scalar.
So representation (\ref{99}) is manifestly relativistic and gauge
invariant.
Our next task is to derive bosonic worldline path integral
representation for Euclidean Green functions. In what follows, we
restrict ourselves only to derivation of such representation for
generating functional $Z(j)$ for vacuum correlators
\begin{equation}
<\psi^{\dag}_{f_1}(x_1)\psi_{f_1}(x_1)\ldots
\psi^{\dag}_{f_n}(x_n)\psi_{f_n}(x_n)>
\label{103}
\end{equation}
\noindent However, our method is quite general and can be easily applied
to derivation of analogues representations for arbitrary Green
functions.
Standard functional integral representation for $Z(j)$ can be
written as
\begin{equation}
Z(j)=\int DA\: e^{S_{YM}}\int D\Psi D\bar{\Psi}
e^{\left\{ \sum_f\int dx \:\left[ i\bar{\Psi}_f \hat{\nabla}\Psi_f+im_f
\bar{\Psi}_f\Psi_f-ij_f\bar{\Psi}_f\Psi_f\right]\right\}}
\label{104}
\end{equation}
Integrating with respect to fermionic fields, we get
\begin{equation}
Z(j)=\int DA\: e^{S_{YM}}\prod_f{\det}(i\hat{\nabla}+im_f-ij_f)
\label{105}
\end{equation}
So our task is reduced to derivation of bosonic path integral
representation for determinant
$$
{\det}(i\hat{\nabla}+im-ij)
$$
The latter problem can be easily solved by applying of the
results obtained in the section 2. Indeed,
\begin{eqnarray}
\lefteqn{
\ln{\det} (i\hat{\nabla}+im-ij)=\frac 1 2
\ln[{\det}(i\hat{\nabla}+im-ij)\gamma^5]^{2}}\nonumber\\
&=&\frac 1 2
\ln{\det}(\nabla_{\mu}\nabla^{\mu}-\sigma^{\mu\nu}F_{\mu\nu}
+\hat{\partial}j+(m-j)^2)\nonumber\\
&=&\dT\mathop{{\rm tr}} e^{-T
(-\nabla_{\mu}\nabla^{\mu}+\sigma^{\mu\nu}F_{\mu\nu}-\hat{\partial}
j+2mj-j^2)}
\label{105'}
\end{eqnarray}
Using representation for path ordered exponent from section 2, one
obtains:
\begin{eqnarray}
\lefteqn{\ln{\det}(i\hat{\nabla}+i(m+j))}\nonumber\\
&=& \dT\dQ \nonumber\\
&&\exp\left\{
\int^1_0 \mathop{dt} \left[ -\frac{\dot q^2}{4T}-z^{\dag} \dot z
-\psi^{\dag}\dot{\psi}+i\dot q z^{\dag} A_{\mu} z\right.\right.\nonumber\\
&&\left.\left.-T(\psi^{\dag}\sigma^{\mu\nu}\psi)(z^{\dag} F_{\mu\nu} z
-\psi^{\dag}\gamma^{\mu}\psi\partial_{\mu}j(q)-2mj(q)+j^2(q) \right]
\right\}
\label{106}
\end{eqnarray}
Substituting (\ref{105'}) in (\ref{106}) for each $f$, we get
worldline bosonic path integral representation for generating
functional $Z(j)$. The corresponding formulae for $n$-point
correlators are rather cumbersome but quite computable. Author
hopes that they can be used for computer simulations on the
lattice.
{\hsize=13.26cm
\section{Bosonic worldline path integral
representation for fermionic determinants and Green
functions in Minkowski space and quasiclassical approximation in QCD}
}
The derivation of bosonic worldline path integral representation for
fermionic determinants in Minkowski space is slightly more involved
then one in Euclidean space. The origin of complications is
non-unitarity of finite dimensional representation of the Lorentz
group.
The analog of the representation (\ref{8}) in Minkowski space can be
written as
\begin{equation}
\ln{\det}(i\hat{\nabla}-m)=\dTM \mathop{{\rm tr}} e^{iT\wH}
\label{107}
\end{equation}
Trace in (\ref{107}) can be represented as functional integral:
\begin{equation}
\mathop{{\rm tr}} e^{iT\wH}=\int_{PBC} Dq \: \mathop{\mbox{tr P}}
e^{i\int^1_0 \mathop{dt} \left( -\frac{\dot q^2}{4T}+\dot q A(q)+T
\sigma^{\mu\nu}F_{\mu\nu}(q)\right)}
\label{108'}
\end{equation}
\noindent eq. (\ref{108'}) is an analog of eq. (\ref{10}). Path ordering in
(\ref{108'}) corresponds to colour and spinor structures.
However, in contrast to Euclidean case, we cannot directly use the
representations of the type (\ref{37}), (\ref{39.4}) to write
ordered exponent in (\ref{108'}) as functional integral. Indeed,
those representation comprise, in particular, the factor
\begin{equation}
\delta(\psi^{\dag}\psi-3)
\label{108''}
\end{equation}
\noindent (see (\ref{99})) that is not Lorentz invariant because spinor
representations of Lorentz group are not unitary.
To obtain manifestly Lorentz invariant representation, we will use,
at first, representation (\ref{37}) for describing of colour degrees
of freedom and representation (\ref{37.1}) for describing of spinor
ones. In such terms eq. (\ref{108'}) can be rewritten as
\begin{eqnarray}
\lefteqn{\mathop{{\rm tr}} e^{i\wH}}\nonumber\\
&=&\lim_{\epsilon \to +0} \int_{PBC}DqD^2zD\bar{\psi}D\psi D\lambda \delta(z^{\dag} z-1-\frac N 2)
\nonumber\\
&&{\cal N}_T \exp\left\{i\int^1_0\mathop{dt}\left[-\frac{\dot q^2}{4T}+iz^{\dag} z
+i\bar{\psi}\dot{\psi}+\right.\right.\nonumber\\
&&\left.\left.z^{\dag} \dot q A(q) z+ T(\bar{\psi}\sigma^{\mu\nu}e^{-\epsilon
\frac d{dt}}\psi)(z^{\dag} F^{\mu\nu}z)+\lambda(\bar{\psi}e^{-\epsilon\frac{d}{dt}}
\psi-1)\right]\right\}
\label{108}
\end{eqnarray}
In eq. (\ref{108}) the measure $D^2z$ is defined by (\ref{100}) but
$\psi$ and
$\bar{\psi}$ are independent complex variables. ${\cal N}_T$ is a
normalization constant that will be computed later.
Integrating over $\bar{\psi},\ \ \psi$ in (\ref{108}), one gets:
\begin{eqnarray}
\lefteqn{\mathop{{\rm tr}} e^{i\wH}}\nonumber\\
&=&\lim_{\epsilon \to +0} \int_{PBC}DqD^2z D\lambda \delta(z^{\dag} z-1-\frac N 2)
\nonumber\\
&&{\cal N}_T
{\det}^{-1}\left[-\frac d{dt}+iT\sigma^{\mu\nu}(z^{\dag} F_{\mu\nu}z)
e^{-\epsilon\frac d{dt}}+i\lambda e^{-\epsilon\frac d{dt}}\right]
\nonumber\\
&& \exp\left\{i\int^1_0\mathop{dt}\left[-\frac{\dot q^2}{4T}+iz^{\dag} \dot z+
z^{\dag} \dot q A(q) z- \lambda \right]\right\}
\label{109}
\end{eqnarray}
But
\begin{eqnarray}
\lefteqn{\lim_{\epsilon \to +0}
{\det}^{-1}\left[-\frac d{dt}+iT\sigma^{\mu\nu}(z^{\dag} F_{\mu\nu}z)
e^{-\epsilon\frac d{dt}}+i\lambda e^{-\epsilon\frac d{dt}}\right]}\nonumber\\
&=&e^{-2i\int^1_0 \mathop{dt} \lambda (t)}
{\det}^{-1}\left[-\frac d{dt}+iT\sigma^{\mu\nu}(z^{\dag} F_{\mu\nu}z)
+i\lambda\right]
\label{110}
\end{eqnarray}
\noindent by virtue of identity (\ref{66}). Further,
\begin{eqnarray}
\lefteqn{
{\det}^{-1}\left[-\frac d{dt}+iT\sigma^{\mu\nu}(z^{\dag} F_{\mu\nu}z)
+i\lambda\right]}
\nonumber\\
&=&{\det}^{-1}\left[-\gamma^0\frac d{dt}+\gamma^0iT\sigma^{\mu\nu}(z^{\dag}
F_{\mu\nu}z) +i\gamma^0\lambda\right]
\label{111}
\end{eqnarray}
\noindent The operator in R.H.S. of (\ref{111}) is {\it anti-Hermitean}.
So we can write
\begin{eqnarray}
\lefteqn{
{\det}^{-1}\left[-\gamma^0\frac d{dt}+\gamma^0iT\sigma^{\mu\nu}(z^{\dag}
F_{\mu\nu}z) +i\gamma^0\lambda\right]}
\nonumber\\
&=&\int_{PBC}D^2\psi e^{i\int^1_0 \mathop{dt} (i\psi^{\dag} \gamma^0\dot{\psi}+T(\psi^{\dag}\gamma^0
\sigma{\mu\nu}\psi)(z^{\dag} F_{\mu\nu}z)+\lambda\psi^{\dag}\gamma^0\psi)}
\label{112}
\end{eqnarray}
In the last formula $\psi^{\dag}$ and $\psi$ are already complex conjugate
variables, measure $D^2\psi$ is defined by eq. (\ref{100.1}), and the
"action" is real. So functional integral (\ref{112}) is well defined.
Introducing standard notations $\bar{\psi}=\psi^{\dag}\gamma^0$ and substituting
(\ref{110})-(\ref{112}) in (\ref{109}), one obtains, after
integration with respect to $\lambda$,
\begin{eqnarray}
\lefteqn{
\mathop{{\rm tr}} e^{iT\wH}}\nonumber\\
&=&\dQM \nonumber\\
&&{\cal N}_T \exp\left\{i\int^1_0\mathop{dt}\left[-\frac{\dot q^2}{4T}+iz^{\dag} z
+i\bar{\psi}\dot{\psi}+\right.\right.\nonumber\\
&&\left.\left.z^{\dag} \dot q A(q) z+ T(\bar{\psi}\sigma^{\mu\nu} \psi)(z^{\dag}
F^{\mu\nu}(q)z)\right]\right\}
\label{113}
\end{eqnarray}
The normalization constant ${\cal N}_T$ can be computed in the same
way as its analog in eq. (\ref{99}):
\begin{eqnarray}
{\cal N}_T^{-1}&=&\left(- \frac{N}{2\pi^2T^2}\right)^{-1}
\int\limits_{q(0)=q(1)=x} Dq\:\int_{PBC}D^2\psi D^2 z\delta\left(
z^{\dag}z -1-\frac N 2 \right)\nonumber\\
&&\delta\left( \bar{\psi}\psi
-3\right)
\exp \left\{ i
\int^1_0 \mathop{dt} \left[ -\frac{\dot q^2}{4T}+iz^{\dag} \dot z
+i\psi^{\dag}\dot{\psi}\right]\right\}
\label{114}
\end{eqnarray}
Substituting (\ref{113}) in (\ref{107}), we obtain desired
representation for fer\-mi\-o\-nic determinant:
{\samepage
\begin{eqnarray}
\lefteqn{
\ln{\det}(i\hat{\nabla}-m)}\nonumber\\
&=&\dTM\dQM\nonumber\\
&&{\cal N}_T \exp\left\{i
\int^1_0\mathop{dt}\left[-\frac{\dot q^2}{4T}+iz^{\dag} z
+i\bar{\psi}\dot{\psi}+\right.\right.\nonumber\\
&&\left.\left.z^{\dag} \dot q A(q) z+ T(\bar{\psi}\sigma^{\mu\nu} \psi)(z^{\dag}
F^{\mu\nu}(q)z)\right]\right\}
\label{115}
\end{eqnarray}
}
The representation (\ref{115}) is manifestly gauge and Lorentz
invariant and comprises only bosonic variables. The "action" in
(\ref{115}) is real. So, as we will see soon, it is convenient for
application of stationary phase method.
Now let us derive bosonic path integral representation for
generating functional $Z(j)$ of gauge invariant Green functions
\begin{equation}
G_{f_1,\ldots,f_n}(x_1,\ldots,x_n)=
<T(\bar{\psi}_{f_1}(x_1)\psi_{f_1}(x_1)\ldots
\bar{\psi}_{f_n}(x_n)\psi_{f_n}(x_n))>
\label{116}
\end{equation}
The derivation is completely analogous to one given in the previous
section for corresponding Euclidean correlators.
For $Z(j)$ there exist standard path integral representation via
anti-commuting variables:
\begin{eqnarray}
Z(j)&=&\int DA e^{iS_{YM}}\int D\bar{\Psi}D\Psi
e^{
i\int dx\: \sum\limits_f \left[\bar{\Psi}_f(i\hat{\nabla}-m_f)\Psi_f+
j_f\bar{\Psi}_f\Psi_f\right]}\nonumber\\
&=&
\int DA e^{iS_{YM}}\prod_f{\det}(i\hat{\nabla}-m_f+j_f)
\label{117}
\end{eqnarray}
Repeating with minor changes the derivation of (\ref{106}), one gets
{\samepage
\begin{eqnarray}
\lefteqn{
\ln{\det}(i\hat{\nabla}-m-j)}\nonumber\\
&=&\dTM\dQM\nonumber\\
&&{\cal N}_T \exp\left\{i\int^1_0\mathop{dt}\left[-\frac{\dot q^2}{4T}+iz^{\dag} z
+i\bar{\psi}\dot{\psi}+
z^{\dag} \dot q A(q) z+ \right.\right.\nonumber\\
&&\left.\left.T(\bar{\psi}\sigma^{\mu\nu} \psi)(z^{\dag}
F^{\mu\nu}(q)z)+
2mTj(q)-iT\bar{\psi}\gamma^{\mu}\psi\partial_{\mu}j(q)-
Tj^2(q)
\right]\right\}\nonumber\\
&&\label{118}
\end{eqnarray}
2}
Substituting (\ref{118}) in (\ref{117}), we obtain worldline pat
integral representation for $Z(j)$.
Our next task is the investigation of quasiclassical approximation in
QCD. To this end, we will formulate a scheme of evaluation of
two-point function
\begin{equation}
G_{f_0}(x,y)=<T(\bar{\psi}_{f_0}(x)\psi_{f_0}(x)\bar{\psi}_{f_0}(y)\psi_{f_0}(y))>
\label{119}
\end{equation}
This scheme can be easily generalized for evaluation of arbitrary
Green functions.
The equations for the stationary point will be interpreted as
quasiclassical equations in QCD. They will be formulated in terms of
particles that have spin and colour degrees of freedom and interecting
with Yang-Mills field.
First of all, we introduce more condenced notations:
\begin{eqnarray}
Q_f&\equiv& \{q_f,z^{\dag}_f,z_f,\bar{\psi}_f,\psi_f,T_f,m^2_f\}
\label{120.0}\\
\int DQ_f(\cdots)&\equiv&
-\frac 1 2 \int_0^{\infty} \frac{dT_f}{T_f} \:
\int_{PBC} DqD^2\psi D^2 z\delta\left( z^{\dag}z
-1-\frac N 2 \right)\nonumber\\
&&\delta\left( \bar{\psi}\psi -3\right)
{\cal N}_T(\cdots)
\label{122.1}\\
S[Q_f,A]&=&
\int^1_0\mathop{dt}\left[-\frac{\dot q_f^2}{4T_f}+iz^{\dag}_f\dot z_f
+i\bar{\psi_f}\dot{\psi_f}+
z^{\dag}_f \dot q A(q) z_f
\right.\nonumber\\
&&\left.+ T_f(\bar{\psi_f}\sigma^{\mu\nu}
\psi_f)(z^{\dag}_f F_{\mu\nu}(q)z_f)-T_fm^2_f\right]
\label{120}
\end{eqnarray}
Further, using eq. (\ref{118}), one can get:
\begin{equation}
\frac{\delta}{\delta j(x)}
\ln\det (i\hat{\nabla} -m_f+j)
\raisebox{-10pt}{\rule{0.4pt}{20pt}${\scriptstyle j=0}$}
=\int DQ_f e^{iS[Q_f,A]}R(x|Q_f)
\label{121}
\end{equation}
{\samepage
\begin{eqnarray}
&&\frac{\delta^2}{\delta j(x)\delta j(y)}
\ln\det (i\hat{\nabla} -m_f+j)
\raisebox{-10pt}{\rule{0.4pt}{20pt}${\scriptstyle j=0}$}
\nonumber\\
&=&\int DQ_f e^{iS[Q_f,A]}R(x|Q_f)R(y|Q_f)\nonumber\\
&+&i\delta(x-y)\int DQ_f e^{iS[Q_f,A]}T_f\int^1_0 dt_1dt_2\:
\delta(q_{f}(t_1)-q_f(t_2))
\label{122}
\end{eqnarray}
}
where
\begin{equation}
R(x|Q_f)= T\int^1_0
\mathop{dt}[2m_f-i\bar{\psi}_f(t)\gamma^{\mu}\psi_f(t)\partial_{\mu}]
\delta(x-q_f(t)) \label{123} \end{equation}
For any functional $W(j)$
\begin{equation}
\frac{\delta^2 W(j)}{\delta j(x)\delta j(y)}=
W(j)\left[
\frac{\delta^2 \ln W(j)}{\delta j(x)\delta j(y)}+
\frac{\delta \ln W(j)}{\delta j(x)}
\frac{\delta \ln W(j)}{\delta j(y)}
\right]
\label{124}
\end{equation}
Using (\ref{117}), (\ref{121}), (\ref{122}) and applying (\ref{124})
for $W(j)=\det(i\hat{\nabla}-m_{f_0}+j)$, one obtains:
\begin{equation}
G_{f_0}(x-y)=G^{(1)}_{f_0}(x-y)+G^{(2)}_{f_0}(x-y)
\label{125}
\end{equation}
\noindent where $G^{(1)}_{f_0}$ and $G^{(2)}_{f_0}$ correspond to the first
and to the second terms in R.H.S. of (\ref{124}) respectively:
\begin{eqnarray}
G^{(1)}_{f_0}(x-y)&=&
\int DADQ_{f_0}\:
R(x,Q_{f_0})R(y,Q_{f_0})
\nonumber\\
&&\exp\left\{
iS_{YM}+iS[Q_{f_0},A]
+ \sum_f\int DQ_f \: e^{iS[Q_f,A]}\right\}\nonumber\\
&&+\delta(x-y)(\cdots)
\label{126}\\
G^{(2)}_{f_0}(x-y)&=&
\int DADQ_{f_0}D{Q'}_{f_0}\:
[R(x,Q_{f_0})R(y,Q_{f_0})\nonumber\\
&&\exp\left\{\vphantom{
\sum_f\int DQ_f \: e^{iS[Q_f,A]}}
iS_{YM}+iS[Q_{f_0},A]+iS[Q^{'}_{f_0},A]\right.\nonumber\\
&&\left.+ \sum_f\int DQ_f \: e^{iS[Q_f,A]}\right\}
\label{127}
\end{eqnarray}
\noindent In (\ref{126}) $(\cdots)$ means the factor at $\delta(x-y)$ in
R.H.S. of (\ref{122}).
At first, we investigate the function $G^{(1)}_{f_0}$.
The function $G_{f_0}(x-y)$, in itself, is defined up to counterterm
$$const\, \delta(x-y)$$
\noindent by virtue of ultraviolet divergences. Then the last term in
R.H.S. of (\ref{126}) only redefines this counterterm and so can be
omitted.
Expanding
\begin{equation}
\exp\left\{\sum_f\int DQ_f\: e^{iS[Q,A]}\right\}
\label{128}
\end{equation}
\noindent in series, we can represent (\ref{126}) as
\begin{eqnarray}
\lefteqn{
G^{(1)}_{f_0}(x-y)}\nonumber\\
&=&\sum_{n_f}\frac 1{\prod_f n_f!} \int DADQ_{f_0}\:
\left( \prod_f \prod_{j_f=1}^{n_f}DQ_{j_f}\right)
R(x,Q_{f_0})R(y,Q_{f_0})
\nonumber\\
&&\exp\left\{iS_{YM}+iS[Q_{f_0},A]+ \sum_f\sum_{j_f=1}^{n_f}
iS[Q_{j_f},A]\right\}
\nonumber\\
&\equiv&\sum_{n_f}G^{(1)}_{f_0}(x-y;\{n_f\})
\label{129}
\end{eqnarray}
Obviously, each function $G^{(1)}_{f_0}(x-y;\{n_f\})$ correspond to
contribution of all diagrams comprising $n_{f_1}$ quark loops of
flavor $f_1$, $n_{f_2}$ quark loops of flavor $f_2$, etc.
We will investigate each term in the series (\ref{129}) separately. At
first we consider the term with $n_f=0$:
\begin{equation}
G^{(1)}_{f_0}(x-y;\{n_f=0\})=
\int DADQ_{f_0}\: e^{iS_{YM}+iS[Q_{f_0},A]}
R(x,Q_{f_0})R(y,Q_{f_0})
\label{130}
\end{equation}
The function $R(x|Q_f)$ can be represented as
\begin{equation}
R(x|Q_f)=\int d^4p\:\int^1_0\mathop{dt}\tilde R (p,t|Q_f)e^{ip(x-q_f(t))}
\label{131}
\end{equation}
\noindent where
\begin{equation}
\tilde R (p,t|Q_f)=T(2m_f+p_{\mu}\bar{\psi}(t)\gamma^{\mu}\psi(t))
\label{132}
\end{equation}
\noindent Then we change variables:
\begin{equation}
q(t)\to q'(t)=q(t)-q_0, \ \ A(x)\to A(x-q_0)
\label{132'}
\end{equation}
\noindent where the function $q'(t)$ obeys the boundary conditions
\begin{equation}
q'(0)=q'(1)=0
\label{132''}
\end{equation}
By virtue of translational invariance we get
\begin{eqnarray}
\lefteqn{
G^{(1)}_{f_0}(x-y;\{n_f=0\})
}\nonumber\\
&=&\int DA\int_{q(0)=q(1)=0}DQ_{f_0}\:\int dq_0\:
e^{iS_{YM}+iS[Q_{f_0},A]}\nonumber\\
&&\int d^4p_1d^4p_2 \int^1_0 dt_1\:\int^1_0 dt_2\:
\tilde R(p_1,t_1|q_{f_0}) R(p_2,t_2|q_{f_0})\nonumber\\
&&e^{ip_1(y-q_{f_0}(t_1))}e^{ip_2(y-q_{f_0}(t_2))}
e^{-i(p_1+p_2)q_0}\nonumber\\
&&
\label{133}
\end{eqnarray}
Integration over $q_0$ gives $\delta(p_1+p_2)$, and we obtain the
following representation for Fourier transformation of
$G^{(1)}_{f_0}(x-y;\{n_f=0\})$:
\begin{eqnarray}
\lefteqn{
G^{(1)}_{f_0}(p;\{n_f=0\})
}\nonumber\\
&\equiv&\int dx\: e^{-ipx}
G^{(1)}_{f_0}(x;\{n_f=0\})
\nonumber\\
&=&\int^1_0 {\mathop{dt}}_1 \int^1_0{\mathop{dt}}_2 \int DA\int_{q_{f_0}(0)=
q_{f_0}(1)=0} DQ_{f_0}\:
\tilde R (p,t_1|Q_{f_0})\tilde R(-p,t_2|Q_{f_0})\nonumber\\
&&\exp \left\{ iS_{YM}+iS[Q_{f_0},A]-ip(q_{f_0}(t_1)-q_{f_0}(t_2))
\right\}
\label{134}
\end{eqnarray}
Now it is already easy to write equations for stationary point for
the action in (\ref{134}). Omitting the index $f_0$, we get:
\begin{eqnarray}
\frac 1{g^2}\nabla_{\nu}F^{a\mu\nu}&=&
\int^1_0 \mathop{dt} I^a(t)\dot q^{\mu}(t)\delta(x-q(t))\nonumber\\
&&+T\nabla_{\nu}\left[ \int^1_0 \mathop{dt} I^a(t) S^{\mu\nu}
\delta(x-q(t))\right]
\label{135}
\end{eqnarray}
\begin{equation}
i\left(
\frac d{dt} -i\dot q^{\mu}A_{\mu}(q)
\right)z + TS^{\mu\nu}F_{\mu\nu}z=0
\label{136}
\end{equation}
\begin{equation}
i\frac d{dt}\psi +TF^a_{\mu\nu}I^a\sigma^{\mu\nu}\psi=0
\label{137}
\end{equation}
\begin{equation}
\frac 1 T \ddot q_{\mu}+\dot q^{\nu}F^a_{\mu\nu}I^a
+\nabla_{\mu}F^a_{\nu\rho}(q)I^a S^{\nu\rho}=p_{\mu}(\delta(t_1)-
\delta(t_2))
\label{138}
\end{equation}
\begin{equation}
\frac 1{4T^2} \int^1_0 \mathop{dt} \dot q^2 +\frac 1 2
\int^1_0 \mathop{dt} S^{\mu\nu}F^a_{\mu\nu}I^a=m^2
\label{139}
\end{equation}
\noindent where
\begin{equation}
I^a=z^{\dag} \lambda^a z,\ \ S^{\mu\nu}=\bar{\psi}\sigma^{\mu\nu}\psi
\label{140}
\end{equation}
Eqs. (\ref{135})-(\ref{138}) can be derived by variation of the
action in (\ref{134}) with respect to $A^a_{\mu},\ z,\ \psi$, and
$q^{\mu}$. In the derivation of (\ref{138}) we used
(\ref{135})-(\ref{137}). The eq. (\ref{139}) is obtained by
differentiation with respect to $T$.
It easy to obtain closed system of equations in terms of
$A^a_{\mu},\ z,\ I^a,$ and $S^{\mu\nu}$.
Let
\begin{equation}
I=\lambda^aI^a
\label{141}
\end{equation}
\noindent Then
\begin{equation}
iDI=TS^{\mu\nu}[F_{\mu\nu},I]
\label{142}
\end{equation}
\begin{equation}
\frac d{dt}S^{\mu\nu} =2TI^a F^{a[\mu}_{\ \ \ \ \rho}S^{\nu]\rho}
\label{142'}
\end{equation}
\noindent where
\begin{equation}
DI\equiv \frac{dI}{dt}-i[\dot q^{\mu}A_{\mu}(q),I]
\end{equation}
Unknown functions in (\ref{135})-(\ref{138}) also obey boundary
conditions
\begin{equation}
q(0)=q(1)=0,\ \ z(0)=z(1),\ \ \psi(0)=\psi(1)
\label{144}
\end{equation}
\noindent They also satisfy the equations
\begin{equation}
z^{\dag} z=1+\frac N 2,\ \ \bar{\psi}\psi=3
\label{145}
\end{equation}
\noindent because of the presence of $\delta$-functions in the
definition (\ref{122.1}) of the measure $DQ$. Apropos, we didn't
introduce Lagrange multipliers to take into account these
$\delta$-functions because the solutions of eqs. (\ref{136}),
(\ref{137}) automatically satisfy the conditions
\begin{equation}
z^{\dag} z=const,\ \ \bar{\psi} \psi=const
\label{146}
\end{equation}
Equations (\ref{135}), (\ref{138}), and (\ref{142}) are nothing but
generalization of well-known Wong's equations \cite{24} that describe
classical spinless particle interacting with Yang-Mills field.
Indeed, if one omits the terms containing the tensor of
spin $S^{\mu\nu}$ in eqs. (\ref{135}), (\ref{138}), and (\ref{142})
one gets just Wong's equations up to term
\begin{equation}
-p_{\mu}(\delta(t_1)-\delta(t_2))
\label{147}
\end{equation}
Eqs. (\ref{135})-(\ref{137}) admit simple interpretation. At "time"
$t_1$ quark--anti-quark pair with momentum $p$ is created. Then quark
and anti-quark move interacting with Yang-Mills field. The union of
quark and anti-quark trajectories forms a closed loop passing the
point $q=0$. (See (\ref{144})). At the "time" $t_2$ the
quark--anti-quark pair annihilates.
An analog of eqs. (\ref{135})-(\ref{139}) for terms with $n_f\ne 0$ in
expansion (\ref{129}) can be derived in the similar way. Let
$S^{\{n_f=0\}}$ be the action in the formula (\ref{134}) and
$J^{a\mu}[Q]$ be the current in R.H.S. of eq. (\ref{135}). Then the
analog of $S^{\{n_f=0\}}$ for the term with non zero numbers
$\{n_f\}$ in (\ref{129}) is
\begin{equation}
S^{\{n_f\}}=S^{\{n_f=0\}}+\sum_f\sum_{j_f=1}^{n_f}S[Q_{j_f},A]
\label{148}
\end{equation}
So instead of (\ref{135}) we have in this case equations
\begin{equation}
\frac 1{g^2}\nabla_{\nu}F^{a\mu\nu}=J^{a\mu}[Q_{f_0}]+
\sum_f\sum_{j_f=1}^{n_f}J^{a\mu}[Q_{j_f}]
\label{150}
\end{equation}
Equations
\begin{equation}
\frac{\delta S^{\{n_f\}}}{\delta \psi_{j_f}}=0, \ \
\frac{\delta S^{\{n_f\}}}{\delta z_{j_f}}=0, \ \
\frac{\delta S^{\{n_f\}}}{\delta T_{f}}=0
\label{150'}
\end{equation}
\noindent coincide in form with (\ref{137}), (\ref{139}), and (\ref{140}),
whereas the equation
\begin{equation}
\frac{\delta S^{\{n_f\}}}{\delta q_{j_f}}=0
\label{151}
\end{equation}
\noindent differs from (\ref{138}) by the term (\ref{147}).
We see that quasiclassical configurations that give the main
contribution in functional integrals (\ref{129}) are defined by
equations of very similar structure. The same statement is valid for
semiclassical configuration that give the main contribution in the
function $G^{(2)}(x-y)$ as well as in any other gauge invariant Green
functions. Therefore proposed scheme seems to be sufficiently general
for application in QCD and other gauge theories.
In this paper we restrict ourselves to formulation of general scheme
of quasiclassical approximation in QCD leaving elaborating of details
as well as applications for forthcoming papers. So at this point we
stop our investigation of quasiclassical approximation in QCD. Brief
discussion of possible applications will be given in the next
section.
\section{Conclusion}
In the present paper we derived, first, new path integral
representations for path ordered exponent (see eqs.
(\ref{35})-(\ref{37.1}), (\ref{39}), (\ref{39.4}), (\ref{58})).
We give two alternative, entirely independent derivation of
these representations. So these results seems quite reliable.
Then we applied these representations to derive new variant of
non-Abelian Stokes theorem. Our result is represented by formula
(\ref{75}). Then we transformed the latter to obtain another
formulation of non-Abelian Stokes theorem that is similar to one
proposed recently by Dyakonov and Petrov \cite{21}. As a result,
we got corrected and generalized version of the theorem proved in
\cite{21}.
Dyakonov-Petrov version of non-Abelian Stokes theorem was already
used in discussion of the role of monopoles in QCD \cite{21} and in
attempts to derive string-like effective action in framework of QCD
\cite{25}. So one can hope that our more general and simple version
of this theorem will be also useful in discussion of various problems
of QCD.
Further, we derived pure bosonic worldline path integral
representations for fermionic determinants as well as fermionic Green
functions in Euclidean QCD. (See
eqs. (\ref{99}), (\ref{105'}), and
(\ref{106})).
This representations comprise only integrations with respect to
bosonic variables. On a finite lattice all integrals are quite simple
and well convergent. (Remind, that domain of integration with respect
to $z$ and $\psi$ in
eqs. (\ref{99}), (\ref{105'}), and
(\ref{106}) is bounded).
So representation derived seem quite appropriate for lattice
simulations.
Our results for fermionic determinant and Green functions can be used
also in another way.
Namely, if one substitute instead of Wilson loop some phenomenological
anzatz, one obtain formulation of the theory purely in terms of point
particles. The most well-known example of such anzatz is Wilson area
law:
\begin{equation}
<\mathop{\mbox{tr P}} \exp\left\{ i\oint \mathop{dx^{\mu}} A_{\mu}\right\}>=
\exp\left\{ -KS_{min}+\left[{\mbox{perturbative}\atop
\mbox{corrections}}\right]\right\}
\label{152}
\end{equation}
This anzatz was applied, in particular, to derivation from QCD a
quark--anti- quark potential used in potential models (see, for
instance, recent papers \cite{26}, \cite{26.1} and references
therein).
Other anzatz for Wilson loops were proposed in framework of Dual QCD
model \cite{27} and stochastic vacuum model \cite{28}. (See also
\cite{26.1} for comparison of results obtained in framework of these
models). Recently a sting-like expression in spirit of stochastic
vacuum model was derived in papers \cite{25}, \cite{29}.Anzatz of
another type, that gives an expression for Wilson loop in terms of
trajectories of monopoles, was proposed in \cite{21}.
All these results could be combined with ours to derive some
effective action in terms of point particles that correctly describe
colour and spin properties of quarks. Obviously, that such theory is
much more simpler than initial quantum field theory and so thus
approach of investigation seems to be rather perspective.
Finally, in the present paper we also derived bosonic worldline path
integral representation for fermionic determinants and Green functions
in Minkowski space and started the investigation of the
quasiclassical approximation in QCD.
The key point in the formulation of quasiclassical approximation is
quasiclassical QCD equations (\ref{135})-(\ref{139}) and
(\ref{148})-(\ref{151}) which arise naturally when one applies the
stationary point method to evaluation of functional integrals that
defines Green functions in QCD. Quasiclassical equations derived
appear to be nothing but a generalization of well-known Wong's
equations.
We formulated only those quasiclassical equations which arise in the
problem of evaluation of the concrete Green function (\ref{119}).
However, our method is quite general and one can easy to derive
analogous equations in generic case.
The next problem in investigation of quasiclassical approximation in
QCD is the solution of quasiclassical equation of motion. Though
these equations are very complicated, this problem doesn't seem
hopeless, at least, in non-re\-la\-ti\-vis\-tic approximation. In
electrodynamics in non-re\-la\-ti\-vis\-tic approximation one can
neglect bremsstrahlung and retarded effects and, as a result,
reformulate initial theory in terms of particles interacting by means
of Coulomb forces. In the same way one may hope to derive potential
of interaction of heavy quarks in QCD. Such approach is alternative
to ones based on various anzatz for Wilson loops.
Another interesting possibility to understand the quark confinement
in framework of quasiclassical approach is connected with existence of
classical solutions of Yang-Mills equations with singularity on the
sphere. Such solution were discussed in the context of the problem of
confinement recently in the papers \cite{29'} though the existence of
such solutions was pointed out by several authors in 70's \cite{30}.
Some solutions with singularity on the torus and cylinder was
discussed in papers \cite{31}.
If solutions with singularity on closed spacelike surface
existed also for eqs. (\ref{135})-(\ref{139}) and their
modifications, then quarks, moving inside such surface and
interacting with Yang-Mills field, couldn't cross the surface. This
would mean the confinement of quarks. So it is interesting to
investigate singular solutions of equations (\ref{135})-(\ref{139})
and, if such solutions exist, to develop the corresponding
quasiclassical perturbative theory.
\section*{Appendix A}
In this appendix we shall prove the validity of the identity
(\ref{55}).
\noindent We will prove (\ref{55}) by induction. For $N=2$ it is easy to
prove (\ref{55}) by direct calculations because both sides of
(\ref{55}) are simple rational functions of variables $\exp\{\frac i
2 \alpha_r\}$.
Let us denote
$$
x_r=e^{i\alpha_r}
\eqno(A1)
$$
\noindent Then L.H.S. of (\ref{55}) can be represented as
$$
(-2i)^{N-1}
\frac{
\prod\limits_{r=1}^N \sqrt x_r}{\prod\limits_{1\le p<q\le N}(x_p-x_q)}
\sum_{j=1}^N(-1)^{j+1}x_j^N
\prod_{{\scriptstyle {r,s\ne j\atop
1\le r<s\le N}}}(x_r-x_s)
\eqno(A2)
$$
\noindent whereas R.H.S. of (\ref{55}) as
$$
(-2i)^{N-1}\left( \prod_{k=1}^N\sqrt x_k\right)
\sum_{j=1}^N x_j
\eqno(A3)
$$
\noindent So eq. (\ref{55}) is equivalent to algebraic identity
$$
\sum_{j=1}^N(-1)^{j+1}x_j^N \prod_{{\scriptstyle {r,s\ne j\atop
1\le r<s\le N}}}(x_r-x_s)
=
\left(\sum_{j=1}^N x_j\right)
\prod\limits_{1\le r<s\le N}(x_r-x_s)
\eqno(A4)
$$
To prove (A4), we check, at first, that L.H.S. of (A4) is vanished if
$x_i=x_j$. Obviously, it is sufficient to consider the case
$x_1=x_2\equiv x$.
If $x_1=x_2$ then only $j=1$ and $j=2$ terms survive in L.H.S. of
(A4). The $j=1$ term is
$$
x_1^N\prod\limits_{2\le r<s\le N}(x_r-x_s) =
x^N\left(\prod_{p=3}^N(x-x_p)
\right)\left(\prod\limits_{3\le p<q\le N}(x_p-x_q)\right)
\eqno(A5)
$$
\noindent whereas the $j=2$ term is equal to
$$
-x_2^N
\prod_{{\scriptstyle {r,s\ne 2\atop
1\le r<s\le N}}}(x_r-x_s)
=-
x^N\left(\prod_{p=3}^N(x-x_p)
\right)\left(\prod\limits_{1\le p<q\le N}(x_p-x_q)\right)
\eqno(A6)
$$
\noindent So terms with $j=1$ and $j=2$ are cancelled.
Thus L.H.S. of (A4) can be represented as
$$
P(x_1,\ldots,x_N)
\prod\limits_{1\le r<s\le N}(x_r-x_s)
\eqno(A7)
$$
\noindent We must prove that
$$
P(x_1,\ldots,x_N)
=\sum_{r=1}^N x_r
\eqno(A8)
$$
L.H.S. of eq. (A4) is polynomial of degree $N$ with respect to each
variable $x_j$. Consequently, polynomial
$P(x_1,\ldots,x_N)$
is linear in each $x_j$. So
$$
P(x_1,\ldots,x_N)
=a(x_2,\ldots,x_N)x_1+b(x_2,\ldots,x_n)
\eqno(A9)
$$
Comparing coefficient at $x_1^N$ in L.H.S. of (A4) and (A7), one find
that
$$a(x,\ldots,x_N)=1
\eqno(A10)
$$
\noindent and so it remains to prove that
$$b(x_2,\ldots,x_N)=\sum_{j=2}^N x_j
\eqno(A11)
$$
To prove (A11), let us compare the sums of $x_1$-independent terms in
L.H.S. of (A4) and in (A7). They can be written as
$$
\sum_{j=2}^N (-1)^{j+1} x_j^N
\prod\limits_{{\scriptstyle {2\le r\le N \atop r\ne j}}}
(-x_r)
\prod\limits_{{\scriptstyle {2\le r<s\le N \atop r,s\ne j}}}
(x_r-x_s)
\eqno(A12)$$
\noindent and
$$b(x_2,\ldots,x_N)\prod\limits_{2\le r\le N}
(-x_r)\prod_{2\le r<s\le N}(x_r-x_s)
\eqno(A13)
$$
\noindent respectively.
So (A11) is equivalent to
$$
\sum_{j=2}^N(-1)^{j}x_j^{N-1} \prod_{{\scriptstyle {r,s\ne j\atop
2\le r<s\le N}}}(x_r-x_s)
=
\left(\sum_{j=2}^N x_j\right)
\prod\limits_{2\le r<s\le N}(x_r-x_s)
$$
But the latter equation is valid by virtue of induction assumption.
Indeed, it can be obtained from (A4) by change $N\to N-1,\ ,x_1\to
x_2, \ldots,x_{N-1}\to x_N$. This finishes the proof of identity
(\ref{55}).
\section*{Appendix B}
In this appendix we prove the formula (\ref{66}). First, one notes
that up to inessential factor
$$\det\left( -\frac d{dt} + iBe^{-\epsilon\frac d{dt}}\right)
=\det
\oeo
\eqno(B1)
$$
\noindent So it is sufficient to prove that
$$\lim_{\epsilon \to +0} \ln\det \oeo-\ln\det\owe=
\frac i 2 \int^1_0\mathop{dt}\mathop{{\rm tr}} B
\eqno(B2)$$
\noindent or, equivalently,
$$
\lim_{\epsilon \to +0} \mathop{{\rm tr}}\ln\oeo-\mathop{{\rm tr}}\ln\owe=\frac i 2 \int^1_0\mathop{dt}\mathop{{\rm tr}} B
\eqno(B3)
$$
Let $\hat{\Pi}$ is orthogonal projector in $L^2(S^1)$ on any
{\it finite} dimensional subspace. Then
$$\lim_{\epsilon \to +0} \left[\mathop{{\rm tr}} \hat{\Pi} \ln\oeo -\mathop{{\rm tr}}\hat{\Pi}\ln\owe
\right]=0
\eqno(B4)$$
\noindent In particular, let ${\cal H}_1$ be subspace in $L^2(S^1)$
that is orthogonal to one dimensional subspace spanned on the
function $\phi_0(t)\equiv 1$. Then
$$\lim_{\epsilon \to +0} \left[\mathop{{\rm tr}} \ln\oeo -\mathop{{\rm tr}}\ln\owe
\right]$$
$$=\lim_{\epsilon \to +0} \left[{\mathop{{\rm tr}}}_{{\cal H}_1} \ln\oeo -{\mathop{{\rm tr}}}_{{\cal H}_1}\ln\owe
\right]
\eqno(B5)$$
In the space ${\cal H}_1$ the operator
$$-\frac d{dt} e^{\epsilon\frac d{dt}}
\eqno(B6)$$
\noindent is invertible. Let $G_{\epsilon}(t-t')$ is the kernel of
$$\left(-\frac d{dt} e^{\epsilon\frac d{dt}}\right)^{-1}
$$
The function $G_{\epsilon}(t-t')$ can be expresseed in terms of
eigenfunctions
$$\phi_n(t)=e^{2\pi int}
,\ \ n\ne 0
\eqno(B7)$$
\noindent of the operator (B6) and its eigenvalues
$$\lambda_n=-2\pi in e^{2\pi in\epsilon},\ \ n\ne 0$$
Obviously,
$$G_{\epsilon}(t-t')=\sum_{n\ne0}\frac{\phi_n(t)\bar{\phi_n}(t')}
{\lambda_n}
=-\frac 1{\pi}\sum_{n=1}^{\infty}
\frac{\sin 2\pi n(t-t'-\epsilon)}{n}
\eqno(B8)$$
Using the formula
$$\sum_{n=1}^{\infty}\frac{\sin 2\pi n\epsilon}{\pi n}=
\frac 1 2 -\epsilon
\eqno(B9)$$
\noindent (that is valid for $0<\epsilon<1$), one finds that
$$\lim_{\epsilon \to +0} G_{\epsilon}(0)=\frac 1 2
\eqno(B10)$$
\noindent whereas
$$G_{0}(0)=0
\eqno(B10')$$
\noindent So
$$G_{0}(0)\ne\lim_{\epsilon \to +0} G_{\epsilon}(0)
\eqno(B11)
$$
But for $-1<t<1,\ \ t\ne0$,
$$\lim_{\epsilon \to +0} G_{\epsilon}(t)=G_{0}(t)
\eqno(B11')$$
We will see soon that R.H.S. of (B2) is not vanished just by virtue
of (B11).
Further, up to inessential constant,
$${\mathop{{\rm tr}}}_{{\cal H}_1}\ln\oeo={\mathop{{\rm tr}}}_{{\cal H}_1}\ln\left[
1+i
\left(-\frac d{dt} e^{\epsilon\frac d{dt}}\right)^{-1}
B\right]$$
$$={\mathop{{\rm tr}}}_{{\cal H}_1}\left[
\left(-\frac d{dt} e^{\epsilon\frac d{dt}}\right)^{-1}
B\right]+
\frac 1 2 {\mathop{{\rm tr}}}_{{\cal H}_1}
\left(-\frac d{dt} e^{\epsilon\frac d{dt}}\right)^{-1}
B
\left(-\frac d{dt} e^{\epsilon\frac d{dt}}\right)^{-1}
+\ldots
\eqno(B12)$$
By virtue of equation
$$\int^1_0 \mathop{dt} G_{\epsilon}(t-t')=0
\eqno(B13)$$
\noindent one can replace ${\mathop{{\rm tr}}}_{{\cal H}_1}$ by $\mathop{{\rm tr}}$ in all terms in the
series (B12). So
$${\mathop{{\rm tr}}}_{{\cal H}_1}\ln \oeo
=
i\int^1_0 \mathop{dt} G_{\epsilon}(0) \mathop{{\rm tr}} B(t)$$
$$+\frac 1 2 \int^1_0\mathop{dt}\int^1_0 dt'\: G_{\epsilon}(t-t')
G_{\epsilon}(t'-t)\mathop{{\rm tr}} B(t)B(t')+\ldots
\eqno(B14)$$
In the limit $\epsilon \to +0$ the first term in R.H.S. of (B14) is equal
to
$$\frac i 2 \int^1_0\mathop{dt} \mathop{{\rm tr}} B(t)$$
\noindent (see (B11)) whereas the sum of all others coincide with
$${\mathop{{\rm tr}}}_{{\cal H}_1}\owe$$
\noindent by virtue of (B10'), (B11'). This proves the validity of the
equation (B2).
To check the formula (B2), let us consider the simplest case of one
dimensional harmonic oscillator. Partion function
$$Z(\beta)=\mathop{{\rm tr}}\exp\{-\beta a^{\dag} a\}
\eqno(B15)$$
\noindent is known exactly:
$$Z(\beta)=\sum_{n=0}^{\infty}e^{-\beta n}=\frac 1{1-e^{-\beta}}
\eqno(B15)$$
On the other hand,
$$Z(\beta)=\lim_{\epsilon \to +0}\int_{PBC}D\bar z Dz\:
\exp\left\{\int^1_0\mathop{dt}(-\bar z\dot z -\beta \bar z e^{-\epsilon\frac d{dt}}z
\right\}$$
$$={\det}^{-1}\left(-\frac d{dt}-\beta e^{-\epsilon\frac d{dt}}
\right)
\eqno(B16)
$$
Putting in (B2) $B=i\beta$, one gets:
$${\det}^{-1}\left(-\frac d{dt}-\beta e^{-\epsilon\frac d{dt}}
\right)
=e^{\frac{\beta}{2}}
{\det}^{-1}\left(-\frac d{dt}-\beta
\right)
\eqno(B17)$$
The latter determinant we have already evaluated in the main text of
the paper :
$$
{\det}\left(-\frac d{dt}-\beta
\right)
=const \,\sinh \frac{\beta}2
\eqno(B18)$$
\noindent (see eq. (\ref{50}) for $N=1,\ \ \eta=\epsilon=0$).
Comparing (B16)-(B19), we get
$$Z(\beta)=const\,\frac{e^{\frac{\beta}2}}{\sinh\frac{\beta}2}
\eqno(B19)$$
Evaluating the constant from the condition
$$Z(\infty)=1$$
\noindent we reproduce the true answer (B15). We would like to stress that
if we had put $\epsilon=0$ in (B16) before evaluating of the functional
integral the result would have been wrong.
| proofpile-arXiv_065-431 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
\setcounter{footnote}{0}
A still challenging question in strong interaction physics is
the derivation of the
low-energy properties of the spectrum from ``QCD first principle",
due to our limited present skill with non-perturbative physics.
At very low energy, where the ordinary
perturbation theory cannot be applied,
Chiral Perturbation Theory~\cite{GaLeut} and Extended Nambu--Jona-Lasinio
models~\cite{NJL,ENJL}~\footnote{For a recent complete review see
\cite{Miransky}.} give a consistent framework
in terms of a set of parameters that have to be fixed from the data;
yet the bridge between those effective parameters and the basic
QCD degrees of freedom remains largely unsolved. Although lattice QCD
simulations
recently made definite progress~\cite{lattcsb} in that direction,
the consistent treatment of dynamical
unquenched quarks and the chiral symmetry remains a serious problem. \\
In this paper,
we investigate a new, semi-analytical method, to
explore {\it how far}
the basic QCD Lagrangian can provide, in a self-consistent
way,
non-zero
dynamical quark
masses, quark condensates, and pion decay constant,
in the limit of vanishing Lagrangian (current) quark masses. Such a
qualitative picture
of chiral symmetry breakdown (CSB) can be made more quantitative
by applying
a new ``variational mass" approach,
recently developed within the framework of the
anharmonic oscillator \cite{bgn}, and in
the Gross--Neveu (GN) model \cite{gn1,gn2}.
The starting point is very similar to
the ideas developed a long time ago and implemented
in various different forms in refs.\cite{pms}, \cite{delta}.
There, it was advocated
that the convergence of conventional perturbation
theory may be improved by a variational procedure in which
the separation of the
action into ``free" and ``interaction" parts
is made to depend on some set of auxiliary parameters.
The results obtained by expanding to finite order
in this redefined perturbation
series are optimal in regions of the space of auxiliary
parameters where they are least sensitive to these parameters.
Recently there appeared
strong evidence that this optimized perturbation
theory may indeed lead to a rigorously convergent
series of approximations even in strong coupling cases
\cite{JONE}. \\
An essential novelty,
however, in \cite{bgn}--\cite{gn2} and the present paper,
is that our construction combines in a specific manner
the renormalization group (RG) invariance
with the properties of an analytically continued, arbitrary mass parameter
$m$.
This, at least in a certain approximation to be motivated,
allows us to reach
{\it infinite} order of
the variational-perturbative expansion, and therefore presumably optimal,
provided it converges. Our main results are a set of
non-perturbative
ansatzs for the relevant CSB quantities,
as functions of the variational mass $m$,
which can be
studied for extrema and optimized.
Quite essentially, our construction also provides a simple and
consistent treatment of the renormalization, reconciling
the variational approach with the inherent infinities
of quantum field theory and the RG properties.
Before proceeding, let us note that there exists
a quite radically different attitude towards CSB in QCD, advocating
that the responsible mechanism is most probably the non-perturbative
effects due to the {\em instanton} vacuum~\cite{Callan},
or even more directly related to confinement~\cite{Cornwall}.
However,
even if the instanton picture of CSB is on general grounds well motivated,
and many fruitful ideas have been developed in that
context\footnote{See e.g
ref.~\cite{Shuryak} for a review and original references.},
as far as we are aware there is at present no
sufficiently rigorous or compelling evidence for it, derived from ``first
principle".
In any event, it is certainly of interest to
investigate quantitatively
the ``non-instantonic" contribution to CSB, and we hope
that our method is a more consistent step in that direction.
\section{Dynamical quark masses}
In what follows we only consider the $SU(n_f)_L \times SU(n_f)_R$
part of the chiral symmetry, realized by the QCD Lagrangian
in the absence of quark mass terms,
and for
$n_f =2$ or $n_f = 3$ as physically relevant applications.
Following the treatment of the anharmonic oscillator~\cite{bgn} and its
generalization to the GN model~\cite{gn1,gn2},
let us consider the following modification of the usual QCD Lagrangian,
\begin{equation}
L_{QCD} \to L^{free}_{QCD}(g_0 =0, m_0=0)
-m_0 \sum^{n_f}_{i=1} \overline{q}_i q_i
+ L^{int}_{QCD}(g^2_0 \to x g^2_0)
+x \; m_0 \sum^{n_f}_{i=1} \overline{q}_i q_i\;,
\label{xdef}
\end{equation}
where $L^{int}_{QCD}$ designates the ordinary QCD interaction terms, and
$x$ is a convenient ``interpolating" expansion parameter.
This formally is equivalent
to substituting everywhere in the bare Lagrangian,
\begin{equation} m_0 \to m_0\; (1-x); ~~~~g^2_0 \to g^2_0\; x,
\label{substitution}
\end{equation}
and therefore
in any perturbative (bare) quantity as well,
calculated in terms of $m_0$ and $g^2_0$.
Since the original massless QCD
Lagrangian is recovered for $x \to 1$, $m_0$ is to be considered as
an {\it arbitrary } mass parameter
after substitution (\ref{substitution}). One expects
to optimize physical quantities with
respect to $m_0$ at different, possibly arbitrary
orders of the expansion parameter $x$,
eventually approaching a stable limit, i.e {\it flattest}
with respect to $m_0$,
at sufficiently high order in $x$. \\
However,
before accessing any physical quantity
of interest for such an optimization, the theory should be
renormalized, and there is an unavoidable mismatch
between the expansion in $x$, as
introduced above, and the ordinary perturbative expansion as dictated by the
mass and coupling counterterms.
Moreover,
it is easy to see that
at any finite order in the $x$ expansion,
one always
recovers a trivial result in the limit $ m \to 0$ (equivalently
$x\to 1$), which
is the limit
in which to identify non-zero
order parameters of CSB.
These
problems can be circumvented by
advocating a specific ansatz, which resums
the (reorganized) perturbation series in $x$ and is such that
the limit $x \to 1$
no longer gives
a trivial zero mass gap.
As was shown in detail in ref.~\cite{gn2}, the ansatz for the dynamical
mass is
most easily derived by following the steps\footnote{See
also ref.~\cite{jlk} for
a detailed derivation in the QCD context.}:\\
{\it i}) Consider first the general
solution for the running mass,
given as
\begin{equation}
m(\mu^{'}) = m(\mu )\;\; {\exp\left\{ -\int^{g(\mu^{'})}_{g(\mu )}
dg {\gamma_m(g) \over {\beta(g)}} \right\} }
\label{runmass}
\end{equation}
in terms of the effective coupling $g(\mu)$, whose RG evolution is given
as $\mu dg(\mu)/d\mu \equiv \beta(g)$, and $\gamma_m(g) \equiv
-(\mu/m)d(m(\mu))/d\mu$.
Solving (\ref{runmass}) imposing the
``fixed point" boundary condition:
\begin{equation}
M \equiv m(M),
\label{RGBC}
\end{equation}
at two-loop RG order we obtain, after some algebra
(we use the normalization
$\beta(g) = -b_0 g^3 -b_1 g^5 -\cdots$,
$\gamma_m(g) = \gamma_0 g^2 +\gamma_1 g^4
+\cdots$):
\begin{equation}
M_2 = \bar m \;\;
\displaystyle{f^{-\frac{ \gamma_0}{2b_0}}\;\; \Bigl[\frac{ 1 +\frac{b_1}{b_0}
\bar g^2 f^{-1}}{ 1+\frac{b_1}{b_0}\bar g^2} \Bigr]^{ -\frac{\gamma_1}{
2 b_1}
+\frac{\gamma_0}{2 b_0} } }\;,
\label{MRG2}
\end{equation}
where
$\bar m \equiv m(\bar\mu)$, $\bar g \equiv g(\bar\mu)$
($\bar \mu \equiv \mu \sqrt{4 \pi} e^{-\gamma_E/2}$), and
$f \equiv \bar g^2/g^2(M_2)$ satisfies
\begin{equation}
f = \displaystyle{ 1 +2b_0 \bar g^2 \ln \frac{M_2}{\bar \mu }
+\frac{b_1}{b_0} \bar g^2
\ln \Bigl[\frac{ 1 +\frac{b_1}{b_0} \bar g^2 f^{-1}}{
1 +\frac{b_1}{b_0} \bar g^2 }\;f\;\Bigr] }\; ;
\label{f2def}
\end{equation}
(note in (\ref{MRG2}) and (\ref{f2def})
the recursivity in both $f$ and $M_2$).
The necessary
non-logarithmic perturbative corrections to those pure RG results
are then consistently
included as
\begin{equation}
M^P_2 \equiv M_2 \;\Bigl(1 +{2\over 3}\gamma_0 {\bar g^2\over f}
+{K \over{(4 \pi^2)^2}}{\bar g^4\over f^2}+{\cal O}(g^6)\;\Bigr)\;,
\label{Mpole}
\end{equation}
where
the complicated
two-loop coefficient $K$ was calculated
exactly in ref.~\cite{Gray}.
Equation.~(\ref{Mpole}) defines the (infrared-convergent, gauge-invariant)
pole mass~\cite{Tarrach}
$M^P_2$, in terms of the
$\overline{MS}$ mass at two-loop order,
and can be
shown~\cite{jlk} to resum the leading (LL)
{\it and} next-to-leading logarithmic (NLL)
dependence in $\bar m$ to all
orders. \\
{\it ii})
Perform in expressions (\ref{MRG2}), (\ref{f2def}), (\ref{Mpole})
the substitution
$
\bar m \to \bar m v $,
and integrate the resulting expression, denoted by $M^P_2(v)$,
according to
\begin{equation}
\frac{1}{2i\pi} \;\oint \frac{dv}{v}\;e^v M^P_2(v)\;,
\label{contgen}
\end{equation}
where the contour is around the negative real $v$ axis. \\
In \cite{gn2} it was shown that the previous steps correspond
(up to a specific renormalization scheme (RS) change,
allowed on general grounds from RG properties) to a resummation
of the $x$ series as generated from the substitution
(\ref{substitution})\footnote{$v$ is related to the original
expansion parameter $x$ as $x = 1-v/q$, $q$ being the order of the expansion.}.
Moreover this is in fact the only way of rendering compatible the
above $x$ expansion and the ordinary perturbative one, thus obtaining
finite results.
Actually the resummation coincides with the exact result in the
large-$N$ limit of the GN model. Now, since the
summation can be formally
extended to arbitrary RG orders~\cite{gn2}, including consistently as many
arbitrary perturbative correction terms as known in a given theory,
in the QCD case we make the assumption that it
gives an adequate ``trial ansatz",
to be subsequently optimized in a way
to be specified next.
After appropriate rescaling of the basic parameters, $\bar g$
and $\bar m$, by introducing the RG-invariant basic scale $\Lambda_{\overline{MS}}$~
\cite{Lambda}
(at two-loop order),
and the convenient scale-invariant dimensionless ``mass" parameter
\begin{equation}
m''\equiv \displaystyle{(\frac{\bar m}{ \Lambda_{\overline{MS}}}) \;
2^{C}\;[2b_0 \bar g^2]^{-\gamma_0/(2b_0)}
\;\left[1+\frac{b_1}{b_0}\bar g^2\right]^{
\gamma_0/(2 b_0)-\gamma_1/(2 b_1)}}
\; ,
\label{msec2def}
\end{equation}
we end up with the following dynamical mass ansatz:
\begin{equation}
{ M^P_2 (m^{''})\over \Lambda_{\overline{MS}}}
= {2^{-C} \over{2 i \pi}} \oint dy {e^{\;y/m^{''}}
\over{F^A(y) [C + F(y)]^B}} {\left(1 +{{\cal M}_{1}\over{F(y)}}
+{{\cal M}_{2}\over{F^2(y)}} \right)},
\label{contour7}
\end{equation}
where $y \equiv m'' v$, and
$F$ is defined as
\begin{equation}
F(y) \equiv \ln [y] -A \; \ln [F(y)] -(B-C)\; \ln [C +F(y)],
\label{Fdef}
\end{equation}
with $A =\gamma_1/(2 b_1)$, $B =\gamma_0/(2 b_0)-\gamma_1/(2 b_1)$,
$C = b_1/(2b^2_0)$, in terms of the RG coefficients
\cite{betagamma}.
Finally the perturbative corrections in (\ref{contour7})
are simply given as
${\cal M}_{1} =(2/3)(\gamma_0/2b_0)$ and ${\cal M}_{2} = K/(2b_0)^2$. \\
Observe in
fact that, were we in a simplified QCD world, where there would be
{\em no} non-logarithmic perturbative contributions (i.e. such that
${\cal M}_{1} = {\cal M}_{2} = \cdots = 0$ in (\ref{contour7})),
the latter ansatz would
then resums exactly the $x$ variational expansion.
In that case, (\ref{contour7}) would have a very simple
behaviour near the origin $m'' \to 0$. Indeed, it is easy to see
that (\ref{Fdef}) admits an expansion
$
F(y) \simeq C^{(B-C)/A}\;y^{1/A}$ for $y \to 0$,
which immediately implies that (\ref{contour7}) would
give a simple pole at $y \to 0$,
with a residue giving $M_2 = (2C)^{-C}\;\Lambda_{\overline{MS}} $.
Moreover one can always
choose
an appropriate renormalization scheme in which $
b_2$ and $\gamma_2$ are set to zero, as well as all
higher order coefficients, so that there are no other corrections
to the simple above relation.
Now, in the realistic world, ${\cal M}_1$, ${\cal M}_2$, etc can
presumably not be neglected.
We can nevertheless expand
(\ref{contour7}) near $m'' \to 0$ for any known
non-zero ${\cal M}_{i}$,
using
\begin{equation}
\label{hankel}
\frac{1}{2i \pi} \oint dy e^{y/m^{''}} y^\alpha =
\frac{(m^{''})^{1+\alpha}}{\Gamma[-\alpha]}\; ,
\end{equation}
and the resulting Laurent expansion in $(m'')^{1/A}$ may be
analysed for extrema and optimized at different, in principle
arbitrary $(m'')^{1/A}$ orders. An important point, however, is
that the
perturbative corrections do depend on the RS choice, as is well known.
Since the pure RG behaviour in (\ref{contour7})
already gives the order of magnitude, $M \simeq {\rm const} \times \Lambda_{\overline{MS}}$,
we can hope that
a perturbative but optimized treatment of the remaining corrections
is justifed.
In other words we shall perform an
``optimized perturbation" with respect to $m''$
around the non-trivial fixed point
of the RG solution. \\
To take into account this RS freedom, we first introduce
in (\ref{contour7}) an arbitrary scale parameter $a$,
from $\bar \mu \to a\; \bar \mu$.
Accordingly the
perturbative coefficients ${\cal M}_{i}$ in (\ref{contour7}) take a
logarithmic dependence in $a$, simply
fixed order by order from (\ref{MRG2})--(\ref{Mpole})
and the requirement that (\ref{contour7})
differs from the original $\overline{MS}$ expression only by higher order terms.
The $a$-dependence will eventually exhibit a non-trivial extrema
structure and we shall also
optimize the result with respect to
$a$\footnote{This procedure indeed gave very good results~\cite{gn2}
in the GN model, where in particular for low values of $N$ the
optimal values found, $a_{opt}$, are quite different from 1.}.
Actually
there are other possible changes of renormalization prescriptions
affecting expression (\ref{contour7}) in addition to the $a$ dependence,
which may be
taken into account as well. More precisely, the second coefficient
of $\gamma_m(g)$, $\gamma_1$, do depend on the RS choice,
even in MS schemes~\cite{Collins}.
As it turns out, this additional RS freedom is
very welcome in our case: in fact,
the previous picture is
invalidated, due to the occurence of extra branch cuts
in the $y$ plane at $Re[y_{cut}] > 0$,
as given by the zeros of $dy/dF$ from
(\ref{Fdef}) (in addition
to the original cut on the negative real $y$ axis).
This prevents using
the expansion near the origin, eq.~(\ref{hankel}),
since it would lead to
ambiguities of ${\cal O}(\exp(Re[y]/m''))$
for $m'' \to 0$\footnote{
The origin of those singularities is rather
similar to the ambiguities related
to
renormalons~\cite{renormalons}.
An essential difference, however, is that the present
singularities occur in the analytic continuation of a mass parameter
rather than a coupling constant, and
it is possible to move those singularities away by an
appropriate RS change,
as we discuss next. See ref.~\cite{jlk} for an extended discussion.}.
The specific contour around the negative real axis
was suggested by the known properties of the large
$N$ limit of the GN model, and it is not surprising
if the analytic structure
is more complicated in QCD.
However, the nice
point is that the actual position of those cuts
do depend on the RS, via
$A(\gamma_1)$ in (\ref{Fdef}).
Defining
$
\gamma^{'}_1 \equiv \gamma_1 +\Delta \gamma_1$,
we can choose
$Re[y_{cut}] \simeq 0$ for $\Delta\gamma_1 \simeq$ 0.00437 (0.00267) for
$n_f =$ 2 ($n_f =$ 3), respectively.
We therefore consider~\cite{jlk} the general RS change
\begin{equation}
m' = \bar m\;(1+B_1 \bar g^2 +B_2 \bar g^4)\;;\;\;\;g^{'2} = \bar g^2\;
(1 +A_1 \bar g^2 +A_2 \bar g^4)\;
\label{RSchange}
\end{equation}
(implying $\Delta\gamma_1 = 2b_0 B_1 -\gamma_0 A_1$),
and optimize with respect
to this new arbitrariness\footnote{
We also impose a further RS choice,
$
b^{'}_2 = 0$,
$\gamma^{'}_2 = 0$,
which fixes $A_2$, $B_2$ in (\ref{RSchange}) and
guarantees that our two-loop convention for $\Lambda_{\overline{MS}}$
remains unaffected. Note, however, that
(\ref{RSchange}) implies $\Lambda_{\overline{MS}} \to \Lambda_{\overline{MS}} \; \exp\{\frac{A_1}{2b_0}\}\equiv
\Lambda'$. In what
follows we express the results in terms of the original $\Lambda_{\overline{MS}}$.}.
However one soon realizes that
our extension of the ``principle of minimal sensitivity" (PMS)~\cite{pms}
defines a rather complicated optimization
problem.
Fortunately, we can study this problem within some approximations,
which we believe are legitimate.
Since
the ansatz
(\ref{contour7}) (with the above RS change understood, to make
it consistent) would indeed
be optimal with respect
to $m^{''}$ for {\em vanishing} perturbative non-logarithmic
corrections, ${\cal M}_{i} =0$,
we shall assume that the
expansion for small $m''$ is as close as possible to an optimum,
and
define the $m^{''} \to 0$ limit by some relatively crude but standard
approximation,
avoiding numerical optimization
with respect to $m^{''}$.
The approximation we are looking for is
not unique: given (\ref{contour7}), one could construct
different
approximations leading to a finite limit
for $m'' \to 0$~\cite{gn2}. Here
we shall only demonstrate
the feasibility of our program in the simplest possible
realization.
In fact, since we shall anyhow optimize with respect to the RS dependence
we assume that it largely
takes into
account this non-uniqueness due to higher order uncertainties.
Pad\'e approximants are known to greatly improve perturbative
results~\cite{pade}
and often have the effect of smoothing the RS dependence.
We thus take a simple Pad\'e approximant
which by construction restitutes a simple pole for $F \to 0$
(i.e $m'' \to 0$) in
(\ref{contour7}), and gives
\begin{equation}
{M^{Pad\acute{e}}(a,\Delta\gamma_1,B_1,m''\to 0) = \Lambda_{\overline{MS}}\;
(2C)^{-C} \;a\; \exp\{\frac{A_1}{2b_0}\}\;\left[
1 -{{\cal M}^2_{1}(a, \Delta\gamma_1, B_1)\over{{\cal M}_{2}(a,
\Delta\gamma_1, B_1)}}\right] }
\label{Mpade}
\end{equation}
We have performed a rather systematic study of the possible extrema
of (\ref{Mpade}) for arbitrary
$a$, $B_1$ (with $\Delta\gamma_1$
fixed so that the extra cuts start at $ Re[y] \simeq 0 $).
We obtain the flattest such extrema for $a \simeq 2.1$, $B_1 \simeq 0.1$,
which leads to the result
\begin{equation}
M^{Pad\acute{e}}_{opt}(m''\to 0) \simeq 2.97\;\Lambda_{\overline{MS}}(2)\;
\label{Mnum}
\end{equation}
for $n_f=2$. Similarly, we obtain $M^{Pad\acute{e}}_{opt}(m''\to 0)
\simeq 2.85 \Lambda_{\overline{MS}}(3)$
for $n_f=3$.
Note that these values of the dynamical quark masses, if
they are to be consistent
with the expected
range~\cite{Miransky}
of $M_{dyn}\simeq$ 300-400 GeV, call for relatively low
$\Lambda_{\overline{MS}} \simeq $ 100-150 GeV, which
is indeed supported by our results in the next section.
\section{Composite operators and $F_\pi$}
We shall now generalize the ansatz
(\ref{contour7})
for the pion decay constant $F_\pi$.
The main idea is to
do perturbation theory around the same RG evolution solution
with the non-trivial fixed point, as specified by the function $F$ in
(\ref{Fdef}),
with perturbative correction terms obviously specific to $F_\pi$.
A definition
of $F_\pi$ suiting all our purposes is~\cite{GaLeut,derafael}
\begin{equation}
i\;\int d^4q e^{iq.x} \langle 0 \vert T\;A^i_\mu(x) A^k_\nu(0) \vert 0
\rangle =
\delta^{ik} g_{\mu \nu} F^2_\pi +{\cal O}(p_\mu p_\nu)
\label{Fpidef}
\end{equation}
where the axial vector current $A^i_\mu \equiv
(\bar q \gamma_\mu
\gamma_5\lambda^i q)/2$ (the $\lambda^i$'s are Gell-Mann $SU(3)$ matrices
or Pauli matrices for $n_f =3$, $n_f=2$, respectively). Note that according
to (\ref{Fpidef}), $F_\pi$ is to be considered as an order
parameter of CSB~\cite{Stern}. \\
The perturbative expansion of (\ref{Fpidef}) for $m \neq 0$ is available
to the three-loop order, as it can be easily
extracted
from the very similar contributions to the electroweak
$\rho$-parameter, calculated at two loops in \cite{abdel}
and three loops in \cite{Avdeev}.
The appropriate generalization of
(\ref{contour7}) for $F_\pi$
now takes the form
\begin{eqnarray}
& \displaystyle{{F^2_\pi \over{\Lambda_{\overline{MS}}^2}} = (2b_0)\;
{2^{-2 C} a^2\over{2 i \pi}} \oint {dy\over y}\; y^2 {e^{y/m^{''}}}
\frac{1}{F^{\;2 A-1} [C + F]^{\;2 B}} }
\; \times \nonumber \\
& \displaystyle{ {\delta_{\pi }
\left(1 +{\alpha_{\pi}(a)\over{F}}+{\beta_{\pi}(a)
\over{F^2}}
\;+\cdots \right)} }
\label{Fpiansatz}
\end{eqnarray}
in terms of $F(y)$ defined by eq.~(\ref{Fdef}) and where
$\delta_\pi$, $\alpha_\pi(1)$ and $\beta_\pi(1)$,
whose complicated expressions
will be given elsewhere~\cite{jlk},
are fixed by matching the perturbative $\overline{MS}$
expansion in a way to be specified next.
Formula (\ref{Fpiansatz})
necessitates some comments: apart from the obvious changes in the powers of
$F$, $y$, etc,
dictated by dimensional analysis,
note that the perturbative expansion of the (composite operator)
$\langle A_\mu A_\nu \rangle$ in (\ref{Fpidef})
starts at one-loop, but zero $g^2$ order.
This leads to the extra $2b_0 F$ factor in (\ref{Fpiansatz}),
corresponding to an expansion
starting at ${\cal O}(1/g^2)$\footnote{
The ${\cal O}(1/g^2)$
first-order term cancels anyhow
after the necessary subtraction discussed
below.}.
Another difference is that
the perturbative expansion
of (\ref{Fpidef}) is ambiguous
due to remaining divergences
after mass and coupling
constant renormalization. Accordingly it necessitates additional
subtractions which, within our framework,
are nothing but the usual
renormalization procedure for a composite operator, which is (perturbatively)
well-defined~\cite{Collins}.
The only consequence is that,
after a consistent treatment
of the subtracted terms (i.e respecting RG invariance),
the unambiguous determination of the $1/F^n$ perturbative
terms in (\ref{Fpiansatz})
necessitates the knowledge of the $(n+1)$ order
of the ordinary perturbative expansion.
The nice thing, however, is that the subtracted terms only affect the
values of $\alpha_\pi$ and $\beta_\pi$, but not the
{\em form} of the ansatz (\ref{Fpiansatz}), as soon as the order of the
variational-perturbative expansion is larger than 1~\cite{gn2}.
The consistency of
our formalism is checked by noting that the re-expansion of
(\ref{Fpiansatz})
do reproduce correctly the LL and NLL dependence in $\bar m$ of the
perturbative expansion of the composite operator to all orders.
The
analyticity range with respect to $\Delta\gamma_1$, discussed in section
2, remains valid for
(\ref{Fpiansatz})
as well, since the branch cuts are determined by the very same
relation (\ref{Fdef}). We can thus proceed to a
numerical optimization with respect to the RS dependence, along the
same line as the mass case in section 2. Using an appropriate Pad\'e
approximant form to define the $F \to 0$ ($m'' \to 0$) limit,
we obtain the optimal
values as
\begin{equation}
F^{Pad\acute{e}}_{\pi ,opt}(m'' \to 0)
\simeq 0.55\;\Lambda_{\overline{MS}}(2)\;\;\;(0.59\;\Lambda_{\overline{MS}}(3)\;)\;,
\end{equation}
for $n_f =$ 2 (3). With $F_\pi \simeq 92$ MeV,
this gives $\Lambda_{\overline{MS}} \simeq $ 157 (168) MeV, for $n_f =$ 3 (2).
\section{$\langle \bar q q \rangle$ ansatz}
As is well known~\cite{Collins,Miransky},
$\langle \bar q q \rangle$ is not RG-invariant, while
$m \langle \bar q q \rangle$ is; this is thus the relevant quantity
to consider
for applying our RG-invariant construction.
A straightforward generalization of the
derivation in section 3 leads to the ansatz
\begin{eqnarray}
{\bar m \langle \bar q q\rangle \over{\Lambda_{\overline{MS}}^4}} =(2b_0)
{2^{-4 C} a^4\over{2 i \pi}} \oint {dy\over y} {e^{y/m^{''}} y^4
\over{(F)^{4 A-1} [C + F]^{4 B
}}} {
\delta_{\langle \bar q q\rangle}
\left(1 +{\alpha_{\langle \bar q q\rangle}(a)\over{F(y)}}
\;\right)}
\label{qqansatz}
\end{eqnarray}
where again the coefficients
$\delta_{\langle\bar q q\rangle}$ and
$\alpha_{\langle\bar q q\rangle}(1)$ are obtained from matching the
ordinary perturbative expansion after a subtraction,
and will be given explicitely elsewhere~\cite{jlk}.
The perturbative expansion, known up to two-loop order
\cite{Spiridonov,jlk}
implies that one only knows unambiguously the first
order correction ${\cal O}(1/F)$ in (\ref{qqansatz}),
as previously discussed.
Apart from that, (\ref{qqansatz})
has all the expected properties (RG invariance, resumming LL
and NLL dependence etc),
but a clear inconvenience
is that $\langle\bar q q\rangle$ cannot be directly accessed, being
screened by tiny explicit symmetry breaking effects due to $m \neq 0$.
This is of course a well-known problem, not specific to our construction.
However, it is not clear how to consistently include explicit
symmetry breaking effects
within our framework.
As amply discussed, in (\ref{qqansatz})
$m^{''}$ is an arbitrary parameter, destined to reach its
chiral limit $m^{''} \to 0$.
Accordingly, $\bar m \to 0$
for $m'' \to 0$,
so that one presumably expects only to recover a trivial result,
$\bar m \langle\bar q q\rangle \to 0$ for $m'' \to 0$. This is
actually the case:
although
(\ref{qqansatz}) potentially gives a non-trivial result in the
chiral limit, namely the simple
pole residue ($\equiv 2b_0(2C)^{-C} \;\delta_{\langle \bar q q\rangle}
\;\alpha_{\langle \bar q q\rangle}(a)$,
upon neglecting unknown higher-order
purely perturbative corrections), when we require
extrema of
this expression with respect to RS changes,
using for the $m'' \to 0$ limit
a Pad\'e approximant similar to the one for $F_\pi$,
we do {\em not} find
non-zero extrema.
Such a result is not conclusive regarding
the actual value of $\langle\bar q q \rangle(\bar\mu)$,
but it may be considered
a consistency cross-check
of our formalism. \\
On the other hand, we should mention that our basic expression
(\ref{qqansatz}) {\em does} possess non-trivial extrema for some
$m''_{opt} \neq 0$. These
we however refrain from interpreating in a more quantitative way
since, within our framework, we cannot
give to $m''\times \Lambda_{\overline{MS}}$ the meaning of a true, explicit quark mass
(whose
input we in principle need in order to extract a $\langle \bar q q\rangle
$ value
from (\ref{qqansatz})).
At least, it strongly indicates that it should be possible
to extract $\langle \bar q q\rangle$ in the chiral limit,
by introducing in a consistent way
a small explicit symmetry-breaking mass,
$-m_{0,exp} \bar q_i q_i$, to the basic Lagrangian (\ref{xdef}).
\section{Summary}
In this paper we have shown that
the variational expansion in arbitrary $m''$, as developed
in the context of the GN model~\cite{gn2}, can be formally
extended to the QCD case, apart from the complication due to the
presence of extra singularities, which can be however removed
by appropriate RS change.
As a result we
obtain in the chiral limit non-trivial relationships between
$\Lambda_{\overline{MS}}$ and the dynamical masses and order parameters, $F_\pi$,
$\bar m \langle\bar q q\rangle$.
The resulting
expressions in a generalized RS have been numerically optimized, using
a well-motivated Pad\'e approximant form, due to the complexity of the
full optimization problem. The optimal values obtained for $M_q$ and $F_\pi$
are quite encouraging, while for $\langle\bar q q\rangle$
they are quantitatively
not conclusive,
due to the inherent screening by an explicit
mass term of this quantity, in the limit $m \to 0$.
A possible extension to include consistently explicit breaking mass terms
in our formalism is explored in ref.~\cite{jlk}.
\vskip .5 cm
{\large \bf Acknowledgements} \\
We are grateful to Eduardo de Rafael for
valuable remarks and discussions.
J.-L. K. also thanks Georges Grunberg, Heinrich Leutwyler, Jan Stern
and Christof Wetterich for useful discussions.
(C.A) is grateful to the theory group of Imperial College
for their hospitality.\\
| proofpile-arXiv_065-432 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{#1}\setcounter{equation}{0}}
\newcommand\Sect[2] {\sect{#1}\label{s.#2}
\ifnum0=1 \query{s.#2} \fi}
\def\semitimes {\begin{picture}(9.4,8)\put(1.57,0.32){\line(0,1)
{4.4}}\put(0,0){$\times$} \end{picture} }
\def\mbox{$^{\scriptscriptstyle(h)}\!S$} {\mbox{$^{\scriptscriptstyle(h)}\!S$}}
\def\mbox{$^{\scriptscriptstyle(hh')}\!S$} {\mbox{$^{\scriptscriptstyle(hh')}\!S$}}
\def\mbox{$^{\scriptscriptstyle(h')}\!S$} {\mbox{$^{\scriptscriptstyle(h')}\!S$}}
\def\mbox{\scriptsize(} {\mbox{\scriptsize(}}
\def\mbox{$^{\scriptscriptstyle(\el h)}\!S$} {\mbox{$^{\scriptscriptstyle(\el h)}\!S$}}
\def\mbox{sign} {\mbox{sign}}
\newcommand\sm[2] {\mbox{\rm s}_{#1,#2}}
\def\mbox{\rm s} {\mbox{\rm s}}
\newcommand\Sm[2] {\mbox{\rm S}_{#1,#2}}
\def\mbox{\rm S} {\mbox{\rm S}}
\def$S$-matrix {$S$-matrix}
\def\mbox{\scriptsize)} {\mbox{\scriptsize)}}
\defsemi-simple {semi-simple}
\def\scriptscriptstyle {\scriptscriptstyle}
\def\scriptscriptstyle(\el\el'h) {\scriptscriptstyle(\ell\el'h)}
\def\scriptscriptstyle(\el h) {\scriptscriptstyle(\ell h)}
\def\scriptscriptstyle(h) {\scriptscriptstyle(h)}
\def\scriptscriptstyle(hh') {\scriptscriptstyle(hh')}
\def\scriptscriptstyle(h') {\scriptscriptstyle(h')}
\def\sumihh {{\displaystyle\sum_{\scriptstyle i\atop
\scriptstyle p_i|h',\,\scriptstyle p_i\not{\:|}\,h}}}
\def\sumieh {{\displaystyle\sum_{\scriptstyle i\ne1\atop
\scriptstyle p_i|h}}}
\def\sumiezh {{\displaystyle\sum_{\scriptstyle i\ne1,2\atop
\scriptstyle p_i|h}}}
\newcommand\sumn[1] {{\displaystyle\sum_{#1=1}^{n}}}
\newcommand\sumne[1] {{\displaystyle\sum_{#1=1}^{n-1}}}
\newcommand\sumnee[1] {{\displaystyle\sum_{#1=1}^{n_1-1}}}
\newcommand\sumneee[1]{{\displaystyle\sum_{#1=n_1}^{n_1'-1}}}
\newcommand\sumnepe[1]{{\displaystyle\sum_{#1=1}^{n_1'-1}}}
\newcommand\sumnie[1] {{\displaystyle\sum_{#1=1}^{n_i-1}}}
\newcommand\sumnipe[1]{{\displaystyle\sum_{#1=1}^{n_i'-1}}}
\def\sump {\sum_{\scriptstyle p\atop \scriptstyle n_p\ne0}}
\newcommand\sumph[1] {{\displaystyle\sum_{#1\in{}^{\scriptscriptstyle(h)\!}P}}}
\newcommand\sumpH[1] {\sum_{#1\in^{\scriptscriptstyle(h)\!}P}}
\newcommand\sumphp[1] {{\displaystyle\sum_{#1\in{}^{\scriptscriptstyle(h')\!}P}}}
\def\sumpl {\sum_{\scriptstyle p\atop \scriptstyle n'_p\ne0}}
\newcommand\sumplh[1] {{\displaystyle\sum_{#1\in{}^{\scriptscriptstyle(\el h)\!}P}}}
\newcommand\sumre[1] {{\displaystyle\sum_{#1=1}^r}}
\newcommand\sumrE[1] {\sum_{#1=1}^r}
\newcommand\sumro[1] {{\displaystyle\sum_{#1=0}^r}}
\newcommand\sumrO[1] {\sum_{#1=0}^r}
\renewcommand{\thesection.\arabic{equation}}{\thesection.\arabic{equation}}
\def\mbox{$^{\sssh\!}T$} {\mbox{$^{\scriptscriptstyle(h)\!}T$}}
\defthree-di\-men\-si\-o\-nal {three-di\-men\-si\-o\-nal}
\def\tilde {\tilde}
\def\topo {limit topology}
\deftwo-di\-men\-si\-o\-nal {two-di\-men\-si\-o\-nal}
\def{}^{\sss(\infty)\!}Q {{}^{\scriptscriptstyle(\infty)\!}Q}
\defuntwisted affine Lie algebra {untwisted affine Lie algebra}
\def{}^{\sss(\infty)\!}Q^- {{}^{\scriptscriptstyle(\infty)\!}Q^-}
\newcommand\version[1] {\ifnum0=1 \typeout{}\typeout{#1}\typeout{}
\vskip3mm \centerline{\fbox{\tt DRAFT -- #1 -- \today}}
\vskip3mm \fi}
\def\mbox{$ {\cal V}ir$} {\mbox{$ {\cal V}ir$}}
\def{\rm vol} {{\rm vol}}
\def\varphi^{} {\varphi^{}}
\def\bar\varphi^{} {\bar\varphi^{}}
\def{}^{\sssh\!}\varphi^{} {{}^{\scriptscriptstyle(h)\!}\varphi^{}}
\def{}^{\sssh\!}\varphi {{}^{\scriptscriptstyle(h)\!}\varphi}
\def{}^{\ssshp\!}\varphi^{} {{}^{\scriptscriptstyle(h')\!}\varphi^{}}
\def{}^{\ssslh\!}\varphi^{} {{}^{\scriptscriptstyle(\el h)\!}\varphi^{}}
\def{}^{\sss(\ho)\!}\varphi^{} {{}^{\scriptscriptstyle({h_\circ})\!}\varphi^{}}
\def\mbox{$^{\sssh\!}W$} {\mbox{$^{\scriptscriptstyle(h)\!}W$}}
\def\mbox{$^{\sss(h')\!}W$} {\mbox{$^{\scriptscriptstyle(h')\!}W$}}
\def{}^{\sssh}W {{}^{\scriptscriptstyle(h)}W}
\def\mbox{$^{\ssslh\!}W$} {\mbox{$^{\scriptscriptstyle(\el h)\!}W$}}
\def\mbox{$^{\sssllh\!}W$} {\mbox{$^{\scriptscriptstyle(\el\el'h)\!}W$}}
\def\mbox{$^{\sss(p)\!}W$} {\mbox{$^{\scriptscriptstyle(p)\!}W$}}
\defwith respect to {with respect to }
\defwith respect to the {with respect to the }
\defWess\hy Zumino\hy Witten {Wess$\mbox{-\hspace{-.66 mm}-}$ Zumino$\mbox{-\hspace{-.66 mm}-}$ Witten}
\defWZW model {WZW model}
\defWZW theory {WZW theory}
\defWZW theories {WZW theories}
\def\hat\xi {\hat\xi}
\def{\dl Z} {{\mathbb Z}}
\def${\dl Z}$ {${\mathbb Z}$}
\def\mbox{${\zet}_{\leq 0}$} {\mbox{${{\dl Z}}_{\leq 0}$}}
\def\mbox{${\zet}_{>0}$} {\mbox{${{\dl Z}}_{>0}$}}
\def\mbox{${\zet}_{\geq 0}$} {\mbox{${{\dl Z}}_{\geq 0}$}}
\setlength{\unitlength}{.1 em}
\def\begin{picture} {\begin{picture}}
\def\begin{picture}(0,0) {\begin{picture}(0,0)}
\def\end{picture} {\end{picture}}
\newcommand\pictriangleupddr[6] {\begin{picture}(100,90)
\put(46,70){#1} \put(-2,0){#2} \put(80,0){#3}
\put(17,38){{#4}} \put(71,38){{#5}} \put(41,-4){{#6}}
\put(47,65){\vector(-1,-2){25}}
\put(53,65){\vector(1,-2){25}}
\put(25,5){\vector(1,0){50}}
\end{picture}}
\newcommand\pictriangledowndd[3] {\begin{picture}(100,90)
\put(44,0){#1} \put(17,38){{\footnotesize #2}} \put(72,38){{\footnotesize #3}}
\put(47,15){\vector(-1,2){25}}
\put(53,15){\vector(1,2){25}}
\end{picture}}
\def\number\month/\number\day/\number\year\ \ \ \hourmin {\number\month/\number\day/\number\year\ \ \ \hourmin }
\def\pagestyle{draft}\thispagestyle{draft} \global\def0{1}{\pagestyle{draft}\thispagestyle{draft} \global\def0{1}}
\global\def0{0}
\catcode`\@=12
\documentclass[12pt]{article} \usepackage{amssymb,amsfonts}
\def{\thesection.\arabic{equation}}{{\thesection.\arabic{equation}}}
\setlength{\textwidth}{17cm} \setlength{\textheight}{23cm}
\hoffset -17mm \topmargin= -18mm \raggedbottom
\begin{document}
\version\versionno
\setlength{\unitlength}{.1 em}
\begin{flushright}
{\sf hep-th/9609124} \\
{\sf DESY 96-196} \\ {\sf IHES/P/96/47} \\
{\sf September 1996}
\end{flushright}
\begin{center} \vskip 1cm
{\Large\bf WZW FUSION RINGS IN THE LIMIT} \vskip 2mm
{\Large\bf OF INFINITE LEVEL} \vskip 12mm
{{\large \ J\"urgen Fuchs} {\special{" 0 0 moveto 45 rotate}}}\\[5mm]
{\small DESY}\\ {\small Notkestra\ss e 85}\\ {\small D -- 22603~~Hamburg}
\\[5mm] and \\[5mm] {\large Christoph Schweigert}\\[5mm] {\small IHES}\\
{\small 35, Route de Chartres}\\ {\small F -- 91440 Bures-sur-Yvette}
\end{center}
\vskip 15mm
\begin{quote} {\bf Abstract.} \\
We show that the WZW fusion rings at finite levels form a projective system
with respect to the partial ordering provided by divisibility of the height, i.e.\ the
level shifted by a constant. {}From this projective system we obtain WZW
fusion rings in the limit of infinite level. This projective limit constitutes
a mathematically well-defined prescription for the `classical limit' of
WZW theories\ which replaces the naive idea of `sending the level to infinity'.
The projective limit can be endowed with a natural topology, which plays
an important r\^ole for studying its structure. The representation theory
of the limit can be worked out by considering the associated fusion algebra;
this way we obtain in particular an analogue of the Verlinde formula.
\end{quote}
\vfill{}{~}\\[1 mm]\noindent ------------------\\[1 mm]{} {\small {\special{" 0 0 moveto 45 rotate}}~~Heisenberg fellow} \newpage
\Sect{Fusion rings}{wzw}
Fusion rings constitute a mathematical structure which emerges in various
contexts, for instance in the analysis of the superselection rules of two-di\-men\-si\-o\-nal\
quantum field theories; they describe in particular the basis independent
contents of the operator product algebra\ of two-di\-men\-si\-o\-nal\ conformal field theories\ (for a review see
\cite{jf24}). By definition, a {\em fusion ring\/} \mbox{${\cal R}$}\ is
a unital commutative associative ring over the integers ${\dl Z}$\ which possesses
the following properties: there is a distinguished basis $\mbox{${\cal B}$}=\{\varphi^{}_a\}$
which contains the unit and in which the structure constants \n abc\ are
non-negative integers, and the evaluation at the unit
induces an involutive automorphism, called the conjugation of \mbox{${\cal R}$}.
A fusion ring is referred to as {\em rational\/} iff it is finite-dimensional.
A rational fusion ring is called {\em modular\/} iff the matrix $S$ that
diagonalizes simultaneously all fusion matrices \N a\ (i.e.\ the matrices
with entries $(\N a)_b^{\ c} =\n abc$)
is symmetric and together with an appropriate diagonal matrix $T$ generates a
unitary representation\ of SL$(2,{\dl Z})$ (see e.g.\ \cite{kawA,jf24}).
In this paper we consider the fusion rings of (chiral, unitary) WZW theories. A
WZW theory\ is a conformal field theory\ whose chiral symmetry algebra\ is the semidirect sum of the
Virasoro algebra\ with an untwisted affine Kac\hy Moo\-dy algebra\ \mbox{$\mathfrak g$}, with the level \mbox{$k_{}^{\scriptscriptstyle\vee}$}\ of the
latter a fixed non-negative integer.
To any untwisted affine Kac\hy Moo\-dy algebra\ \mbox{$\mathfrak g$}\ we can thus associate a family of
fusion rings, parametrized by the level \mbox{$k_{}^{\scriptscriptstyle\vee}$}. The issue that we address in this
paper is to construct an analogue of the WZW fusion ring for infinite level,
which is achieved by giving a prescription for `sending the level to infinity'
in an unambiguous manner.
In view of the Lagrangian realization of WZW theories\ as sigma models, this
procedure may be regarded as taking the `classical limit' of WZW theories.
Performing a classical limit of a parametrized family of quantum field theories\
is a rather common concept in the path integral formulation of quantum
theories; it simply corresponds to sending Planck's constant to zero, and
hence provides a kind of correspondence principle. In the
Lagrangian description of WZW theories\ as principal sigma models with
Wess$\mbox{-\hspace{-.66 mm}-}$ Zumino terms, the r\^ole of Planck's constant is played by the inverse
of the level \mbox{$k_{}^{\scriptscriptstyle\vee}$}\ of the underlying affine Lie algebra\ \mbox{$\mathfrak g$}. However, it is known that
the path integral of a WZW sigma model strictly makes sense only if the level
\mbox{$k_{}^{\scriptscriptstyle\vee}$}\ is an integer. In contrast to the path integral description,
in the framework of algebraic approaches to quantum theory so far almost
no attempts have been made to investigate limits of quantum field theories. In this paper
we address this issue for the case of WZW theories. Now in an algebra ic treatment of
WZW theories\ the integrality requirement just mentioned is immediately manifest.
Namely, one observes that the structure of the theory
depends sensitively on the value \mbox{$k_{}^{\scriptscriptstyle\vee}$}\ of the level. For non-negative integral
\mbox{$k_{}^{\scriptscriptstyle\vee}$}\ the state space is a direct sum of unitary irreducible highest weight module s of the algebra \mbox{$\mathfrak g$}, but
its structure changes quite irregularly when going from \mbox{$k_{}^{\scriptscriptstyle\vee}$}\ to $\mbox{$k_{}^{\scriptscriptstyle\vee}$}+1$; at
intermediate, non-integral, values of the level there do not even exist any
unitarizable highest weight representation s.
These observations indicate that it is a rather delicate issue to define what is
meant by the classical limit of a WZW theory, and it seems mandatory to perform this
limit in a manner in which the level \mbox{$k_{}^{\scriptscriptstyle\vee}$}\ is manifestly kept integral (actually,
treating the level formally as a continuous
variable is potentially ambiguous even in situations where one deals with
expressions which superficially make sense also at non-integral level).
It must also be noted that a priori it is by no means clear whether the so
obtained limit will be identical with or at least closely resemble the
structures which originally served to define the quantum theory in terms of
some classical field theory; in the case of WZW fusion rings, this
underlying classical structure is the representation\ ring of the finite-dimensional\ simple Lie algebra\
$\mbox{$\bar{\mathfrak g}$}$ that is generated by the zero modes of the affine Lie algebra\ \mbox{$\mathfrak g$}. Indeed, it seems to be a quite generic feature of quantum theory that
the classical limit does not simply reproduce the classical structure one
started with. (Compare for instance the fact that in the path integral
formulation of quantum field theory the classical paths are typically of
measure zero in the space of all paths that contribute to the path integral.
Similar phenomena also show up when the continuum limit of a lattice theory
is constructed as a projective limit; see e.g.\ \cite{asle2,bellst}.)
However, it is still reasonable to expect that the original
classical structure is, in a suitable manner, contained in the classical limit;
as we will see, this is indeed the case for our construction.
The desire of being able to perform a limit in which the level tends to
infinity stems in part from the fact
that WZW theories\ and their fusion rings can be used to define a regularization
of various systems (such as two-di\-men\-si\-o\-nal\ gauge theories or the Ponzano$\mbox{-\hspace{-.66 mm}-}$ Regge
theory of simplicial three-di\-men\-si\-o\-nal\ gravity), with the unregularized system
corresponding, loosely speaking, to the classical theory. As removing the
regulator is always a subtle issue, it is mandatory that the limit of
the regularized theory is performed in a well-defined, controllable manner,
which, in addition,
should preserve as much of the structure as possible.
\medskip
The basic idea which underlies our construction of the limit of WZW fusion rings
is to interpret the collection of WZW fusion rings as a category ${\cal F}\!us\mbox{\scriptsize(}\mbox{$\mathfrak g$}\mbox{\scriptsize)}$\ within
the category of all commutative rings and identify inside this category a
projective system. By a standard category theoretic construction we can then
obtain the limit (also known as the projective limit) of this projective system.
The partial ordering underlying the projective system is based on a divisibility
property of the parameter $\mbox{$k_{}^{\scriptscriptstyle\vee}$}+\mbox{$g_{}^{\scriptscriptstyle\vee}$}$ that together with the choice of the horizontal subalgebra\
\mbox{$\bar{\mathfrak g}$}\ characterizes the WZW theory\ (\mbox{$g_{}^{\scriptscriptstyle\vee}$}\ denotes the dual Coxeter number of \mbox{$\bar{\mathfrak g}$};
the sum $\mbox{$k_{}^{\scriptscriptstyle\vee}$}+\mbox{$g_{}^{\scriptscriptstyle\vee}$}$ is called the {\em height\/}).
In contrast, in the literature often a purely
formal prescription `\,$\mbox{$k_{}^{\scriptscriptstyle\vee}$}\to\infty$\,' is referred to as the classical
limit of WZW theories. In that terminology it is implicit that the standard ordering
on the set of levels is chosen to give it the structure of a directed set.
Now the projective limit is associated to a projective system as a whole,
not just to the collection of objects that appear in the system; in particular
it depends on the underlying directed set and hence on the choice of partial
ordering. Our considerations show, as a by-product, that it is not possible to
associate to the standard ordering any well-defined limit of the fusion rings.
The rest of this paper is organized as follows. We start in subsection
\ref{swz} by introducing the category ${\cal F}\!us\mbox{\scriptsize(}\mbox{$\mathfrak g$}\mbox{\scriptsize)}$\ of WZW fusion rings
associated to an untwisted affine Lie algebra\ \mbox{$\mathfrak g$};
in subsection \ref{s.qdim} conditions for the existence of
non-trivial morphisms of this category are obtained.
In subsection \ref{s.ps} we define the projective system, and in the remainder
of section \ref{s.PS} we check that the morphisms introduced by this definition
possess the required properties. The projective limit of the so obtained
projective system is a unital commutative associative ring of countably
infinite dimension. This ring \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ is constructed in \secref{pl}; there
we also gather some basic properties of \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ and introduce a natural
topology on \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}. In \secref{binf} a concrete description of a
distinguished basis \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\ for the projective limit is obtained. This
basis is similar to the distinguished bases
of the fusion rings at finite level; in order to show that \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ is indeed
generated by \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}, the topology on \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ plays an essential r\^ole.
In \secref{gb} we demonstrate that the representation\ ring of the horizontal subalgebra\
$\mbox{$\bar{\mathfrak g}$}\subset\mbox{$\mathfrak g$}$ is contained in the projective limit \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ as a proper subring.
In the final \secref{rep} we study the representation\ theory of \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}, respectively\ of the
associated fusion algebra over ${\mathbb C}$\,. In particular,
we determine all irreducible representation s and show that \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\
possesses a property which is the topological analogue of semi-simplicity,
namely that any continuous \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}-module is the closure
of a direct sum of irreducible modules. To obtain these results it is again
crucial to treat the projective limit as a topological space.
Finally, we establish an analogue of the Verlinde formula for \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}.
\Sect{The projective system of WZW fusion rings}{PS}
\subsection{WZW fusion rings}\label{swz}
The primary fields of a unitary WZW theory\
are labelled by integrable highest weights of the relevant affine Lie algebra\
\mbox{$\mathfrak g$}, or what is the same, by the value \mbox{$k_{}^{\scriptscriptstyle\vee}$}\ of the level and by
dominant integral weights $\Lambda$ of \mbox{$\bar{\mathfrak g}$}\ (the horizontal subalgebra of \mbox{$\mathfrak g$})
whose inner product with the highest coroot of \mbox{$\bar{\mathfrak g}$}\ is not larger than \mbox{$k_{}^{\scriptscriptstyle\vee}$}.
We denote by \mbox{$g_{}^{\scriptscriptstyle\vee}$}\ the dual Coxeter number of the simple Lie algebra\ \mbox{$\bar{\mathfrak g}$}\ and define
\begin{equation} \mbox{$I$}:= \{ i\in{\dl Z} \mid i\ge\mbox{$g_{}^{\scriptscriptstyle\vee}$} \} \,. \labl i
Thus \mbox{$I$}\ is the set of possible values of the {\em height\/} $h\equiv \mbox{$k_{}^{\scriptscriptstyle\vee}$}+\mbox{$g_{}^{\scriptscriptstyle\vee}$}$
of the WZW theory\ based on \mbox{$\mathfrak g$}. For any $h\iN\II$ the fusion rules of a WZW theory\ at
height $h$ define a modular fusion ring, with the elements of the distinguished
basis corresponding to the primary fields. We denote this ring by \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\ and its
distinguished basis by \mbox{$^{\scriptscriptstyle(h)}\!{\cal B}$}, and the
corresponding generators of SL$(2,{\dl Z})$ by \mbox{$^{\scriptscriptstyle(h)}\!S$}\ and \mbox{$^{\sssh\!}T$}.
The distinguished basis \mbox{$^{\scriptscriptstyle(h)}\!{\cal B}$}\ of the ring \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\ can be labelled as
\begin{equation} \mbox{$^{\scriptscriptstyle(h)}\!{\cal B}$}=\{{}^{\sssh\!}\varphi^{}_a \,|\,a\!\in\!\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}\} \end{equation}
by the set
\begin{equation} \mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$} := \{ a\in\mbox{$\overline L^{\rm w}$} \mid a^i\ge1\ {\rm for}\ i=1,2,...\,,r;\;
(a,\theta^{\scriptscriptstyle\vee})<h \} \labl{ph}
of integral weights in the interior of (the horizontal projection of) the
fundamental Weyl chamber of \mbox{$\mathfrak g$}\ at {\em level\/} $h$; here $r$,
$\theta^{\scriptscriptstyle\vee}$ and \mbox{$\overline L^{\rm w}$}\ denote the rank, the highest coroot
and the weight lattice of \mbox{$\bar{\mathfrak g}$}, respectively. Note that from
here on we use shifted \mbox{$\bar{\mathfrak g}$}-weights $a=\Lambda+\rho$, which have level $h
=\mbox{$k_{}^{\scriptscriptstyle\vee}$}+\mbox{$g_{}^{\scriptscriptstyle\vee}$}$, in place of unshifted weights $\Lambda$ which are at level \mbox{$k_{}^{\scriptscriptstyle\vee}$}.
Here $\rho$ is the Weyl vector of \mbox{$\bar{\mathfrak g}$}; in particular, $a=\rho$ is the label of
the unit element
of \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}. This convention will simplify various formul\ae\ further on.
The ring product of \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\ will be denoted by the symbol `\,$\star$\,'; thus
the fusion rules are written as
\begin{equation} {}^{\sssh\!}\varphi^{}_a \star {}^{\sssh\!}\varphi^{}_b = \sumph c \nh abc\,{}^{\sssh\!}\varphi^{}_c \,. \end{equation}
The collection $(\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$})_{h\in\II}$ of such WZW fusion rings forms a
category, more precisely a subcategory of the category of commutative rings,
which we denote by ${\cal F}\!us\mbox{\scriptsize(}\mbox{$\mathfrak g$}\mbox{\scriptsize)}$. The objects of ${\cal F}\!us\mbox{\scriptsize(}\mbox{$\mathfrak g$}\mbox{\scriptsize)}$\ are the rings \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$},
and the morphisms (arrows) are those ring homomorphisms (which are automatically
unital and
compatible with the conjugation) which map the basis \mbox{$^{\scriptscriptstyle(h')}\!{\cal B}$}\ up to sign factors
to \mbox{$^{\scriptscriptstyle(h)}\!{\cal B}$}. These are the natural requirements to be imposed on morphisms. Namely,
one preserves precisely all structural properties of the fusion ring,
except for the positivity of the structure constants; the latter
is not an algebraic property, so that one should be prepared to give it up.
\subsection{Existence of morphisms}\label{s.qdim}
It is not a priori clear whether the category ${\cal F}\!us\mbox{\scriptsize(}\mbox{$\mathfrak g$}\mbox{\scriptsize)}$\ as defined above has any
non-trivial
morphisms at all. To analyze this issue, we consider the quotients ${\mbox{$^{\scriptscriptstyle(h)}\!S$}_{a,b}}
/{\mbox{$^{\scriptscriptstyle(h)}\!S$}_{a,\rho}}$ ($a,b\!\in\!\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}$) of $S$-matrix\ elements. These are known as the
(generalized) quantum dimension s, or more precisely, as the $a$th quantum dimension\ of the element
${}^{\sssh\!}\varphi^{}_b$, of the modular fusion ring \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}. The generalized quantum dimension s
furnish precisely all inequivalent irreducible representation s of \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\ \cite{kawA}. We denote by
\begin{equation} \qd ha :\quad \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\to{\mathbb C}\,, \quad {}^{\sssh\!}\varphi^{}_b\mapsto
\frac{\mbox{$^{\scriptscriptstyle(h)}\!S$}_{a,b}}{\mbox{$^{\scriptscriptstyle(h)}\!S$}_{a,\rho}} \labl{qd}
the irreducible representation\ of \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\ which associates to any element its $a$th generalized quantum dimension.
Assume now that $f\!:\;\mbox{$^{\scriptscriptstyle(h')\!}{\cal R}$}\to\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}$ is a non-trivial morphism, i.e.\ a ring
homomorphism which maps the distinguished basis \mbox{$^{\scriptscriptstyle(h')}\!{\cal B}$}\ of \mbox{$^{\scriptscriptstyle(h')\!}{\cal R}$}\
to the basis \mbox{$^{\scriptscriptstyle(h)}\!{\cal B}$}\ of \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}. Then the composition $\qd ha\circ f$ provides us
with a one-dimensional, and hence irreducible, representation\ of \mbox{$^{\scriptscriptstyle(h')\!}{\cal R}$}, i.e.\ we have
$\qd ha\circ f=\qd {h'}{a'}$ for some $a'\!\in\!\mbox{$_{}^{\scriptscriptstyle(h')\!\!}P$}$.
Let now \mbox{$^{\scriptscriptstyle(h)\!\!}L$}\ denote the extension of the field ${\mathbb Q}$ of rational numbers
by the quantum dimension s\ $\mbox{$^{\scriptscriptstyle(h)}\!S$}_{a,b}/\mbox{$^{\scriptscriptstyle(h)}\!S$}_{a,\rho}$ of all elements of \mbox{$^{\scriptscriptstyle(h)}\!{\cal B}$};
the observation just made then implies that
\begin{equation} \mbox{$^{\scriptscriptstyle(h)\!\!}L$} \subseteq \mbox{$^{\scriptscriptstyle(h')}\!L$} \labl{ll}
(when $f$ is surjective, one gets in fact the whole field $\mbox{$^{\scriptscriptstyle(h)\!\!}L$}$).
As we will see, this result puts severe constraints on the existence of
morphisms from \mbox{$^{\scriptscriptstyle(h')\!}{\cal R}$}\ to \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}. It follows from the Kac\hy Peterson formula\ \cite{KAc3} for the
$S$-matrix\ that
\begin{equation} \mbox{$^{\scriptscriptstyle(h)\!\!}L$}\subseteq\qzm h\,, \labl q
with $\zeta_m:=\exp(2\pi{\rm i}/m)$ and
$M$ the smallest positive integer for which all entries of the metric on the
weight space of \mbox{$\bar{\mathfrak g}$}\ are integral multiples of $1/M$
(except for $\mbox{$\bar{\mathfrak g}$}=A_r$ where $M=r+1$, $M$ satisfies $M\leq4$).
The inclusion \erf{ll} therefore implies that \mbox{$^{\scriptscriptstyle(h)\!\!}L$}\ lies in the intersection
$\qzm h \cap \qzm{h'} = \qzm{\,\ggt h{h'}}$,
and that this intersection is strictly larger than ${\mathbb Q}$
unless $\mbox{$^{\scriptscriptstyle(h)\!\!}L$}={\mathbb Q}$. Here $\ggt mn$ stands for the largest common divisor of
$m$ and $n$. In the specific case that $h$ and $h'$ are coprime, $\ggt h{h'}=1$,
it follows that
\begin{equation} \mbox{$^{\scriptscriptstyle(h)\!\!}L$} \,\subseteq\, \mbox{$^{\scriptscriptstyle(h')}\!L$} \cap \qzm h \,\subseteq\, \qzm{} \,. \end{equation}
Now typically the field \mbox{$^{\scriptscriptstyle(h)\!\!}L$}\ is quite a bit smaller than \qzm h,
i.e.\ the inequality \erf q is not saturated (e.g.\ if the ring is
self-conjugate, \mbox{$^{\scriptscriptstyle(h)\!\!}L$}\ is already contained in the maximal real subfield of \qzm
h); nevertheless,
inspection shows that the requirement $\mbox{$^{\scriptscriptstyle(h)\!\!}L$}\subseteq\qzm{}$ is fulfilled only
in very few cases (for instance, for \mbox{$\bar{\mathfrak g}$}\ of type $B_{2n},\,C_r,\,D_{2n},\,
E_7,\,E_8$ or $F_4$, one has $M\le2$ so that the requirement is just $\mbox{$^{\scriptscriptstyle(h)\!\!}L$}=
{\mathbb Q}$). In addition, the main quantum dimension s $\mbox{$^{\scriptscriptstyle(h)}\!S$}_{a,\rho}/\mbox{$^{\scriptscriptstyle(h)}\!S$}_{\rho,\rho}$ lie
in fact in \qz{2h}, and hence the above requirement would restrict them to
lie in $\qz{2h}\cap\qzm{}=\qz{\ggt{2h}M}$, and thus to be rational whenever
$2h$ and $M$ are coprime.
It follows that for almost all pairs $h,\,h'$ of coprime heights there
cannot exist any morphism from \mbox{$^{\scriptscriptstyle(h')\!}{\cal R}$}\ to \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}. The same arguments also
show that the existence of non-trivial morphisms becomes the more probable
the larger the value of $\ggt h{h'}$ is. The most favourable situation is when
$h'$ is a multiple of $h$; in the next section we will show that in this
case a whole family of morphisms from \mbox{$^{\scriptscriptstyle(h')\!}{\cal R}$}\ to \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\ (with $h\iN\II$ arbitrary)
can be constructed in a natural way.
The considerations above indicate in particular that the naive way of
taking the limit `\,$k\to\infty$\,' with the standard
ordering on the set \mbox{$I$}\ cannot correspond to any well-defined limit of the
WZW fusion rings. In contrast, as we will show, when replacing the standard
ordering by a suitable partial ordering, a limit can indeed be constructed,
namely as the projective limit of a projective system that is associated to
that partial ordering.
Let us also mention that the required ring homomorphism property implies
that any morphism of ${\cal F}\!us\mbox{\scriptsize(}\mbox{$\mathfrak g$}\mbox{\scriptsize)}$\ maps simple currents to simple currents.
(By definition, simple currents are those elements
$\varphi_a$ of the distinguished basis \mbox{$^{\scriptscriptstyle(h)}\!{\cal B}$}\ which have inverses in the
fusion ring; they satisfy $\sum_c\n abc=1$ for all $b$. Such elements are
sometimes also called units of the ring, not to be confused with the unit
element of the fusion ring.)
\subsection{The projective system}\label{s.ps}
On the set \mbox{$I$}\ \erf i of heights one can define a partial ordering `\,$\preceq$\,' by
\begin{equation} i \preceq j \ :\Leftrightarrow\ i\,|\,j \,, \labl l
where the vertical bar stands for divisibility.
For any two elements $i,i'\iN\II$ there then exists a $j\iN\II$ (for example, the
smallest common multiple of $i$ and $i'$) such that $i\preceq j$ and $i'\preceq j$.
Thus the partial ordering \erf l endows \mbox{$I$}\ with the structure of a
{\em directed set}.
We will now show that
when the set \mbox{$I$}\ is considered as a directed set via the partial ordering
\erf l, the collection $(\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$})_{h\in\II}$ of WZW fusion rings can be made into a
{\em projective system}, that is,
for each pair $i,j\iN\II$ satisfying $i\preceq j$ there exists a morphism
\begin{equation} \ff ji:\quad \mbox{$^{\scriptscriptstyle(j)\!}{\cal R}$}\to\mbox{$^{\scriptscriptstyle(i)\!}{\cal R}$}\,, \labl f
such that \ff ii is the identity for all $i\!\in\! I$ and
such that for all $i,j,k\iN\II$ which satisfy $i\preceq j\preceq k$, the diagram
\begin{equation} \begin{array}{l} {}\\[-2.2em]
\pictriangleupddr {$\!\!\!$\mbox{$^{\scriptscriptstyle(k)\!}{\cal R}$}}{\mbox{$^{\scriptscriptstyle(j)\!}{\cal R}$}~~}{\mbox{$^{\scriptscriptstyle(i)\!}{\cal R}$}}{\fF kj}{\fF ki}{\fF ji}
\\[.3em] \end{array} \labl1
commutes.
We have to construct the maps \ff ij for all pairs $i,\,j$ with $i|j$. Writing
$i=h$ and $j=\ell h$ with $\ell\!\in\!{\dl N}$, the construction goes as follows.
The horizontal projection \mbox{$^{\sssh\!}W$}\ of the affine Weyl group at height $h$ has
the structure of a semidirect product $\mbox{$^{\sssh\!}W$} = \overline W\semitimes h\mbox{$\overline L^{\scriptscriptstyle\vee}$}$,
with $\overline W$ the Weyl group and \mbox{$\overline L^{\scriptscriptstyle\vee}$}\ the coroot lattice of \mbox{$\bar{\mathfrak g}$}, so that
in particular \mbox{$^{\sssh\!}W$}\ is contained
as a finite index subgroup in \mbox{$^{\ssslh\!}W$}, the index having the value $\ell^{\,r}_{}$.
Thus any orbit of \mbox{$^{\sssh\!}W$}\ decomposes into orbits of \mbox{$^{\ssslh\!}W$}, and each Weyl chamber
at height $\ell h$ is the union of $\ell^{\,r}_{}$ Weyl chambers at height $h$.
As a consequence, we find that the following statement holds for the set \mbox{$_{}^{\scriptscriptstyle(\el h)\!\!}P$}\
defined according to \erf{ph}.
To any $a\!\in\!\mbox{$_{}^{\scriptscriptstyle(\el h)\!\!}P$}$ there either exists a unique element $w_a\!\in\!\mbox{$^{\sssh\!}W$}$ such that
\begin{equation} a':=w_a(a) \labl{a'}
belongs to the set \mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}, or else $a$ lies on the boundary of some affine Weyl
chamber at height $h$. In the former case we define
\begin{equation} \ff{\el h}h({}^{\ssslh\!}\varphi^{}_a):= \epsilon_{\el}(a)\cdot {}^{\sssh\!}\varphi^{}_{a'} \labl'
with $\epsilon_{\el}(a)=\mbox{sign}(w_a),$ while in the latter case we set
$\ff{\el h}h({}^{\ssslh\!}\varphi^{}_a):=0$. It is convenient to include this latter case
into the formula \erf', which is achieved by setting
\begin{equation} \epsilon_{\el}(a):= \left\{ \begin{array}{ll}
0 & \mbox{if $a$ lies on the boundary of an}\\
& \mbox{affine Weyl chamber at height $h$\,,}
\\[1mm] \mbox{sign}(w_a) & {\rm else} \,.
\end{array}\right. \labl{eps}
\subsection{Proof of the morphism properties}
We have to prove that $\ff ij$ defined this way is a
ring homomorphism and that it satisfies the composition property \erf1.
It is obvious from the definition that $\ff ii={\sf id}$ (and also that
$\ff ij$ is surjective). To show the homomorphism property, we write $\ff ij$
in matrix notation, and for convenience use
capital letters for the fusion ring \mbox{$^{\scriptscriptstyle(\el h)\!}{\cal R}$}\ and lower case letters for the
fusion ring \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}. Thus the elements of the basis \mbox{$^{\scriptscriptstyle(\el h)}\!{\cal B}$}\ of \mbox{$^{\scriptscriptstyle(\el h)\!}{\cal R}$}\ are denoted
by $\phi^{}_A\equiv{}^{\ssslh\!}\varphi^{}_{\!A}$ with $A\!\in\!\mbox{$_{}^{\scriptscriptstyle(\el h)\!\!}P$}$, while for the elements
of \mbox{$^{\scriptscriptstyle(h)}\!{\cal B}$}\ we just write $\varphi^{}_a$ with $a\!\in\!\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}$, and we use the notation \mbox{\rm S}\
and \mbox{\rm s}\ for the $S$-matrices in place of \mbox{$^{\scriptscriptstyle(\el h)}\!S$}\ and \mbox{$^{\scriptscriptstyle(h)}\!S$}, respectively.
The mapping is then defined on the preferred basis $\mbox{$^{\scriptscriptstyle(\el h)}\!{\cal B}$}$ as
\begin{equation} \ff{\el h}h(\phi^{}_A) = \sumph b \P Ab\, \varphi^{}_b \, \labl3
with
\begin{equation} \P Ab \equiv {}^{\scriptscriptstyle(\ell h,h)\!\!}_{}\P Ab := \epsilon_{\el}(A)\,
\delta^{}_{w_A(A),b} \,, \labl P
and extended linearly to all of \mbox{$^{\scriptscriptstyle(\el h)\!}{\cal R}$}. As has been established in
\cite{fusS2}, the matrix \erf P satisfies the relations
\futnote{In \cite{fusS2}, mappings of the type \erf' were encountered as
so-called quasi-Galois scalings. In that setting, the level of the WZW theory\ is
not changed, while the weights $A$ are scaled by a factor of $\ell$, followed by
an appropriate affine Weyl transformation to bring the weight $\ell A$ back to
the Weyl chamber \mbox{$_{}^{\scriptscriptstyle(\el h)\!\!}P$}\ or to its boundary. Since what matters is only the
relative
`size' of weights and the translation part of the Weyl group, these mappings
are effectively the same as in the present setting where there is no scaling
of the weights but the extension from \mbox{$^{\ssslh\!}W$}\ to \mbox{$^{\sssh\!}W$}\ scales the translation
lattice down by a factor of $\ell$.
\\[.2em]Note that in \cite{fusS2} the letter $P$ was used for the matrix \erf P
in place of $F$, and $D$ was defined as the transpose of the matrix \erf D.}
\begin{equation} \mbox{\rm S}\,F = \ell^{\,r/2}_{}\,D\,\mbox{\rm s} \,, \qquad
F\,\mbox{\rm s} = \ell^{\,r/2}_{}\,\mbox{\rm S}\,D \,, \labl{sS}
with
\begin{equation} \D Ab \equiv {}^{\scriptscriptstyle(\ell h,h)\!\!}_{}\D Ab:= \delta_{A,\ell b}\,. \labl D
Furthermore, from the Kac$\mbox{-\hspace{-.66 mm}-}$ Peterson formula \cite{KAc3} for the modular
matrix $S$, one deduces the identity
\begin{equation} \sm ab = \ell^{\,r/2}_{}\, \Sm{\ell a}b \labl K
for all $a,b\!\in\!\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}$.
Combining the relations \erf P -- \erf K and the
Verlinde formula \cite{verl2}, we obtain for any pair $A,B\!\in\!\mbox{$_{}^{\scriptscriptstyle(\el h)\!\!}P$}$
\begin{equation} \begin{array}{l} \!\!
\ff{\el h}h(\phi^{}_A \star \phi^{}_B) = \sumplh C \nlh ABC\, \ff{\el h}h(\phi^{}_C)
= \sumplh{C,D}\;\sumph e \displaystyle\frac {\Sm AD^{} \Sm BD^{} \Sm CD^*}{\Sm\rho D}
\, \P Ce\, \varphi^{}_e
\\{}\\[-2.6mm] \hsp{23.2}
= \!\sumplh{D}\;\sumph e \displaystyle\frac {\Sm AD \Sm BD (\mbox{\rm S}^*\!F)_{D,e}}{\Sm\rho D}
\, \varphi^{}_e
= \ell^{\,r/2}_{}\cdot \!\!\!\sumplh{D}\;\sumph e \displaystyle\frac {\Sm AD^{} \Sm BD^{}
(D\mbox{\rm s}^*)_{D,e}}{\Sm\rho D} \, \varphi^{}_e
\\{}\\[-2.6mm] \hsp{23.2}
= \ell^{\,r/2}_{}\cdot \sumph{d,e} \displaystyle\frac {\Sm A{\ell d}^{} \Sm B{\ell d}^{}
\,\sm de^*} {\Sm \rho{\ell d}}\, \varphi^{}_e
= \ell^{\,r}_{}\cdot \sumph{d,e} \displaystyle\frac {\Sm A{\ell d}^{} \Sm B{\ell d}^{}
\,\sm de^*} {\sm \rho d}\, \varphi^{}_e
\\{}\\[-2.6mm] \hsp{23.2}
= \ell^{\,r}_{}\cdot \sumph{d,e} \displaystyle\frac {(\SMD)_{A,d}^{} (\SMD)_{B,d}^{} \,
\sm de^*} {\sm \rho d}\, \varphi^{}_e
= \sumph{d,e} \displaystyle\frac {(F\mbox{\rm s})_{A,d}^{} (F\mbox{\rm s})_{B,d}^{} \,\sm de^*}
{\sm \rho d}\, \varphi^{}_e
\\{}\\[-2.6mm] \hsp{23.2}
= \sumph{a,b,d,e} \P Aa \P Bb\, \displaystyle\frac {\sm ad^{} \,\sm bd^{} \,\sm ed^*}
{\sm \rho d}\, \varphi^{}_e
= \sumph{a,b,c} \P Aa \P Bb\, \nh abc \, \varphi^{}_c
\\{}\\[-2.2mm] \hsp{23.2}
= \sumph{a,b} \P Aa \P Bb\, \varphi^{}_a \star \varphi^{}_b
= \ff{\el h}h(\phi_A) \star \ff{\el h}h(\phi_B)
\end{array} \end{equation}
Thus $\ff{\el h}h$ is indeed a homomorphism.
As a side remark, let us mention that an analogous situation arises for the
conformal field theories\ which describe a free boson compactified on a circle of rational radius
squared. These theories are labelled by
an (even) positive integer $h$, and for each value of $h$ the
fusion ring is just the group ring ${\dl Z}\zet_h$ of the abelian group
${\dl Z}_h={\dl Z}/h{\dl Z}$. The modular $S$-matrix\ is given by
$\mbox{$^{\scriptscriptstyle(h)}\!S$}_{p,q} = h^{-1/2} \exp(2\pi{\rm i} p q /h)$, where the labels $p$ and $q$
which correspond to the primary fields are integers
which are conveniently considered as defined modulo $h$. It is straightforward
to check that the identities \erf{sS} are again valid (with $r$ set to 1)
if one defines ${}^{\scriptscriptstyle(\ell h,h)\!}_{}\P Ab:=\delta^{(h)}_{A,b}$ and
${}^{\scriptscriptstyle(\ell h,h)\!}_{}\D Ab:=\delta^{(\ell h)}_{A,\ell b}$, where the
superscript on the $\delta$-symbol $\delta^{(m)}_{a,b}$ indicates that
equality needs to hold only modulo $m$. As a consequence, this way we obtain
again a projective system based on the divisibility of $h$ (the composition
property is immediate). Moreover, precisely as in the case of WZW theories,
with a different partial ordering of
the set $\{h\}=\mbox{${\zet}_{>0}$}$ it is not possible to define a projective system.
\subsection{Proof of the composition property}
Finally, the composition property \erf1 of the homomorphisms \erf' is
equivalent to the relation
\begin{equation} \sumplh B
{}^{\scriptscriptstyle(\ell\el'h,\ell h)\!}_{}\P {\sf A}B \,
{}^{\scriptscriptstyle(\ell h,h)\!}_{}\P Bc =
{}^{\scriptscriptstyle(\ell\el'h,h)\!}_{}\P {\sf A}c \labl{ppp}
among the projection matrices $F$ that involve the three different
heights $h$, $\ell h$ and $\ell\el'h$.
Here as before the elements of \mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}\ and \mbox{$_{}^{\scriptscriptstyle(\el h)\!\!}P$}\ are denoted by lower
case and capital letters, respectively, while for the elements of \mbox{$_{}^{\scriptscriptstyle(\el\el'h)\!\!}P$}\
we use sans-serif font. The relation \erf{ppp} is in fact an immediate
consequence of the definition of
the homomorphisms \ff ij. The explicit proof is not very illuminating;
the reader who wishes to skip it should proceed directly to \secref{pl}.
To prove \erf{ppp}, let us first assume that the left hand side does not
vanish. Then there exist unique Weyl transformations $\overline w_1,\overline w_2
\!\in\!\overline W$ and unique vectors $\beta_1,\beta_2\!\in\! h\mbox{$\overline L^{\scriptscriptstyle\vee}$}$ in the
coroot lattice scaled by $h$, and a unique weight $B\!\in\!\mbox{$_{}^{\scriptscriptstyle(\el h)\!\!}P$}$, such that
\begin{equation} \overline w_1({\sf A}) + \ell\beta_1 = B \,, \qquad
\overline w_2(B) +\beta_2 = c \,, \labl{ww}
and the left hand side of \erf{ppp} takes the value
\begin{equation} \epsilon_{\el\el'}({\sf A})\,\epsilon_{\el}(B) = \mbox{sign}(\overline w_1)\,\mbox{sign}(\overline w_2)
= \mbox{sign}(\overline w_1\overline w_2) \,. \labl{sww}
By combining the two relations \erf{ww}, it follows that
\begin{equation} \overline w_2\overline w_1({\sf A}) + \beta = c \,, \labl{beta}
where $\beta=\ell\,\overline w_2(\beta_1)+\beta_2$. Since $\beta$ is again an
element of $h\mbox{$\overline L^{\scriptscriptstyle\vee}$}$, this means that \erf{beta} describes, up to sign, the
mapping corresponding to the right hand side of \erf{ppp}. Further, the
sign of the right hand side is then given by $\mbox{sign}(\overline w_2\overline w_1)$
and hence equal to \erf{sww}; thus \erf{ppp} indeed holds.
We still have to analyze \erf{ppp} when its left hand side vanishes.
Then either the $\sf A$\,th row of ${}^{\scriptscriptstyle(\ell\el'h,\ell h)\!}_{}F$
or the $c$\,th column of ${}^{\scriptscriptstyle(\ell h,h)\!}_{}F$ must be zero.
In the former case, the weight ${\sf A}\!\in\!\mbox{$_{}^{\scriptscriptstyle(\el\el'h)\!\!}P$}$ belongs to the
boundary of some Weyl chamber with respect to \mbox{$^{\ssslh\!}W$}, and thus there exist $\overline w\!\in\!\overline W$
and $\beta\!\in\! h\mbox{$\overline L^{\scriptscriptstyle\vee}$}$ such that $\overline w({\sf A})+\ell\beta={\sf A}$.
But this means that ${\sf A}\!\in\!\mbox{$_{}^{\scriptscriptstyle(\el\el'h)\!\!}P$}$ also lies on the
boundary of some Weyl chamber with respect to $\mbox{$^{\sssh\!}W$}\supset\mbox{$^{\ssslh\!}W$}$, and hence also
the right hand side of \erf{ppp} vanishes as required.
In the second case, there are unique elements $w_1\!\in\!\mbox{$^{\ssslh\!}W$}$ and $w_2\!\in\!\mbox{$^{\sssh\!}W$}$
satisfying $w_1({\sf A})=B$ and $w_2(B)=B$.
Because of $\mbox{$^{\ssslh\!}W$}\subset\mbox{$^{\sssh\!}W$}$, $w_1$
can also be considered as an element of the Weyl group \mbox{$^{\sssh\!}W$}\ at height $h$.
By assumption, $w_2$ is a non-trivial element of \mbox{$^{\sssh\!}W$}. The product
$w_0:=w_1^{-1}w_2^{}w_1^{}\!\in\!\mbox{$^{\sssh\!}W$}$ is then non-trivial, too, and satisfies
\begin{equation} w_0({\sf A}) =w_1^{-1}w_2^{}w_1^{}({\sf A})
=w_1^{-1}w_2^{}(B)=w_1^{-1}(B)={\sf A} \,. \end{equation}
Thus the weight $\sf A$ is invariant under a non-trivial element of \mbox{$^{\sssh\!}W$}\
and hence lies on the boundary of some Weyl chamber with respect to \mbox{$^{\sssh\!}W$}; this implies
again that the right hand side of \erf{ppp} vanishes as required.
This concludes the proof of \erf{ppp}, and hence of the claim that
$(\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$})_{h\in\II}$ together with the maps \ff ij constitutes a projective system.
\Sect{The projective limit \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}}{pl}
We are now in a position to construct the projective limit \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ of
the projective system that we introduced in subsection \ref{s.ps}.
\subsection{Projective limits and coherent sequences}
A projective system $(\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$})_{h\in I}$\, in some category \mbox{${\cal C}$}\ is said to
possess a {\em projective limit\/} $(\mbox{$\cal L$},f)$ (also called
the inverse limit, or simply the limit) if there exist an object \mbox{$\cal L$}\
as well as a family $f$ of morphisms $\ef h\!:\ \mbox{$\cal L$}\to\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}$ (for all $h\!\in\!\mbox{$I$}$)
which satisfy the following requirements (see e.g.\ \cite{ENCY}).
First, for all $h,h'\!\in\!\mbox{$I$}$ with $h\preceq h'$ the diagram
\begin{equation} \begin{array}{l} {}\\[-2.2em]
\pictriangleupddr {\mbox{$\cal L$}}{\mbox{$^{\scriptscriptstyle(h')\!}{\cal R}$}~~}{\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}}{\eF {h'}}{\eF h}{\fF{h'}h}
\\[.3em] \end{array} \labl I
commutes; and second, the following {\em universal property\/} holds:
for any object \mbox{$\cal O$}\ of the category for which a family of morphisms
$\,\eg h\!:\;\mbox{$\cal O$}\to\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\,$ ($h\!\in\!\mbox{$I$}$) exists which possesses
a property analogous to \erf I, i.e.\
\begin{equation} \ff {h'}h \circ \eg{h'} = \eg h \quad {\rm for}\ \, h\preceq h' \, , \end{equation}
there exists a unique morphism $\,g\!:\;\mbox{$\cal O$}\to\mbox{$\cal L$}$ such that the diagram
\begin{equation} \begin{array}{l} \begin{picture}(120,170)
\put(0,90){ \pictriangleupddr {\mbox{$\cal L$}}{\mbox{$^{\scriptscriptstyle(h')\!}{\cal R}$}~~}{\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}}{\eF{h'}}{\eF h}{\fF{h'}h} }
\put(0,18){ \pictriangledowndd {\,\mbox{$\cal O$}}{\eG{h'}}{\eG h} }
\put(113,92){\begin{picture}(0,0) \put(0,0){\oval(30,142)[r]} \put(19.8,3.5){$\scsg$}
\put(0,71){\vector(-1,0){45}} \put(0,-71){\line(-1,0){45}} \end{picture}}
\end{picture} \\[-1.8em] \end{array} \labl{II}
commutes for all $h,h'\!\in\!\mbox{$I$}$ with $h\preceq h'$.
To be precise, in the above characterization of the projective limit $(\mbox{$\cal L$},f)$
it is implicitly assumed that \mbox{$\cal L$}\ is an object in \mbox{${\cal C}$}\ and that the \ef i are
morphisms of \mbox{${\cal C}$}. But in fact such an object and such morphisms need not exist.
In the general case one must rather employ a definition of the projective
limit as a certain functor from the category \mbox{${\cal C}$}\ to the category of sets,
and then the question arises whether this functor is `representable' through
an object \mbox{$\cal L$}\ and morphisms \ef i as described above. In this language
the crucial issue is the existence of a representing object \mbox{$\cal L$}\
(see e.g.\ \cite{ARti,PAre,HIst}).
\futnot{PAre': sec. 2.5; HIst: chap. VIII.5}
Now one and the same projective system can frequently
be regarded as part of various different categories.
For instance, when describing the projective system of our interest one can
restrict oneself to the category ${\cal F}\!us\mbox{\scriptsize(}\mbox{$\mathfrak g$}\mbox{\scriptsize)}$. As we will see, when doing so
a projective limit of the projective system does not exist. But
one can also consider it, say, in the category of commutative rings, or in the
still bigger category of vector spaces, or even in the category of
sets. The existence and the precise form of the projective limit usually
depend on the choice of category.
In our case, however, the category \mbox{${\cal C}$} \,=\,${\cal F}\!us\mbox{\scriptsize(}\mbox{$\mathfrak g$}\mbox{\scriptsize)}$\ we start with
is small, i.e.\ its objects are sets, and as a consequence
there exists a natural construction by which the object \mbox{$\cal L$}\ and
the morphisms \ef i can be obtained in a concrete manner (in particular, \mbox{$\cal L$}\
is again a set). Moreover, it turns out that the projective limit we obtain in
the category of sets is exactly the same as the limit
that we obtain in the category of commutative rings or vector spaces,
which also indicates that this way of performing the limit is a quite natural.
This construction proceeds as follows.
Given a projective system of objects \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\ and morphisms \ff{h'}h of a
small category \mbox{${\cal C}$},
one regards the objects $\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\!\in\!\mbox{${\cal C}$}$ as sets and considers the infinite direct
product $\prod_{h\inI}\!\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}$ of all objects of \mbox{${\cal C}$}.
The elements of this set are those maps
\begin{equation} \psi:\ \; \mbox{$I$} \;\to\; \bigcup_{h\inI}^. \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$} \labl{psi}
from the index set \mbox{$I$}\ to the disjoint union of all objects \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\
which obey $\psi(h)\!\in\!\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}$ for all $h\!\in\!\mbox{$I$}$;
they are sometimes called `generalized sequences' (ordinary sequences can be
formulated in this language by considering the index set ${\mathbb N}$ with the
standard ordering $\le$\,). The subset $\mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$} \subset\prod_{h\inI}\!\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}$
consisting of {\em coherent sequences}, i.e.\ of those generalized sequences
for which
\begin{equation} \ff {h'}h \circ \psi(h') = \psi(h) \labl j
for all $h,h'\!\in\!\mbox{$I$}$ with $h\preceq h'$, is isomorphic to the projective limit.
More precisely, \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ is isomorphic to \mbox{$\cal L$}\ as a set, and the morphisms
$\ef h$ are the projections to the components, i.e.\
\begin{equation} \ef h(\psi):=\psi(h) \,. \end{equation}
For the projective system introduced in subsection \ref{s.ps} where \mbox{${\cal C}$}\ is
the (small) category ${\cal F}\!us\mbox{\scriptsize(}\mbox{$\mathfrak g$}\mbox{\scriptsize)}$, the projective limit is clearly {\em not\/}
contained in the original category, because no object \mbox{$^{\scriptscriptstyle(i)\!}{\cal R}$}\ of ${\cal F}\!us\mbox{\scriptsize(}\mbox{$\mathfrak g$}\mbox{\scriptsize)}$\
can possess morphisms to {\em all\/} objects \mbox{$^{\scriptscriptstyle(j)\!}{\cal R}$}. In order to identify
nevertheless a projective limit associated to the projective system
defined by \erf', it is therefore necessary to consider
the set \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ of coherent sequences. In accordance with the remarks above,
for definiteness from now on we will simply refer to \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ as `the'
projective limit of the system \erf1 of WZW fusion rings.
\subsection{Properties of \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}}
Let us list a few simple properties of the projective limit \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}. First, \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\
is a ring over ${\dl Z}$. The product $\psi_1\star \psi_2$ in \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ is defined
pointwise, i.e.\ by the requirement that
\begin{equation} (\psi_1\star \psi_2)(h):= \psi_1(h)\star \psi_2(h) \labl{pcp}
for all $h\iN\II$. This definition makes sense, i.e.\
for all $\psi_1,\psi_2\!\in\!\mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}$ also their product is in \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}, because
\begin{equation} \begin{array}{ll}
\ff{h'}h\circ(\psi_1\star \psi_2)(h') \!\!&=
(\ff{h'}h\circ\psi_1(h'))\star (\ff{h'}h\circ\psi_2(h')) \\[1.9mm]&
= \psi_1(h) \star \psi_2(h) = (\psi_1\star \psi_2)(h) \,; \end{array}\end{equation}
here in the first line the morphism property of the maps $\ff{h'}h$ is used.
{}From the definition \erf{pcp} it is clear that the product of \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ is
commutative and associative, and that \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ is unital, with the unit element
being the element $\psi_\circ\!\in\!\mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}$ that satisfies
\begin{equation} \psi_\circ(h)={}^{\sssh\!}\varphi^{}_{\!\rho} \end{equation}
for all $h\iN\II$.
Second, a conjugation $\psi\mapsto \psi^+$ can be defined on \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ by setting
\begin{equation} \psi^+(h):=(\psi(h))^+ \end{equation}
for all $h\iN\II$. The conjugation ${}^{\sssh\!}\varphi^{}\mapsto({}^{\sssh\!}\varphi)^+$ on the rings \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\
commutes with the projections $\ff{h'}h$. As a consequence, indeed
$\psi^+\!\in\!\mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}$ whenever $\psi\!\in\!\mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}$, conjugation is an involutive
au\-to\-morphism of \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}, and the unit element $\psi_\circ$ is self-conjugate.
In \secref{binf} we will construct a countable basis \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\ of the ring \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$};
this basis contains in particular the unit element $\psi_\circ$.
For any $\psi\!\in\!\mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}$ and any $h\iN\II$, $\psi(h)$ is either zero or, up to
possibly a sign, an element of the basis \mbox{$^{\scriptscriptstyle(h)}\!{\cal B}$}\ of \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}.
Also, while by construction the structure constants in the basis \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\
are integers, there seems to be no reason why they should be non-negative.
Accordingly, an interpretation of the limit \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\
as the representation\ ring of some underlying algebra ic structure is even less obvious
than in the case of the fusion rings \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}.\,%
\futnote{The latter can e.g.\ be regarded as the representation\ rings of the `quantum
symmetry' of the associated WZW theories. However, so far there is no agreement on
the precise nature of those quantum symmetries.}
In particular, in \secref{gb} we will see that \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ does not coincide
with the representation\ ring \mbox{$\overline{\mbox{${\cal R}$}}$}\ of the simple Lie algebra\ $\mbox{$\bar{\mathfrak g}$}\subset\mbox{$\mathfrak g$}$, but rather
that it contains \mbox{$\overline{\mbox{${\cal R}$}}$}\ as a tiny proper subring.
As it turns out, the fusion product of two elements of \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\ is generically
{\em not\/} a finite linear combination of elements of \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}, or in other
words, \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\ does not constitute an ordinary basis of \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}. Rather, it must
be regarded as a topological basis. For this interpretation to make sense,
a suitable topology on \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ must be defined. This will be achieved in
the next subsection.
\subsection{The \topo\ of \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}}\label{topo}
The fusion rings \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\ can be considered as topological spaces by simply
endowing them with the discrete topology, i.e.\ by declaring every subset to be
open (and hence also every subset to be closed). The projective limit \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\
then becomes a topological space in a natural manner, namely by defining its
topology as the coarsest topology in which all projections $\ef h\!:\;
\mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\to\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}$ are continuous; this will be called the {\em\topo\/} on \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}.
\futnot{This is the natural adaptation of the product topology on the
direct product $\prod_{h\inI}\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}$.}
More explicitly, the \topo\ on \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ is described by the property that
any open set in \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ is an (arbitrary, i.e.\ not necessarily finite nor
even countable) union of elements of
\begin{equation} \mbox{$\Omega$}:= \{ \efm h(M) \mid h\!\in\!\mbox{$I$},\; M\subseteq\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$} \} \,, \labl{Om}
i.e.\ of the set of all pre-images of all sets in any of the fusion rings \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}.\\
Note that we need not require to take also finite intersections of these
pre-images. This is because \mbox{$\Omega$}\ is closed under taking finite intersections,
as can be seen as follows. Let $\omega_i\!\in\!\mbox{$\Omega$}$ for $i=1,2,...\,,N$; by definition,
each of the $\omega_i$ can be written as $\omega_i=\efm{h_i}(M_i)$ for some heights
$h_i\iN\II$ and some subsets $M_i\subseteq\mbox{$^{\scriptscriptstyle(h_i)\!}{\cal R}$}$. Denote then by $h$ the smallest
common multiple of the $h_i$ for $i=1,2,...\,,N$. Because of \erf I we have
$\ef{h_i}=\ff h{h_i}\!\circ\!\ef h$, so that
\begin{equation} \efm{h_i}(M_i)=\efm h(\ffm h{h_i}(M_i))=\efm h(\tilde M_i) \,, \end{equation}
where for all $i=1,2,...\,,N$ the sets $\tilde M_i:=\ffm h{h_i}(M_i)$
are subsets of \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}.
Because of $\bigcap_{i=1}^N\tilde M_i\subseteq \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}$, it thus
follows that
\begin{equation} \bigcap_{i=1}^N \omega_i =\bigcap_{i=1}^N \efm h(\tilde M_i)
= \efm h\mbox{\large[}\bigcap_{i=1}^N \tilde M_i\mbox{\large]} \end{equation}
is an element of the set \mbox{$\Omega$}\ \erf{Om}. Thus \mbox{$\Omega$}\ is closed under taking
finite intersections, as claimed.\\
As a consequence of this property of \mbox{$\Omega$}, in particular any non-empty open
set in \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ contains a subset which is of the form $\efm h(M)$ for some
$h\iN\II$ and some $M\subseteq\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}$; for later reference, we call this fact
the `pre-image property' of the non-empty open sets in \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}.
Note that the \topo\ on \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ is {\em not\/} the discrete one, but finer.
To see this, suppose the \topo\ were the discrete one. Then for any $\psi\!\in\!
\mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}$ the one-element set $\{\psi\}$ would be open and hence a
union of sets in $\Omega$ \erf{Om}; but as $\{\psi\}$ just contains one single
element, this means that it even has to belong itself to $\Omega$.
This in turn means that there would exist $h\iN\II$ and $M\subseteq\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}$ such that
$\{\psi\}=f_h^{-1}(M)$, and hence simply $M=\{\psi(h)\}$. This, however, would
imply that each element $\psi\!\in\!\mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}$ would already be determined uniquely by
the value $\psi(h)$ for a single height $h$. {}From the explicit description
of \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ as a space of coherent generalized sequences,
it follows that this is definitely not true. Thus the
assumption that the \topo\ is the discrete one leads to a contradiction.
Whenever two elements $\psi,\psi'\!\in\!\mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}$ are distinct, there exists some
height $h\iN\II$ such that $\psi(h)\ne\psi'(h)$. The open subsets
$\omega:=f_h^{-1}(\{\psi(h)\})$ and $\omega':=f_h^{-1}(\{\psi'(h)\})$ then
satisfy $\psi\!\in\!\omega$ and $\psi'\!\in\!\omega'$ as well as
$\omega\cap\omega'=\emptyset$. This means that when endowed with the \topo,
\mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ is a Hausdorff space.
\Sect{A distinguished basis \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\ of \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}}{binf}
In this section we construct a (topological)
basis \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\ of the projective limit \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ of WZW fusion rings.
\subsection{A linearly independent subset of \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}}\label{s1}
We start by defining the subset $\mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\subset\mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}$ as the set of all those
elements $\psi\!\in\!\mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}$ which for every $h\iN\II$ satisfy
\begin{equation} \psi(h)= \epsilon_h\cdot{}^{\sssh\!}\varphi^{}_a \labl{bd}
for some
\begin{equation} a\!\in\!\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$} \qquad{\rm and}\qquad \epsilon_h\!\in\!\{0,\pm1\} \end{equation}
(i.e.\ for each height
$h$ the fusion ring element $\psi(h)\!\in\!\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}$ is either zero or, up to a sign, an
element of the distinguished basis \mbox{$^{\scriptscriptstyle(h)}\!{\cal B}$}), and for which in addition not all of
the prefactors $\epsilon_h$ vanish and $\epsilon_h=1$ for the smallest $h\iN\II$ for which
$\epsilon_h\ne0$. The latter requirement ensures that $-\psi\not\in\mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}$ for
all $\psi\!\in\!\mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}$.
Note that at this point we cannot tell yet whether the set \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\ is large
enough to generate the whole ring \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}; in fact, it is even not yet clear
whether \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\ is non-empty. These issues will be dealt with in subsections
\ref{s2} to \ref{sinw} below, where we will in particular see that the set
\mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\ is countably infinite. However, what we already can see is
that the set \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\ is linearly independent. To prove this, consider any set
of finitely many distinct elements $\psi_i$, $i=1,2,...\,,N$ of \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}.
We first show that to any pair $i,j\in\{1,2,...\,,N\}$ there exists a height
$h_{ij}\iN\II$ such that\\[.2em]
\mbox{~~~~~}(i)~~$\psi_i(h_{ij})\ne 0$ \,and\, $\psi_j(h_{ij})\ne 0$ \,\ \
and\\[.2em]
\mbox{~~~~~}(ii)~$\psi_i(h_{ij})\ne \pm\psi_j(h_{ij})$\,.\\[.2em]
To see this, assume that the statement is wrong, i.e.\ that for each height
$h$ either one of the elements $\psi_i(h)$ and $\psi_j(h)$ of \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\
vanishes, or one has $\psi_i(h)= \pm\psi_j(h)$. Now because of $\psi_i\ne0$
and $\psi_j\ne0$ there exists heights $h_i$ and $h_j$ with $\psi_i(h_i)\ne0$
and $\psi_j(h_j)\ne0$. This implies that also $\psi_i(\tilde h_{ij})\ne 0$
and $\psi_j(\tilde h_{ij})\ne 0$ for $\tilde h_{ij}:=h_ih_j$.
By our assumption it then follows that $\psi_j(\tilde h_{ij})= \pm
\psi_i(\tilde h_{ij})$, which in turn implies that
$\psi_j(h_i)= \pm \psi_i(h_i)\ne0$.
Now this conclusion actually extends to arbitrary heights $h$.
Namely, from the previous result we know that for any $h$ the elements
$\psi_i(h\tilde h_{ij})$ and $\psi_j(h\tilde h_{ij})$ must both be non-zero.
By our assumption this implies that $\psi_j(h \tilde h_{ij})=\pm\psi_i(h \tilde
h_{ij})$. Projecting this equation down to the height $h$, it follows that
$\psi_j(h)=\pm\psi_i(h)$. Since $h$ was arbitrary,
it follows that in fact $\psi_j= \pm\psi_i$,
and hence (as $-\psi_i$ is not in \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}) that $\psi_j=\psi_i$. This is in
contradiction to the requirement that all $\psi_i$ should be distinct.
Thus our assumption must be wrong, which proves that (i) and (ii) are fulfilled.
Applying now the properties (i) and (ii) for any pair $i,j\in\{1,2,...\,,N\}$
with $i\ne j$, it follows that at the height $h:=\prod_{i,j;i<j}h_{ij}$
\futnot{It is already sufficient to take $h$ as the s.c.m.\ of the $h_{ij}$.}
we have\\[.2em]
\mbox{~~~~~}(i)~~$\psi_i(h)\ne 0$ \,for all\, $i=1,2,...\,,N$ \,\ \ and\\[.2em]
\mbox{~~~~~}(ii)~$\psi_i(h)\ne \pm\psi_j(h)$ \,for all $i,j\in\{1,2,...\,,N\}$,
$i\ne j$\,.\\[.2em]
Thus all the elements $\psi_i(h)$ of \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\ are distinct and, up to sign,
elements of the distinguished basis \mbox{$^{\scriptscriptstyle(h)}\!{\cal B}$}. This implies in particular that
the only solution of the equation $\sum_{i=1}^N\xi_i\psi_i(h)=0$ is
$\xi_i=0$ for $i=1,2,...\,,N$, which in turn shows that also the equation
$\sum_{i=1}^N\xi_i\psi_i=0$ has only this solution.
Thus, as claimed, the $\psi_i$ are linearly independent elements of \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}.
\subsection{\mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\ generates all of \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}}\label{s2}
Next we claim that the set \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\ spans \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ in the sense
that the closure (in the \topo) of the linear span of \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\ in \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}, i.e.\
of the set
\begin{equation} \mbox{$\langle^{\scriptscriptstyle(\infty)\!\!}{\cal B}\rangle$} \equiv {\rm span}_{\dl Z}^{}(\mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}) \end{equation}
of finite ${\dl Z}$-linear combinations of elements of \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}, is already all of
\mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}.
To prove this, assume that the statement is wrong, or in other words, that the
set
\begin{equation} {\cal S} := \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\,\setminus\,\mbox{$\overline{\langle^{\scriptscriptstyle(\infty)\!\!}{\cal B \labl{mw}
is non-empty. By definition, the set ${\cal S}$ is open, and hence because of the
pre-image property\ it contains a subset ${\cal M}\subseteq{\cal S}$ of the form ${\cal M}=\efm h(M)$
for some $h\iN\II$ and some $M\subseteq\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}$. Further,
as an immediate consequence of the construction that we will present
in the subsections \ref{siw} and \ref{sinw}, for each $a\!\in\!\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}$ there exists
an element (in fact, infinitely many elements) $\psi_a\!\in\!\mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}$ such that
$\ef h(\psi_a)={}^{\sssh\!}\varphi^{}_a$ (namely, we need to prescribe the value of
$\psi_a(p)$ only for the finitely many prime factors $p$ of $h$).
Now choose some $y\!\in\! M$, decompose it with respect to the basis \mbox{$^{\scriptscriptstyle(h)}\!{\cal B}$}\ of \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}, i.e.
$y=\sumpH a n_a\,{}^{\sssh\!}\varphi^{}_a$ with $n_a\!\in\!{\dl Z}$, and define
$\eta:=\sumpH a n_a\,\psi_a$. Then, on one hand, by construction we have
$\ef h(\eta)=y$, i.e.\ $\eta\!\in\!{\cal M}$, and hence $\eta\in{\cal S}$, while on the other
hand $\eta$ is a finite linear combination of elements of \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\
(since $y$ is a finite linear combination of elements of \mbox{$^{\scriptscriptstyle(h)}\!{\cal B}$}), and hence
$\eta\in\mbox{$\langle^{\scriptscriptstyle(\infty)\!\!}{\cal B}\rangle$}\subseteq\mbox{$\overline{\langle^{\scriptscriptstyle(\infty)\!\!}{\cal B$. By the definition \erf{mw} of ${\cal S}$, this
is a contradiction, and hence our assumption must be wrong.
Together with the result of the previous subsection we thus see that
\mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\ is a (topological) basis of \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}.
\subsection{Distinguished sequences of integral weights}\label{siw}
We will now construct all elements of the projective limit \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\
which belong to the subset \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\ as introduced in subsection \ref{s1}.
These are obtained as generalized sequences $\psi$ satisfying both \erf j
and the defining relation \erf{bd} of \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}.
More specifically, we construct sequences ${({a_h^{}})}_{h\in\II}$ of labels
${a_h^{}}\!\in\!\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}$ and associated signs $\eta({a_h^{}})$ such that
all those $\psi$ which are of the form
\begin{equation} \psi(h)=\eta({a_h^{}})\,{}^{\sssh\!}\varphi^{}_{{a_h^{}}} \labl Q
belong to the subset $\mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\subset\mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}$. When applied to \erf Q,
the requirement \erf j amounts to
\begin{equation} \ff{\el h}h({}^{\ssslh\!}\varphi^{}_{a_{\ell h}^{}}) = \eta({a_{\ell h}^{}})\eta({a_h^{}})\cdot{}^{\sssh\!}\varphi^{}_{a_h^{}} \,, \labl k
which in view of the definition \erf' of \ff{\el h}h\ is equivalent to
\begin{equation} {a_{\ell h}^{}}=w({a_h^{}}) \qquad{\rm for\ some}\ w\!\in\!\mbox{$^{\sssh\!}W$} \labl m
and
\begin{equation} \eta({a_{\ell h}^{}})\eta({a_h^{}})=\epsilon_{\el}({a_{\ell h}^{}}) \,. \labl M
To start the construction of the elements of \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}, we first concentrate our
attention to integral weights of the height $h$ theory which are not
necessarily integrable and which are considered as defined only modulo
$h\mbox{$\overline L^{\scriptscriptstyle\vee}$}$; we denote these weights by ${b_h^{}}$. Suppose then that we prescribe for
each prime $p$ such a weight ${b_p^{}}$ and that these weights satisfy in
addition the restriction that for any two primes $p,\,p'$ they differ by
an element of the coroot lattice,
\begin{equation} {b_p^{}}-{b_{p'}^{}} \in \mbox{$\overline L^{\scriptscriptstyle\vee}$} \,. \labl A
We claim that there then exists a sequence ${({b_h^{}})}_{h\in\II}$ which for
prime heights takes the prescribed values ${b_p^{}}$ and for which the relation
\begin{equation} {b_{h'}^{}} = {b_h^{}} \ {\rm mod}\; h\mbox{$\overline L^{\scriptscriptstyle\vee}$} \labl F
holds for all $h,\,h'$ with $h\preceq h'$.
To prove this assertion, we display such a sequence explicitly. To this end, let
\begin{equation} h=:\prod_{\scriptstyle j\atop \scriptstyle p_j|h}{p_j^{n_j}} \labl G
denote the decomposition of $h$ into prime factors, and define
\begin{equation} {h_i}:=\frac{h}{p_i^{n_i}} \labl H
and
\begin{equation} \invmod\hi{p_i^{n_i}}:= ({h_i})^{-1}_{} \ {\rm mod}\; {p_i^{n_i}} \,. \labl J
Then we set
\begin{equation} {b_h^{}} := {b_{p_1^{}}^{}} + \sumieh {h_i}\,\invmod\hi{p_i^{n_i}}\,({b_{p_i^{}}^{}}-{b_{p_1^{}}^{}}) \,. \labl{bh}
Recall that ${b_h^{}}$ is defined only modulo $h\mbox{$\overline L^{\scriptscriptstyle\vee}$}$.
In \erf{bh} $p_1$ is any of the prime divisors of $h$; it has been singled out
only in order to make the formula for ${b_h^{}}$ to look as simple as possible, and
in fact ${b_h^{}}$ does not depend (modulo $h\mbox{$\overline L^{\scriptscriptstyle\vee}$}$) on the choice of $p_1$.
To see this, let ${b_h^{\scriptscriptstyle(2)}}$ denote the number obtained analogously as in
\erf{bh}, but with $p_1$ replaced by some other prime factor $p_2$ of $h$. Then
\begin{equation} \begin{array}{l} {b_h^{}}-{b_h^{\scriptscriptstyle(2)}}={b_{p_1^{}}^{}}-{b_{p_2^{}}^{}} + \sumiezh {h_i}\,\invmod\hi{p_i^{n_i}}\,({b_{p_2^{}}^{}}-{b_{p_1^{}}^{}})
\\[1.5mm] \hsp{19.1} + {h_2}\,\invmod\hz{p_2^{n_2}}\,({b_{p_2^{}}^{}}-{b_{p_1^{}}^{}}) - {h_1}\,\invmod\he{p_1^{n_1}}\,({b_{p_1^{}}^{}}-{b_{p_2^{}}^{}})
\,. \end{array}\end{equation}
Using the fact that ${h_i}$ is divisible by ${p_j^{n_j}}$ for all primes $p_j$
dividing $h$ except for $j=i$, and that ${h_i}\invmod\hi{p_i^{n_i}}=1\ {\rm mod}\;{p_i^{n_i}}$,
it is easily checked that the right hand side of this expression
vanishes modulo ${p_j^{n_j}}\mbox{$\overline L^{\scriptscriptstyle\vee}$}$ for all $p_j$ dividing $h$, and hence, using \erf A,
also vanishes modulo $h\mbox{$\overline L^{\scriptscriptstyle\vee}$}$.
To establish the coherence property \erf F, we now consider
two heights $h,\,h'$ such that $h|h'$. Then we set
\begin{equation} h'=:\prod_{\scriptstyle j\atop \scriptstyle p_j|h'}{p_j^{n'_j}} \end{equation}
and ${h_{i}'}:=h'/{p_i^{n'_i}}$, and without loss of generality we can assume that $p_1$
divides $h$ as well as $h'$. By the definition \erf{bh} we then have
\begin{equation} {b_{h'}^{}}-{b_h^{}}= \sumieh \mbox{\Large\{} {h_{i}'}\,\invmod\hip{p_i^{n'_i}} - {h_i}\,\invmod\hi{p_i^{n_i}} \mbox{\Large\}} \,({b_{p_i^{}}^{}}-{b_{p_1^{}}^{}})
+\!\!\! \sumihh {h_{i}'}\,\invmod\hip{p_i^{n'_i}} \,({b_{p_i^{}}^{}}-{b_{p_1^{}}^{}}) \,; \end{equation}
again it is straightforward to verify that this
vanishes modulo ${p_j^{n_j}}\mbox{$\overline L^{\scriptscriptstyle\vee}$}$ for all $p_j$ dividing $h$. This shows that the
property \erf F is satisfied for the sequence defined by \erf{bh} as claimed.
Next we note that we did not require that the prescribed values ${b_p^{}}$
lie on the Weyl orbit of an integrable weight at height $p$, but rather
they may also lie on the boundary of some Weyl chamber of \mbox{$^{\sss(p)\!}W$}. However,
if ${b_p^{}}$ does belong to the Weyl orbit of an integrable weight, then also
each weight ${b_h^{}}$ with $p|h$ is on the Weyl orbit of an integrable weight at
height $h$. Namely, because of the property \erf F we have in particular
\begin{equation} {b_h^{}} = {b_p^{}} + p^n\,\beta \end{equation}
for some $\beta\!\in\!\mbox{$\overline L^{\scriptscriptstyle\vee}$}$. Hence, assuming that ${b_h^{}}$ is left invariant by
some $w\!\in\!\mbox{$^{\sssh\!}W$}$, i.e.\ that ${b_h^{}}=w({b_h^{}})\equiv\overline w({b_h^{}})+h\gamma$ for some
element $\gamma$ of the coroot lattice, it follows that
\begin{equation} \begin{array}{l} \overline w({b_p^{}})= \overline w({b_h^{}}-p^n\beta) =
\overline w({b_h^{}})-p^n\,\overline w(\beta) \\[1.5mm] \hsp{9.9}
= {b_h^{}} - h\gamma -p^n\,\overline w(\beta)
= {b_p^{}} +p^n\,(\beta-\overline w(\beta))- h\gamma \,. \end{array}\end{equation}
Since by assumption the only element of \mbox{$^{\sss(p)\!}W$}\ which leaves the weight
${b_p^{}}$ invariant is the identity, it follows that $\overline w={\sf id}$ and
$\gamma=0$, implying that also the only element of \mbox{$^{\sssh\!}W$}\ that leaves
${b_h^{}}$ invariant is the identity, which is equivalent to the claimed property.
Our next task is to investigate to what extent the sequence ${({b_h^{}})}_{h\in\II}$
is characterized by the prescribed values ${b_p^{}}$ at prime heights and
by the requirement \erf F. To this end let ${({\tilde b_h^{}})}_{h\in\II}$ be another
such sequence, i.e.\ a sequence such that ${\tilde b_p^{}}={b_p^{}}$ for all primes
$p$ and ${\tilde b_{h'}^{}}-{\tilde b_h^{}}\in h\mbox{$\overline L^{\scriptscriptstyle\vee}$}$ for $h|h'$.
First we observe that for $h$ and $h'$ coprime, the properties
\begin{equation} {\tilde b_{hh'}^{}} = {\tilde b_h^{}} \ {\rm mod}\; h\mbox{$\overline L^{\scriptscriptstyle\vee}$} \qquad{\rm and}\qquad {\tilde b_{hh'}^{}} = {\tilde b_{h'}^{}} \ {\rm mod}\; h'\mbox{$\overline L^{\scriptscriptstyle\vee}$} \end{equation}
fix ${\tilde b_{hh'}^{}}$ already uniquely (modulo $hh'\mbox{$\overline L^{\scriptscriptstyle\vee}$}$), so that the whole freedom
is parametrized by the freedom in the choice of ${\tilde b_h^{}}$ at heights which
are a prime power. Concerning the latter freedom, we claim that for any prime
$p$ there is a sequence of elements $\beta_p^{{\scriptscriptstyle(}j{\scriptscriptstyle)}}$ of the coroot lattice $\mbox{$\overline L^{\scriptscriptstyle\vee}$}$ which
are defined modulo $p\mbox{$\overline L^{\scriptscriptstyle\vee}$}$ such that the most general choice of ${\tilde b_{p^n}^{}}$ reads
\begin{equation} {\tilde b_{p^n}^{}} = {b_{p^n}^{}} + \sumne j \beta_p^{{\scriptscriptstyle(}j{\scriptscriptstyle)}}\,p^j \labl{bb}
with ${b_{p^n}^{}}$ defined according to \erf{bh}, i.e.\ simply ${b_{p^n}^{}}={b_p^{}}$. This
statement is proven by induction. For $n=1$ it is trivially fulfilled.
Further, assuming that \erf{bb} is satisfied for some $n\ge1$ and setting
$\gamma:={\tilde b_{p^{n+1}}^{}}-{b_{p^{n+1}}^{}}$ (defined modulo $p^{n+1}\mbox{$\overline L^{\scriptscriptstyle\vee}$}$), one has
\begin{equation} {\tilde b_{p^{n+1}}^{}}-{\tilde b_{p^n}^{}}= \gamma-\sumne j \beta_p^{{\scriptscriptstyle(}j{\scriptscriptstyle)}}\,p^j \,. \end{equation}
By the required properties of the sequence $({\tilde b_h^{}})$, the left hand side
of this formula must vanish modulo $p^n\mbox{$\overline L^{\scriptscriptstyle\vee}$}$, and hence we have
\begin{equation} \gamma = \beta_p^{{\scriptscriptstyle(}n{\scriptscriptstyle)}}\,p^n + \sumne j \beta_p^{{\scriptscriptstyle(}j{\scriptscriptstyle)}}\,p^j = \sumn j \beta_p^{{\scriptscriptstyle(}j{\scriptscriptstyle)}}\,p^j \end{equation}
for some $\beta_p^{{\scriptscriptstyle(}n{\scriptscriptstyle)}}\!\in\!\mbox{$\overline L^{\scriptscriptstyle\vee}$}$.
This shows that ${\tilde b_{p^{n+1}}^{}}$ is again of the form described by \erf{bb};
furthermore, as $\gamma$ is defined modulo $p^{n+1}\mbox{$\overline L^{\scriptscriptstyle\vee}$}$, $\beta_p^{{\scriptscriptstyle(}n{\scriptscriptstyle)}}$ is defined
modulo $p\mbox{$\overline L^{\scriptscriptstyle\vee}$}$ as required, and hence the proof of the formula \erf{bb} is
completed.
With these results we are now in a position to give a rather explicit
description of the allowed sequences ${\tilde b_h^{}}$. Namely, we can parametrize the
general form of ${\tilde b_h^{}}$ in terms of the freedom in ${\tilde b_{p^n}^{}}$ according to
\begin{equation} \begin{array}{l} \hsp{-3}{\tilde b_h^{}}= {\tilde b_{p_1^{n_1}}^{}} + \sumieh{h_i}\,\invmod\hi{p_i^{n_i}}\,({\tilde b_{p_i^{n_i}}^{}}-{\tilde b_{p_1^{n_1}}^{}})
\\[1.5mm] \hsp{.7}
= {b_{p_1^{}}^{}} + \sumnee j \beta_{p_1}^{{\scriptscriptstyle(}j{\scriptscriptstyle)}}\,p_1^j + \sumieh {h_i}\,\invmod\hi{p_i^{n_i}}\,({b_{p_i^{}}^{}}-{b_{p_1^{}}^{}})
+ \sumieh {h_i}\,\invmod\hi{p_i^{n_i}}\,\mbox{\large[} \sumnie j \beta_{p_i}^{{\scriptscriptstyle(}j{\scriptscriptstyle)}}\,p_i^j - \sumnee j \beta_{p_1}^{{\scriptscriptstyle(}j{\scriptscriptstyle)}}\,p_1^j
\mbox{\large]}
\\[1.5mm] \hsp{.7}
= {b_h^{}} + \sumieh {h_i}\,\invmod\hi{p_i^{n_i}}\,\mbox{\large[} \sumnie j \beta_{p_i}^{{\scriptscriptstyle(}j{\scriptscriptstyle)}}\,p_i^j - \sumnee j
\beta_{p_1}^{{\scriptscriptstyle(}j{\scriptscriptstyle)}}\,p_1^j \mbox{\large]} + \sumnee j \beta_{p_1}^{{\scriptscriptstyle(}j{\scriptscriptstyle)}}\,p_1^j \,.
\end{array} \labl X
Further, any such sequence fulfills the consistency requirement that
${\tilde b_{h'}^{}}-{\tilde b_h^{}}\in h\mbox{$\overline L^{\scriptscriptstyle\vee}$}$ for heights $h,\,h'$ with $h|h'$. Namely, in this case the
formula \erf X yields
\begin{equation} \begin{array}{l} \hsp{-3.8} {\tilde b_{h'}^{}}-{\tilde b_h^{}} = \sumieh \mbox{\Large\{}
{h_{i}'}\,\invmod\hip{p_i^{n'_i}}\, \mbox{\large[} \sumnipe j \beta_{p_i}^{{\scriptscriptstyle(}j{\scriptscriptstyle)}}\,p_i^j - \sumnepe j \beta_{p_1}^{{\scriptscriptstyle(}j{\scriptscriptstyle)}}\,p_1^j \mbox{\large]}
- {h_i}\,\invmod\hi{p_i^{n_i}}\, \mbox{\large[} \sumnie j \beta_{p_i}^{{\scriptscriptstyle(}j{\scriptscriptstyle)}}\,p_i^j - \sumnee j \beta_{p_1}^{{\scriptscriptstyle(}j{\scriptscriptstyle)}}\,p_1^j \mbox{\large]} \mbox{\Large\}}
\\[1.5mm] \hsp{16.4}
+ \sumihh
{h_{i}'}\,\invmod\hip{p_i^{n'_i}}\, \mbox{\large[} \sumnipe j \beta_{p_i}^{{\scriptscriptstyle(}j{\scriptscriptstyle)}}\,p_i^j - \sumnepe j \beta_{p_1}^{{\scriptscriptstyle(}j{\scriptscriptstyle)}}\,p_1^j \mbox{\large]}
+ \sumneee j \beta_{p_1}^{{\scriptscriptstyle(}j{\scriptscriptstyle)}}\,p_1^j \ \ {\rm mod}\; h\mbox{$\overline L^{\scriptscriptstyle\vee}$} \,.
\end{array} \labl Y
Once more one can easily check that this expression vanishes modulo ${p_j^{n_j}}\mbox{$\overline L^{\scriptscriptstyle\vee}$}$
for all primes $p_j$ that divide $h$, and hence vanishes modulo $h\mbox{$\overline L^{\scriptscriptstyle\vee}$}$. Thus
the consistency requirement indeed is satisfied.
\subsection{Distinguished sequences of integrable weights and the basis \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}}
\label{sinw}
What we have achieved so far is a characterization of all sequences
${({b_h^{}})}_{h\in\II}$ of integral weights defined modulo $h\mbox{$\overline L^{\scriptscriptstyle\vee}$}$ that satisfy \erf F.
We now use this result to construct sequences ${({a_h^{}})}_{h\in\II}$ of
highest weights which satisfy the requirement \erf m and of which infinitely
many are {\em integrable\/} weights, with all non-integrable weights being
equal to zero.
We start by prescribing integrable weights ${a_p^{}}\!\in\!\mbox{$_{}^{\scriptscriptstyle(p)\!\!}P$}$ for all primes $p$
with $p\ge\mbox{$g_{}^{\scriptscriptstyle\vee}$}$, and set ${b_p^{}}={a_p^{}}$ for $p\ge\mbox{$g_{}^{\scriptscriptstyle\vee}$}$, while for all primes
$p<\mbox{$g_{}^{\scriptscriptstyle\vee}$}$ we choose arbitrary weights ${b_p^{}}$ (which are necessarily
non-integrable). Next we employ the previous results to find the sequences
${({b_h^{}})}_{h\in\II}$. Finally we define ${a_h^{}}$ for any arbitrary height $h$ as
follows. If ${b_h^{}}$ lies on the boundary of a Weyl chamber with respect to \mbox{$^{\sssh\!}W$}, then
we set ${a_h^{}}=0$. Otherwise there are a unique element $\overline w_h\in\overline W$ and a
unique
\futnote{To be precise, because the weights ${a_h^{}}$ are defined only modulo $h\mbox{$\overline L^{\scriptscriptstyle\vee}$}
$, $\gamma_h$ is only unique once a definite representative of the equivalence
class of weights that is described by ${a_h^{}}$ is chosen.}
element $\gamma_h\!\in\!\mbox{$\overline L^{\scriptscriptstyle\vee}$}$ such that $\overline w_h({b_h^{}})+h\gamma_h$ is integrable, and
in this case we set
\begin{equation} {a_h^{}}:= \overline w_h({b_h^{}})+h\gamma_h \,. \end{equation}
By construction, the weights ${a_h^{}}$ have the following properties.
If ${a_{h'}^{}}=0$ for some height $h'$, then ${b_{h'}^{}}$ is on the boundary of a Weyl
chamber with respect to \mbox{$^{\sss(h')\!}W$}; for any $h$ dividing $h'$, it then follows from
${b_{h'}^{}}-{b_h^{}}\!\in\! h\mbox{$\overline L^{\scriptscriptstyle\vee}$}$ that ${b_h^{}}$ is on the boundary of a Weyl chamber with respect to \mbox{$^{\sssh\!}W$},
and hence we also have ${a_h^{}}=0$. On the other hand, if
${b_{h'}^{}}$ is equivalent with respect to \mbox{$^{\sss(h')\!}W$}\ to an integrable weight ${a_{h'}^{}}\!\in\!\mbox{$_{}^{\scriptscriptstyle(h')\!\!}P$}$, then it
is a fortiori equivalent to ${a_{h'}^{}}$ with respect to the larger group \mbox{$^{\sssh\!}W$}, and then the property
${b_{h'}^{}}-{b_h^{}}\!\in\! h\mbox{$\overline L^{\scriptscriptstyle\vee}$}$ implies that also ${b_h^{}}$ is equivalent with respect to \mbox{$^{\sssh\!}W$}\ to ${a_{h'}^{}}$,
and hence that the associated weight ${a_h^{}}$ is integrable at height $h$ and is
equivalent with respect to \mbox{$^{\sssh\!}W$}\ to ${a_{h'}^{}}$, too. Thus ${a_h^{}}$ and ${a_{h'}^{}}$ are on the same orbit
with respect to \mbox{$^{\sssh\!}W$}\ whenever $h$ divides $h'$, and hence \erf m holds as promised.
Note that by construction for all \mbox{$\bar{\mathfrak g}$}\ except $\mbox{$\bar{\mathfrak g}$}=A_1$ the sequences so
obtained contain some zero weights. However, any sequence which contains at
least one non-zero weight contains in fact infinitely many non-zero (and
hence integrable) weights.
The final step is now to define
\begin{equation} \psi(h):=\eta({a_h^{}})\,{}^{\sssh\!}\varphi^{}_{{a_h^{}}} \labl x
as in \erf Q, where ${a_h^{}}$ is as constructed above, and where
\begin{equation} \eta({a_h^{}}):= \left\{ \begin{array}{ll} \mbox{sign}(\overline w_h) & {\rm for}\
{a_h^{}}\!\in\!\mbox{$^{\sssh\!}W$}\,, \\[1mm] 0 & {\rm for}\ {a_h^{}}=0 \,.
\end{array} \right. \labl{ets}
To show that $\psi$ is an element of the projective limit, it only remains
to check the property \erf M of the prefactor $\eta({a_h^{}})$. For ${a_h^{}}=0$ \erf M
just reads $0=0$ and is trivially satisfied. For ${a_h^{}}\!\in\!\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}$, the previous
results show that ${a_{\ell h}^{}}=w_{\ell h}^{}\circ w_0\circ w_h^{-1}({a_h^{}})$, where $w_0$
is the Weyl translation relating ${b_h^{}}$ and ${b_{\ell h}^{}}$, so that
\begin{equation} \epsilon_{\el}({a_{\ell h}^{}}) = \mbox{sign}(w_{\ell h}^{}\circ w_0\circ w_h^{})
= \mbox{sign}(\overline w_{\ell h}^{})\cdot\mbox{sign}(\overline w_h^{}) \,. \end{equation}
In view of the definition \erf{ets} of $\eta({a_h^{}})$, this is precisely the
required relation \erf M. We conclude that the basis \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\ of \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ precisely
consists of the elements \erf x. In particular, \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\ is countably infinite.
\Sect{The fusion ring of \mbox{$\bar{\mathfrak g}$}}{gb}
As already pointed out in the introduction, it is expected that in the
limit of infinite level of WZW theories\ somehow the simple Lie algebra\ \mbox{$\bar{\mathfrak g}$}\ which is the
horizontal subalgebra\ of \mbox{$\mathfrak g$}\ and its representation\ theory should play a r\^ole. More specifically,
one might think that the representation ring \mbox{$\overline{\mbox{${\cal R}$}}$}\ of \mbox{$\bar{\mathfrak g}$}\ emerges. As
we will demonstrate below, indeed this ring shows up, but it is only a proper
subring of the projective limit \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ we constructed, and almost all
elements of \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ are {\em not\/} contained in the ring \mbox{$\overline{\mbox{${\cal R}$}}$}.
Let us describe \mbox{$\overline{\mbox{${\cal R}$}}$}\ and its connection with the category ${\cal F}\!us\mbox{\scriptsize(}\mbox{$\mathfrak g$}\mbox{\scriptsize)}$\ in some
detail. \mbox{$\overline{\mbox{${\cal R}$}}$}\ is defined as the ring
over ${\dl Z}$\ of all isomorphism classes of finite-dimensional\ \mbox{$\bar{\mathfrak g}$}-representation s, with the ring
product the ordinary tensor product of \mbox{$\bar{\mathfrak g}$}-representation s (or, equivalently, the
pointwise product of the characters of these representation s).
This ring \mbox{$\overline{\mbox{${\cal R}$}}$}\ is a fusion ring with an infinite basis.
The elements $\bar\varphi^{}_a$ of a distinguished basis of \mbox{$\overline{\mbox{${\cal R}$}}$}\ are labelled by
the (shifted) highest weights of irreducible finite-dimensional\ \mbox{$\bar{\mathfrak g}$}-representation s, i.e.\
by elements of the set
\begin{equation} \mbox{$\bar P$} := \{ a\in\mbox{$\overline L^{\rm w}$} \mid 0<a^i\ {\rm for}\, \ i=1,2,...\,,r \} \,.
\labl{pb}
Now for any $h\!\in\!\mbox{$I$}$ let us define the map $\fb h\!:\ \mbox{$\overline{\mbox{${\cal R}$}}$}\to\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\,$ as follows.
If $a\!\in\!\mbox{$\bar P$}$ lies on the boundary of some Weyl chamber with respect to \mbox{$^{\sssh\!}W$}, we set
$\fb h(\bar\varphi^{}_a):=0$; otherwise there exist a unique $a'\!\in\!\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}$
and a unique $w\!\in\!\mbox{$^{\sssh\!}W$}$ such that $w(a)=a'$, and in this case we set
\begin{equation} \fb h(\bar\varphi^{}_a):= \epsilon(a)\cdot {}^{\sssh\!}\varphi^{}_{a'} \labl`
with $\epsilon(a)=\mbox{sign}(w)$. As in the case of the maps $\ff{\el h}h$ \erf', we
will consider \erf` as covering all cases, i.e.\ set $\epsilon(a)=0$ if
$a$ lies on the boundary of a Weyl chamber at height $h$.
To analyze the relation between the ring \mbox{$\overline{\mbox{${\cal R}$}}$}\ and the category ${\cal F}\!us\mbox{\scriptsize(}\mbox{$\mathfrak g$}\mbox{\scriptsize)}$,
we first recall the expressions
\begin{equation} \nb abc = \sum_{\overline w\in\overline W} \mbox{sign}(\overline w)\,\mult b(\overline w(c)-a)
\labl{nb}
for the Litt\-le\-wood\hy Ri\-chard\-son coefficient s (or tensor product coefficients) of \mbox{$\bar{\mathfrak g}$}\ \cite{raca2,spei} and
\begin{equation} \nh abc = \sum_{w\in{}^{\sssh}W} \mbox{sign}(w)\,\mult b(w(c)-a)
\labl{nh}
for the fusion rule coefficient s, i.e.\ the structure constants of the WZW fusion ring \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\
\cite{KAc3,walt3,fugp2,fuva2}. Here $\mult a(b)$ denotes the multiplicity of
the (shifted) weight $b$ in the \mbox{$\bar{\mathfrak g}$}-representation\ with (shifted) highest weight $a$.
It will be convenient to extend the validity of \erf{nb} by adopting it
as a definition of \nb abc\ for arbitrary (i.e., not necessarily lying in
\mbox{$\bar P$}) integral weights $a$ and $c$, and also extend it to arbitrary integral
weights $b$ that do
not lie on the boundary of any Weyl chamber with respect to $\overline W$ by setting
\begin{equation} \mult b(c):=\mbox{sign}(\overline w_b^{})\,\mult{\overline w_b^{}(b)}(c) \,, \end{equation}
with $\overline w_b$ the unique element of $\overline W$ such that $\overline w_b(b)\!\in\!\mbox{$\bar P$}$.
The multiplicities $\mult a(b)$ are invariant under the Weyl group $\overline W$,
i.e.\ $\mult a(\overline w(b))=\mult a(b)$ for all $\overline w\!\in\!\overline W$.
As a consequence, the numbers \nb abc\ and \nh abc\ are related by
\futnote{In the formulation of \cite{KAc3,walt3,fugp2,fuva2} the factor of
$|\overline W|^{-1}$ is absent because there \nb abc\ is taken to be non-zero only
if $a,b\!\in\!\mbox{$\bar P$}$.}
\begin{equation} \nh abc = \frac1{|\overline W|}\, \sum_{w\in{}^{\sssh}W} \mbox{sign}(w)\,\nb ab{w(c)} \,.
\labl B
The invariance of $\mult a(b)$ under $\overline W$ also implies
that for arbitrary integral weights $a,\,b$ and $c$ the symmetry property
\begin{equation} \nb abc = \nb bac \labl{sy}
follows from the analogous property of the Litt\-le\-wood\hy Ri\-chard\-son coefficient s with $a,b,c\in\mbox{$\bar P$}$, and that
\begin{equation} \begin{array}{l} \nb{\overline w_1(a)}b{\,\ \overline w_2(c)}
= \displaystyle\sum_{\overline w\in\overline W} \mbox{sign}(\overline w)\,\mult b(\overline w\,\overline w_2(c)-\overline w_1(a))
\\{}\\[-3mm] \hsp{16.2}
= \displaystyle\sum_{\overline w\in\overline W} \mbox{sign}(\overline w)\,\mult b(\overline w_1^{-1}\overline w\:\overline w_2(c)-a)
= \mbox{sign}(\overline w_1\overline w_2)\cdot\nb abc \,. \end{array} \end{equation}
When combined with the symmetry property \erf{sy}, the latter formula yields
\begin{equation} \nb{\overline w_1(a)}{\overline w_2(b)}{\ \ \ \ \ \ \overline w_3(c)}
= \mbox{sign}(\overline w_1\overline w_2\overline w_3)\cdot\nb abc \,. \end{equation}
To obtain information about the effect of affine Weyl transformations on
the labels of \nb abc, we consider an alternating sum over the Weyl group \mbox{$^{\sssh\!}W$}.
We have
\begin{equation} \begin{array}{l} \displaystyle\sum_{w_2\in{}^{\sssh}W}\!\!\mbox{sign}(w_2)\,\nb{w_1(a)}b{\ \,w_2(c)}
= \!\!\displaystyle\sum_{\scriptstyle\overline w,\overline w_2\in\overline W \atop\scriptstyle\beta_2\in\bar L^{\scriptscriptstyle\vee}}
\!\!\mbox{sign}(\overline w)\mbox{sign}(\overline w_2)
\,\mult b(\overline w\,\overline w_2(c)+h\overline w(\beta_2)-\overline w_1(a)-h\beta_1)
\\{}\\[-3mm] \hsp{43.7}
= \displaystyle\sum_{\overline w,\overline w_2\in\overline W} \displaystyle\sum_{\beta\in\bar L^{\scriptscriptstyle\vee}} \mbox{sign}(\overline w\,\overline w_2)
\,\mult b(\overline w_1^{-1}\overline w\,\overline w_2(c)+h\overline w_1^{-1}\overline w(\beta)-a)
\\{}\\[-3mm] \hsp{43.7}
= \mbox{sign}(w_1)\cdot\displaystyle\sum_{w_2\in{}^{\sssh}W}\mbox{sign}(w_2)\,\nb ab{w_2(c)} \,.
\end{array}\end{equation}
Here $\beta:=\beta_2-\overline w^{-1}(\beta_1)$.
Together with the symmetry property \erf{sy} it then follows that
\begin{equation} \sum_{w_3\in{}^{\sssh}W}\mbox{sign}(w_3)\,\nb{w_1(a)}{w_2(b)}{\ \ \ \ \ w_3(c)}
= \mbox{sign}(w_1)\,\mbox{sign}(w_2)\cdot\sum_{w_3\in{}^{\sssh}W}\mbox{sign}(w_3)\,\nb ab{w_3(c)}
\labl C
for all $w_1,w_2\!\in\!\mbox{$^{\sssh\!}W$}$. We can rewrite this as
\begin{equation} \sum_{w\in{}^{\sssh}W}\epsilon_{\el}(A)\epsilon_{\el}(B)\,\mbox{sign}(w)\,\nb{w_A(A)}{w_B(B)}
{\ \ \ \ \ \ \ \ \ \ w(c)}
= \displaystyle\sum_{w\in{}^{\sssh}W} \mbox{sign}(w)\,\nb AB{w(c)} \, \labl{515}
which by interpreting $A$ and $B$ as elements of \mbox{$\bar P$}\ rather than \mbox{$_{}^{\scriptscriptstyle(\el h)\!\!}P$}\
yields, after summation over $c\!\in\!\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}$,
\begin{equation} \fb h(\bar\varphi^{}_A)\star\fb h(\bar\varphi^{}_B)=\fb h(\bar\varphi^{}_A\star \bar\varphi^{}_B) \,, \end{equation}
and hence shows that the maps \fb h defined by \erf` are ring homomorphisms.
Now for all $A\!\in\!\mbox{$_{}^{\scriptscriptstyle(\el h)\!\!}P$}$ we have $\ff{\el h}h(\phi_A)=\epsilon_{\el}(A)(\phi_{w_A(A)})
=:\epsilon_{\el}(A)\cdot \varphi^{}_a$. Then owing to \erf B we obtain, after dividing
\erf{515} by $|\overline W|$, the relation
\begin{equation} \nh{\!\FF{\el h}h(A)}{\FF{\el h}h(B)}{\ \ \ \ \ \ \ \ \ \ \ \ \ c}
= \epsilon_{\el}(A)\,\epsilon_{\el}(B)\,\nh abc
= \displaystyle\sum_{C:\ \phi_C\in\FF{\el h}h^{-1}(\varphi_c)} \epsilon_{\el}(C)\;\nlh ABC \labl:
(on the left hand side, we use the short hand notation $\ff{\el h}h(A)\!\in\!\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}$ to indicate
the label that corresponds to the element $\epsilon_{\el}(A)\ff{\el h}h(\phi_A)$ of \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}).
Summation over $c\!\in\!\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}$
then yields $\ff{\el h}h(\phi_A)\star\ff{\el h}h(\phi_B)=\ff{\el h}h(\phi_A\star \phi_B)$,
so that \erf: is just the homomorphism property of the maps \ff ij which were
defined by \erf' in terms of the fusion rule coefficient s. (Thereby we have also obtained
an alternative proof of the homomorphism property of those maps.)
To investigate further the relation between $\mbox{$\overline{\mbox{${\cal R}$}}$}$ and the projective limit
\mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}, we introduce the linear mappings
\begin{equation} \begin{array}{llll} \jb h:& \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\to\mbox{$\overline{\mbox{${\cal R}$}}$}\,,& {}^{\sssh\!}\varphi^{}_a\mapsto\bar\varphi^{}_a \,,
\\[1.6mm] \jj h{h'}:& \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\to\mbox{$^{\scriptscriptstyle(h')\!}{\cal R}$}\,,& {}^{\sssh\!}\varphi^{}_a\mapsto{}^{\ssshp\!}\varphi^{}_a &
\end{array} \labl{jj}
which map each basis element ${}^{\sssh\!}\varphi^{}_a$ of \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\ to that basis element of \mbox{$\overline{\mbox{${\cal R}$}}$}\
and \mbox{$^{\scriptscriptstyle(h')\!}{\cal R}$}\ ($h'\ge h$), respectively, which is labelled by the same weight
$a\in\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}\subseteq\mbox{$_{}^{\scriptscriptstyle(h')\!\!}P$}\subset\mbox{$\bar P$}$.
For $h\preceq h'\preceq h''$, these maps satisfy
\begin{equation} \jb{h'}\circ\jj h{h'} =\jb h \,, \quad
\jj{h'}{h''}\circ\jj h{h'} =\jj h{h''}\, \end{equation}
as well as
\begin{equation} \fb{h}\circ\jb h ={\sf id}_h \,, \quad
\ff{h'}{h}\circ\jj h{h'} ={\sf id}_{h} \end{equation}
and
\begin{equation} \fb{h}\circ\jb{h'} =\ff{h'}h \,, \quad
\ff{h''}{h}\circ\jj{h'}{h''} =\ff{h'}h \,. \end{equation}
We say that a generalized sequence $\psi$ in the projective limit \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ is
{\em ultimately constant\/} iff there exists a ${h_\circ}\iN\II$ such that
\begin{equation} \psi(h) = \jj{h_\circ} h\circ\psi({h_\circ}) \labl{const}
(and hence for basis elements in particular ${a_h^{}}= a_{h_\circ}$) for all heights
$h\ge{h_\circ}$. Now assume that $\psi_1$ and $\psi_2$ are elements of \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\
which are ultimately constant, with associated heights ${h_{\circ,1}}$ and ${h_{\circ,2}}$, respectively.
Then in particular for all heights $h$ larger than ${h_\circ}:=2\,{\rm max}
({h_{\circ,1}},{h_{\circ,2}})$ the fusion product $\psi_1(h)\star\psi_2(h)$ in \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\
is isomorphic to the product $\overline\psi_1\star\overline\psi_2$ in \mbox{$\overline{\mbox{${\cal R}$}}$}, where $\overline\psi_1
:=\jb{h_\circ}\circ\psi_1({h_\circ})$, and analogously for $\overline\psi_2$. This implies that
\begin{equation} (\jj{h_\circ} h\circ\psi_1({h_\circ})) \star ((\jj{h_\circ} h\circ\psi_2({h_\circ}))
= \jj{h_\circ} h\circ (\psi_1({h_\circ}) \star\psi_2({h_\circ})) \labl{labl}
even though $\jj{h_\circ} h$ is not a ring homomorphism, and hence
$(\psi_1\star\psi_2)(h)\equiv\psi_1(h)\star\psi_2(h)=\jj{h_\circ} h\circ
(\psi_1\star\psi_2)({h_\circ})$ for all $h\ge{h_\circ}$. Thus the product
$\psi_1\star\psi_2$ is again ultimately constant. Also, the property of being
ultimately constant is preserved upon taking (finite) linear transformations and
conjugates. The set of ultimately constant elements therefore constitutes
a subring of \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}.
\futnot{positivity of the $\n abc$ in the subring is guaranteed once the over-all
sign for the basis elements is chosen properly.}
The following consideration shows that this subring is isomorphic to the
fusion ring \mbox{$\overline{\mbox{${\cal R}$}}$}. First, any ultimately constant element is a linear
combination of ultimately constant elements $\psi_{}^{\scriptscriptstyle(a)}$ for which $\psi_{}^{\scriptscriptstyle(a)}({h_\circ})$
is an element of the canonical basis of \mbox{$^{\scriptscriptstyle(\ho)\!}{\cal R}$}, $\psi_{}^{\scriptscriptstyle(a)}({h_\circ})={}^{\sss(\ho)\!}\varphi^{}_a$ for
some $a\in\mbox{$_{}^{\scriptscriptstyle(\ho)\!\!}P$}\subset\mbox{$\bar P$}$.
But there is a unique element $\psi$ of \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ with the latter property,
because at all heights $h$ smaller than ${h_\circ}$ the value $\psi(h)$ is
already fixed by imposing the requirement \erf k. Thus there is a bijective
linear map between the subring of ultimately constant elements and the
fusion ring \mbox{$\overline{\mbox{${\cal R}$}}$}, defined by $\bar\varphi^{}_a\mapsto\psi_{}^{\scriptscriptstyle(a)}$ for $a\!\in\!\mbox{$\bar P$}$.
Moreover, the same argument which led to \erf{labl} shows that this
map is in fact an isomorphism of fusion rings. As this map is provided
in a canonical manner, we can actually identify the two rings.
A generic element of \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ is {\em not\/} ultimately constant, so that
the subring of ultimately constant elements is a proper subring of \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}.
Thus what we have achieved is to identify the fusion ring \mbox{$\overline{\mbox{${\cal R}$}}$}\ as a
proper sub-fusion ring of the projective limit ring \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}.
To conclude this section, let us remark that of course we could have enlarged
by hand the category ${\cal F}\!us\mbox{\scriptsize(}\mbox{$\mathfrak g$}\mbox{\scriptsize)}$\ to a larger category ${\cal F}\!u\!\!\overline{\!\!\mbox{~~}s\mbox{\scriptsize(}\mbox{$\mathfrak g$}\mbox{\scriptsize)}}$\ by just including
one additional object into the category, namely the ring \mbox{$\overline{\mbox{${\cal R}$}}$}, together with
the morphisms \fb h. This essentially amounts to cutting the category
of rings in such a way that one is able to
identify the ring \mbox{$\overline{\mbox{${\cal R}$}}$}\ as the projective limit of this category ${\cal F}\!u\!\!\overline{\!\!\mbox{~~}s\mbox{\scriptsize(}\mbox{$\mathfrak g$}\mbox{\scriptsize)}}$.
We do not regard this as a viable alternative to our construction, though,
since when doing so one performs manipulations which are suggested merely
by one's prejudice on what the limit should look like. (Also, phenomena like
level-rank dualities in fusion rings require to consider various rings for
different algebra s \mbox{$\mathfrak g$}\ on the same footing; the category ${\cal F}\!u\!\!\overline{\!\!\mbox{~~}s\mbox{\scriptsize(}\mbox{$\mathfrak g$}\mbox{\scriptsize)}}$\
cannot accommodate such phenomena.)
In contrast, our construction of the limit employs only the description in
terms of coherent sequences, which is a natural procedure for any small
category, and does not presuppose any desired features of the limit.
\Sect{Representation\ theory of $\mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}$}{rep}
A basic tool in the study of fusion rings is their representation\ theory. Of particular
importance are the irreducible representation s, which lead in particular to the notion of
(generalized) quantum dimensions. In this section
we show that an analogous representation\ theory exists for the projective limit as
well. In our considerations the \topo\ will again play an essential r\^ole.
\subsection{One-dimensional\ representation s}
Let us consider for any two $h,h'\iN\II$ with $h'=\ell h$ the
injection of the label set \mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}\ (defined as in \erf{ph}) into the label set
\mbox{$_{}^{\scriptscriptstyle(h')\!\!}P$}\ that is defined by multiplying the weights $a$ by a factor of $\ell$:
\begin{equation} a \;\mapsto\; \ell a \end{equation}
for all $a\!\in\!\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}$. (This induces an injection ${}^{\sssh\!}\varphi^{}_a\mapsto{}^{\ssslh\!}\varphi^{}_{\ell a}$
of the distinguished basis $\mbox{$^{\scriptscriptstyle(h)}\!{\cal B}$}$ of $\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}$ into the distinguished basis $\mbox{$^{\scriptscriptstyle(\el h)}\!{\cal B}$}$
of $\mbox{$^{\scriptscriptstyle(\el h)\!}{\cal R}$}$. However, when this map is extended linearly to all of \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}, it does
{\em not\/} provide a homomorphism of fusion rings.)
We can use these injections to perform an {\em inductive\/} limit
of the set $(\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$})_{h\in I}$ of label sets, where the set $I$ \erf i is again
considered as directed via the partial ordering \erf l. We denote this
inductive limit by ${}^{\scriptscriptstyle(\infty)\!\!}P$.
An element $\alpha$ of ${}^{\scriptscriptstyle(\infty)\!\!}P$ can be characterized by an integrable
weight ${\alpha(h)}\!\in\!\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}$ at some suitable height $h$; at any multiple $\ell h$
of this height, the same element $\alpha$ of ${}^{\scriptscriptstyle(\infty)\!\!}P$ is then represented
by the weight ${\alpha(\el h)}=\ell\,{\alpha(h)}$. In particular, quite unlike as in
the case of the projective limit, each element of the inductive limit ${}^{\scriptscriptstyle(\infty)\!\!}P$
is already determined by its representative at a single height. Also note that
an element $\alpha\!\in\!{}^{\scriptscriptstyle(\infty)\!\!}P$ is {\em not\/} defined at all heights $h$;
in particular, for any $h\iN\II$ the set of those $\alpha\!\in\!{}^{\scriptscriptstyle(\infty)\!\!}P$
which have a representative at height $h$ is in one-to-one correspondence with
the elements of \mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}, and hence is in particular finite.
We will use the notation $\alpha\!\downarrow\!\!\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}$ to indicate that $\alpha\!\in\!{}^{\scriptscriptstyle(\infty)\!\!}P$ has a
representative ${\alpha(h)}\!\in\!\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}$ at height $h$.
We claim that any element of ${}^{\scriptscriptstyle(\infty)\!\!}P$ gives rise to a one-dimensional representation\ of
the projective limit \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ of the fusion rings. To see this, we choose
for a given $\alpha\!\in\!{}^{\scriptscriptstyle(\infty)\!\!}P$ a suitable height $h\iN\II$ such that $\alpha\!\downarrow\!\!\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}$.
To any coherent sequence $(\psi(l))_{l\in I}$ in the projective
limit $\mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}$ we then associate the number
\begin{equation} {\cal D}_\alpha(\psi) :=
\frac{\mbox{$^{\scriptscriptstyle(h)}\!S$}_{\psi(h),{\alpha(h)}}}{\mbox{$^{\scriptscriptstyle(h)}\!S$}_{\rho,{\alpha(h)}}} \,, \labl{cald}
i.e.\ the ${\alpha(h)}$th quantum dimension of the element $\psi(h)$ of the ring
\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}. Here we use the short-hand notation
\begin{equation} \mbox{$^{\scriptscriptstyle(h)}\!S$}_{\psi(h),b}:=\sumph a \zeta_a\, \mbox{$^{\scriptscriptstyle(h)}\!S$}_{a,b} \qquad {\rm for}\;\
\psi(h)=\sumph a \zeta_a\,{}^{\sssh\!}\varphi^{}_a \end{equation}
for linear combinations of $S$-matrix\ elements.
Using the identities \erf D and \erf{sS} as well as ${\alpha(\el h)}=\ell{\alpha(h)}$
and the defining properties of $\psi$, we have
\begin{equation} \frac{\mbox{$^{\scriptscriptstyle(\el h)}\!S$}_{\psi(\ell h),{\alpha(\el h)}}}{\mbox{$^{\scriptscriptstyle(\el h)}\!S$}_{\rho,{\alpha(\el h)}}}
= \frac{(\mbox{$^{\scriptscriptstyle(\el h)}\!S$}\,D)_{\psi(\ell h),{\alpha(h)}}}{(\mbox{$^{\scriptscriptstyle(\el h)}\!S$}\,D)_{\rho,{\alpha(h)}}}
= \frac{(F\,\mbox{$^{\scriptscriptstyle(h)}\!S$})_{\psi(\ell h),{\alpha(h)}}}{(F\,\mbox{$^{\scriptscriptstyle(h)}\!S$})_{\rho,{\alpha(h)}}}
= \frac{\mbox{$^{\scriptscriptstyle(h)}\!S$}_{\psi(h),{\alpha(h)}}}{\mbox{$^{\scriptscriptstyle(h)}\!S$}_{\rho,{\alpha(h)}}} \,; \labl{s/s}
this shows that the formula \erf{cald} yields a well-defined map from \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\
to ${\mathbb C}$\,, i.e.\ it does not depend on the particular choice of $h$.
Using the knowledge about the representation\ theory of the rings \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}, it
then follows immediately that
\begin{equation} {\cal D}_\alpha(\psi)\,{\cal D}_\alpha(\psi') = \sumph{a,b} \zeta_a^{}
\zeta_b'\,\nh abc\, \frac{\mbox{$^{\scriptscriptstyle(h)}\!S$}_{c,{\alpha(h)}}}{\mbox{$^{\scriptscriptstyle(h)}\!S$}_{\rho,{\alpha(h)}}}
= {\cal D}_\alpha(\psi\star\psi') \,. \end{equation}
Thus the prescription \erf{cald} indeed provides us with a one-dimensional\ representation\ of
$\mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}$.
Let us now associate to any element $\psi$ of $\mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}$ the infinite sequence of
quantum dimensions \erf{cald}, labelled by ${}^{\scriptscriptstyle(\infty)\!\!}P$; this way we obtain a map
\begin{equation} {\cal D}:\quad \psi \,\mapsto\, ({\cal D}_\alpha(\psi))_{\alpha\in{}^{\scriptscriptstyle(\infty)\!\!}P}
\labl{ca}
from the ring \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ to the algebra
\begin{equation} {\cal X} := \{ (\xi_\alpha)_{\alpha\in{}^{\scriptscriptstyle(\infty)\!\!}P} \!\mid\! \xi_\alpha\!\in\!{\mathbb C}\,
\} \end{equation}
of all countably infinite sequences of complex numbers.
Since we are now dealing with complex numbers rather than only integers, it
is natural to consider instead of the fusion ring \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ the corresponding
algebra over ${\mathbb C}$\,, to which we refer as the {\em fusion algebra\/} \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}.
(For simplicity we regard \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\ as an algebra over ${\mathbb C}$\,.
In principle it would be sufficient to consider it over a certain
subfield of ${\mathbb C}$\ generated by appropriate roots of unity.) It is then
evident that the map ${\cal D}\!:\,\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\to{\cal X}$ defined by \erf{ca} is an
algebra homomorphism. (We continue to use the symbol ${\cal D}$. More generally,
below we will always assume that the various maps to be used, such as
the projection \erf3, are continued ${\mathbb C}$\,-linearly from the fusion rings
\mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\ to the associated fusion algebras \mbox{$^{\scriptscriptstyle(h)}\!\!{\cal A}$}, and use the same symbols for
these extended maps as for the original ones.)
\subsection{An isomorphism between \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\ and ${\cal X}$}
In this subsection we show that the map ${\cal D}$ introduced above
even constitutes an
{\em iso\/}morphism between the complex algebras \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\ and ${\cal X}$:
\begin{equation} {\cal D}:\quad \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$} \;\stackrel{\cong}{\longrightarrow}\; {\cal X} \,.
\labl{dax}
Injectivity of ${\cal D}$ is easy to check. Suppose we have ${\cal D}(\psi)=0$. Fix
any $h\!\in\! I$; then all quantum dimensions of the element $\psi(h)$ of \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\
vanish. From the properties of the fusion ring \mbox{$^{\scriptscriptstyle(h)\!}{\cal R}$}\ it then follows
immediately that $\psi(h)=0$. This is true for all $h\!\in\! I$, and hence we have
$\psi=0$. This proves injectivity.
To show also surjectivity requires more work. We first need to introduce the
elements
\begin{equation} {\rm e}_a \equiv {}^{\scriptscriptstyle(h)\!}{\rm e}_a := \mbox{$^{\scriptscriptstyle(h)}\!S$}_{\rho,a} \sumph b \mbox{$^{\scriptscriptstyle(h)}\!S$}^*_{a,b}\, {}^{\sssh\!}\varphi^{}_b
\;\in \mbox{$^{\scriptscriptstyle(h)}\!\!{\cal A}$} \labl{ea}
of the fusion algebras at height $h$. These elements are idempotents, i.e.\ obey
\begin{equation} {\rm e}_a\star {\rm e}_b = \delta_{a,b}\, {\rm e}_{a} \,. \labl{eee}
Owing to the unitarity of the modular
transformation matrix $S$, the idempotents $\{{}^{\scriptscriptstyle(h)\!}{\rm e}_a\!\mid\! a\!\in\!\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}\}$ form a
basis of the fusion algebra\ \mbox{$^{\scriptscriptstyle(h)}\!\!{\cal A}$}, and they constitute a partition of the unit
element, in the sense that
\begin{equation} \sumph a {}^{\scriptscriptstyle(h)\!}{\rm e}_a = {}^{\sssh\!}\varphi^{}_\rho \,. \labl{pou}
Also, for any element $\psi\!\in\!\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}$ with
$\psi(h)={\rm e}_a$ and any $\alpha\!\in\!{}^{\scriptscriptstyle(\infty)\!\!}P$ with $\alpha\!\downarrow\!\!\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}$ we have
\begin{equation} {\cal D}_\alpha(\psi) = \delta_{a,{\alpha(h)}} \,. \labl{pea}
We now study how the idempotents ${\rm e}_{{\alpha(h)}}$ behave
under the projection \erf3. First, when $\alpha\!\in\!{}^{\scriptscriptstyle(\infty)\!\!}P$ has a
representative ${\alpha(h)}$ at height $h$, then for every positive
integer $\ell$ we have, using the first of the identities \erf{sS},
\begin{equation} \begin{array}{ll} \ff{\el h}h({\rm e}_{{\alpha(\el h)}}) \!\!
&= {\mbox{$^{\scriptscriptstyle(\el h)}\!S$}_{\rho,{\alpha(\el h)}}} \sumplh A \!\!\mbox{$^{\scriptscriptstyle(\el h)}\!S$}^*_{{\alpha(\el h)},A}\ff{\el h}h(\phi^{}_A)
= {\mbox{$^{\scriptscriptstyle(\el h)}\!S$}_{\rho,\ell{\alpha(h)}}} \sumplh A\, \sumph b \! \mbox{$^{\scriptscriptstyle(\el h)}\!S$}^*_{{\alpha(\el h)},A}
\P Ab\, \varphi^{}_b \\{}\\[-.3em]
&= \ell^{-r/2}\cdot {\mbox{$^{\scriptscriptstyle(h)}\!S$}_{\rho,{\alpha(h)}}} \sumph b\!(\mbox{$^{\scriptscriptstyle(\el h)}\!S$}^* F)_{{\alpha(\el h)},b}
\, \varphi^{}_b
= {\mbox{$^{\scriptscriptstyle(h)}\!S$}_{\rho,{\alpha(h)}}} \sumph{b,c} \! D_{\ell{\alpha(h)},c}\mbox{$^{\scriptscriptstyle(h)}\!S$}^*_{c,b}\, \varphi^{}_b
\\{}\\[-.7em]
&= \mbox{$^{\scriptscriptstyle(h)}\!S$}_{\rho,{\alpha(h)}} \sumph b \mbox{$^{\scriptscriptstyle(h)}\!S$}^*_{{\alpha(h)},b}\, \varphi^{}_b
= {\rm e}_{{\alpha(h)}} \, . \end{array}\labl{ee}
On the other hand, when $\alpha$ has a representative at height $h$, but
not at height $h'$, we can compute as follows. Since $hh'$ is a multiple of
$h$, $\alpha$ has a representative ${\alpha(hh')}$ at height $hh'$. Thus
we can repeat the previous calculation to deduce that
\begin{equation} \begin{array}{ll} \ff{hh'}{h'}({\rm e}_{{\alpha(hh')}}) \!\!&
= \mbox{$^{\scriptscriptstyle(hh')}\!S$}_{\rho,{\alpha(hh')}} \cdot h^{r/2} \sumphp b (D\,\mbox{$^{\scriptscriptstyle(h')}\!S$})_{{\alpha(hh')},b} \,
{}^{\ssshp\!}\varphi^{}_b \\{}\\[-.7em] & = (h/h')^{r/2}\cdot \mbox{$^{\scriptscriptstyle(h)}\!S$}_{\rho,{\alpha(h)}}
\sumphp{b,c} \delta_{h'{\alpha(h)},hc} \mbox{$^{\scriptscriptstyle(h')}\!S$}_{c,b}\, {}^{\ssshp\!}\varphi^{}_b \,. \end{array} \end{equation}
Now in the sum over $c$ on the right hand side\ one has a contribution only if
$c=h'{\alpha(h)}/h={\alpha(hh')}/h$ is an element of the label set \mbox{$_{}^{\scriptscriptstyle(h')\!\!}P$}\ at height $h'$.
But in this case we would conclude that $\alpha$ has in fact a representative
at height $h'$, namely ${\alpha(h')}=c$, which contradicts our assumption.
Therefore we conclude that in the case under consideration we have
$\ff{hh'}{h'}({\rm e}_{{\alpha(hh')}})=0.$ Together with the result \erf{ee}
it follows that by setting
\begin{equation} {\rm e}_\alpha(h):= \left\{ \begin{array}{ll} 0 & \mbox{if $\alpha\!\in\!{}^{\scriptscriptstyle(\infty)\!\!}P$ has no
representative at height $h$}\,, \\[.1em] {\rm e}_{{\alpha(h)}} & {\rm else}\,,
\end{array} \right. \labl{eal}
we obtain an element ${\rm e}_\alpha$ of the projective limit \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}.
Moreover, according to the relation \erf{pea} the map \erf{dax} acts on
${\rm e}_\alpha\!\in\!\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}$ as
\begin{equation} \mbox{\large(} {\cal D}({\rm e}_\alpha) \mbox{\large)}_\beta = \delta_{\alpha,\beta} \labl{peb}
for all $\alpha,\beta\!\in\!{}^{\scriptscriptstyle(\infty)\!\!}P$, and the ${\rm e}_\alpha$ provide a partition of the unit
element, analogously as in \erf{pou},
\begin{equation} \sum_{\alpha\in{}^{\scriptscriptstyle(\infty)\!\!}P} {\rm e}_\alpha = \psi_\circ \,; \end{equation}
here the sum is to be understood as a limit of finite sums in the \topo.
Now for each $h\iN\II$ let us define the map \,$g_h\!:\, {\cal X}\to\mbox{$^{\scriptscriptstyle(h)}\!\!{\cal A}$}$ by
$(\xi_\alpha)_{\alpha\in{}^{\scriptscriptstyle(\infty)\!\!}P}\mapsto \sum_{\scriptstyle\beta\in{}^{\scriptscriptstyle(\infty)\!\!}P
\atop \scriptstyle\beta\downarrow_{}^{\scriptscriptstyle(h)\!}P} \xi_\beta\, {\rm e}_{{\beta(h)}}$.
Since $\beta\!\downarrow\!\!\mbox{$_{}^{\scriptscriptstyle(\el h)\!\!}P$}$ \,if\, $\beta\!\downarrow\!\!\mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}$, we then have
\begin{equation} \ff{\el h}h \circ g_{\ell h}^{}((\xi_\alpha)) =
\sum_{\scriptstyle\beta\in{}^{\scriptscriptstyle(\infty)\!\!}P \atop \scriptstyle\beta\downarrow_{}^{\scriptscriptstyle(\el h)\!}P}\xi_\beta\,
{\rm e}_{\beta}(h) =
\sum_{\scriptstyle\beta\in{}^{\scriptscriptstyle(\infty)\!\!}P \atop \scriptstyle\beta\downarrow_{}^{\scriptscriptstyle(h)\!}P}\xi_\beta\,
{\rm e}_{{\beta(h)}} = g_h((\xi_\alpha)) \end{equation}
for all positive integers $\ell$. Analogously we can define a map
\begin{equation} g:\quad {\cal X} \to \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\,,\qquad (\xi_\alpha)_{\alpha\in{}^{\scriptscriptstyle(\infty)\!\!}P}^{}\mapsto
\!\!\sum^{}_{\beta\in{}^{\scriptscriptstyle(\infty)\!\!}P} \xi_\beta\,{\rm e}_\beta \end{equation}
with similar properties. As a consequence of the relation \erf{peb} one finds
that this map satisfies
\begin{equation} {\cal D} \circ g = {\sf id}_{{\cal X}}^{} \,. \end{equation}
This implies that the injective map ${\cal D}$ is also surjective (and that
$g$ is injective). Thus we have proven the isomorphism \erf{dax}.
\subsection{Semi-simplicity}
It is known \cite{kawA} that the fusion algebra s \mbox{$^{\scriptscriptstyle(h)}\!\!{\cal A}$}\ at finite heights $h$
are semi-simple\ associative algebra s. In this subsection we show that in a suitable
topological sense the same statement holds for the projective limit \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}, too.
We first combine the identity \erf{eee} and the definition \erf{eal}
of the element ${\rm e}_\alpha$ of \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\ with the fact that the idempotents
${\rm e}_a$ form a basis of \mbox{$^{\scriptscriptstyle(h)}\!\!{\cal A}$}. This way we learn that for all $\psi\!\in\!
\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}$ and all heights $h\iN\II$ the fusion product $({\rm e}_\alpha\star\psi)(h)=
{\rm e}_\alpha(h)\star\psi(h)$ is proportional to ${\rm e}_\alpha(h)$.
Thus for each $\alpha\!\in\!{}^{\scriptscriptstyle(\infty)\!\!}P$ the span
\begin{equation} {\cal I}_\alpha := \langle{\rm e}_\alpha\rangle \labl{Id}
of $\{{\rm e}_\alpha\}$ is a one-dimensional\ twosided ideal of the projective limit, i.e.\ we
have $\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\,{\cal I}_\alpha={\cal I}_\alpha\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\subseteq{\cal I}_\alpha$.
We claim that when we endow the algebra\ with the \topo, then in fact \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\
is the closure of the direct sum of the ideals \erf{Id} in this topology:
\begin{equation} \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$} = \overline{\bigoplus_{\alpha\in{}^{\scriptscriptstyle(\infty)\!\!}P} {\cal I}_\alpha} \,. \labl{clos}
(In particular, the idempotents ${\rm e}_\alpha$ form a topological basis of \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}.)
To prove this, we first recall from subsection \ref{topo} that in the \topo\
each open set in \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\ is a union of elements of the set
$\mbox{$\Omega$}=\{ \efm h(M) \!\mid\! h\!\in\!\mbox{$I$},\; M\;{\rm open\;in}\;\mbox{$^{\scriptscriptstyle(h)}\!\!{\cal A}$}\}$
of all pre-images of all open sets in any of the fusion algebra s \mbox{$^{\scriptscriptstyle(h)}\!\!{\cal A}$}.
Here we assume that we have already chosen a topology on each of the fusion
algebra s \mbox{$^{\scriptscriptstyle(h)}\!\!{\cal A}$}. (Actually the choice of this topology on \mbox{$^{\scriptscriptstyle(h)}\!\!{\cal A}$}\ will not be important;
for definiteness, we may take the discrete one, as in the case of fusion rings,
or also the metric topology of \mbox{$^{\scriptscriptstyle(h)}\!\!{\cal A}$}\ as a finite-dimensional\ complex vector space.)
Consider now an arbitrary element $\xi\!\in\!\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}$, which because of the
isomorphism $\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\cong{\cal X}$ we can write as $\xi=(\xi_\alpha)_{\alpha\in
{}^{\scriptscriptstyle(\infty)\!\!}P}$. Adopting some definite numbering ${}^{\scriptscriptstyle(\infty)\!\!}P=\{{\alpha_m}\!\mid\! m\!\in\!{\dl N}\}$
of the countable set ${}^{\scriptscriptstyle(\infty)\!\!}P$, for $n\!\in\!{\dl N}$ we define
\begin{equation} \hat\xi_n := \sum_{m\le n} \xi_{\alpha_m} {\rm e}_\alpham \ \in\; \bigoplus_{\alpha\in
{}^{\scriptscriptstyle(\infty)\!\!}P} {\cal I}_\alpha \,. \end{equation}
To prove our assertion, we must then show that
for every $h\iN\II$ and every open set $M\subseteq\mbox{$^{\scriptscriptstyle(h)}\!\!{\cal A}$}$ which satisfy
$f_h(\xi)\!\in\! M$ we have $\hat\xi_n\!\in\! f_h^{-1}(M)$, i.e.\
\begin{equation} f_h(\hat\xi_n) \in M \,, \end{equation}
for all but finitely many $n$. Now by direct calculation we obtain
\begin{equation} f_h(\hat\xi_n) = \sum_{\scriptstyle m\le n \atop\ \scriptstyle{\alpha_m}\downarrow_{}^{\scriptscriptstyle(h)\!}P}
\!\! \xi_{\alpha_m} {}^{\scriptscriptstyle(h)\!}{\rm e}_{\alpham(h)} \,; \end{equation}
this is a finite sum, and for sufficiently large $n$ it becomes independent
of $n$ because only finitely many $\alpha\!\in\!{}^{\scriptscriptstyle(\infty)\!\!}P$ have a representative
in \mbox{$_{}^{\scriptscriptstyle(h)\!\!}P$}. In fact, for sufficiently large $n$ we simply have
\begin{equation} f_h(\hat\xi_n) = \sum_{\alpha\downarrow_{}^{\scriptscriptstyle(h)\!}P} \xi_\alpha\, {}^{\scriptscriptstyle(h)\!}{\rm e}_{\alpha(h)}
\equiv f_h(\xi) \,. \end{equation}
Since $f_h(\xi)\!\in\! M$, this immediately shows that indeed $f_h(\hat\xi_n) \!\in\! M$
for almost all $n$, and hence the proof is completed. (Note that the fact that
$f_h(\hat\xi_n)$ ultimately becomes equal to $f_h(\xi)$ holds for any chosen
topology of \mbox{$^{\scriptscriptstyle(h)}\!\!{\cal A}$}, and hence the conclusion is indeed independent of that
topology.)
\subsection{Simple and semi-simple\ modules}\label{Sss}
The representation\ theory of \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\ can now be developed by following the
same steps as in the representation\ theory of semi-simple\ algebras.
However, when considering modules $V$ over \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$},
it is natural to restrict one's attention from the outset to
{\em continuous\/} modules, i.e.\ to modules which
are topological vector spaces and on which the representation\ of \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\ is continuous
(in particular, every element of \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\ is represented by a continuous map).
We will do so, and suppress the qualification `continuous' from now on.
The one-dimensional\ ideals ${\cal I}_\alpha$ are simple modules over \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\ under the (left
or right) regular representation. Our first result is now that these one-dimensional\ modules
already provide us
with all simple modules, i.e.\ that every simple \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}-module $L$ satisfies
\begin{equation} L \,\cong\,{\cal I}_\alpha \end{equation}
for some $\alpha\!\in\!{}^{\scriptscriptstyle(\infty)\!\!}P$.
To show this, we first observe that if $L\not\cong{\cal I}_\alpha$, then
${\cal I}_\alpha L = 0$. Namely, since ${\cal I}_\alpha$ is an ideal of \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}, we have
$\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\,{\cal I}_\alpha L\subseteq{\cal I}_\alpha L$; thus ${\cal I}_\alpha L$ is a submodule
of $L$, which by the simplicity of $L$ implies that either
${\cal I}_\alpha L = L$ or ${\cal I}_\alpha L=0$. In the former case,
${\cal I}_\alpha L=L$, we can find a vector $y\!\in\! L$ such that the space
${\cal I}_\alpha y$ is not zero-dimensional. Indeed, because of
$\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\,{\cal I}_\alpha\,y\subseteq {\cal I}_\alpha\,y\subseteq L$
this space is a submodule of $L$, and hence by the simplicity of $L$ it must
be equal to $L$. It follows that the map from ${\cal I}_\alpha$ to $L$ defined by
$\lambda \mapsto \lambda y$ is surjective. Since $L$ is simple, by Schur's lemma
this implies that it is even an isomorphism. This shows that $L\cong{\cal I}_\alpha$
when ${\cal I}_\alpha L=L$, and hence ${\cal I}_\alpha L=0$ when $L\not\cong{\cal I}_\alpha$.\\
Suppose now that $L$ is a non-zero simple module and is not isomorphic to any
${\cal I}_\alpha$. Then $\bigoplus_\alpha{\cal I}_\alpha L = 0$; since $L$ is a
continuous module, we can take the closure of this relation, so as to find that
$\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\,L =0$. But we have $L\subseteq\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\,L$, and hence
this would imply that $L=0$, which
is a contradiction. Hence we learn that indeed, up to isomorphism,
the ideals ${\cal I}_\alpha$ of \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\ exhaust all the simple modules over \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}.
Next we consider modules $V$ over \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\ which can be obtained
from families of simple modules. Similarly as in \cite[\S XVII.2]{LAng}
one can show that the following conditions are equivalent:
\begin{quote}
$(i)$~~~~$V$ is the closure of the sum of a family of simple submodules.\\[.3em]
$(ii)$~~~$V$ is the closure of the direct sum of a family of simple submodules.
\\[.3em]
$(iii)$\hsp{2.85}Every closed submodule $W$ of $V$ is a direct summand, i.e.\
there exists\\\hsp{10.48}a closed submodule $W'$ such that $V = W \oplus W'$.
\end{quote}
Any (continuous) module fulfilling these equivalent conditions
will be referred to as a {\emsemi-simple\/} module.
The equivalence of $(i)$\,$\mbox{-\hspace{-.66 mm}-}$\,$(iii)$ is proven as follows.
First, if $V=\overline{\sum_{i\in J}L_i}$ is the closure of a (not necessarily
direct) sum of simple submodules $L_i$, denote by $J'$ a maximal subset of
$J$ such that $V':=\sum_{j\in J'}L_j$ is a direct sum. Since the
intersection of $V'$
with any of the simple modules $L_i$ is a submodule of $L_i$, the maximality
of $J'$ implies that $i\!\in\! J'$ and hence in fact $J'=J$.
Thus $(i)$ implies $(ii)$. \\
Second, if $W$ is a submodule of $V$, let $J''$ be the maximal subset of
$J$ such that the sum $W+\sum_{j\in J''}L_j$ is direct. Then the same
arguments as before show that $V = \overline{W \oplus \bigoplus_{j\in J''}L_j}$.
If, furthermore, $W$ is closed, then it follows that $V = \overline{W}\oplus
\overline{\bigoplus_{j\in J''}L_j} = W\oplus\overline{\bigoplus_{j\in J''}L_j}$.
This shows that$(ii)$ implies $(iii)$.\,%
\futnote{It is indeed necessary to require $W$ to be closed. Consider e.g.\
the case $V=\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}$ and $W=\bigoplus_\alpha{\cal I}_\alpha$.
The submodule $W$ is neither closed nor does it have a complement.}
\noindent Third, assume that $V$ is a non-zero module which satisfies $(iii)$,
and let $v$ be a non-zero vector in $V$.
The kernel of the homomorphism $\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\to\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\, v$ is a closed ideal of \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$},
which in turn is contained in a maximal closed ideal ${\cal J}\!\subset\!\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}$
that is strictly contained in \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}. One then has $V={\cal J}v \oplus W$
and $\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\,v={\cal J}v \oplus (W\cap\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\, v)$ with some submodule
$W\subset V$. Now $W\cap\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\, v$ is simple because ${\cal J}v$ is maximal
in $\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\, v$; thus $V$ contains a simple submodule. Next let $V' \ne 0 $ be
the submodule of $V$ that is the closure of the sum of all simple
submodules of $V$. If $V'$ were not all of $V$, then one would have
$V=V'\oplus V''$ with $V''\ne0$; but by the same reasoning as before,
$V''$ then would contain a simple submodule, in contradiction to the definition
of $V'$. Thus $V'=V$, so we see that $(iii)$ implies $(i)$.
\subsection{Arbitrary modules}
With the characterization of semi-simple\ modules above, we are now in a position
to study arbitrary modules of \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}, in an analogous manner as
in \cite[\S XVII.4]{LAng}. Let us first assume that $W$ is a closed submodule
of a semi-simple\ module $V$, and denote by $W'$ the closure of the
direct sum of all simple submodules of $W$. Then there is a submodule
$V'$ of $V$ such that $V= W'\oplus V'$. Every $w\!\in\! W$ can be uniquely
written as $w=w'+v'$ with $w'\!\in\! W'$ and $v'\!\in\! V'$. Because of
$v'=w-w'\!\in\! W$ we thus have $W=W'\oplus(W\cap V')$. The module
$W\cap V'$ is a closed submodule of $W$. If it were non-zero, it would
therefore (by the same reasoning as in the proof of `$(iii)\to(i)$' in
subsection \ref{Sss}) contain a simple submodule, in contradiction with
the definition of $W$. Thus we learn that $W=W'$, or in other words:
\begin{quote}
Every closed submodule of a semi-simple\ \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}-module is semi-simple.
\end{quote}
\noindent
Next we consider again a closed submodule $W$ of a semi-simple\ module $V$, and
investigate the quotient module $V/W$.
There is a closed submodule $W'$ such that $V$ is the direct sum
$V=W\oplus W'$. Now the projection $V \to V/W$ induces a continuous
isomorphism from $W'$ to $V/W$. Furthermore, according to the result
just obtained, $W'$ is semi-simple. Thus we have shown:
\begin{quote}
Every quotient module of a semi-simple\ \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}-module with respect to \\
a closed submodule is semi-simple.
\end{quote}
\noindent Now
any arbitrary \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}-module can be regarded as a quotient module of a suitable
free module modulo a closed submodule. Moreover, every free \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}-module is the
closure of a direct sum of countably many copies of \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\ and hence is a semi-simple\
module. The two previous results therefore imply:
\begin{quote}
Every \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}-module is semi-simple.
\end{quote}
Finally we consider again an arbitrary \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}-module $V$. We denote by $V_\alpha$
the closure of the direct sum of all those submodules of $V$ which are
isomorphic to the simple \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}-module ${\cal I}_\alpha$. Since each
simple module over \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\ is isomorphic to some
${\cal I}_\alpha$, any simple submodule of $V$ is contained in $V_\beta$ for some
$\beta\!\in\!{}^{\scriptscriptstyle(\infty)\!\!}P$. Now every \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}-module is semi-simple\ and hence the closure
of the direct sum of its simple submodules. Thus we learn that
\begin{equation} V = \overline{ \bigoplus_{\alpha\in{}^{\scriptscriptstyle(\infty)\!\!}P} V_\alpha } \, . \end{equation}
Moreover, we have ${\rm e}_\beta V_\alpha=\delta_{\alpha,\beta}V_\alpha$ for
all $\alpha,\beta\!\in\!{}^{\scriptscriptstyle(\infty)\!\!}P$, and hence
\begin{equation} V_\alpha = {\rm e}_\alpha V = {\cal I}_\alpha V \, . \end{equation}
As a consequence, we see that:
\begin{quote}
Every \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}-module $V$ can be written as
\begin{equation} V = \overline{\bigoplus_{\alpha\in{}^{\scriptscriptstyle(\infty)\!\!}P} {\cal I}_\alpha V} =
\overline{\bigoplus_{\alpha\in{}^{\scriptscriptstyle(\infty)\!\!}P} {\rm e}_\alpha V} \,, \end{equation}
and for each $\alpha\!\in\!{}^{\scriptscriptstyle(\infty)\!\!}P$ the submodule ${\cal I}_\alpha V$ is the closure of
the direct sum of all submodules of $V$ that are isomorphic to ${\cal I}_\alpha$.
\end{quote}
\noindent
We can conclude that the structure of any arbitrary (continuous)
module over \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\ is known explicitly, i.e.\ we have developed the full
(topological) representation\ theory of \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}.
\subsection{Diagonalization}
{}From the definition \erf{eal} of ${\rm e}_\alpha\!\in\!\mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}$ and the basic property
\erf{eee} of the idempotents ${\rm e}_a\!\in\!\mbox{$^{\scriptscriptstyle(h)}\!\!{\cal A}$}$ it follows that the elements
${\rm e}_\alpha$ of \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\ are again idempotents:
\begin{equation} {\rm e}_\alpha \star {\rm e}_\beta = \delta_{\alpha,\beta}\,{\rm e}_\alpha \end{equation}
for all $\alpha,\beta\!\in\!{}^{\scriptscriptstyle(\infty)\!\!}P$.
In other words, by the basis transformation from the distinguished basis
\mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\ of the fusion algebra\ \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\ to the basis of idempotents one
diagonalizes the fusion rules of \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$},
precisely as in the case of the algebra s \mbox{$^{\scriptscriptstyle(h)}\!\!{\cal A}$}\ at finite level.
Indeed, by combining the definitions \erf{ea} and \erf{eal} we can describe
the transformation from \mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\ to the basis of idempotents ${\rm e}_\alpha$ explicitly.
Namely, for any $\psi=(\psi(h))_{h\inI}\!\in\!\mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}$ we have
\begin{equation} \psi = \sum_{\alpha\in{}^{\scriptscriptstyle(\infty)\!\!}P}\! {}^{\sss(\infty)\!}Q_{\psi,\alpha} \, {\rm e}_\alpha \,, \end{equation}
with
\begin{equation} {}^{\sss(\infty)\!}Q_{\psi,\alpha}:= \frac{\mbox{$^{\scriptscriptstyle(h)}\!S$}_{\psi(h),{\alpha(h)}}}{\mbox{$^{\scriptscriptstyle(h)}\!S$}_{\rho,{\alpha(h)}}}
\,, \end{equation}
where $h\iN\II$ is a height at which $\alpha$ has a representative. Note that
owing to the relation \erf{s/s} the quotient ${}^{\sss(\infty)\!}Q_{\psi,\alpha}$
does not depend on the particular choice of $h$.
(This just rephrases the fact that the map $\cal D$ is an isomorphism.)
For any $\psi,\psi'\!\in\!\mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}$ we thus have
\begin{equation} \psi\star\psi'= \sum_{\alpha\in{}^{\scriptscriptstyle(\infty)\!\!}P}
\frac{\mbox{$^{\scriptscriptstyle(h)}\!S$}_{\psi(h),{\alpha(h)}}} {\mbox{$^{\scriptscriptstyle(h)}\!S$}_{\rho,{\alpha(h)}}}
\frac{\mbox{$^{\scriptscriptstyle(h)}\!S$}_{\psi'(h),{\alpha(h)}}}{\mbox{$^{\scriptscriptstyle(h)}\!S$}_{\rho,{\alpha(h)}}} \, {\rm e}_\alpha
= \sum_{\chi\in {}^{\scriptscriptstyle(\infty)\!}{\cal B}} \mbox{\Large(} \!\!\sum_{\alpha\in{}^{\scriptscriptstyle(\infty)\!\!}P}
{}^{\sss(\infty)\!}Q_{\psi,\alpha} {}^{\sss(\infty)\!}Q_{\psi',\alpha} \, {}^{\sss(\infty)\!}Q^-_{\alpha,\chi}\, \mbox{\Large)} \, \chi \,, \end{equation}
where ${}^{\sss(\infty)\!}Q^-$ is the matrix for the inverse basis transformation,
\begin{equation} {\rm e}_\alpha = \sum_{\psi\in {}^{\scriptscriptstyle(\infty)\!}{\cal B}} {}^{\sss(\infty)\!}Q^-_{\alpha,\psi}\, \psi \,. \end{equation}
In other words, the fusion rule coefficient s of the projective limit \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}\ can be written as
\begin{equation} _{}^{\scriptscriptstyle(\infty)}\!\!\n\psi{\psi'}{\psi''} = \sum_{\alpha\in{}^{\scriptscriptstyle(\infty)\!\!}P}
{}^{\sss(\infty)\!}Q_{\psi,\alpha}\, {}^{\sss(\infty)\!}Q_{\psi',\alpha}\, {}^{\sss(\infty)\!}Q^-_{\alpha,\psi''} \, . \end{equation}
This is nothing but the analogue of the Verlinde formula \cite{verl2} that
is valid for the fusion rule coefficient s of the fusion algebra s \mbox{$^{\scriptscriptstyle(h)}\!\!{\cal A}$}.
Note that already at finite height $h$ the two indices which label the rows
and columns, respectively, of the matrix $\mbox{$^{\scriptscriptstyle(h)}\!S$}$ which diagonalizes the fusion rules
are a priori of a rather different nature. Namely, one of them labels the
elements of the distinguished basis \mbox{$^{\scriptscriptstyle(h)}\!{\cal B}$}, while the other labels the
inequivalent one-dimensional\ irreducible representation s of \mbox{$^{\scriptscriptstyle(h)}\!\!{\cal A}$}. It is a quite non-trivial
\futnot{and not well understood}
property of the fusion algebra s which arise in rational conformal field theory\
(and is a prerequisite for the modularity of those fusion algebra s)
that nevertheless the diagonalizing matrix can be chosen such that it is
symmetric, so that in particular the two kinds of labels can be treated on an
equal footing \cite{jf24}.
Our results clearly display that this nice feature of the finite height
fusion algebra s \mbox{$^{\scriptscriptstyle(h)}\!\!{\cal A}$}\ is not shared by their non-rational limit \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}; in the
case of \mbox{$_{}^{\scriptscriptstyle(\infty)\!\!\!}{\cal A}$}, there seems to be no possibility to identify the two sets
\mbox{$^{\scriptscriptstyle(\infty)\!\!}{\cal B}$}\ and ${}^{\scriptscriptstyle(\infty)\!\!}P$ which label the elements of the distinguished basis and
the one-dimensional\ irreducible representation s, respectively, with each other.
On the other hand, our results show that the projective limit \mbox{$^{\scriptscriptstyle(\infty)\!}{\cal R}$}\ that
we constructed in this paper still possesses all those structural properties
of a modular fusion ring which can reasonably be expected to survive in
the limit of infinite level.
\def\,\linebreak[0]} \def\wB {$\,$\wb{\,\linebreak[0]} \def\wB {$\,$\,\linebreak[0]} \def\wB {$\,$\wb}
\newcommand\Bi[1] {\bibitem{#1}}
\newcommand\Erra[3] {\,[{\em ibid.}\ {#1} ({#2}) {#3}, {\em Erratum}]}
\newcommand\BOOK[4] {{\em #1\/} ({#2}, {#3} {#4})}
\newcommand\J[5] {\ {\sl #5}, {#1} {#2} ({#3}) {#4} }
\newcommand\Prep[2] {{\sl #2}, preprint {#1}}
\defJ.\ Fuchs {J.\ Fuchs}
\defFortschr.\wb Phys. {Fortschr.\,\linebreak[0]} \def\wB {$\,$\wb Phys.}
\defHelv.\wb Phys.\wB Acta {Helv.\,\linebreak[0]} \def\wB {$\,$\wb Phys.\wB Acta}
\newcommand\npbF[5] {{\sl #5}, Nucl.\wb Phys.\ B\ {#1} [FS{#2}] ({#3}) {#4}}
\defNucl.\wb Phys.\ B (Proc.\wb Suppl.) {Nucl.\,\linebreak[0]} \def\wB {$\,$\wb Phys.\ B (Proc.\,\linebreak[0]} \def\wB {$\,$\wb Suppl.)}
\defNuovo\wB Cim. {Nuovo\wB Cim.}
\defNucl.\wb Phys.\ B {Nucl.\,\linebreak[0]} \def\wB {$\,$\wb Phys.\ B}
\defPhys.\wb Lett.\ B {Phys.\,\linebreak[0]} \def\wB {$\,$\wb Lett.\ B}
\defJ.\wb Math.\wb Phys. {J.\,\linebreak[0]} \def\wB {$\,$\wb Math.\,\linebreak[0]} \def\wB {$\,$\wb Phys.}
\defCom\-mun.\wb Math.\wb Phys. {Com\-mun.\,\linebreak[0]} \def\wB {$\,$\wb Math.\,\linebreak[0]} \def\wB {$\,$\wb Phys.}
\defAlgebra {Algebra}
\def{Academic Press} {{Academic Press}}
\def{Addi\-son\hy Wes\-ley} {{Addi\-son$\mbox{-\hspace{-.66 mm}-}$ Wes\-ley}}
\defalgebra {algebra}
\def{Berlin} {{Berlin}}
\def{Birk\-h\"au\-ser} {{Birk\-h\"au\-ser}}
\def{Cambridge} {{Cambridge}}
\def{Cambridge University Press} {{Cambridge University Press}}
\deffusion rule {fusion rule}
\def{Gordon and Breach} {{Gordon and Breach}}
\newcommand{\inBO}[7] {in:\ {\em #1}, {#2}\ ({#3}, {#4} {#5}), p.\ {#6}}
\defInfinite-dimensional {Infinite-dimensional}
\def{Kluwer Academic Publishers} {{Kluwer Academic Publishers}}
\def{New York} {{New York}}
\defQuantum\ {Quantum\ }
\defquantum group {quantum group}
\def{Sprin\-ger Verlag} {{Sprin\-ger Verlag}}
\defsym\-me\-tries {sym\-me\-tries}
\deftransformation {transformation}
\defWZW\ {WZW\ }
\vskip5em
\small
\noindent{\bf Acknowledgement.} \ We are grateful to I.\ Kausz and B.\ Pareigis
for helpful comments.
\newpage
| proofpile-arXiv_065-433 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
This paper is devoted to the spectral decomposition of the space
of sections $L^2(Y,V_Y(\gamma))$ of a locally homogenous bundle $V_Y(\gamma)$ over a locally symmetric
space $Y$ of rank one of infinite volume with respect to locally invariant
differential operators.
Let $G$ be a real semisimple linear connected Lie group of real rank one
with finite center. Let $K\subset G$ be a maximal compact subgroup.
Then $X:=G/K$ is a Riemannian symmetric space of negative curvature.
Let $\Gamma\subset G$ be a convex-cocompact, torsion-free, discrete
subgroup and $Y:=\Gamma\backslash X$ be the corresponding locally symmetric space.
Let $(\gamma,V_\gamma)$ be a finite-dimensional unitary representation
of $K$. Then we form the (locally)
homogeneous vector bundle $V(\gamma):=G\times_KV_\gamma$ ($V_Y(\gamma):=\Gamma\backslash G\times_KV_\gamma$).
Let ${\bf g}$ denote the Lie algebra of $G$,
${\cal U}({\bf g})$ the universal enveloping algebra of ${\bf g}$ and ${\cal Z}$ its center.
Through the left regular action of ${\cal U}({\bf g})$ on $C^\infty(X,V(\gamma))$
any $A\in {\cal Z}$ gives rise to an $G$-invariant differential
operator $A_\gamma$. This operator descends to $C^\infty(Y,V_Y(\gamma))$.
Let ${\cal Z}_\gamma$ denote the algebra $\{A_\gamma\:|\: A\in {\cal Z}\}$.
By $\nabla$ we denote the canonical invariant connection
of $V(\gamma)$.
The algebra ${\cal Z}_\gamma$ is a finite extension of the algebra ${\bf C}(\Delta)$, where $\Delta=\nabla^*\nabla$ is the
Bochner Laplace operator on $V(\gamma)$.
We employ a suitable invariant scalar product on ${\bf g}$ in order to normalize the Riemannian metric of $X$ and the Casimir operators
$\Omega_G$ and $\Omega_K$ of $G$ and $K$. The Casimir operators are related with the Laplacian by $\Delta=-\Omega_G+\gamma(\Omega_K)$.
We view ${\cal Z}_\gamma$ as an algebra of unbounded operators
on the Hilbert spaces $L^2(X,V(\gamma))$ (resp. $L^2(Y,V_Y(\gamma))$) with common domain $C_c^\infty(X,V(\gamma))$ (resp. $C_c^\infty(Y,V_Y(\gamma))$).
Since $X$ (resp. $Y$) are complete any formally selfadjoint locally invariant elliptic operator is essentially selfadjoint on the domain $C^\infty_c(X,V(\gamma))$ (resp. $C_c^\infty(Y,V_Y(\gamma))$).
Note that $\Omega_G$ is elliptic and formally selfadjoint.
Using this it is easy to see that the algebra ${\cal Z}_\gamma$ can be generated
by essentially selfadjoint elements with commuting resolvents.
Thus there exist spectral decompositions
of $L^2(X,V(\gamma))$ (resp. $L^2(Y,V_Y(\gamma))$) with respect to ${\cal Z}_\gamma$
(which can equivalently be considered as a spectral decompositions with respect to ${\cal Z}$ as we will do).
We will also consider spectral decompositions of these Hilbert spaces
with respect to finite abelian extensions of ${\cal Z}_\gamma$ which are obtained
by adjoining further essentially selfadjoint differential operators.
Since ${\cal Z}_\gamma$ is a quotient of ${\cal Z}$ we can parametrize
characters of ${\cal Z}_\gamma$ using the Harish-Chandra isomorphism.
Let ${\bf h}\subset {\bf g}$ denote a Cartan algebra of ${\bf g}$,
$W=W({\bf g},{\bf h})$ the Weyl group, and ${\bf h}_{\bf C}^*$ the complexified
dual of ${\bf h}$.
The Harish-Chandra isomorphism identifies characters of
${\cal Z}$ with points in ${\bf h}^*_{\bf C}/W$. Let $\lambda\in{\bf h}^*_{\bf C}$
represent some $W$-orbit. Then we denote the corresponding character
of ${\cal Z}$ by $\chi_\lambda$.
The abstract spectral decomposition gives a measurable
field of Hilbert spaces $\{H_\lambda\}_{\lambda\in {\bf h}^*_{\bf C}/W}$,
a measure $\kappa$ on ${\bf h}^*_{\bf C}/W$, and an isometry
$$\alpha:L^2(Y,V_Y(\gamma))\cong \int_{{\bf h}^*_{\bf C}/W} H_\lambda \kappa(d\lambda)\ .$$
If we let ${\cal Z}$ act on $H_\lambda$ by the character $\chi_\lambda$,
then $\alpha$ is compatible with the action of ${\cal Z}$.
It is of course apriori known that $\kappa$ is supported
on the set of $\lambda$ with the property that $\chi_\lambda$
factors through ${\cal Z}_\gamma$.
We want to describe this set in greater detail.
The sphere bundle of $X$ can be identified with the homogeneous space
$G/M$, where $M\subset K$.
Let ${\bf m}$ denote the Lie algebra of $M$.
We choose a Cartan algebra ${\bf t}$ of ${\bf m}$ and let ${\bf a}$ be a one-dimensional subspace of the orthogonal complement of ${\bf k}$ in ${\bf g}$.
Then ${\bf a}\oplus{\bf t}=:{\bf h}$ is a Cartan algebra of ${\bf g}$.
We further choose a positive root system of ${\bf t}$.
Let $\rho_m$ denote half of the sum of the positive roots of $({\bf m},{\bf t})$.
For $\sigma\in \hat{M}$ let $\mu_\sigma\in{\bf t}^*$ be its highest weight.
Let ${\aaaa_\C^\ast}$ denote the complexification of the dual of ${\bf a}$.
A pair $(\sigma\in \hat{M},\lambda\in{\aaaa_\C^\ast})$ determines the character
$\chi_{\mu_\sigma+\rho_m-\lambda}$ of ${\cal Z}$.
Then representation theory of $G$ implies that
$${\mbox{\rm supp}}(\kappa)\subset \{\chi_{\mu_\sigma+\rho_m-\lambda}|[\gamma_{|M}:\sigma]\not=0,\lambda\in{\aaaa_\C^\ast}\}\ .$$
In fact there are more restrictions since
characters contributing to the spectral decomposition must be selfadjoint.
This restriction implies that if
$\chi_{\mu_\sigma+\rho_m-\lambda}\in{\mbox{\rm supp}}(\kappa)$, then $\lambda$ has to be either real or
imaginary. Thus we apriori know that the support of $\kappa$ is contained
in the projection to ${\bf h}^*_{\bf C}/W$ of a finite union of lines in ${\bf h}^*_{\bf C}$.
Speaking about the absolute-continuous part of the spectrum, we have in mind that the measure $\kappa$ restricted to the corresponding one-dimensional
set is absolute continuous with respect to the one-dimensional Lebesgue measure.
We do not employ this apriory knowledge about the support of $\kappa$ in the proofs, it rather follows from our arguments.
The goal of this paper is to describe
in detail spectral decomposition of $L^2(Y,V_Y(\gamma))$ with respect to
the algebra ${\cal Z}$.
The Eisenstein series is used to
identify the absolute continuous
spectrum. We then show that $\kappa$ has no singular continuous
component. Finally obtain the finiteness of the point spectrum
and a description of all infinite-dimensional eigenspaces.
In analogy to the spectral decomposition in the finite volume case
we obtain a decomposition
$$L^2(Y,V_Y(\gamma))=L^2(Y,V_Y(\gamma))_c\oplus L^2(Y,V_Y(\gamma))_{res}\oplus L^2(Y,V_Y(\gamma))_{cusp}\oplus L^2(Y,V_Y(\gamma))_{scat} .$$
Here $L^2(Y,V_Y(\gamma))_c$ is the continuous part given by wave packets
of Eisenstein series. The space $L^2(Y,V_Y(\gamma))_{res}$ is the finite-dimensional residual part which is essentially
generated by the residues of Eisenstein series. The cuspidal
part $L^2(Y,V_Y(\gamma))_{cusp}$ consists of a finite number of
infinite dimensional eigenspaces and is related to the discrete
series representations of $G$ occuring in $L^2(X,V(\gamma))$.
The scattering part $L^2(Y,V_Y(\gamma))_{scat}$ consists
of finite-dimensional eigenspaces at the boundary of the continuous spectrum.
In contrast to the finite-volume case this part can be non-trivial
as we demonstrate by an example.
The motivation for studying the spectral decomposition with respect to ${\cal Z}$
(and larger commutative algebras)
instead of $\Delta$ is that these algebras encode additional symmetries.
If only the spectral decomposition of $L^2(Y,V_Y(\gamma))$ with respect
to the Laplacian $\Delta$ is considered, then one encounters
embedded eigenvalues. Their "stability"
is explained by the additional symmetries since
they are isolated with respect to the larger algebras.
For locally symmetric manifolds of the sort considered in present paper
the spectral decomposition of $L^2(Y,V_Y(\gamma))$ with respect to the Laplacian
(respectively partial results) were obtained by
\begin{itemize}
\item Patterson \cite{patterson75} for trivial $\gamma$ and surfaces
\item Lax-Phillips \cite{laxphillips82},
\cite{laxphillips841}, \cite{laxphillips842}, \cite{laxphillips85}, Perry \cite{perry87} for higher dimensional hyperbolic manifolds and trivial $\gamma$
\item Mazzeo-Phillips
\cite{mazzeophillips90} for differential forms on hyperbolic manifolds
\item Epstein-Melrose-Mendoza \cite{epsteinmelrosemendoza91}, Epstein-Melrose \cite{epsteinmelrose90} for differential forms on complex-hyperbolic manifolds.
\end{itemize}
There is related work on Eisenstein series and the scattering matrix in the real hyperbolic case for trivial $\gamma$ (e.g.
\cite{patterson76}, \cite{patterson761}, \cite{patterson89}, \cite{mandouvalos86}, \cite{mandouvalos89}, \cite{perry89}).
At the end of this introduction let us make some remarks concerning the methods.
Once the Eisenstein series are constructed the
realization of the absolute continuous part of the spectrum is almost standard.
One of the most important steps in proving the spectral decomposition is
to show the absence of the singular continuous spectrum.
Usually, the limiting absorption principle (e.g. \cite{perry87}) or commutator methods (see e.g. \cite{froesehislopperry91}) are employed at this point. Here we use a completely different method (proposed in \cite{bernstein88}) which is based on an apriori knowledge of all relevant generalized eigenfunctions of ${\cal Z}$. Our discussion of the point spectrum is based on the asymtotic expansion
of eigenfunctions and boundary value theory.
Before starting with the main topic of the paper in Section \ref{esess}
we analyse the boundary values of generalized eigenfunctions along
the geodesic boundary $\partial X$ of $X$. In particular we are interested in the
space of $\Gamma$-invariant distributional sections of homogeneous
bundles over $\partial X$ with support in the limit set of $\Gamma$.
Sections \ref{fiert} to \ref{invvv} are devoted to the analysis
on $\partial X$.
Our results also have a more representation theoretic interpretation.
Let $L^2(\Gamma\backslash G)_K$ denote the space of all $K$-finite
vectors on $L^2(\Gamma\backslash G)$. Then combining our results for
all $\gamma\in K$ one can obtain a decomposition of
$L^2(\Gamma\backslash G)_K$ into unitarizable $({\bf g},K)$-modules
and a decomposition of $L^2(\Gamma\backslash G)$ as a direct
integral of unitary representations of $G$. The classification of the unitary dual of $G$ then leads to further restrictions of the location of the residual part of the spectrum.
\noindent
{\it Acknowledgement: We thank R. Mazzeo and P. Perry for discussing
of parts of this work.}
\section{Geometric preparations}\label{fiert}
Let $G$ be a connected, linear, real semisimple Lie group of rank one, $G=KAN$ be an Iwasawa decomposition
of $G$, ${\bf g}={\bf k}\oplus{\bf a}\oplus{\bf n}$ be the corresponding Iwasawa decomposition of the Lie algebra ${\bf g}$,
$M:=Z_K(A)$ be the centralizer of $A$ in $K$ and $P:=MAN$
be a minimal parabolic subgroup.
The group $G$ acts isometrically on the rank-one symmetric space $X:=G/K$.
Let $\partial X:=G/P=K/M$ be its geodesic boundary. We consider $X\cup\partial X$ as a compact manifold with boundary.
By the classification of symmetric spaces with strictly negative
sectional curvature $X$ is one of the following spaces:
\begin{itemize}
\item a real hyperbolic space,
\item a complex hyperbolic space,
\item a quaternionic hyperbolic space,
\item or the Cayley hyperbolic plane,
\end{itemize}
and $G$ is a linear group finitely covering of the orientation-preserving isometry
group of $X$.
Let $\Gamma \subset G$ be a torsion-free, discrete subgroup.
\begin{ass}\label{asss}
We assume that there is a $\Gamma$-invariant partition $\partial X =\Omega\cup \Lambda$, where $\Omega\not=\emptyset$ is open and
$\Gamma$ acts freely and cocompactly on
$X\cup\Omega$.
\end{ass}
The locally symmetric space $Y:=\Gamma\backslash X$ is a complete Riemannian manifold of infinite volume without cusps. It can be compactified by adjoining
the geodesic boundary $B:=\Gamma\backslash \Omega$.
We call $\Lambda$ the limit set of $\Gamma$.
A group $\Gamma$ satisfying \ref{asss} is also called convex-cocompact
since it acts cocompactly on the convex hull of the limit set $\Lambda$.
The quotient $Y$ can be called a Kleinian manifold in generalizing
the corresponding notion for three-dimensional hyperbolic manifolds.
We now consider some geometric consequences of \ref{asss}
which eventually allow us to define the exponent $\delta_\Gamma$ of $\Gamma$.
Let $g=\kappa(g)a(g)n(g)$, $\kappa(g)\in K$, $a(g)\in A$, $n(g)\in N$ be
defined with respect to the given Iwasawa decomposition.
By ${\aaaa_\C^\ast}$ we denote the comlexified dual of ${\bf a}$.
If $\lambda\in {\aaaa_\C^\ast}$, then we set $a^\lambda:={\rm e}^{\langle \lambda,\log(a)\rangle}\in{\bf C}$.
The roots of ${\bf a}$ corresponding to ${\bf n}$ distinguish a positive cone ${\bf a}^*_+$.
Define $\rho\in {\bf a}_+^*$ as usual by $\rho(H):=\frac{1}{2}{\mbox{\rm tr}}({\mbox{\rm ad}}(H)_{|{\bf n}})$, $\forall H\in{\bf a}$.
We adopt the following conventions about the notation for points of $X$ and $\partial X$.
A point $x\in \partial X$ can equivalently be denoted by a subset $kM\subset K$
or $gP\subset G$ representing this point in $\partial X=K/M$ or $\partial X=G/P$.
If $F\subset \partial X$, then $FM:=\bigcup_{kM\in F}kM\subset K$.
Analogously, we can denote $b\in X$ by $gK\subset G$, where $gK$ represents
$b$ in $X=G/K$.
\begin{lem}\label{no}
For any compact $F\subset \Omega$ we have $\sharp(\Gamma\cap FMA_+K)<\infty$.
\end{lem}
{\it Proof.$\:\:\:\:$}
The compact set $FMA_+K\cup F\subset X\cup\Omega$ contains at most a finite
number of points of the orbit $\Gamma K$ of the origin of $X$. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
Let $A_+:=\exp({\bf a}_+)$.
Any element $g\in G$ has a decomposition $g=ka_gh$, $k,h\in K$, $a_g\in A_+\cup \{1\}$, where $a_g$ is uniquely determined by $g$.
\begin{lem}\label{no1}
Let $k_0M \in \partial X$. For any compact $W\subset (\partial X\setminus k_0M)M$ there exists a neighbourhood $U\subset K$ of $k_0M$ and constants $c>0$, $C<\infty$, such that for all $g=ha_gh^\prime \in WA_+K$ and $k\in U$
\begin{equation}\label{unm}
c a_g \le a(g^{-1}k) \le C a_g \ .
\end{equation}
\end{lem}
{\it Proof.$\:\:\:\:$}
The set $W^{-1}k_0M$ is compact and disjoint from $M$.
Let $w\in N_K(M)$ represent the non-trivial element of the Weyl group of $({\bf g},{\bf a})$.
Set $\bar{{\bf n}}=\theta({\bf n})$,
where $\theta$ is the Cartan involution of $G$ fixing $K$ and define $\bar{N}:=exp(\bar{{\bf n}})$.
There is a precompact open $V\subset \bar{N}$
such that $W^{-1}k_0M\subset w \kappa(V)M$.
By enlarging $V$ we can assume that $V$ is $A_+$-invariant, where $A$ acts on $\bar{N}$
by $(a,\bar{n})\mapsto a\bar{n}a^{-1}$.
Moreover, there exists an open neighbourhood $U\subset K$ of $k_0M$
such that $w^{-1}W^{-1}U M\subset \kappa(V)M$.
Let $k\in U$ and $g=ha_gh^\prime \in WA_+K$.
Then we have $h^{-1}k=w\kappa(\bar{n})m $ for $\bar{n}\in V$, $m\in M$.
Furthermore,
\begin{eqnarray*}
a(g^{-1}k)&=&a(h^{\prime -1}a_g^{-1}h^{-1}k)\\
&=& a(a_g^{-1}w\kappa(\bar{n})m) \\
&=&a(a_g \kappa(\bar{n}))\\
&=&a(a_g\bar{n}n(\bar{n})^{-1}a(\bar{n})^{-1})\\
&=&a(a_g\bar{n}a_g^{-1}) a(\bar{n})^{-1} a_g\ .
\end{eqnarray*}
Now $a_g\bar{n}a_g^{-1}\in V$. Set
\begin{eqnarray*}
c&:=&\inf_{\bar{n}\in V} a(\bar{n}) \inf_{\bar{n}\in V} a(\bar{n})^{-1}\\
C&:=&\sup_{\bar{n}\in V} a(\bar{n}) \sup_{\bar{n}\in V} a(\bar{n})^{-1}\ .
\end{eqnarray*}
Since $V$ is precompact we have $0 < c \le C<\infty$.
It follows that $c a_g \le a(g^{-1}k )\le Ca_g\ .$
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{lem}\label{poi}
The series
$$\sum_{g\in\Gamma} a_g^{-2\rho}$$
converges.
\end{lem}
{\it Proof.$\:\:\:\:$}
We represent $\partial X=G/P$. There is a $P$-invariant splitting ${\bf g}=\bar{{\bf n}}\oplus{\bf p}$
and an identification $\bar{{\bf n}}^*={\bf n}$ by the invariant bilinear form on ${\bf g}$. In particular $\Lambda^{max}\bar{{\bf n}}^*=\Lambda^{max}{\bf n}$ as a
$P$-module.
It follows that \begin{equation}\label{sad}\Lambda^{max}T^*\partial X=G\times_P \Lambda^{max}{\bf n} \ . \end{equation}
This bundle is trivial as a $K$-homogeneous bundle.
Fix an orientation of $\partial X$ and let
$\omega\in C^\infty(\partial X, \Lambda^{max}T^*\partial X)$
be a positive $K$-invariant volume form,
which is unique up to a positive scalar factor.
Let $g\in G$. Then $g$ acts as a diffeomorphism on $\partial X$.
Using (\ref{sad}) and the $K$-invariance of $\omega$ we obtain $(g^*\omega)(kM)=a(g^{-1}k)^{-2\rho}\omega(kM)$.
Let $F\subset \Omega$ be a compact set with non-trivial interior such that
$gF\cap F=\emptyset$ for all $1\not=g\in\Gamma$. Such $F$ exists by Assumption \ref{asss}.
Then
\begin{eqnarray*}
\infty&>& \int_{\partial X} \omega
> \int_{\cup_{g\in\Gamma}\: gF} \omega\\
&=& \sum_{g\in\Gamma} \int_{gF} \omega
= \sum_{g\in\Gamma} \int_F g^*\omega\\
&=& \sum_{g\in\Gamma} \int_F a(g^{-1}k)^{-2\rho} \omega(kM).
\end{eqnarray*}
Let $F_1\subset \Omega$ be a compact neighbourhood of $F$.
By Lemma \ref{no} the set $\Gamma\cap F_1MA_+K$ is finite.
We apply the Lemma \ref{no1} taking for $W$ the closure of
$\Gamma\setminus (\Gamma\cap F_1MA_+K)$. Then we can cover $FM$
with finitly many sets $U$ the existence of which was asserted
in that lemma.
Thus here is a constant $C\in A$ such that for all $g\in \Gamma\setminus (\Gamma\cap F_1MA_+K)$ and $k\in FM$
$$a(g^{-1}k)\le Ca_g\ .$$
It follows that
\begin{eqnarray*}
\sum_{g\in\Gamma} a_g^{-2\rho} \int_F \omega&=& \sum_{g\in\Gamma} \int_F a_g^{-2\rho} \omega\\&
\le& C^{2\rho} \sum_{g\in\Gamma\setminus(\Gamma\cap F_1 MA_+K)} \int_F a(g^{-1}kM)^{-2\rho} \omega(kM)\ \ + \sum_{g\in\Gamma\cap F_1MA_+K} a_g^{-2\rho}\\[0.5cm]
&<& \infty\ .
\end{eqnarray*}
This implies the lemma since
$$ \int_F \omega\not=0\ .$$
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{ddd}
Let $\delta_\Gamma\in {\bf a}^*$ be the smallest element such that $\sum_{g\in\Gamma} a_g^{- \rho-\lambda}$ converges for all $\lambda\in{\bf a}^*$ with $\lambda>\delta_\Gamma$.
\end{ddd}
$\delta_\Gamma$ is called the exponent of
$\Gamma$. By Proposition \ref{poi} we have $\delta_\Gamma \le \rho$.
If $\Gamma$ is non-trivial, then $\delta_\Gamma\ge -\rho$.
For the trivial group we have $\delta_{\{1\}}=-\infty$.
If $X$ is the $n$-dimensional hyperbolic space and we identify ${\bf a}^*$ with ${\bf R}$ such that $\rho=\frac{n-1}{2}$, then it was shown by Patterson \cite{patterson762} and Sullivan \cite{sullivan79}, that $\delta_\Gamma+\frac{n-1}{2}=\dim_H(\Lambda)$,
where $\dim_H$ denotes the Hausdorff dimension
(the Hausdorff dimension of the empty set is
by definition $-\infty$). It was also shown in \cite{sullivan79} that in the real hyperbolic case $\sum_{g\in\Gamma} a_g^{- \rho-\delta_\Gamma}$ diverges.
Hence $\delta_\Gamma<\rho$.
In the proof of the meromorphic continuation of the Eisenstein
series we employ at a certain place
that $X$ belongs to a series of symmetric
spaces with increasing dimensions.
It is there where we need the following assumption
\begin{ass}\label{caly}
If $X$ is the Cayley hyperbolic plane, then we assume that $\delta_\Gamma<0$.
\end{ass}
We believe that this assumption is only of technical nature
and can be dropped using other methods for the meromorphic continuation
of the Eisenstein series.
\section{Analytic preparations}\label{anaprep}
The goal of the following two sections is to construct
$\Gamma$-invariant vectors in principal series representations of $G$.
For this reason we introduce the extension map $ext$ and the
scattering matrix, and construct their meromorphic continuations.
The principal series representations of $G$ are realized
on spaces of distribution sections of bundles over $\partial X$.
Roughly speaking the extension map extends a $\Gamma$-invariant
distribution sections on $\Omega$ across the limit set $\Lambda$.
The space of $\Gamma$-invariant distributions on $\Omega$
is easily described as the space of distribution sections
on bundles over $B$. The majority of the $\Gamma$-invariant vectors
of the principal series is then obtained by extension.
Consider a finite-dimensional unitary representation $\sigma$ of $M$ on $V_\sigma$. If $w\in W({\bf g},{\bf a})$ is the non-trivial element
of the Weyl group, then we can form
the representation
$\sigma^w$ of $M$ on $V_\sigma$ by conjugating
the argument of $\sigma$ with a representative of $w$ in $N_K(M)$.
In this section we will assume that $\sigma$ is irreducible.
Note that we will change this convention later in the case that $\sigma$ and $\sigma^w$ are non-equivalent by considering the sum $\sigma\oplus \sigma^w$ instead.
For $\lambda\in {\aaaa_\C^\ast}$ we form the representation $\sigma_\lambda$
of $P$ on $V_{\sigma_\lambda}:=V_\sigma$, which is given by
$\sigma_\lambda(man):=\sigma(m)a^{\rho-\lambda}$.
Let $V(\sigma_\lambda):=G\times_P V_{\sigma_\lambda}$ be the associated
homogeneous bundle.
Set $V_B(\sigma_\lambda):=\Gamma\backslash V(\sigma_\lambda)$.
Let $\tilde{\sigma}$ be the dual representation to $\sigma$.
Then there are natural pairings
\begin{eqnarray*}
V(\tilde{\sigma}_{-\lambda})\otimes V(\sigma_\lambda)&\rightarrow& \Lambda^{max}T^*\partial X\\
V_B(\tilde{\sigma}_{-\lambda})\otimes V_B(\sigma_\lambda)&\rightarrow& \Lambda^{max}T^*\partial B\ .
\end{eqnarray*}
The orientation of $\partial X$ induces one of $B$.
Employing these pairings and integration with respect to the fixed
orientation we obtain identifications
\begin{eqnarray*}
C^{-\infty}(\partial X,V(\sigma_\lambda))&=&C^{\infty}(\partial X,V(\tilde{\sigma}_{-\lambda}))^\prime\\
C^{-\infty}(B,V_B(\sigma_\lambda))&=&C^{\infty}(B,V_B(\tilde{\sigma}_{-\lambda}))^\prime\ .
\end{eqnarray*}
As a $K$-homogeneous bundle we have a canonical identification $V(\sigma_\lambda)\cong K\times_M V_\sigma$. Thus $\bigcup_{\lambda\in{\aaaa_\C^\ast}} V(\sigma_\lambda)\rightarrow {\aaaa_\C^\ast}\times \partial X$
has the structure of a trivial holomorphic family of bundles.
Let $\pi^{\sigma,\lambda}$ denote the representation of $G$ on the space of sections of $V(\sigma_\lambda)$ given by the left-regular representation.
Then $\pi^{\sigma,\lambda}$ is called a principal series representation
of $G$. Note that there are different globalizations of this
representation which are distinguished by the regularity
of the sections (smooth, distribution e.t.c.).
For any small open subset $U\subset B$
and diffeomorphic lift $\tilde{U}\subset \Omega$ the restriction $V_B(\sigma_\lambda)_{|U}$
is canonically isomorphic to $V(\sigma_\lambda)_{|\tilde{U}}$.
Let $\{U_\alpha\}$ be a cover of $B$ by open sets as above.
Then
$$\bigcup_{\lambda\in{\aaaa_\C^\ast}} V_B(\sigma_\lambda)\rightarrow {\aaaa_\C^\ast}\times B$$
can be given the structure of a holomorphic family of bundles by glueing the trivial families
$$\bigcup_{\lambda\in{\aaaa_\C^\ast}}V_B(\sigma_\lambda)_{|U}\cong \bigcup_{\lambda\in{\aaaa_\C^\ast}}V(\sigma_\lambda)_{|\tilde{U}}$$ together
using the holomorphic families of glueing maps induced by $\pi^{\sigma,\lambda}(g)$, $g\in\Gamma$.
Thus it makes sense to speak of holomorphic or smooth or continuous families
of sections
${\aaaa_\C^\ast}\ni\mu\mapsto f_\mu\in C^{\pm\infty}(B,V_B(\sigma_\mu))$.
When dealing with holomorphic families of vectors in topological vector
spaces we will employ the following functional analytic facts.
Let ${\cal F},{\cal G},{\cal H} \dots$ be complete locally convex
topological vector spaces.
A locally convex vector space is called a Montel space if its
closed bounded subsets are compact.
A Montel space is reflexive, i.e., the canonical map into its bidual is an isomorphism.
Moreover, the dual space of a Montel space is again a Montel space.
\begin{fact} The space of smooth sections of a vector bundle
and its topological dual are Montel spaces.
\end{fact}
We equip ${\mbox{\rm Hom}}({\cal F},{\cal G})$ with the topology of uniform convergence
on bounded sets.
Let $V\subset {\bf C}$ be open.
A map $f:V\rightarrow {\mbox{\rm Hom}}({\cal F},{\cal G})$ is called holomorphic
if for any $z_0\in V$ there is a sequence $f_i\in {\mbox{\rm Hom}}({\cal F},{\cal G})$ such that
$f(z)=\sum_{n=0}^\infty f_i (z-z_0)^i$ converges for all $z$ close to $z_0$.
Let $f:V\setminus \{z_0\} \rightarrow {\mbox{\rm Hom}}({\cal F},{\cal G})$ be holomorphic
and $f(z)=\sum_{n=-N}^\infty f_i (z-z_0)^i$ for all $z\not=z_0$ close to $z_0$.
Then we say that $f$ is meromorphic and has a pole of order $N$ at $z_0$.
If $f_i$, $i=-N,\dots,-1$, are finite dimensional, then $f$
has, by definition, a finite-dimensional singularity.
We call a subset $A\subset {\cal F}\times {\cal G}^\prime$ sufficient if for
$B\in {\mbox{\rm Hom}}({\cal F},{\cal G})$ the condition
$<\phi,B \psi >=0$, $ \forall (\psi,\phi)\in A$, implies $B=0$.
\begin{fact}\label{holla}
The following assertions are equivalent :
\begin{enumerate}
\item (i) $\:\:f:V\rightarrow {\mbox{\rm Hom}}({\cal F},{\cal G})$ is holomorphic.
\item (ii) $\:\:\: f$ is continuous and
there is a sufficient set $A\subset {\cal F}\times {\cal G}^\prime$
such that for all $(\psi,\phi)\in A$ the function $V\ni z\mapsto \langle \phi,f(z)\psi\rangle $
is holomorphic.
\end{enumerate}
\end{fact}
\begin{fact}\label{seq}
Let $f_i:V\rightarrow Hom({\cal F},{\cal G})$ be a sequence
of holomorphic maps. Moreover let $f :V\rightarrow Hom({\cal F},{\cal G})$ be continuous such that for a sufficient set
$A \subset {\cal F}\times {\cal G}^\prime$ the functions $\langle \phi,f_i \psi\rangle $,
$(\psi,\phi)\in A$,
converge locally uniformly
in $V$ to $\langle\phi,f \psi\rangle$. Then $f$ is holomorphic, too.
\end{fact}
\begin{fact}\label{adjk}
Let $f:V\rightarrow {\mbox{\rm Hom}}({\cal F},{\cal G})$
be continuous.
Then the adjoint $f^\prime:V\rightarrow {\mbox{\rm Hom}}({\cal G}^\prime,{\cal F}^\prime)$
is continuous.
If $f$ is holomorphic, then so is $f^\prime$.
\end{fact}
\begin{fact}\label{comp}
Assume that ${\cal F}$ is a Montel space.
Let $f:V\rightarrow {\mbox{\rm Hom}}({\cal F},{\cal G})$
and $f_1:V\rightarrow {\mbox{\rm Hom}}({\cal G},{\cal H})$
be continuous.
Then $f_1\circ f : V\rightarrow {\mbox{\rm Hom}}({\cal F},{\cal H})$
is continuous.
If $f,f_1$ are holomorphic, so is $f_1\circ f$.
\end{fact}
The following lemma will be employed in Section \ref{extttttt}.
Since it is of purely functional analytic nature we consider it at this place.
Let ${\cal H}$ be a Hilbert space and ${\cal F}\subset {\cal H}$
be a Fr\'echet space such that the embedding is continuous and compact.
In the application we have in mind ${\cal H}$ will be some $L^2$-
space of sections of a vector bundle over a compact closed manifold
and ${\cal F}$ be the Fr\'echet space of smooth sections of this bundle.
The continuous maps ${\mbox{\rm Hom}}({\cal H},{\cal F})$ will be called smoothing operators.
Let $V\subset {\bf C}$ be open and connected, and $V\ni z\rightarrow R(z)\in {\mbox{\rm Hom}}({\cal H},{\cal F})$
be a meromorphic family of smoothing operators with at most finite-dimensional singularities.
Note that $R(z)$ is a meromorphic family of compact operators on ${\cal H}$ in a natural way.
\begin{lem} \label{merofred}
If $1-R(z)$ is invertible at some point $z\in V$ where $R(z)$ is regular,
then
$$(1-R(z))^{-1}=1-S(z)\ ,$$
where $V\ni z\rightarrow S(z)\in {\mbox{\rm Hom}}({\cal H},{\cal F})$
is a meromorphic family of smoothing operators with at most finite dimensional singularities.
\end{lem}
{\it Proof.$\:\:\:\:$}
We apply Reed-Simon IV \cite{reedsimon78}, Theorem XIII.13
in order to conclude that $(1-R(z))^{-1}$ is a meromorphic
family of operators on ${\cal H}$ having at most finite-dimensional singularities.
Making the ansatz $(1-R(z))^{-1}= 1-S(z)$, where apriori $S(z)$ is a meromorphic
familiy of bounded operators on ${\cal H}$ with finite dimensional singularities,
we obtain $S=-R-R\circ S$. This shows that $S$ is a meromorpic family in ${\mbox{\rm Hom}}({\cal H},{\cal F})$.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
This finishes the functional analytic preparations and we now turn to the construction of the extension map. In fact, we first introduce its adjoint
which is the push-down
$$\pi_\ast:C^\infty(\partial X,V(\sigma_\lambda))\rightarrow C^\infty(B,V_B(\sigma_\lambda))\ .$$
Using the identification $C^\infty(B,V_B(\sigma_\lambda))= {}^\Gamma C^\infty(\Omega,V(\sigma_\lambda))$
we define $\pi_\ast$ by
\begin{equation}\label{summ}\pi_*(f)(kM)= \sum_{g\in\Gamma} (\pi(g)f)(kM),\quad kM\in\Omega \ ,\end{equation}
if the sum converges. Here $\pi(g)$ is the action
induced by $\pi^{\sigma,\lambda}(g)$.
Note that the universal enveloping algebra
${\cal U}({\bf g})$ is a filtered algebra. Let ${\cal U}({\bf g})_m$, $m\in{\bf N}_0$,
be the space of elements of degree less or equal than $m$.
For any $m$ and bounded subset $A\subset {\cal U}({\bf g})_m$
we define the seminorm $\rho_{m,A}$ on $C^\infty(\partial X,V(\sigma_\lambda))$
by $$\rho_{m,A}(f):=\sup_{X\in A, k\in K}|f(\kappa(kX))|\ .$$
These seminorms define the Fr\'echet topology of $C^\infty(\partial X,V(\sigma_\lambda))$
(in fact a countable set of such seminorms is sufficient).
In order to define the Fr\'echet topology on $C^\infty(B,V_B(\sigma_\lambda))$
we fix an open cover $\{U_\alpha\}$ of $B$ such that each $U_\alpha$ has a diffeomorphic
lift $\tilde{U}_\alpha \subset\Omega$.
Then we have canonical isomorphisms
$$C^\infty(\tilde{U}_\alpha ,V(\sigma_\lambda))\cong C^\infty(\ U_\alpha,V_B(\sigma_\lambda))\ .$$
For any $U\in \{U_\alpha\}$ we define the topology of $C^\infty(\tilde{U},V(\sigma_\lambda))$
using the seminorms
$$\rho_{U,m,A}(f):=\sup_{X\in A, k\in \tilde{U}M }|f(\kappa(kX))|\ ,$$
where $m\in{\bf N}_0$ and $A\subset {\cal U}({\bf g})_m$ is bounded.
Since $C^\infty(B,V_B(\sigma_\lambda))$ maps to $C^\infty( U_\alpha,V_B(\sigma_\lambda))$
by restriction for each $\alpha$ we obtain a system of seminorms defining the Fr\'echet topology of $C^\infty(B,V_B(\sigma_\lambda))$.
\begin{lem}\label{anal1}
If ${\rm Re }(\lambda)<-\delta_\Gamma$, then the sum (\ref{summ}) converges for $f\in C^\infty(\partial X,V(\sigma_\lambda))$ and defines
a continuous map $$\pi_\ast: C^\infty(\partial X,V(\sigma_\lambda))\rightarrow C^\infty(B,V_B(\sigma_\lambda))\ .$$
Moreover, $\pi_*$ depends holomorphically on $\lambda$.
\end{lem}
{\it Proof.$\:\:\:\:$}
Consider $U\in \{U_\alpha\}$. We want to estimate
$$C^\infty(\partial X,V(\sigma_\lambda))\ni f\mapsto res_{\tilde{U}}\circ \pi(g) f \in
C^\infty(\tilde{U},V(\sigma_\lambda))\ .$$
Let $\Delta:{\cal U}({\bf g})\rightarrow {\cal U}({\bf g})\otimes {\cal U}({\bf g})$
be the coproduct and write $\Delta(X)=\sum_i X_i\otimes Y_i$.
Fix $l\in {\bf N}_0$ and a bounded set $A\in {\cal U}({\bf g})_l$.
Then there is another bounded set $A_1\subset {\cal U}({\bf g})_l$
depending on $A$ such that
\begin{eqnarray*}
\rho_{U,l,A}(res_{\tilde{U}M}\circ \pi(g) f)&=&\sup_{X\in A,k\in \tilde{U}}|(\pi(g)f)(\kappa(kX))|\\
&=& \sup_{X\in A,k\in \tilde{U}M}|\sum_ i a(g^{-1}\kappa(kX_i))^{\lambda-\rho}f(\kappa(g^{-1}\kappa(kY_i)))|\\
&\le &
\sup_{X\in A_1,k\in \tilde{U}M}| a(g^{-1}kX)^{\lambda-\rho}| \sup_{X\in A_1,k\in \tilde{U}M}| f(\kappa(g^{-1}kX))| \ .
\end{eqnarray*}
The Poincar\'e-Birkhoff-Witt theorem gives a decomposition ${\cal U}({\bf g})={\cal U}(\bar{{\bf n}}){\cal U}({\bf m}){\cal U}({\bf a})\oplus {\cal U}({\bf g}){\bf n}$.
Let $q:{\cal U}({\bf g})\rightarrow {\cal U}(\bar{{\bf n}}){\cal U}({\bf m}){\cal U}({\bf a})$
be the associated projection. Then for $g\in G$ and $X\in{\cal U}({\bf g})$ we have
$\kappa(gX)=\kappa(gq(X))$, $a(gX)=a(gq(X))$.
Let $U_1\subset \Omega$ be an open neighbourhood of $\tilde{U}$.
Then by Lemma \ref{no} the intersection $\Gamma\cap U_1MA_+K$ is finite.
Let $W:=(\partial X\setminus U_1)M$. Then by Lemma \ref{no1}
we can find a compact $A_+$-invariant set $V\subset \bar{{\bf n}}$
such that $W^{-1}\tilde{U}M\subset w\kappa(V)M$.
For $g=ha_gh^\prime\in WA_+K$ and $k\in \tilde{U}M$ we obtain $h^{-1}k=w\kappa(\bar{n})m$ for some $\bar{n}\in V$, $m\in M$.
Let $X\in {\cal U}({\bf g})$. Then
\begin{eqnarray*}
\kappa(g^{-1}kX)&=&\kappa(h^{\prime -1}a_g^{-1}h^{-1}kX)\\
&=&h^{\prime -1}\kappa(a_g^{-1}w\kappa(\bar{n})mX)\\
&=&h^{\prime -1}w\kappa(a_g\bar{n}n(\bar{n})^{-1}a(\bar{n})^{-1}mX) \\
&=&h^{\prime -1}w\kappa(a_g\bar{n}a_g^{-1} a_g[n(\bar{n})^{-1}a(\bar{n})^{-1}mXm^{-1}a(\bar{n})n(\bar{n})]a_g^{-1})m\\
&=&h^{\prime -1}w\kappa(a_g\bar{n}a_g^{-1} a_gq(n(\bar{n})^{-1}a(\bar{n})^{-1}mXm^{-1}a(\bar{n})n(\bar{n}))a_g^{-1})m\ .
\end{eqnarray*}
Since $V$ is compact the sets $n(V)^{-1}a(V)^{-1}MA_1Ma(V)n(V)=:A_2\subset {\cal U}({\bf g})_l$
and $q(A_2)$ are bounded.
Conjugating $q(A_2)$ with $A_+$ gives
clearly another bounded set $A_3\subset {\cal U}({\bf g})_l$.
We can find a bounded set $A_4\subset {\cal U}({\bf g})_l$ such that
$\kappa(a_g\bar{n}a_g^{-1}A_3)\subset \kappa(\kappa(a_g\bar{n}a_g^{-1})A_4)$
for all $a_g\in A_+$.
This implies for $g\in WA_+K$ that
\begin{equation}\label{kkll1}\sup_{X\in A_1,k\in \tilde{U}}| f(\kappa(g^{-1}kX))|\le \rho_{l,A_4}(f)\ .\end{equation}
We also have
\begin{eqnarray*}
a(g^{-1}kX)&=&a(h^{\prime -1}a_g^{-1}h^{-1}kX)\\
&=& a(a_g^{-1}w\kappa(\bar{n})mX)\\
&=&a(a_g \kappa(\bar{n})mXm^{-1})\\
&=&a(a_g\bar{n}n(\bar{n})^{-1}a(\bar{n})^{-1}mXm^{-1})\\
&=&a(a_g\bar{n}a_g^{-1}a_g n(\bar{n})^{-1}a(\bar{n})^{-1}mXm^{-1}a(\bar{n})n(\bar{n})a_g^{-1}) a(\bar{n})^{-1} a_g\ .\\
&=&a(a_g\bar{n}a_g^{-1}a_g q(n(\bar{n})^{-1}a(\bar{n})^{-1}mXm^{-1}a(\bar{n})n(\bar{n}))a_g^{-1}) a(\bar{n})^{-1} a_g\ .
\end{eqnarray*}
Again there is a constant $C<\infty$ such that
$$ |a(a_g\bar{n}a_g^{-1}a_g q(n(\bar{n})^{-1}a(\bar{n})^{-1}mXm^{-1}a(\bar{n})n(\bar{n}))a_g^{-1})^{\lambda-\rho}| a(\bar{n})^{ \rho- \lambda } <C$$
for all $a_g\in A_+$, $\bar{n}\in V$, $m\in M$, and $X\in A_1$.
It follows that
\begin{equation}\label{hj1}\sup_{X\in A_1,k\in \tilde{U}M}| a(g^{-1}kX)^{\lambda-\rho}|\le C a_g^{\lambda-\rho}\end{equation}
for almost all $g\in \Gamma$.
The estimates (\ref{kkll1}) and (\ref{hj1}) together imply that the sum
$$C^l(\partial X,V(\sigma_\lambda))\ni f\mapsto \sum_{g\in\Gamma} res_{\tilde{U}}\circ \pi(g) f \in C^l(\tilde{U},V(\sigma_\lambda))$$
converges for ${\rm Re }(\lambda)<-\delta_\Gamma$
and defines a continuous map of Banach spaces.
This map depends holomorphically on $\lambda$
by Fact \ref{seq}.
Combining these considerations for all
$U\in \{U_\alpha\}$ and $l\in{\bf N}_0$ we obtain that
$$\pi_*:C^\infty(\partial X,V(\sigma_\lambda))\rightarrow C^\infty(B,V_B(\sigma_\lambda))$$
is defined and continuous for ${\rm Re }(\lambda)<-\delta_\Gamma$.
Moreover it is easy to see that $\pi_*$ depends holomorphically
on $\lambda$. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
Still postponing the introduction of the extension map we
now consider its left-inverse, the restriction
$$res:{}^\Gamma C^{-\infty}(\partial X,V(\sigma_\lambda))\rightarrow C^{-\infty}(B,V_B(\sigma_\lambda))\ .$$
In fact the space ${}^\Gamma C^{-\infty}(\Omega,V(\sigma_\lambda))$
of $\Gamma$-invariant vectors in $C^{-\infty}(\Omega,V(\sigma_\lambda))$
can be canonically identified with the corresponding space
$C^{-\infty}(B,V_B(\sigma_\lambda))$.
Composing this identification with the restriction
$res_\Omega:C^{-\infty}(\partial X,V(\sigma_\lambda))\rightarrow C^{-\infty}(\Omega,V(\sigma_\lambda))$
we obtain the required restriction map $res$.
\begin{lem}\label{lopi}
There exists a continous map
$$\widetilde{res}: C^{-\infty}(\partial X ,V(\sigma_\lambda))\rightarrow C^{-\infty}(B,V_B(\sigma_\lambda))\ ,$$
which depends holomorphically on $\lambda$ and coincides with $res$ on
${}^\Gamma C^{-\infty}(\Omega,V(\sigma_\lambda))$.
\end{lem}
{\it Proof.$\:\:\:\:$}
We exhibit $\widetilde{res}$ as the adjoint of a continuous map
$$\pi^*:C^\infty(B,V_B(\tilde{\sigma}_{-\lambda}))\rightarrow C^\infty (\partial X ,V(\tilde{\sigma}_{-\lambda}))$$
which depends holomorphically on $\lambda$. Then the lemma follows from Fact
\ref{adjk}.
Let $\{U_\alpha\}$ be a finite open cover of $B$ such that each $U_\alpha$ has a diffeomorphic
lift $\tilde{U}_\alpha \subset\Omega$. Choose a subordinated partition of unity
$\{\chi_\alpha\}$. Pulling $\chi_\alpha$ back to $\tilde{U}_\alpha$ and extending the resulting function by $0$ we obtain a function $\tilde \chi_\alpha\in C^\infty (\partial X ,V(\tilde{\sigma}_{-\lambda}))$. We define
$$\pi^*(f):=\sum_\alpha \tilde\chi_\alpha f, \quad f\in C^\infty(B,V_B(\tilde{\sigma}_{-\lambda}))\ ,$$
where we consider $f$ as an element of
${}^\Gamma C^{-\infty}(\partial X,V(\tilde{\sigma}_{-\lambda}))$.
Then we set $\widetilde{res}:=(\pi^*)^\prime$. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
The extension map $ext$ will be defined as an right inverse to $res$.
\begin{ddd}\label{defofext}
For ${\rm Re }(\lambda)>\delta_\Gamma$ we define
the extension map
$$ext: C^{-\infty}(B,V_B(\sigma_\lambda))\rightarrow {}^\Gamma C^{-\infty}(\partial X,V(\sigma_\lambda))$$
to be the adjoint of
$$\pi_*:C^{ *}(\partial X,V(\tilde{\sigma}_{-\lambda})) \rightarrow C^{*}(B,V_B(\tilde{\sigma}_{-\lambda}))\ .$$
\end{ddd}
This definition needs a justification. In fact,
by Lemma \ref{anal1} the extension exists, is continuous, and by Fact \ref{adjk} it depends holomorphically on $\lambda$. Moreover, it is easy to see that the range of the adjoint of $\pi_*$ consists of $\Gamma$-invariant vectors.
\begin{lem}\label{iden}
We have
$res\circ ext = {\mbox{\rm id}}$. \end{lem}
{\it Proof.$\:\:\:\:$}
Recall the definition of $\pi^*$ from the proof of Lemma \ref{lopi}. Then
$\pi_*\pi^*$ is the identity on $C^\infty(B,V_B(\tilde{\sigma}_{-\lambda}))$.
We obtain
$$res \circ ext=\widetilde{res}\circ ext= (\pi^*)^\prime \circ (\pi_*)^\prime
=(\pi_*\pi^*)^\prime={\mbox{\rm id}}\ .$$
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
Let $C^{-\infty}(\Lambda,V(\sigma_\lambda))$ denote the space of distribution sections of $V(\sigma_\lambda)$ with support in the limit set $\Lambda$.
Since $\Lambda$ is $\Gamma$-invariant
$C^{-\infty}(\Lambda,V(\sigma_\lambda))$ is a subrepresentation of
the representation $\pi^{\sigma,\lambda}$ of $\Gamma$ on $C^{-\infty}(\partial X,V(\sigma_\lambda))$.
Note that ${}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))$ is exactly the kernel of $res$.
\begin{lem}\label{mainkor}
If ${}^\Gamma C^{-\infty}(\Lambda ,V(\sigma_\lambda))=0$ and if $ext$ is defined, then we have $ext\circ res ={\mbox{\rm id}}$.
\end{lem}
{\it Proof.$\:\:\:\:$}
The assumption implies that $res$ is injective.
By Lemma \ref{iden} we have
$res(ext\circ res - {\mbox{\rm id}})=0$. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
In order to apply this lemma we have to check its assumption.
In the course of the paper we will prove several vanishing results for
${}^\Gamma C^{-\infty}( \Lambda ,V(\sigma_\lambda))$.
One of these is available at this point.
\begin{lem}\label{pokm}
If ${\rm Re }(\lambda)>0$ and ${\rm Im}(\lambda)\not=0$, then
${}^\Gamma C^{-\omega}( \Lambda ,V(\sigma_\lambda))=0$.
\end{lem}
{\it Proof.$\:\:\:\:$}
We employ the Poisson transform.
Let $\gamma$ be a finite-dimensional representation of $K$ on $V_\gamma$
such that there exists an injective $T\in {\mbox{\rm Hom}}_M(V_\sigma,V_\gamma)$.
We will view sections of $V(\gamma)$ as functions from $G$ to
$V_\gamma$ satisfying the usual $K$-invariance condition.
\begin{ddd}\label{defofpoi} The Poisson transform
$$P:=P^T_\lambda:C^{-\infty}(\partial X,V(\sigma_\lambda))\rightarrow C^\infty(X,V(\gamma))$$
is defined by
$$(P \phi)(g):=\int_K a(g^{-1}k)^{-(\rho+\lambda)}\gamma(\kappa(g^{-1}k)) T \phi(k) dk\ .$$
Here $\phi\in C^{-\infty}(\partial X, V(\sigma_\lambda))$ and the integral is a formal notation
meaning that the distribution $\phi$ has to be applied to the smooth integral kernel.
\end{ddd}
In \cite{olbrichdiss} it is shown that the Poisson transform $P$ intertwines
the left-regular representations of $G$ on $C^{-\infty}(\partial X,V(\sigma_\lambda))$ and $C^\infty(X,V(\gamma))$.
For ${\rm Im}(\lambda)\not=0$ the principal series representation $\pi^{\sigma,\lambda}$ of $G$ on $C^{-\infty}(\partial X, V(\sigma_\lambda))$
is topologically irreducible. Since the Poisson transform $P$ does not vanish
and is continuous it is injective.
There is a real constant $c_\sigma$
(see \cite{bunkeolbrich955})
such that $(-\Omega_G+c_\sigma+\lambda^2)P \phi =0$
(where $\lambda^2 :=\langle \lambda,\lambda\rangle $ with respect to the ${\bf C}$-linear scalar product induced on ${\aaaa_\C^\ast}$ by the the invariant form on ${\bf g}$).
Let $V\subset \partial X$ and $U\subset X$ such that ${\rm clo}(U)\cap V=\emptyset$,
where we take the closure of $U$ in $X\cup\Omega$.
Then for ${\rm Re }(\lambda)>0$ the integral kernel of the Poisson transform
$(g,k)\rightarrow p_\lambda(g,k):= a(g^{-1}k)^{-(\rho+\lambda)} \gamma(\kappa(g^{-1}k))T$
is a smooth function from $VM$ to $L^2(U,V(\gamma))\otimes V_{\tilde{\sigma}}$
(a much more detailed analysis of the Poisson kernel is given below in the proof of Lemma \ref{th43}).
If $\phi\in {}^\Gamma C^{-\infty}( \Lambda ,V(\sigma_\lambda)) $,
then $P \phi $ is $\Gamma$-invariant, and since ${\rm Re }(\lambda)>0$
it descends to a section in $L^2(Y,V_Y(\gamma))$.
Moreover it is annihilated by $(-\Omega_G+c_\sigma+\lambda^2)$.
Since $Y$ is complete $\Omega_G$ is essentially selfadjoint on the domain $C_c^\infty(Y,V_Y(\gamma))$.
Its selfadjoint closure
has the domain of definition
$\{f\in L^2(Y,V_Y(\gamma))| \Omega_Gf\in L^2(Y,V_Y(\gamma))\}$.
In particular, $\Omega_G$ can not have non-trivial eigenvectors in $L^2(Y,V_Y(\gamma))$
to eigenvalues with non-trivial imaginary part.
Since ${\rm Im}(\lambda^2)\not=0$ we conclude that
$P\phi =0$ and hence $\phi=0$ by the injectivity of the Poisson transform.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\section{Meromorphic continuation of $ext$}\label{extttttt}
The extension $ext$ forms a holomorphic
family of maps depending on $\lambda\in{\aaaa_\C^\ast}$
(we have omitted this dependence in order to simplify
the notation) which is defined now for ${\rm Re }(\lambda)>\delta_\Gamma$.
In the present section we consider the meromorphic continuation of $ext$
to (almost) all of ${\aaaa_\C^\ast}$. Our main tool is the scattering matrix
which we introduce below. The scattering matrix for the trivial
group $\Gamma=\{1\}$ is the Knapp-Stein intertwining operator
of the corresponding principal series representation.
We first recall basic properties of the Knapp-Stein intertwining operators.
Then we define the scattering matrix
using the extension and the Knapp-Stein intertwining operators.
We simultaneously obtain the meromorphic continuations of the scattering matrix and the extension map.
If $\sigma$ is a representation of $M$, then
we define its Weyl-conjugate $\sigma^w$ by $\sigma^w(m):=\sigma( w^{-1} mw)$,
where $w\in N_K(M)$ is a representative of the non-trivial element of the Weyl group $\cong {\bf Z}_2$ of $({\bf g},{\bf a})$.
The Knapp-Stein intertwining operators form meromorphic families of $G$-equivariant
operators (see \cite{knappstein71})
$$\hat{J}_{\sigma,\lambda}:C^{*}(\partial X,V(\sigma_\lambda)) \rightarrow C^{*}(\partial X ,V(\sigma^w_{-\lambda})),\quad *= -\infty,\infty\ .$$
Here $\:\hat{}\:$ indicates that $\hat{J}_{\sigma,\lambda}$ is unnormalized.
In order to fix our conventions we give a definition of $\hat{J}_{\sigma,\lambda}$
as an integral operator acting on smooth functions for ${\rm Re }(\lambda)<0$.
Its continuous extension to distributions is obtained by duality.
For ${\rm Re }(\lambda)\ge 0$ it is defined by meromorphic continuation.
Consider $f\in C^{\infty}(\partial X,V(\sigma_\lambda))$
as a function on $G$ with values
in $V_{\sigma_\lambda}$ satisfying the usual invariance condition with respect to $P$.
For ${\rm Re }(\lambda)<0$ the intertwining operator is defined by the
convergent integral
\begin{equation}\label{furunkel}
(\hat{J}_{\sigma,\lambda}f)(g):=\int_{\bar{N}} f(g w\bar{n}) d\bar{n}\ .
\end{equation}
For all irreducible $\sigma\in\hat{M}$ we fix a minimal $K$-type (see \cite{knapp86}, Ch. XV for all that) of the principal series representation $C^{\infty}(\partial X,V(\sigma_\lambda))$.
Let $c_{\sigma}(\lambda)$ be the value of $\hat{J}_{\sigma^w,-\lambda }$
on this minimal $K$-type.
Then $c_{ \sigma}(\lambda)$ is a meromorphic function on ${\aaaa_\C^\ast}$ and we define
the normalized intertwining operators by
$$J_{ \sigma,\lambda}:= c_{ \sigma^w}^{-1}(-\lambda )\hat{J}_{\sigma,\lambda}\ .$$
Let $P_\sigma(\lambda):=c_\sigma(\lambda)^{-1}c_\sigma(-\lambda)^{-1}$
be the Plancherel density.
Then the intertwining operators satisfy the following functional equation.
\begin{equation}\label{spex} \hat{J}_{\sigma,\lambda}\circ \hat{J}_{\sigma^w,-\lambda}=\frac{1}{P_\sigma(\lambda)},\quad \quad J_{\sigma^{w },-\lambda }\circ J_{ \sigma,\lambda}={\mbox{\rm id}}\ .\end{equation}
Our next goal is to show that the intertwining operators form
a meromorphic family of operators in the sense defined in Section \ref{anaprep}.
This is an easy application of the approach to the intertwining operators
developed by Vogan-Wallach (see \cite{wallach92}, Ch. 10). The additional point
we have to verify is that in the domain of convergence of (\ref{furunkel}) the
operators $\hat{J}_{\sigma,\lambda}$ indeed form a continuous family.
\begin{lem}\label{iny1}
For ${\rm Re }(\lambda)<0$ the intertwining operators
$$\hat{J}_{\sigma,\lambda}:C^{\infty}(\partial X,V(\sigma_\lambda)) \rightarrow C^{\infty}(\partial X ,V(\sigma^w_{-\lambda}))$$
form a holomorphic family of continuous operators.
\end{lem}
{\it Proof.$\:\:\:\:$}
Let $X_i$, $i=1,\dots,\dim({\bf k})$, be an orthonormal base of ${\bf k}$.
For any multiindex $r=(i_1,\dots,i_{\dim({\bf k})})$ we set $X_r=\prod_{l=1}^{\dim({\bf k})} X_l^{i_l}$, $|r|=\sum_{l=1}^{\dim({\bf k})} i_l$, and for $f\in C^{\infty}(K,V_{\sigma_\lambda})$
we define the seminorm
$$\|f\|_r=\sup_{k\in K} |f(X_rk)|\ .$$
It is well known that the system $\{\|.\|_r\}$, $r$ running over all multiindices,
defines the Fr\'echet topology of $C^{\infty}(K,V_{\sigma_\lambda})$ and by restriction the topology
of $C^{\infty}(\partial X,V(\sigma_\lambda))$.
We extend $f\in C^{\infty}(K,V_{\sigma_\lambda})$ to a function $f_\lambda$ on $G$ by setting
$f_\lambda(kan):=f(k)a^{\lambda-\rho}$.
Then we can define
$$\hat{J}_{\sigma,\lambda} (f)(k)=\int_{\bar{N}} f_\lambda (k w\bar{n}) d\bar{n}\ .$$
For any $\lambda_0\in {\aaaa_\C^\ast}$ with ${\rm Re }(\lambda)<0$ and $\delta>0$ we can find
an $\epsilon>0$ such that for $|\lambda-\lambda_0|<\epsilon $
$$ \int_{\bar{N}} |a(\bar{n})^{\lambda_0-\rho}-a(\bar{n})^{\lambda-\rho}| d\bar{n} <\delta\ .$$
We then have
\begin{eqnarray*}
\|\hat{J}_{\sigma, \lambda_0} f -\hat{J}_{\sigma,\lambda} f \|_r&=&\sup_{k\in K} \int_{\bar{N}} (f_{\lambda_0} (X_rk w\bar{n}) - (f_{\lambda } (X_rk w\bar{n}) ) d\bar{n}\\
&=& \sup_{k\in K} \int_{\bar{N}} f(X_rkw\kappa(\bar{n})) (a(\bar{n})^{\lambda_0-\rho}-a(\bar{n})^{\lambda-\rho})d\bar{n}\\
&\le& \|f\|_r \int_{\bar{N}} |a(\bar{n})^{\lambda_0-\rho}-a(\bar{n})^{\lambda-\rho}|d\bar{n}\\
&\le& \delta \|f\|_r
\end{eqnarray*}
This immediately implies that $\lambda\mapsto \hat{J}_{\sigma,\lambda}$
is a continuous family of operators on the space of smooth functions.
The fact that the family $\hat{J}_{\sigma,\lambda}$, ${\rm Re }(\lambda)<0$, depends holomorphically on $\lambda$
is now easy to check (apply \cite{wallach92}, Lemma 10.1.3 and Fact \ref{holla}).
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{lem}\label{jjj}
The family of intertwining operators
$$\hat{J}_{\sigma,\lambda}:C^{\infty}(\partial X,V(\sigma_\lambda)) \rightarrow C^{\infty}(\partial X ,V(\sigma^w_{-\lambda}))$$
extends meromorphically to all of ${\aaaa_\C^\ast}$.
\end{lem}
{\it Proof.$\:\:\:\:$}
We employ \cite{wallach92}, Thm. 10.1.5, which states that there are polynomial
maps $b:{\aaaa_\C^\ast}\rightarrow {\bf C}$ and $D:{\aaaa_\C^\ast}\rightarrow {\cal U}({\bf g})^K$,
such that
\begin{equation}\label{shifty}
b(\lambda)\hat{J}_{\sigma,\lambda}=\hat{J}_{\sigma, \lambda-4\rho}\circ \pi^{\sigma,\lambda-4\rho}(D(\lambda))\ .
\end{equation}
This formula requires some explanation.
We identify $$C^{\infty}(\partial X, V(\sigma_\lambda))\cong C^\infty(K,V_\sigma)^M$$
canonically. Then all operators act on the same space $C^\infty(K,V_\sigma)^M$.
If we know that $\hat{J}_{\sigma,\lambda}$ is meromorphic up to ${\rm Re }(\lambda)<\mu$,
then we conclude that
$$\hat{J}_{\sigma,\lambda}= \frac{1}{b(\lambda) } \hat{J}_{\sigma,\lambda-4\rho}\circ \pi^{\sigma,\lambda-4\rho}(D(\lambda))$$
is meromorphic up to ${\rm Re }(\lambda)<\mu+4\rho$.
Thus the lemma follows from Lemma \ref{iny1}.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
We call $\lambda\in {\aaaa_\C^\ast}$ integral if
$$ 2 \frac{\langle \lambda,\alpha\rangle }{\langle\alpha,\alpha\rangle} \in{\bf Z}\quad \mbox{for some root $\alpha$ of $({\bf g},{\bf a})$}\ ,$$
and non-integral otherwise.
The integral points form a discrete lattice in ${\bf a}^*$.
It is known that the poles of $\hat{J}_{\sigma,\lambda}$ have at most order one and appear only at integral $\lambda$ (\cite{knappstein71}, Thm. 3 and Prop. 43).
\begin{lem}\label{off}
Let $\chi,\phi\in C^\infty(\partial X)$ such that ${\mbox{\rm supp}}(\phi)\cap {\mbox{\rm supp}}(\chi)=\emptyset$.
Then $\chi\hat{J}_{\sigma,\lambda}\phi$ is a holomorphic family of smoothing operators.
In particular, the residues of $\hat{J}_{\sigma,\lambda}$ are differential
operators.
\end{lem}
{\it Proof.$\:\:\:\:$}
Since ${\mbox{\rm supp}}(\phi)\cap{\mbox{\rm supp}}(\chi)=\emptyset$,
there exists a compact set $V\subset \bar{N}$ such that
$$\kappa({\mbox{\rm supp}}(\chi)M w (\bar{N}\setminus V)M \subset (\partial X\setminus {\mbox{\rm supp}}(\phi))M\ .$$
For ${\rm Re }(\lambda)<0$ and $f\in C^\infty(\partial X, V(\sigma_\lambda))$
we have (viewing $f$ as a function on $K$ with values in $V_{\sigma_\lambda})$
\begin{eqnarray*}
(\chi \hat{J}_{\sigma,\lambda} \phi f)(k) &=& \int_{\bar{N}} \chi(k)f(\kappa(kw\bar{n}))\phi(\kappa(kw\bar{n})) a(\bar{n})^{ \lambda-\rho}d\bar{n}\\
&=& \int_{ V }\chi(k) f(\kappa(kw\bar{n})) \phi(\kappa(kw\bar{n})) a(\bar{n})^{ \lambda-\rho}d\bar{n}\ .
\end{eqnarray*}
The right-hand side of this equation extends to all of ${\aaaa_\C^\ast}$
and defines a holomorphic family of operators.
This proves the first part of the lemma.
It in particular implies that the residues of $\hat{J}_{\sigma,\lambda}$ are local operators.
Hence the second assertion follows.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
We need the following consequence of Lemma \ref{off}.
Let $W\subset \partial X$ be a closed
subset and let
$${\cal G}_\lambda:=\{f\in C^{-\infty}(\partial X, V(\sigma_\lambda))|
f_{|\partial X\setminus W}\in C^{\infty}(\partial X\setminus W, V(\sigma_\lambda))\}\ .
$$
We equip ${\cal G}_\lambda$ with the weakest topology such that the maps
${\cal G}_\lambda\hookrightarrow C^{-\infty}(\partial X, V(\sigma_\lambda))$
and ${\cal G}_\lambda\rightarrow C^{\infty}(\partial X\setminus W, V(\sigma_\lambda))$
are continuous.
Let $U\subset \bar{U}\subset \partial X\setminus W$ be open.
\begin{lem}\label{dissmo}
The composition
$$res_U \circ \hat{J}_{\sigma,\lambda}:{\cal G}_\lambda\rightarrow C^\infty(U,V(\sigma^w_{-\lambda}))$$
is well-defined and depends meromorphically on $\lambda$.
\end{lem}
We introduce a notational convention concerning $\sigma$.
Below $\sigma$ shall always denote a Weyl-invariant representation
of $M$ which is either irreducible or of the
form $\sigma^\prime\oplus \sigma^{\prime w}$ with $\sigma^\prime$ irreducible
and not Weyl-invariant.
In the latter case $c_\sigma(\lambda):=c_{\sigma^\prime}(\lambda)=c_{\sigma^{\prime w}}(\lambda)$, $P_\sigma(\lambda):=P_{\sigma^\prime}(\lambda)=P_{\sigma^{\prime w} }(\lambda)$.
We omit the subscript $\sigma$ in the notation of the intertwining operators.
We now turn to the definition of the (normalized) scattering matrix
as a family of operators
$$\hat{S}_\lambda \:( S_{\lambda})\: :C^{ *}(B,V_B(\sigma_\lambda))\rightarrow C^{*}(B,V_B(\sigma_{-\lambda })),\quad *=\infty,-\infty \ .$$
\begin{ddd}\label{scatdef}
For ${\rm Re }(\lambda)>\delta_\Gamma$ we define
\begin{equation}\label{scatde}
\hat{S}_{ \lambda} :=res\circ \hat{J}_{\lambda}\circ ext\quad, \quad
S_{ \lambda} :=res\circ J_{\lambda}\circ ext \ .\end{equation}
\end{ddd}
\begin{lem}\label{kkk}
For ${\rm Re }(\lambda)>\delta_\Gamma$
the scattering matrix forms a meromorphic family of operators
$$C^{\pm\infty}(B,V_B(\sigma_\lambda))\rightarrow C^{\pm\infty}(B,V_B(\sigma_{-\lambda }))\ .$$ If $\hat{S}_\lambda$ is singular and ${\rm Re }(\lambda)>\delta_\Gamma$, then $\lambda$ is integral and the residue of $\hat{S}_\lambda$ is a differential operator.
\end{lem}
{\it Proof.$\:\:\:\:$}
The assertion for the scattering matrix acting on distributions follows from
Lemma \ref{jjj}, Lemma \ref{off} and Fact \ref{comp}.
The fact that the scattering matrix restricts to smooth sections follows
from Lemma \ref{dissmo}.\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{lem}\label{lok}
If ${\rm Re }(\lambda)>\delta_\Gamma$, then the adjoint
$${}^tS_{ \lambda}: C^{\infty}(B,V_B(\tilde{\sigma}_\lambda))\rightarrow C^{\infty}(B,V_B(\tilde{\sigma}_{-\lambda }))$$
of
$$S_{ \lambda}: C^{-\infty}(B,V_B(\sigma_\lambda))\rightarrow C^{-\infty}(B,V_B(\sigma_{-\lambda }))$$
coincides with the restriction of $S_\lambda$ to $C^{\infty}(B,V_B(\tilde{\sigma}_\lambda))$.
\end{lem}
{\it Proof.$\:\:\:\:$}
We employ the fact that the corresponding relation holds for the intertwining operators (step (\ref{zx2}) below).
Let $F\subset \Omega$ be a fundamental domain of $\Gamma$ and $\chi\in C_c^\infty(\Omega)$
be a cut-off function with $F\subset {\mbox{\rm supp}}(\chi)$ and $\sum_{g\in\Gamma} g^*\chi\equiv 1$ on $\Omega$.
Let $\phi\in C^{\infty}(B,V_B(\sigma_\lambda))$,
$f\in C^{\infty}(B,V_B(\tilde{\sigma}_\lambda))$,
and consider $\phi$ as a distribution section.
Then
\begin{eqnarray}
\langle \phi ,{}^tS_\lambda f\rangle &=&\langle S_\lambda\phi , f\rangle
= \langle res\circ J_\lambda\circ ext \phi , f\rangle\nonumber\\
&=&\langle J_\lambda\circ ext \phi , \chi f \rangle
= \langle ext \phi , {}^t J_\lambda \chi f \rangle \label{zx1}\\
&=&\langle ext \phi , J_\lambda \chi f \rangle\label{zx2}\\
&=&\langle \chi \phi , \sum_{g\in\Gamma } res_\Omega \circ \pi(g) J_\lambda \chi f\rangle
= \sum_{g\in\Gamma } \langle \chi \phi , res_\Omega \circ \pi(g) J_\lambda \chi f\rangle\label{zx3}\\
&=& \sum_{g\in\Gamma } \langle \chi \phi , J_\lambda \pi(g) \chi f \rangle
= \sum_{g\in\Gamma } \langle \pi(g) J_\lambda \chi \phi , \chi f\rangle\nonumber\\
&=& \langle \phi , S_\lambda f\rangle\ . \label{zx4}
\end{eqnarray}
Here in (\ref{zx1}) and (\ref{zx3}) we view $\chi f$ and $\chi \phi$ as
sections over $\Omega$, respectively, and $\pi(g)$ is induced by the corresponding principal
series representations.
In order to obtain (\ref{zx4}) from the preceding line we do the transformations
backwards with the roles of $\phi$ and $f$ interchanged.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{lem}\label{pfun}
If $|{\rm Re }(\lambda)|<- \delta_\Gamma$,
then the scattering matrix satisfies the functional equation
(viewed as an identity of meromorphic families of operators)
$$S_{-\lambda}\circ S_\lambda = {\mbox{\rm id}}\ .$$
\end{lem}
{\it Proof.$\:\:\:\:$}
We employ Lemmas \ref{mainkor}, \ref{pokm}, and (\ref{spex}) in order to compute for ${\rm Im}(\lambda)\not= 0$, ${\rm Re }(\lambda)\not=0$,
\begin{eqnarray*}
S_{ -\lambda }\circ S_{ \lambda}&=&res\circ J_{ -\lambda }\circ ext\circ res \circ J_{ \lambda}\circ ext\\
&=&res\circ J_{ -\lambda }\circ J_{ \lambda}\circ ext \\
&=&res\circ ext\\
&=& {\mbox{\rm id}}\ .
\end{eqnarray*}
This identity now extends meromorphically to all of $\{|{\rm Re }(\lambda)|< -\delta_\Gamma\}$.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
Now we start with the main topic of the present section, the meromorphic continuation of $ext$.
We first invoke the meromorphic Fredholm theory Lemma \ref{merofred}
in order to provide a meromorphic continuation of the scattering matrix
to almost all of ${\aaaa_\C^\ast}$ under the condition $\delta_\Gamma<0$.
We then use this continuation of the scattering matrix in order to define
the meromorphic continuation of $ext$.
Finally, if $X$ is not the Cayley hyperbolic plane, we drop the assumption
$\delta_\Gamma<0$.
To be more precise our method for the meromorphic continuations
breaks down at a countable number of points.
Therefore we introduce the set
\begin{equation}\label{acagood}{\aaaa_\C^\ast}(\sigma):=\{\lambda\in{\aaaa_\C^\ast}\:|\: {\rm Re }(\lambda)>\delta_\Gamma\:\mbox{or}\:
P_\sigma(\lambda) \hat{J}_{-\lambda}\:\mbox{is regular}\}\ .\end{equation}
Not only the poles of $\hat{J}_{-\lambda}$, but also the poles of $P_\sigma(\lambda)$ are located at integral $\lambda$ (see e.g. the explicit formla (\ref{blanche}) given in the proof of Lemma \ref{casebycase}). Therefore, ${\aaaa_\C^\ast}(\sigma)$ contains all non-integral
points, and many of the integral points, too.
If $\delta_\Gamma<0$, then we show the meromorphic continuation of $ext$
and of the scattering matrix for $\lambda\in {\aaaa_\C^\ast}(\sigma)$.
This result is employed in the proof of Proposition \ref{mystic}.
If $\delta_\Gamma\ge 0$, then for simplicity
we show the meromorphic continuation of
$ext$ and the scattering matrix to the set of all
non-integral $\lambda\in{\aaaa_\C^\ast}$, only.
\begin{prop}\label{part1}
The scattering matrix
$$\hat{S}_\lambda :C^{\pm\infty}(B,V_B(\sigma_\lambda))\rightarrow C^{\pm\infty}(B,V_B(\sigma_{-\lambda }))$$
and the extension map
$$ext: C^{-\infty}(B,V_B(\sigma_\lambda))\rightarrow {}^\Gamma C^{-\infty}(\partial X,V(\sigma_\lambda))$$
have meromorphic continuations to
the set of all non-integral $\lambda\in{\aaaa_\C^\ast}$.
In particular, we have
\begin{equation}\label{forme}
ext = J_{-\lambda} \circ ext \circ S_{\lambda}, \quad S_{-\lambda}\circ S_\lambda = {\mbox{\rm id}}\ .\end{equation}
Moreover,
$ext$ and $S_\lambda$ have at most finite-dimensional singularities
at non-integral $\lambda$.
\end{prop}
{\it Proof.$\:\:\:\:$}
We first assume that $\delta_\Gamma<0$.
We construct the meromorphic continuation of
$$S_\lambda:C^{\infty}(B,V_B(\sigma_\lambda))\rightarrow C^{\infty}(B,V_B(\sigma_{-\lambda }))\ ,$$ and then we extend this continuation to distributions
by duality using Lemma \ref{lok}. The idea is to set $S_\lambda:=S_{-\lambda}^{-1}$ for ${\rm Re }(\lambda)<-\delta_\Gamma$ and to show that $S_{-\lambda}^{-1}$ forms
a meromorphic family.
Let $\{U_\alpha\}$ be a finite open covering of $B$ and let $\tilde{U}_\alpha$ be
diffeomorphic lifts of $U_\alpha$.
Choose a subordinated partition of unity $\phi_\alpha$.
We view $\phi_\alpha$ as a smooth compactly supported function on $\tilde{U}_\alpha$. For $h\in\Gamma$ we set $\phi_\alpha^h(x):=\phi_\alpha(h^{-1}x)$.
Let $1\in L\subset \Gamma$ be a finite subset. Then we define
$\chi\in {}^\Gamma C^\infty(\Omega\times\Omega)$ by
$$\chi(x,y):=\sum_{g\in\Gamma,h\in L,\alpha} \phi_\alpha(gx)\phi_\alpha^h(gy)\ .$$
Let
$$\hat{J}^{diag}_{\lambda}:C^\infty(B, V_B(\sigma_\lambda))\rightarrow C^\infty(B, V_B(\sigma_{-\lambda}))$$ be the meromorphic family
of operators obtained by multiplying
the distributional kernel of $\hat{J}_{ \lambda}$ by $\chi$.
If $f\in C^\infty(B, V_B(\sigma_\lambda))$, then
$$(\hat{J}^{diag}_{ \lambda})f =\sum_{\alpha,h\in L} \phi_\alpha \hat{J}_\lambda(\phi^h_\alpha f)$$
using the canonical identifications.
Below we shall employ the fact that $\hat{J}^{diag}_{ \lambda}$ depends
on $L$.
Let $$U:=\{\lambda\in{\aaaa_\C^\ast}\:|\:{\rm Re }(\lambda)>\delta_\Gamma,\: -\lambda\in{\aaaa_\C^\ast}(\sigma),\: \lambda\:\mbox{non-integral if}\:{\rm Re }(\lambda)\le 0\}\ .$$ Then $U$ is open and
connected.
For $\lambda\in U$ define
\begin{equation}\label{klio}R(\lambda):=P_\sigma(\lambda) \hat{J}^{diag}_{-\lambda} \circ \hat{S}_\lambda - {\mbox{\rm id}}\ .\end{equation}
The inverse of the unnormalized scattering matrix for $\lambda\in U$
should be given by
\begin{equation}\label{finit}\hat{S}_{ \lambda}^{-1}=P_\sigma(\lambda) ({\mbox{\rm id}}+ R(\lambda))^{-1}\circ \hat{J}^{diag}_{ -\lambda }\ .\end{equation}
It exists as a meromorphic family if $({\mbox{\rm id}}+ R(\lambda))^{-1}$ does.
We want to apply the meromorphic Fredholm theory (Proposition \ref{merofred})
in order to invert ${\mbox{\rm id}} +R(\lambda)$ for $\lambda\in U$
and to conclude that $({\mbox{\rm id}}+R(\lambda))^{-1}$
is meromorphic.
We check the assumption of Proposition \ref{merofred}.
We choose a Hermitian metric on $V_B(\sigma_0)$ and a volume form on $\Omega$.
The Hilbert space ${\cal H}$ of Proposition \ref{merofred}
is $L^2(B,V_B(\sigma_0))$ defined using these choices.
The Fr\'echet space ${\cal F}$ is just $C^\infty(B, V_B(\sigma_0))$.
Implicitly, we identify the spaces $C^\infty(B, V_B(\sigma_\lambda))$ with $C^\infty(B, V_B(\sigma_0))$ using a trivialization of the holomorphic
family of bundles $\{V_B(\sigma_\lambda)\}$, $\lambda\in{\aaaa_\C^\ast}$.
We claim that $R(\lambda)$ is a holomorphic family
of smoothing operators on $U$.
If $\lambda\in U$, then
$\hat{J}^{diag}_{-\lambda}$ as well as
$$P_\sigma(\lambda)\hat{S}_\lambda=res\circ P_\sigma(\lambda)\hat{J}_\lambda\circ ext$$ are regular.
Moreover, $R(\lambda)$ is smoothing by Lemma \ref{off}.
Hence $R(\lambda)$ is indeed a holomorphic family of smoothing operators
and this proves the claim.
Next we show that if $L\subset \Gamma$ is sufficiently exhausting, then
${\mbox{\rm id}}+ R(\lambda)$ is injective for some $\lambda\in U$.
Since ${\mbox{\rm id}}+R(\lambda)$ is Fredholm of index zero it is then invertible at this point.
Here is one of the two instances where we assume $\delta_\Gamma<0$.
We fix some non-integral $\lambda\in {\aaaa_\C^\ast}$ with $|{\rm Re }(\lambda)|<-\delta_\Gamma$.
Define $\hat{J}^{off}_{-\lambda}:=res\circ\hat{J}_{-\lambda}-\hat{J}^{diag}_{-\lambda}
\circ res$. By Lemma \ref{off} the composition
$R(\lambda) = - P_\sigma(\lambda) \hat{J}^{off}_{-\lambda} \circ \hat{J}_{\lambda}\circ ext$
is a bounded operator on $C^k(B,V_B(\sigma_\lambda))$.
The proof of
Lemma \ref{anal1} shows that for ${\rm Re }(\lambda)>\delta_\Gamma$
the push down is a continuous map
$\pi_\ast:C^0(\partial X,V(\tilde{\sigma}_{-\lambda}))\rightarrow
C^0(B,V_B(\tilde{\sigma}_{-\lambda}))$.
Thus the adjoint $ext$ restricts to a continuous map between
the Banach spaces of measures of bounded variation
$$ext: M_b(B,V_B(\sigma_\lambda))\rightarrow M_b(\partial X,V(\sigma_\lambda))\ .$$
The operator $\hat{J}_\lambda$ is a singular
integral operator composed with a differential operator
(as explained in Lemma \ref{jjj}).
Thus there is a $k\in{\bf N}_0$ such that
$$\hat{J}_\lambda:M_b(\partial X,V(\sigma_\lambda))\rightarrow C^{k}(\partial X,V(\sigma_{\lambda}))^\prime$$
is continuous.
The scattering matrix $\hat{S}_\lambda$
extends to a continuous operator
$$\hat{S}_\lambda:M_b(B,V_B(\sigma_\lambda))\rightarrow C^{k}(B,V_B(\sigma_{\lambda}))^\prime$$
and, by dualization and Lemma \ref{lok}, to a continuous map
$$\hat{S}_\lambda:C^k(B,V_B(\sigma_\lambda))\rightarrow C^{0}(B,V_B(\sigma_{-\lambda})) \ .$$
\begin{lem}\label{converg1}
If $L$ runs over an increasing sequence of subsets
exhausting $\Gamma$, then
$(\hat{J}^{diag}_{-\lambda}-\hat{S}_{-\lambda})\to 0$
in the sense of bounded operators from $C^{0}(B,V_B(\sigma_{-\lambda}))$
to $C^{k}(B,V_B(\sigma_{\lambda}))$.
\end{lem}
{\it Proof.$\:\:\:\:$}
It follows from the estimates proved in Lemma \ref{anal1} that
$(f\mapsto \sum_{h\in L } \phi^h_\alpha f)$ tends to $(f\mapsto ext(f))$
in the sense of bounded operators between the
Banach spaces $C^0(B,V_B(\sigma_\lambda))$ and
$M_b(\partial X, V(\sigma_\lambda ))$.
We apply Lemma \ref{dissmo} letting $W$ be a compact neighbourhood
of $\Lambda$ and $U\subset \Omega$ contain $\tilde{U}_\alpha$ for all
$\alpha$.
It implies that
$$\sum_{h\in \Gamma\setminus L }\phi_\alpha \hat{J}_{-\lambda} \phi^h_\alpha\to 0$$
in the sense of bounded operators from $C^0(B,V_B(\sigma_{-\lambda}))$
to $C^{k}(\partial X,V(\sigma_\lambda))$ for all $\alpha$.
The assertion of the lemma now follows.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
By Lemmas \ref{converg1}, \ref{pfun} and Equation (\ref{spex}) the operator $R(\lambda)$ tends to zero in the sense of bounded operators on $C^k(B,V_B(\sigma_\lambda))$
when $L$ runs over an increasing sequence of subsets exausting $\Gamma$.
Thus if $L$ is large enough, then ${\mbox{\rm id}}+R(\lambda)$ is injective.
We now have verified the assumptions of Proposition \ref{merofred}.
We conclude that
the family $\hat{S}_{\lambda}^{-1}$ is meromorphic for $\lambda\in U$.
If $\hat{S}_{\lambda}^{-1}$ has a singularity and $\lambda$ is non-integral,
then this singularity is finite-dimensional.
Here is the second and main instance, where we need the assumption $\delta_\Gamma<0$.
Namely, it implies that
$$\{\lambda\in{\aaaa_\C^\ast}\:|\:-\lambda\in U\}\cup \{\lambda\in{\aaaa_\C^\ast}\:|\:{\rm Re }(\lambda)>\delta_\Gamma\}={\aaaa_\C^\ast}(\sigma)\ .$$
Furthermore, by Lemma \ref{pfun} we have $S_\lambda=S_{-\lambda}^{-1}$ on $\{\lambda\in{\aaaa_\C^\ast}\:|\:-\lambda\in U\}\cap \{\lambda\in{\aaaa_\C^\ast}\:|\:{\rm Re }(\lambda)>\delta_\Gamma\}$. Thus, setting
$S_\lambda:=S_{-\lambda}^{-1}$ for $-\lambda\in U$ we obtain a well-defined continuation of $S_\lambda$ to all of ${\aaaa_\C^\ast}(\sigma)$.
By duality this continuation extends to distributions still having the same finite-dimensional
singularities at non-integral points.
It remains to consider the extension map.
We employ the scattering matrix in order to define for
$\lambda\in{\aaaa_\C^\ast}(\sigma)$,
${\rm Re }(\lambda)<-\delta_\Gamma$
$$
ext_1 := J_{-\lambda} \circ ext \circ S_{\lambda} \ .$$
We claim that $ext=ext_1$. In fact since $res$ is injective on an open subset of $\{|{\rm Re }(\lambda)| < -\delta_\Gamma\}$, the computation
\begin{eqnarray*}
res\circ ext_1&=& res\circ J_{-\lambda} \circ ext \circ S_\lambda\\
&=& S_{-\lambda}\circ S_\lambda \\
&=&{\mbox{\rm id}}
\end{eqnarray*}
implies the claim.
We now have constructed a meromorphic continuation of $ext$ to all of
${\aaaa_\C^\ast}(\sigma)$. The relation (\ref{forme})
between the scattering matrix and $ext$ follows by meromorphic continuation. This equation also implies that $ext$ has at most finite-dimensional singularities.
We have finished the proof of Proposition \ref{part1} assuming $\delta_\Gamma<0$.
The identities
$$\hat{S}_\lambda =res\circ \hat{J}_\lambda \circ ext, \quad S_\lambda\circ S_{-\lambda}={\mbox{\rm id}} $$ extend to all of ${\aaaa_\C^\ast}(\sigma)$ by meromorphic continuation.
We now show how to drop the assumption $\delta_\Gamma<0$ using the embedding trick
and Assumption \ref{caly}.
If $X$ is the Cayley hyperbolic plane, then by assumption
$\delta_\Gamma<0$ and the proposition is already proved.
Thus we can assume that $X$ belongs to a series of rank-one symmetric
spaces.
Let $\dots \subset G^n\subset G^{n+1}\subset \dots$ be the corresponding sequence
of real, semisimple, linear Lie groups inducing embeddings of the corresponding
Iwasawa constituents $K^n\subset K^{n+1}$,
$N^n \stackrel{\scriptstyle \subset}{ \scriptstyle \not=} N^{n+1}$, $ M^n\subset M^{n+1}$
such that $A=A^n = A^{n+1}$.
Then we have totally geodesic embeddings of the symmetric spaces
$X^n\subset X^{n+1}$
inducing embeddings of their boundaries
$\partial X^n\subset \partial X^{n+1}$.
If $\Gamma\subset G^n$ satisfies \ref{asss} then it keeps satisfying \ref{asss}
when viewed as a subgroup of $G^{n+1}$.
We obtain embeddings
$\Omega^n\subset \Omega^{n+1}$ inducing
$B^n\subset B^{n+1}$
while the limit set $ \Lambda^n$ is identified with $ \Lambda^{n+1}$.
Let $\rho^n(H)=\frac{1}{2}{\mbox{\rm tr}}({\mbox{\rm ad}}(H)_{|{\bf n}^n})$, $H\in{\bf a}$.
The exponent of $\Gamma$ now depends on $n$ and is denoted by
$\delta_\Gamma^n$.
We have the relation $\delta_\Gamma^{n+1}=\delta_\Gamma^n-\rho^{n+1}+\rho^n$.
Thus $\delta_\Gamma^{n+m}\to -\infty$ as $m\to 0$ and hence
taking $m$ large enough we can satisfy $\delta_\Gamma^{n+m}<0$.
The aim of the following discussion is to show how the meromorphic continuation $ext^{n+1}$ leads to the continuation of $ext^n$.
Let $\sigma^{n+1}$ be a Weyl-invariant representation of $M^{n+1}$. Then it
restricts to a Weyl-invariant representation of $M^n$.
For any given finite-dimensional representation $\sigma^n$ of $M^n$ we can find
a Weyl-invariant representation $\sigma^{n+1}$ of $M^{n+1}$ such that $\sigma^{n+1}_{|M^n}$
contains $\sigma^n$ as a subrepresentation.
The representation $\sigma^{n+1}_\lambda$ of $P^{n+1}$ restricts to the representation
$(\sigma^{n+1}_{|P^n})_{\lambda+\rho^n-\rho^{n+1}}$ of $P^n$.
This induces an isomorphism of bundles
$$V_{B^{n+1}}(\sigma^{n+1}_\lambda)_{|B^n}=V_{B^n}((\sigma^{n+1}_{|P^n})_{\lambda+\rho^n-\rho^{n+1}})\ .$$
We will omit the subscript ${}_{|P^n}$ and the superscript ${}^{n+1}$ of $\sigma$ in the following discussion.
We obtain a push forward of distributions
$$i_*:C^{-\infty}(B^n,V_{B^n}(\sigma_\lambda))\rightarrow C^{-\infty}(B^{n+1},V_{B^{n+1}}(\sigma_{\lambda+\rho^n-\rho^{n+1}}))\ .$$
For $\phi\in C^{-\infty}(B^n,V_{B^n}(\sigma_\lambda))$ the push forward
$i_\ast(\phi)$ has support in $B^n\subset B^{n+1}$.
Then also
${\mbox{\rm supp}}(ext(\phi))\subset \partial X^n$.
We try to define a pull back
$ext^n(\phi):=i^*\circ ext^{n+1}\circ i_*(\phi)$ as follows.
For $f\in C^\infty(\partial X^{n},V(\tilde{\sigma}_{-\lambda}))$ let $\tilde{f}\in C^\infty(\partial X^{n+1},V(\tilde{\sigma}_{-\lambda-\rho^n+\rho^{n+1}}))$
be an arbitrary extension. Then we set
$$\langle ext^n(\phi),f\rangle := \langle ext^{n+1}\circ i_*(\phi_\lambda),\tilde{f}\rangle\ .$$
\begin{lem}\label{well}
$ext^n$ is well defined.
\end{lem}
We must show that this definition does not depend on the choice
of the extension of $\tilde{f}$.
It is sufficient to show that if $h\in C^\infty(\partial X^{n+1},V(\tilde{\sigma}_{-\lambda-\rho^n+\rho^{n+1}}))$
vanishes on $\partial X^{n}$, then $\langle ext^{n+1}\circ i_*(\phi_\lambda),h\rangle=0$.
Embed $\phi,h$ into holomorphic families $\phi_\mu, h_\mu$
, $\phi_\mu\in C^{-\infty}(B^n,V_{B^n}(\sigma_\mu))$, $h_\mu\in C^\infty(\partial X^{n+1},V(\tilde{\sigma}_{-\mu-\rho^n+\rho^{n+1}}))$
such that $(h_\mu)_{|\partial X^{n} }=0$.
Then for ${\rm Re }(\mu)$ large enough\linebreak[4] $\pi^{n+1}_*(h_{-\mu})_{|B^n}=0$
and thus $$\langle ext^{n+1}\circ i_*(\phi_\mu),h_{-\mu} \rangle= \langle i_*(\phi_\mu),\pi^{n+1}_*h_{-\mu}\rangle=0\ .$$
By meromorphic continuation this identity holds for all $\mu$, in particular at $\mu=\lambda$.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
If $\lambda\in{\aaaa_\C^\ast}$ is non-integral, then so is $\lambda+\rho^n-\rho^{n+1}$.
We deduce the properties of $ext^{n}$ from the corresponding
properties of $ext^{n+1}$.
In particular, $ext^n$ is continuous, meromorphic, and has at most finite-dimensional singularities at $\lambda$ if $ext^{n+1}$ has these properties at $\lambda+\rho^n-\rho^{n+1}$.
We define the meromorphic continuation of the scattering matrix by
(\ref{scatde}).
Then it is easy to see that the scattering matrix has the properties
as asserted.
This finishes the proof of Proposition \ref{part1}.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{lem}\label{lead}
If $ext$ is meromorphic at $\lambda\in {\aaaa_\C^\ast}$ and ${}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))=0$, then $ext$\
is regular at $\lambda$.
\end{lem}
{\it Proof.$\:\:\:\:$}
If $ext$ is meromorphic, then for any holomorphic family $\mu\to\phi_\mu\in C^{-\infty}(B,V_B(\sigma_\mu))$ the leading singular part of
$ext(\phi_\mu)$ at $\mu=\lambda$ belongs to ${}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))$.
In fact $res\circ ext(\phi_\mu)=\phi_\mu$ has no singularity.
If ${}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))=0$,
then $ext(\phi_\mu)$ is regular at $\mu=\lambda$ for any holomorphic family $\mu\to \phi_\mu$.\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
In Lemma \ref{ghu} below we consider the converse of Lemma \ref{lead}
for non-integral $\lambda\in{\aaaa_\C^\ast}$ with ${\rm Re }(\lambda)>0$.
\begin{lem}\label{firtu}
If ${\rm Re }(\lambda)>0$ and $ext:C^{-\infty}(B,V_B(\sigma_\mu))\rightarrow {}^\Gamma C^{-\infty}(\partial X,V(\sigma_\mu))$ is meromorphic at $\mu=\lambda$,
then the order of a singularity of $ext$ at $\lambda$ is at most $1$.
\end{lem}
{\it Proof.$\:\:\:\:$}
Let $\gamma\in \hat{K}$ be such that there exists an injective $T\in {\mbox{\rm Hom}}_M(V_\sigma,V_\gamma)$.
We also can and will require that
the Poisson transform $P_\lambda^T=:P$ is injective
(e.g. by taking $\gamma$ to be the minimal $K$-type of the
principal series representation $\pi^{\sigma,\lambda}$).
Let $f_\mu\in C^\infty(B,V_B(\sigma_\mu))$, $\mu\in {\aaaa_\C^\ast}$,
be a holomorphic family such that $ext(f_\mu)$ has a pole of order
$n\ge 1$ at $\mu=\lambda$, ${\rm Re }(\lambda)>0$.
We assume that $n\ge 2$ and argue by contradiction.
Let $0\not=\phi \in {}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))$
be the leading singular part of $ext(f_\mu)$ at $\mu=\lambda$.
Then $(\lambda^2-\mu^2)^{n-1} ext(f_\mu)$ and hence
$(\lambda^2-\mu^2)^{n-1} P_\mu^T ext(f_\mu)$
have first-order poles, the latter with residue $-(2\lambda)^{n-1} P\phi $.
Since $res_\Omega\circ ext(f_\mu)$ is smooth if $kM\in\Omega M$ we have $P^T_\mu\circ ext(f_\mu) (ka)=O(a^{\mu-\rho})$.
Moreover $P\phi (ka)=O(a^{-\lambda-\rho})$ and both estimates
hold uniformly for $kM$ in compact subsets of $\Omega$, $\mu$ near $\lambda$, and large $a\in A_+$.
This justifies the following computation using partial integration:
\begin{eqnarray*}
\infty&=&\lim_{\mu\to\lambda,\: {\rm Re }(\mu)<{\rm Re }(\lambda)} \langle
(\lambda^2-\mu^2)^{n-1} P_\mu^T ext(f_\mu), P\phi \rangle_{L^2(Y)}\\
&=&\lim_{\mu\to\lambda,\: {\rm Re }(\mu)<{\rm Re }(\lambda)} \langle
(-\Omega_G+c_\sigma+\lambda^2)^{n-1} P_\mu^T ext(f_\mu), P\phi \rangle_{L^2(Y)}\\
&=& 0
\end{eqnarray*}
This is a contradiction and thus $n=1$.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
Since $\sigma$ is an unitary representation of $M$, we have
for $\lambda\in \imath{\bf a}^*$ a positive conjugate linear
pairing $V_{\sigma_\lambda}\otimes V_{\sigma_\lambda} \rightarrow V_{1_{-\rho}}$
and hence a natural $L^2$-scalar product $C^\infty(B,V_B(\sigma_\lambda))\otimes C^\infty(B,V_B(\sigma_\lambda))\rightarrow {\bf C}$.
Let $L^2(B,V_B(\sigma_\lambda))$ be associated Hilbert space.
Using Lemma \ref{lok} we see that the adjoint $S^*_\lambda$
with respect to this Hilbert space structure is just $S_{-\lambda}$.
\begin{lem}\label{unitary}
If ${\rm Re }(\lambda)=0$, $\lambda\not=0$, then $S_\lambda$ is regular and unitary.
\end{lem}
{\it Proof.$\:\:\:\:$}
The scattering matrix $S_\lambda$ is meromorphic at non-zero imaginary points $\lambda$. Let now $\lambda$ be imaginary, $S_{\pm\lambda}$ be regular and
$f\in C^\infty(B,V_B(\sigma_\lambda))$. Then by the functional equation (\ref{forme})
$$\|S_\lambda f\|_{L^2(B,V_B(\sigma_\lambda))}^2=\langle S_{-\lambda}\circ S_\lambda f, f \rangle_{L^2(B,V_B(\sigma_\lambda))} = \|f\|^2_{L^2(B,V_B(\sigma_\lambda))}\ .$$
This equation remains valid at all non-zero imaginary points. Hence, $S_\lambda$
is regular and unitary there.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\section{Invariant distributions on the limit set}\label{invvv}
In present section we study the space
${}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))$ of
invariant distributions which are
supported on the limit set, mainly for ${\rm Re }(\lambda)\ge 0$.
We show that nontrivial distributions
of this kind can only exist for a countable set of parameters $\lambda\ge 0$ with possibly finitely many accumulation points.
In particular, we show that
${}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))=0$ if $ext$ is regular at $\lambda$, ${\rm Re }(\lambda)\ge 0$, and $\chi_{\mu_\sigma+\rho_m-\lambda}$ is non-integral.
In the course of this paper we will prove the finiteness of the discrete spectrum of ${\cal Z}$ on
$L^2(Y,V_Y(\gamma))$, ore equivalently, that nontrivial invariant distributions
with support on the limit set (with ${\rm Re }(\lambda)\ge 0$) can in fact only exist for a finite set of parameters $\lambda\ge 0$. The proof of the finiteness of the point spectrum
is essentially based on the spectral comparison Proposition \ref{esspec}. But this proposition is not applicable in order to exclude that eigenvalues accumulate at the boundary of the continuous spectrum.
Here we will employ Corollary \ref{nahenull} instead,
and it is important to show that $ext$ is meromorphic at $\lambda=0$
for certain $\sigma$.
First we show a variant of Green's formula.
We need nice cut-off functions which exist by the following lemma.
\begin{lem}\label{lll}
There exists a cut-off function $\chi$ such that
\begin{enumerate}
\item $\chi> 0$ on a fundamental domain $F\subset X$,
\item ${\mbox{\rm supp}}(\chi)\subset \cup_{g\in L} gF$ for some finite subset
$L\subset \Gamma$,
\item $\sum_{g\in\Gamma}g^*\chi = 1$,
\item $\sup_{k\in ({\rm clo}(F)\cap\Omega)M ,\: a\in A_+} a\:|\nabla^i\chi(ka)|<\infty$, $i\in{\bf N}$,
\item the function $\chi_\infty$ on $K$ defined by $\chi_\infty(k):=\lim_{a\to\infty}\chi(ka)\chi_\infty(k)$ is smooth.
\end{enumerate}
\end{lem}
{\it Proof.$\:\:\:\:$}
Let $W\subset \Omega$ be compact such that ${\rm clo}(F)\cap \Omega\subset W$.
Let $\psi\in C^\infty(\partial X)$ be a cut-off function such that
$\psi_{|{\rm clo}(F)\cap\Omega}=1$ and $\psi_{|\partial X\setminus W}=0$.
Let $B_R\subset X$, $R\in{\bf R}$ be the $R$-ball in $X$ centered at the origin
and choose $R>1$ so large that $F\subset B_R\cup WA_+$.
Let $\sigma\in C^\infty(A_+)$ be a cut-off function such that
$\sigma(r)=1$ for $r>1$ and $\sigma(r)=0$ for $r<1/2$. Finally let $\phi\in C_c^{\infty}(X)$
be a cut-off
function such that $\phi_{|B_R}=1$ and $\phi_{|\partial X\setminus B_{R+1}}=0$.
Then we set $\tilde{\chi}(ka):=\phi(ka) + \psi(k)\sigma(a)$,
$k\in K$, $a\in A_+$, $ka\in X$.
If we define
$$\chi:=\frac{\tilde{\chi}}{\sum_{g\in \Gamma }g^*\tilde{\chi}}\ ,$$
then $\chi$ obviously satisfies (1),(2), (3), and (5).
It remains to verify (4).
Note that by construction of $\tilde{\chi}$ for all $l\in {\bf N}$ there exists a constant $C<\infty$ such that for all $k\in K$, $a\in A_+$
$$|\nabla^l \tilde{\chi}(ka)| < C a^{-1}\ .$$
Hence for any finite subset $L\subset \Gamma$ and
$l\in{\bf N}$ there exists a constant $C<\infty$ such that
for all $k\in K$,
$g\in L$, $a\in A_+$
$$|\nabla^l g^\ast \tilde{\chi}(ka)| < C a^{-1}\ .$$
This implies (4).
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
Now we consider the variant of Green's formula.
Let $\phi\in {}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))$. If ${\rm Re }(\lambda)\ge 0$, then
$res\circ \hat{J}_\lambda(\phi)\in C^{\infty}(B,V_B(\sigma_{-\lambda}))$ is well defined
even if $\hat{J}_\mu$ has a pole at $\mu=\lambda$. In the latter case the residue
of $\hat{J}_\mu$ at $\mu=\lambda$ is a differential operator $D_\lambda$ (see Lemma \ref{off})
and $res\circ D_\lambda(\phi)=0$.
\begin{prop}\label{green}
Assume that
${\rm Re }(\lambda)\ge 0$ and $\lambda\not=0$.
If
$\phi\in {}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))$ and
$f\in {}^\Gamma C^{-\infty}(\partial X,V(\tilde{\sigma}_\lambda))$
is such that $f_{|\Omega}$ is smooth,
then $$\langle res\circ J_\lambda (\phi),res(f)\rangle =0\ .$$
\end{prop}
{\it Proof.$\:\:\:\:$}
Let $(\gamma,V_\gamma) \in \hat{K}$ be a minimal $K$-type of the principal series representation of $G$
on $C^{\infty}(\partial X, V(\sigma_\lambda))$. Then there is an injective
$T\in {\mbox{\rm Hom}}_M(V_\sigma,V_\gamma)$.
For ${\rm Re }(\lambda)>0$ we define the endomorphism valued function
\begin{equation}\label{cgamma} c_{\gamma}(\lambda):=\int_{\bar{N}} a(\bar{n})^{-(\lambda+\rho)}
\gamma(\kappa(\bar{n}))d\bar{n}\in {\mbox{\rm End}}_M(V_\gamma)\ .\end{equation}
The function $c_{\gamma}(\lambda)$ extends meromorphically to all of ${\aaaa_\C^\ast}$.
If $c_\gamma(\lambda)$ is singular, then $\lambda$ is integral.
We choose $\tilde{T}\in {\mbox{\rm Hom}}_M(V_{\tilde{\sigma}},V_{\tilde{\gamma}})$
such that $\langle \gamma(w) T v, c_\gamma(\lambda) \tilde{T}u\rangle =\langle v,u\rangle$
for all $v\in V_\sigma$, $u\in V_{\tilde{\sigma}}$.
This is possible since $\gamma$ is a minimal $K$-type and hence $c_\gamma(\lambda)$ is invertible for ${\rm Re }(\lambda)\ge 0$, $\lambda\not=0$.
Let $A=-\Omega_G+c_\sigma+\lambda^2$,
$P=P_\mu$ be the Poisson transform
(associated to $T$ or $\tilde{T}$,
respectively),
and $\chi$ be the cut-off function constructed
in Lemma \ref{lll}.
Note that $A=-\nabla^*\nabla + {\cal R}$ for some selfadjoint endomorphism ${\cal R}$
of $V(\gamma)$, where $-\nabla^*\nabla=\Delta$ is the Bochner Laplacian associated to the
invariant connection $\nabla$ of $V(\gamma)$.
By $B_R$ we denote the metric $R$-ball centered at the origin of $X$.
The following is an application of Green's formula:
\begin{eqnarray}
0&=&\langle \chi A P\phi,Pf\rangle_{L^2(B_R)}- \langle \chi P\phi,A Pf \rangle_{L^2(B_R)}\label{limiz}\\
&=& \langle A \chi P\phi,Pf\rangle_{L^2(B_R)}- \langle \chi P\phi,A Pf \rangle_{L^2(B_R)} - \langle [ A, \chi] P\phi,Pf\rangle_{L^2(B_R)}\nonumber \\
&=& - \langle \nabla_n \chi P\phi,Pf\rangle_{L^2(\partial B_R)}+ \langle \chi P\phi,\nabla_n Pf \rangle_{L^2(\partial B_R)} - \langle [ A, \chi] P\phi,Pf\rangle_{L^2(B_R)}\ , \nonumber
\end{eqnarray}
where $n$ is the exterior unit normal vector field at $\partial B_R$.
For the following discussion we distinguish between the two cases
${\rm Re }(\lambda)>0$ and ${\rm Re }(\lambda)=0$, $\lambda\not=0$.
We first consider the case ${\rm Re }(\lambda)>0$.
We employ the following asymptotic behaviour of the Poisson transforms
of $f$ and $\psi$.
For $kM\in \Omega$ we have
\begin{eqnarray}
P(f)(ka)&=&a^{\lambda-\rho} c_\gamma(\lambda) \tilde{T} f(k)+O(a^{\lambda-\rho-\epsilon})\label{k0o}\\
P(\phi)(ka)&=&a^{-\lambda-\rho} \gamma(w)T (\hat{J}_\lambda \phi)(k) + O(a^{-\lambda-\rho-\epsilon})\label{k0o1}
\end{eqnarray}
where $\epsilon>0$.
While (\ref{k0o}) follows from the fact that $f_{|\Omega}$ is smooth,
(\ref{k0o1}) is shown in Lemma \ref{eee}.
The estimate can be differentiated with respect to $a$ and holds locally uniformly on $\Omega$.
By property (4) of $\chi$ we see that
$\langle [ A, \chi] P\phi,Pf\rangle$
is integrable and by property (3) and the $\Gamma$-invariance
of $f$ and $\phi$ we have
$\langle [ A, \chi] P\phi,Pf\rangle_{L^2(X)}=0$.
Taking the limit $R\to\infty$ in (\ref{limiz}) we obtain
\begin{eqnarray*}
0&=& (\lambda+\rho)\int_{\partial X} \chi_\infty(k) \langle\gamma(w)T (\hat{J}_\lambda \phi )(k), c_\gamma(\lambda)\tilde{T} f(k)\rangle \\
&&+(\lambda-\rho)\int_{\partial X} \chi_\infty(k) \langle\gamma(w)T (\hat{J}_\lambda \phi )(k),c_\gamma(\lambda)\tilde{T}f(k)\rangle \\
&=&2\lambda \int_{\partial X} \chi_\infty(k) \langle (\hat{J}_\lambda \phi )(k),f(k)\rangle \\
&=&2\lambda \langle res\circ\hat{J}_\lambda (\phi),res( f)\rangle \ .
\end{eqnarray*}
This is the assertion of the proposition for ${\rm Re }(\lambda)>0$.
Now we discuss the case ${\rm Re }(\lambda)=0$ and $\lambda\not=0$.
In this case we have the following asymptotic behaviour
$$P(f)(ka) = a^{\lambda-\rho} c_\gamma(\lambda) \tilde{T} f(k)+ a^{-\lambda-\rho}\gamma(w) \tilde{T} \hat{J}_\lambda f(k) + O(a^{ -\rho-\epsilon})\ .$$
Instead of taking the limit $R\to\infty$ in (\ref{limiz}) we apply
$\lim_{r\to\infty}\frac{1}{r}\int_0^r dR$.
Again we have
$$\lim_{r\to\infty}\frac{1}{r}\int_0^r \langle [ A, \chi] P\phi,Pf\rangle_{L^2(B_R)} dR=0\ . $$
Moreover, the asymptotic term $a^{-\lambda-\rho} \tilde{T} \hat{J}_\lambda f(k)$
does not contribute to the limit because of
$$-2\lambda \lim_{r\to\infty}\frac{1}{r}\int_0^r R^{-2\lambda} \langle \chi(.R) \gamma(w)T \hat{J}_\lambda \phi ,\gamma(w) \tilde{T} \hat{J}_\lambda f\rangle_{L^2(\partial X)} dR=0\ .$$
The contribution of the term $a^{\lambda-\rho} c_\gamma(\lambda) \tilde{T} f(k)$
leads to
$$0=2\lambda \langle res\circ\hat{J}_\lambda (\phi),res( f)\rangle$$
as in the case ${\rm Re }(\lambda)>0$.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
By the Harish-Chandra isomorphism characters of ${\cal Z}$ are
parametrized by elements of ${\bf h}_{\bf C}^*/W$.
A character $\chi_\lambda$, $\lambda\in {\bf h}_{\bf C}^*$, is called integral,
if $$2\frac{\langle \lambda,\alpha\rangle}{\langle\alpha,\alpha\rangle}\in{\bf Z}$$
for all roots $\alpha$ of $({\bf g},{\bf h})$.
\begin{lem}\label{th43}
Let $\lambda\in {\aaaa_\C^\ast}$ be such that
\begin{itemize}
\item ${\rm Re }(\lambda)\ge 0$ and $\chi_{\mu_\sigma+\rho_m-\lambda}$ is a non-integral character of ${\cal Z}$ or
\item ${\rm Re }(\lambda)<0$ and $\lambda$ is non-integral.
\end{itemize}
Let and $U\subset \partial X$ be open.
If $\phi\in C^{-\infty}(\partial X, V(\sigma_\lambda))$ satisfies
$\phi_{|U}= 0$,
then $(\hat{J}_\lambda\phi)_{|U}= 0$ implies $\phi=0$.
\end{lem}
Before turning to the proof note that Lemma \ref{off} implies that $(\hat{J}_\lambda\phi)_{|U}$
is well-defined even if $\hat{J}_\lambda$ has a pole.\\[0.5cm]\noindent
{\it Proof.$\:\:\:\:$}
We modify an argument given by van den Ban-Schlichtkrull \cite{vandenbanschlichtkrull89}
for the case $\sigma=1$.
If $\bar{N}$ is two-step nilpotent, then $\bar{{\bf n}}=\bar{{\bf n}}_1\oplus \bar{{\bf n}}_2$, $[\bar{{\bf n}}_1,\bar{{\bf n}}_1]=\bar{{\bf n}}_2$, where
$\bar{{\bf n}}_1$ corresponds to the negative of the shorter root $\alpha_1\in{\bf a}^*_+$.
\begin{lem}\label{eee}
Let $\lambda\in{\aaaa_\C^\ast}$, $\gamma$ be a representation of $K$, and
$\phi\in C^{-\infty}(\partial X,V(\sigma_\lambda))$ satisfy $M \not\in {\mbox{\rm supp}}(\phi)$.
Then for $T\in {\mbox{\rm Hom}}(V_\sigma,V_\gamma)$, $k\in K\setminus{\mbox{\rm supp}}(\phi)M$ the Poisson transform
$P\phi:=P^T_\lambda\phi$ has an asymptotic expansion as $a\to\infty$ of the form
\begin{equation}\label{epan}
(P\phi)(ka)=a^{-(\lambda+\rho)}\gamma(w)T(\hat{J}_\lambda\phi)(k)
+\sum_{n\ge 1} a^{-(\lambda+\rho)-n\alpha_1}\psi_n(k)\ .
\end{equation}
Here $\psi_n$ are smooth functions on $K\setminus{\mbox{\rm supp}}(\phi)M$.
The expansion converges uniformly for $a>>0$ and $k$ in compact subsets
of $K\setminus {\mbox{\rm supp}}(\phi)M$.
\end{lem}
{\it Proof.$\:\:\:\:$}
In the following computation we write the pairing of a distribution
with a smooth function as an integral.
\begin{eqnarray}
(P\phi)(ka)&=&\int_K a(a^{-1}k^{-1}h)^{-(\lambda+\rho)}\gamma(\kappa(a^{-1}k^{-1}h)) T \phi(h) dh\nonumber\\
&=& \int_{\bar{N}} a( a^{-1}w\kappa(\bar{n}))^{-(\lambda+\rho)} \gamma(\kappa( a^{-1}w\kappa(\bar{n}))) T\phi(kw\kappa(\bar{n}))
a(\bar{n})^{-2\rho} d\bar{n} \nonumber\\
&=&\int_{\bar{N}} a(a \bar{n} a^{-1})^{-(\lambda+\rho)} a^{-(\lambda+\rho)} a(\bar{n})^{\lambda+\rho} \gamma(w) \gamma(\kappa(a\bar{n}a^{-1})) T\phi(kw\kappa(\bar{n}))
a(\bar{n})^{-2\rho} d\bar{n} \nonumber\\
&=&a^{-(\lambda+\rho)}\gamma(w) \int_{\bar{N}} a(\bar{n})^{\lambda-\rho}
a(a\bar{n}a^{-1})^{-(\lambda+\rho)} \gamma(\kappa(a\bar{n}a^{-1})) T \phi(kw\kappa(\bar{n})) d\bar{n}\label{tyh}\ .
\end{eqnarray}
For $z\in {\bf R}^+$ define $a_z\in A$ through $z=a_z^{-\alpha_1}$. We consider
the map
$\Phi:(0,\infty)\times \bar{N}\ni (z,\bar{n})\mapsto a_z\bar{n}a_z^{-1}\in \bar{N}$ which is can also be written as
$$\Phi(z,exp(X+Y)):= exp(zX+z^2Y), \quad X\in{\bf n}_1, Y\in{\bf n}_2\ .$$
Thus $\Phi$ and hence $ (z,\bar{n})\mapsto a(a_z\bar{n}a_z^{-1})^{-(\lambda+\rho)} \gamma(\kappa(a_z\bar{n}a_z^{-1}))$ extend analytically to ${\bf R}\times\bar{N}$.
Taking the Taylor expansion with respect to $z$ at $z=0$ we obtain
$$a(a_z\bar{n}a_z^{-1})^{-(\lambda+\rho)} \gamma(\kappa(a_z\bar{n}a_z^{-1}))={\mbox{\rm id}} + \sum_{n\ge 1}A_n(\bar{n}) z^n\ .$$
Here $A_n:\bar{N}\rightarrow {\mbox{\rm End}}(V_\gamma)$ are analytic and the series converges
in the spaces of smooth functions on $\bar{N}$ with values in ${\mbox{\rm End}}(V_\gamma)$.
Inserting this expansion into (\ref{tyh}) we obtain
$$(P\phi)(ka)=a^{-(\lambda+\rho)} \gamma(w) T (\hat{J}_\lambda \phi)(k) + \sum_{n\ge 1} a^{-(\lambda+\rho)-n\alpha_1} \psi_n(k)\ ,$$
where
$$\psi_n(k):=\gamma(w)\int_{\bar{N}} A_n(\bar{n}) T a(\bar{n})^{\lambda-\rho }\phi(kw\kappa(\bar{n})) d\bar{n}\ .$$
Note that $k\mapsto \phi(kw\kappa(.))$ is a smooth family of distributions
with compact support in $\bar{N}$.
Thus $\psi_n$ is smooth. This finishes the proof of the lemma. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
We now continue the proof of Lemma \ref{th43}.
If ${\rm Re }(\lambda)<0$, then we reduce to the case ${\rm Re }(\lambda)>0$ replacing $\phi$ by $\hat{J}_\lambda(\phi)$. We can do this because $\lambda$ is then non-integral and $\hat{J}_\lambda$ is regular and bijective.
Note that in this case $\chi_{\mu_\sigma+\rho_m-\lambda}$ is non-integral.
Thus assume that ${\rm Re }(\lambda)\ge 0$.
We choose the representation $\gamma\in \hat{K}$ and $T\in {\mbox{\rm Hom}}_M(V_\sigma,V_\gamma)$ such that $P:=P^T_\lambda$ is injective.
The range of $P$ can be identified with the kernel of a certain invariant
differential operator $D:C^{\infty}(X,V(\gamma))\rightarrow C^{\infty}(X,V(\gamma^\prime))$
for a suitable representation $\gamma^\prime$ of $K$ (see \cite{bunkeolbrich947}, Sec.3). In particular, $P\phi$ is real-analytic.
We now assume $\phi\not\equiv 0$.
Moreover, without loss of generality we can assume that $ M\in U$.
Since $P\phi$ is real analytic, the expansion (\ref{epan}) does not vanish.
Let $m$ be the smallest integer such that $\psi_m\not\equiv 0$ near $M$
(where $\psi_0:=\gamma(w)T\hat{J}_\lambda \phi)$.
We argue that $m=0$ and thus obtain a contradiction.
To prove that $m=0$ we again argue by contradiction. Assume that $m>0$.
With respect to the coordinates $k,a$ the operator $D$ has the form
$D=D_0+a^{-\alpha_1}R(a,k)$, where $D$ is a constant coefficient
operator on $A$ and $R$ remains bounded if $a\to\infty$ (see \cite{warner721}, Thm. 9.1.2.4).
Moreover, it is known that $D_0$ coincides with the $\bar{N}$-radial part of $D$.
Choose $k\in K$ near $1$ and $\sigma^\prime\subset \gamma_{|M}$ such that
that there exists an orthogonal projection $S\in {\mbox{\rm Hom}}_M(V_\gamma,V_{\sigma^\prime})$ with
$S \gamma(k)\psi_m(k) =:v \not= 0$.
Consider the $\bar{N}$-invariant section $f\in C^{\infty}(X,V(\gamma))$
defined by
$$f(\bar{n}a):=a^{-(\lambda+\rho+m\alpha_1)} S^* v\ .$$
Since $D$ annihilates the asymptotic expansion (\ref{epan}), one can check
that $Df=D_0f=0$ and thus $f=P\phi_1$ for some $\bar{N}$-invariant $\phi_1\in C^{-\infty}(\partial X,V(\sigma_\lambda))$.
Now $f=P^S_{\lambda+m\alpha_1}\delta v$, where $\delta v\in C^{-\infty}(\partial X,V(\sigma^\prime_{\lambda+m\alpha_1}))$
is the delta distribution at $1$ with vector part $v$.
Since $D$ and $P^S_{\lambda+m\alpha_1}$ are $G$-equivariant and $\delta v$ generates the $G$-module $C^{-\infty}(\partial X,V(\sigma^\prime_{\lambda+m\alpha}))$, we obtain
a non-trivial
intertwining operator $I$ from $C^{-\infty}(\partial X,V(\sigma^\prime_{\lambda+m\alpha}))$
to the kernel of $D$, hence to $C^{-\infty}(\partial X,V(\sigma_\lambda))$.
Thus principal series representations $\pi^{\sigma,\lambda}$ and $\pi^{\sigma^\prime,\lambda+m\alpha_1}$ have the same infinitesimal character
$\chi_{\mu_\sigma+\rho_m-\lambda}$. Since by assumption this character is non-integral both principal series are irreducible (see \cite{collingwood85}, 4.3.3).
Hence $I$ is an isomorphism. Because of $m\not=0$
we have $\sigma\not=\sigma^\prime$. This implies that $\pi^{\sigma,\lambda}$
and $\pi^{\sigma^\prime,\lambda+m\alpha_1}$ can not be isomorphic.
This is the contradiction we aimed at.
We conclude that $m=0$.\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
The above argument can be extended to cover some cases
of $\lambda\in{\bf a}^*$, ${\rm Re }(\lambda)\ge 0$, with $\chi_{\mu_\sigma+\rho_m-\lambda}$ integral.
This would lead to stronger vanishing results
for ${}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))$ below.
For example, if $\sigma$ is the trivial $M$-type, then the
assertion of Lemma \ref{th43} holds true for all $\lambda\in {\aaaa_\C^\ast}$
with ${\rm Re }(\lambda)\ge 0$.
But there exist examples of $\sigma\in\hat{M}$
and $\lambda\in{\bf a}^*$ with ${\rm Re }(\lambda)\ge 0$ and
$\chi_{\mu_\sigma+\rho_m-\lambda}$ integral where
the assertion of Lemma \ref{th43} is false.
The possible failure of the lemma at $\lambda=0$ is connected
with the existence of $L^2(Y,V_Y(\gamma))_{scat}$.
\begin{kor}\label{ujn}
Assume that $\lambda\in {\aaaa_\C^\ast}$, that ${\rm Re }(\lambda)\ge 0$, and that
$\chi_{\mu_\sigma+\rho_m-\lambda}$ is non-integral.
Then
$$res_\Omega\circ \hat{J}_\lambda:C^{-\infty}(\Lambda,V(\sigma_\lambda))\rightarrow C^{\infty}(\Omega,V(\sigma_{-\lambda}))$$
is injective.
\end{kor}
\begin{lem}\label{ghu}
Assume that $\lambda\in {\aaaa_\C^\ast}$, that ${\rm Re }(\lambda)\ge 0$, that
$\chi_{\mu_\sigma+\rho_m-\lambda}$ is non-integral,
and that
$$ext:C^{\infty}(B,V_B(\sigma_\lambda))\rightarrow {}^\Gamma C^{-\infty}(\partial X,V(\sigma_\lambda))$$
is regular. Then ${}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))=0$ .
\end{lem}
{\it Proof.$\:\:\:\:$}
The assumption of the lemma implies that
$$ext:C^{\infty}(B,V_B(\tilde{\sigma}_\lambda))\rightarrow {}^\Gamma C^{-\infty}(\partial X,V(\tilde{\sigma}_\lambda))$$
is regular.
Indeed, if this extension is singular, then $\lambda$ is real and
$$ext:C^{\infty}(B,V_B(\sigma_\lambda))\rightarrow {}^\Gamma C^{-\infty}(\partial X,V(\sigma_\lambda))$$
is singular since there is a conjugate linear isomorphism $\sigma\cong\tilde{\sigma}$.
Let $\phi\in{}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))$.
Then for all $f\in C^{\infty}(B,V_B(\tilde{\sigma}_\lambda))$ we have
by Lemma \ref{green}
$$0=\langle res\circ \hat{J}_\lambda(\phi),res\circ ext(f)\rangle= \langle res\circ \hat{J}_\lambda(\phi), f \rangle \ .$$
Thus $res\circ \hat{J}_\lambda(\phi)=0$ and by Corollary \ref{ujn} we conclude
$\phi=0$.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{lem}\label{extregatim}
If ${\rm Re }(\lambda)= 0$ and ${\rm Im}(\lambda)\not=0$, then
$$ext:C^{-\infty}(B,V_B(\sigma_\lambda))\rightarrow {}^\Gamma C^{-\infty}(\partial X,V(\sigma_\lambda))$$
is regular.
\end{lem}
{\it Proof.$\:\:\:\:$}
Note that $\chi_{\mu_\sigma+\rho_m-\lambda}$ is a non-integral
character of ${\cal Z}$. Since $\lambda\in{\aaaa_\C^\ast}(\sigma)$, the extension
is meromorphic at $\lambda$. Assume that $ext$ has a pole.
The singular part of $ext$ maps
to distributions which are supported on the limit set $\Lambda$.
Then by Corollary
\ref{ujn} the scattering matrix
$S_\lambda=res\circ J_\lambda\circ ext$ has a pole at $\lambda$, too.
But this contradicts Lemma \ref{unitary}.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
In the following proposition we formulate
the most complete vanishing result for\linebreak[4] ${}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))$, ${\rm Re }(\lambda)\ge 0$,
which is stated in the present paper.
In general, it is not optimal for integral infinitesimal characters.
We give upper bounds depending on $\delta_\Gamma$ for the parameters $\lambda$ with ${}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))\not=0$.
One can then deduce corresponding bounds for the discrete spectrum
of ${\cal Z}$ on $L^2(Y,V_Y(\gamma))$. If one is only interested in
the finiteness of the discrete spectrum, then it is sufficient to
know the rough bound $\lambda \le \rho$, which is implied by the classification
of the unitary representations of $G$ (see the proof of Corollary \ref{nahenull}).
\begin{prop}\label{upperbound}
Let ${\rm Re }(\lambda)\ge 0$. Then any of the following conditions
implies that $${}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))=0\ .$$
\begin{enumerate}
\item ${\rm Im}(\lambda)\not=0$,
\item ${\rm Re }(\lambda)>\delta_\Gamma$ and $\chi_{\mu_\sigma+\rho_m-\lambda}$ is
a non-integral character of ${\cal Z}$,
\item $$\hspace{-2cm}{\rm Re }(\lambda)>\frac{\rho^2+\rho\delta_\Gamma}{3\rho-\delta_\Gamma}$$
\end{enumerate}
\end{prop}
{\it Proof.$\:\:\:\:$}
We consider $1$. If ${\rm Re }(\lambda)>0$, then the assertion follows
from Lemma \ref{pokm}. If ${\rm Re }(\lambda)=0$, then we apply Lemmas
\ref{ghu} and \ref{extregatim}.
Sufficiency of condition $2.$ follows from Lemma \ref{ghu} and
\ref{defofext}.
Condition $3.$ is important for $\lambda$ with
$\chi_{\mu_\sigma+\rho_m-\lambda}$ integral. In this
case the relation of the space ${}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))$ with the singularities
of $ext$ is quite unclear.
Let $\gamma$ denote a minimal $K$-type of the principal
series representation $\pi^{\sigma,\lambda}$ of $G$ on $C^\infty(\partial X,V(\sigma_\lambda))$,
and let $T\in {\mbox{\rm Hom}}_M(V_\sigma,V_\gamma)$ be an isometric
embedding. Then the Poisson transform
$$P^T_\lambda :C^{-\infty}(\partial X,V(\sigma_\lambda))\rightarrow C^\infty(X,V(\gamma))$$ is injective.
Let $\psi\in {}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))$
and $f:=P^T_\lambda\psi\in C^\infty(Y,V_Y(\gamma))$.
Since ${\rm Re }(\lambda)>0$ we have for any test function $\phi\in C^\infty(\partial X,V(\sigma_{\bar{\lambda}}))$
$$\bar{c}_\sigma(\lambda)(\psi,\phi)=\lim_{a\to\infty} a^{\rho-\bar{\lambda}}\int_K (f(ka), T \phi(k)) dk \ .$$
We will show that condition $3.$ implies that $(\psi,\phi)=0$.
Since $\phi$ was arbitrary, it follows $\psi=0$.
In the following argument we assume that ${\rm Re }(\lambda)<\rho$.
An easy modification works for ${\rm Re }(\lambda)\ge\rho$.
The asymptotic expansion of $f$ near $\Omega$ obtained
in Lemma \ref{eee} implies that $f\in L^p(Y,V_Y(\gamma))$ for
$p=\frac{2\rho}{\rho+{\rm Re }(\lambda)}+\epsilon$, $\forall \epsilon>0$.
If we set $\tilde{f}(ka)=a^{-(\rho+\delta_\Gamma+\epsilon)/p}f(ka)$,
then $\tilde{f}\in L^p(X,V(\gamma))$.
In fact we can estimate
\begin{eqnarray*}
\|\tilde{f}\|_{L^p(X,V(\gamma))}&=&\sum_{g\in\Gamma} \int_{gF} |\tilde{f}(g)|^p dg\\
&\le & \sum_{g\in \Gamma} \min\{a_h\:|\: h\in gF\}^{-(\rho+\delta_\Gamma+\epsilon)}
\int_{gF} |f(g)|^p dg\\
&\le & C \sum_{g\in \Gamma} a_g^{-(\rho+\delta_\Gamma+\epsilon)}\\
&<& \infty \ .
\end{eqnarray*}\
Let $\chi\in C^\infty_c(0,1)$ be such that $\int_0^1\chi(t) dt=1$.
We extend $\chi$ to ${\bf R}$ by zero.
Then we define $\chi_n\in C_c^\infty(A_+)$ by
$\chi_n(a):=a^{-\rho-\bar{\lambda}}\chi(|\log(a)|-n)$.
If we set $\phi_n(ka)= T\chi_n(a)\phi(k)$,
then we can write
$$\bar{c}_\sigma(\lambda)(\psi,\phi )
=\lim_{n\to \infty} (f,\phi_n) \ .$$
Now let $\tilde{\phi}_n(ka):= a^{(\rho+\delta_\Gamma+\epsilon)/p} \phi_n(ka)$.
Then
\begin{equation}\label{thishere} \bar{c}_\sigma(\lambda)(\psi,\phi)
=\lim_{n\to \infty} (\tilde{f},\tilde{\phi}_n) \ .\end{equation}
We consider $\Psi(ka):=a^{(\rho+\delta_\Gamma+\epsilon)/p}a^{-\rho-\bar{\lambda}}T \phi(k)$
and let $q\in (1,\infty)$ be the dual exponent to $p$.
Note that $\bar{c}_\sigma(\lambda)\not=0$.
If we assume that
$\Psi\in L^q(X,V(\gamma))$, then $(\psi,\phi)=0$ follows from
(\ref{thishere}).
The following inequality implies that $\Psi\in L^q(X,V(\gamma))$:
\begin{equation}\label{ineq}q\left(\frac{\rho+\delta_\Gamma+\epsilon}{p}-(\rho+{\rm Re }(\lambda))\right)<-2\rho \ .\end{equation}
A simple computation shows that (\ref{ineq}) indeed follows from $3$.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
If $\chi_{\mu_\sigma+\rho_m-\lambda}$ is integral, then
one can combine the argument of Lemma \ref{th43}, or
the understanding of its failure, respectively, with the
above argument in order to choose a better exponent $p$.
This leads to a stronger vanishing result.
We omit this rather involved discussion,
partly because we do not know how to formulate a general result.
This omission causes some loss of information about the
discrete spectrum of ${\cal Z}$ on $L^2(Y,V_Y(\gamma))$.
In general, we do not know whether $ext$ is meromorphic at $\lambda=0$.
But we can show the following result.
\begin{prop}\label{mystic}
If $\hat J_\lambda$ has a pole at $\lambda=0$, then $ext$ is meromorphic in
a neighbourhood of $0$.
\end{prop}
The proof of the proposition will occupy the remainder of this section.
First we want to fix an important corollary which applies to all $M$-types
$\sigma$.
\begin{kor}\label{nahenull}
There exists $\epsilon>0$ such that
$ext$ is regular and ${}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))=0$ for $\lambda\in (0,\epsilon)$.
\end{kor}
{\it Proof.$\:\:\:\:$}
If contrary to the assertion there is a
sequence $\lambda_\alpha>0$ of poles of $ext$ with $\lambda_\alpha\to 0$,
then the residues of $ext$ give elements of ${}^\Gamma C^{-\infty}(\Lambda,V(\sigma_{\lambda_\alpha}))$. Applying to these elements a suitable Poisson
transform $P^T_{\lambda_\alpha}$, $T\in {\mbox{\rm Hom}}_M(V_\sigma,V_\gamma)$, we obtain smooth sections in $L^2(Y,V_Y(\gamma))$. Considered as
elements of $L^2(\Gamma\backslash G)\otimes V_\gamma$ they generate unitary
representations of $G$ whose underlying Harish-Chandra modules are dual
to the underlying Harish-Chandra modules of $C^{\infty}(\partial X,V(\sigma_{\lambda_\alpha}))$, at least for $\lambda_\alpha$ sufficiently small.
Here we have used that principal series representations with non-integral infinitesimal character are irreducible. Since the dual of a unitary representation is
unitary too, we conclude that the principal series representations
$C^{\infty}(\partial X,V(\sigma_{\lambda_\alpha}))$ carry invariant
Hermitian scalar products or, in representation theoretic language, belong
to the complementary series. But, by a result of Knapp-Stein \cite{knappstein71}, Par. 14, there is a complementary series for $\sigma$ iff
$P_\sigma(0)=0$ or, equivalently, iff $\hat J_\lambda$ has a pole at $\lambda=0$. But in this case $ext$ is meromorphic at $\lambda=0$ by Proposition
\ref{mystic}. This is in contradiction with the existence of the sequence $\lambda_\alpha$ of poles of $ext$ with $\lambda_\alpha\to 0$.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
We now turn to the proof of Proposition \ref{mystic}. If $\delta_\Gamma<0$, there
is nothing to show. For $\delta_\Gamma\ge 0$ we are going to use the embedding
trick as in the proof of Proposition \ref{part1}.
Let $G=G^n\subset G^{n+1}\subset\dots$ be the corresponding series of groups.
Choose $k\in{\bf N}$ such that $\delta_\Gamma^{n+k}=\delta_\Gamma^{n}-\rho^{n+k}+\rho^n<0$. If there is a representation $\sigma^{n+k}$ of $M^{n+k}$ such that $\sigma^n:=\sigma\subset \sigma^{n+k}_{\ |M^n}$ and $-\rho^{n+k}+\rho^n \in {\aaaa_\C^\ast} (\sigma^{n+k})$,
then $ext^{n+k}$ is meromorphic at $\lambda=-\rho^{n+k}+\rho^n$, which in
turn implies the meromorphy of $ext^n$ at $\lambda=0$.
Recall the definition (\ref{acagood}) of ${\aaaa_\C^\ast} (\sigma^{n+k})$.
Since $\hat J_\lambda$ has at most first-order poles
$$P_{\sigma^{n+k}}(-\rho^{n+k}+\rho^n)=0$$ implies that $-\rho^{n+k}+\rho^n \in {\aaaa_\C^\ast} (\sigma^{n+k})$. It follows from the functional equation (\ref{spex}) that the intertwining operator $\hat J_\lambda$
has a pole at $\lambda=0$ iff
$P_{\sigma^n}(0)=0$. In this case $\sigma^n$ is irreducible and
Weyl-invariant.
So the discussion above shows that the
following lemma implies Proposition \ref{mystic}.
\begin{lem}\label{casebycase}
Let $G^n$ one of the following four series of groups :
\begin{eqnarray*}
Spin(1,2n)&&n\ge 1\ ,\\
Spin(1,2n+1)&&n\ge 1\ ,\\
SU(1,n)&&n\ge 2\ ,\\
Sp(1,n)&&n\ge 2\ .
\end{eqnarray*}
If $\sigma^n \in \hat M^n$ (i.e. $\sigma^n$ is irreducible) is
Weyl invariant and satisfies
$P_{\sigma^n}(0)=0$, then for any $k\in{\bf N}$
there exists $\sigma^{n+k} \in \hat M^{n+k}$ with $\sigma^n\subset \sigma^{n+k}_{\ |M^n}$ and $P_{\sigma^{n+k}}(-\rho^{n+k}+\rho^n)=0$.
\end{lem}
{\it Proof.$\:\:\:\:$}
First we need explicit expressions of the Plancherel densities $P_{\sigma^n}$ as
given e.g. in \cite {knapp86}, Prop. 14.26. Consider the Cartan subalgebra
${\bf h}_{\bf C}^n={\bf a}_{\bf C}\oplus{\bf t}_{\bf C}^n$, ${\bf t}^n$ being a Cartan
subalgebra of ${\bf m}^n$, and the subset $\Delta_r^n$ of the roots $\Delta^n$
of ${\bf h}_{\bf C}^n$ in ${\bf g}^n_{\bf C}$ given by
$$ \Delta_r^n:=\{\alpha \in \Delta^n\:|\:\alpha_{|{\bf a}} \mbox{ is a root of
${\bf a}$ in }{\bf n}^n\}\ .$$
For $G^n\not=Spin(1,2n+1)$ there is exactly on real root $\beta_r\in \Delta_r^n$
distinguished by $\beta_{r\:|{\bf t}^n}=0$. Furthermore, in this case we consider
a special element $m^n$ of the center of $M^n$. For $G=Spin(1,2n)$ the element $m^n$ is the non-trivial element in the kernel of the projection $Spin(1,2n)\rightarrow SO_0(1,2n)$. For $G^n=SU(1,n)$ $(Sp(1,n))$ we use the standard representation
in $Gl(n+1,{\bf C})$ $(Gl(n+1,\bf H))$ in order to fix $m^n$ by
$$ m^n:= \left(
\begin{array}{rrc}
-1&0&0\\
0&-1&0\\
0&0&{\mbox{\rm id}}_{n-1}
\end{array}
\right)\ . $$
Since $(m^n)^2=1$ we have $\sigma^n(m^n)=\pm {\mbox{\rm id}}$. The embedding
$M^n\hookrightarrow M^{n+k}$ sends $m^n$ to $m^{n+k}$.
There are nontrivial constants $C(\sigma^n)$ such that
\begin{equation}\label{blanche}
P_{\sigma^n}(\lambda)=C(\sigma^n)f_{\sigma^n}(\lambda)\prod_{\beta\in \Delta_r^n} \langle \mu_{\sigma^n}+\rho_m^n-\lambda,\beta\rangle\ ,
\end{equation}
where $\langle.,.\rangle$ is a Weyl-invariant bilinear scalar product on
$({\bf h}_{\bf C}^n)^*$ and
$$ f_{\sigma^n}(\lambda)=\left\{
\begin{array}{ccl}
1&G^n=Spin(1,2n+1)&\\
\tan \left(\pi \frac{\langle\lambda,\beta_r\rangle}{\langle\beta_r,\beta_r\rangle}\right), &G^n\not=Spin(1,2n+1),&
\sigma^n(m^n)=-{\rm e}^{2\pi\imath\frac{\langle\rho^n,\beta_r\rangle}{\langle\beta_r,\beta_r \rangle}}{\mbox{\rm id}}\\
\cot \left(\pi \frac{\langle\lambda,\beta_r\rangle}{\langle\beta_r,\beta_r\rangle}\right), &G^n\not=Spin(1,2n+1),&
\sigma^n(m^n)={\rm e}^{2\pi\imath\frac{\langle\rho^n,\beta_r\rangle}{\langle\beta_r,\beta_r \rangle}}{\mbox{\rm id}}
\end{array}
\right. \ . $$
According to these three possibilities of the form of the Plancherel density we
distinguish between the odd-dimensional, the tan and the cot case. We shall
construct the representation $\sigma^{n+k}$ case by case. The easiest one is
\noindent
{\bf The tan case}\newline
For any representation $\sigma^n$ fitting into this case we have
$$P_{\sigma^n}(0)=\tan(0)=0\ .$$
Let $\sigma^{n+k}\in \hat M^{n+k}$ be an arbitrary representation satisfying
$\sigma^n\subset \sigma^{n+k}_{\ |M^n}$. Then $\sigma^{n+k}(m^{n+k})=-{\mbox{\rm id}}$
iff $\sigma^n(m^n)=-{\mbox{\rm id}}$. Thus $\sigma^{n+k}$ belongs to the $\tan$-case iff
$\rho^{n+k}-\rho^n$ is an integer multiple of $\beta_r$ (this is always the case
except when $G^n=SU(1,n)$ and $k$ is odd). We conclude that
$$ P_{\sigma^{n+k}}(-\rho^{n+k}+\rho^n)=\left\{
\begin{array}{c}
C\tan(\pi\frac{\langle\-\rho^{n+k}+\rho^n,\beta_r\rangle}{\langle\beta_r,\beta_r \rangle})\\
C^\prime\cot(\pi\frac{\langle\-\rho^{n+k}+\rho^n,\beta_r\rangle}{\langle\beta_r,\beta_r \rangle})
\end{array}\right\}
=0\ .$$
\noindent
{\bf The odd-dimensional case}\newline
As a Cartan subalgebra of ${\bf g}^n$ we choose
$${\bf h}^n:= \left\{ T_\nu:= \left(
{\scriptsize
\begin{array}{cccc}
\begin{array}{cc}
0&\nu_0\\ \nu_0&0
\end{array}&&&\\
&\begin{array}{cc}
0&-\nu_1\\ \nu_1&0
\end{array}&&\\
&&\ddots&\\
&&&\begin{array}{cc}
0&-\nu_n\\ \nu_n&0
\end{array}
\end{array}
}
\right)\:\Bigg|\:\nu_i\in{\bf R} \right\}\ \ ,$$
where ${\bf a}=\{T_\nu\:|\:\nu_i=0,\:i=1,\dots,n\}$ and ${\bf t}^n=\{T_\nu\:|\:\nu_0=0\}$.
Define $e_i\in ({\bf h}_{\bf C}^n)^*$ by $e_0(T_\nu):=\nu_0$ and
$e_i(T_\nu):=\imath \nu_i$,
$i=1,\dots,n$. We normalize $\langle.,.\rangle$ such that $\{e_i\}$ becomes an
orthonormal basis of $({\bf h}_{\bf C}^n)^*$. Sometimes we will write elements of $\imath ({\bf t}^n)^*$ as $n$-tuples of reals with respect to the basis $e_1,\dots,e_n$. We choose positive roots of ${\bf t}^n$ in ${\bf m}^n_{\bf C}$ as follows:
$$\Delta_m^{n,+}:=\{e_i\pm e_j \:|\: 1\le i<j\le n\}\ .$$
Then $\rho^n_m=(n-1,n-2,\dots,1,0)$ and the irreducible representations of
$M^n$ correspond to the highest weights
$$ \hat M^n \cong \{\mu_{\sigma^n}=(m_1,\dots,m_n)\:|\:m_1\ge\dots\ge m_{n-1}\ge |m_n|,\: m_i-m_j\in{\bf Z},\: m_n\in\frac{1}{2}{\bf Z}\}\ .$$
Furthermore
$$\Delta^n_r=\{e_0\pm e_i\:|\:i=1,\dots,n\}\ .$$
We obtain $\rho^n=ne_0$ and for $\lambda=ze_0$
$$P_{\sigma^n}(\lambda)= C(\sigma^n)\prod_{i=1}^{n}(z^2-(m_i+n-i)^2)\ .$$
We see that $P_{\sigma^n}(0)=0$ iff $m_n=0$. In this case set
$$\mu_{\sigma^{n+k}}:=(m_1,\dots,m_{n-1},m_n=0,\underbrace{0,\dots,0}
_{\mbox{\scriptsize $k$ times}})\ .$$
Then $\sigma^n\subset \sigma^{n+k}_{\ |M^n}$ and the $n$-th factor of $P_{\sigma^{n+k}}$ is given by $z^2-(m_n+n+k-n)^2 = z^2- k^2$. Thus
$P_{\sigma^{n+k}}(-\rho^{n+k}+\rho^n)=P_{\sigma^{n+k}}(-ke_0)=0$.
\noindent
{\bf The cot case}\\
First we observe that if $G=Spin(1,2n)$ and $\sigma^n$ belongs to the $\cot$-case, then $P_{\sigma^n}(0)\not=0$. In fact, in this case $\sigma^n$ is a faithful representation of $M^n=Spin(2n-1)$ and the only root $\beta\in\Delta^n_r$
perpendicular to $\mu_{\sigma^n}+\rho_m^n$ is $\beta_r$. We leave the simple verification to the reader. Since $\cot$ has a pole at $0$ the observation follows
from (\ref{blanche}). An alternative proof is given in \cite{knappstein71}, Prop. 55.
We are left with the discussion of $G^n=SU(1,n)$ and $G^n=Sp(1,n)$.
We start with $G^n=SU(1,n)$.
The group $M^n$ has the form
$$M^n= \left\{\left(\begin{array}{ccc}
z&0&0\\0&z&0\\0&0&B \end{array}\right)\:|\:\:
z\in U(1),B\in U(n-1), z^2\det B=1\right\}\ .$$
We consider the Cartan subalgebra of ${\bf u}(1,n)$
$$\tilde{\bf h}^n:=
\left\{ T_\nu:= \left(
{\scriptsize
\begin{array}{ccccc}
\imath \nu_1&\nu_0&&&\\
\nu_0&\imath \nu_1&&&\\
&&\imath\nu_2&&\\
&&&\ddots&\\
&&&&\imath\nu_n
\end{array}
}
\right)\:\Bigg|\:\nu_i\in{\bf R} \right\}\ \ ,$$
their subalgebras
${\bf a}:=\{T_\nu\:|\:\nu_i=0,\:i=1,\dots,n\}$ and $\tilde{\bf t}^n:=\{T_\nu\:|\:\nu_0=0\}$.
Then ${\bf h}^n:=\{T_\nu\in\tilde{\bf h}^n\:|\:2\nu_1+\sum_{i=2}^n\nu_i=0\}$ and
${\bf t}^n:=\{T_\nu\in\tilde{\bf t}^n\:|\:2\nu_1+\sum_{i=2}^n\nu_i=0\}$ are Cartan subalgebras
of ${\bf g}^n$ and ${\bf m}^n$, respectively.
We define elements $\alpha,\beta,e_i\in (\tilde{\bf h}_{\bf C}^n)^*$, $i=2,\dots,n$, by $\alpha(T_\nu):=\nu_0$, $\beta(T_\nu)=\imath\nu_1$ and
$e_i(T_\nu):=\imath \nu_i$. We extend $\langle.,.\rangle$ to $(\tilde{\bf h}^n_{\bf C})^*$ such that $\{\alpha,\beta,e_2,\dots,e_n\}$ becomes an orthogonal basis of $(\tilde{\bf h}_{\bf C}^n)^*$ with
$|\alpha|^2=|\beta|^2=1$ and $|e_i|^2=2$.
Sometimes we will write elements of $\imath (\tilde{\bf t}^n)^*$ as $n$-tuples of reals with respect to the basis $\beta,e_2,\dots,e_n$. We choose the positive roots of ${\bf t}^n$ in ${\bf m}^n_{\bf C}$ as follows:
$$\Delta_m^{n,+}:=\{e_i-e_j \:|\: 2\le i<j\le n\}\ .$$
Then $\rho^n_m=\frac{1}{2}(0,n-2,n-4,\dots,4-n,2-n)$.
Furthermore we have
$$\Delta^n_r=\{2\alpha,\alpha\pm(\beta-e_i)\:|\:i=2,\dots,n\}$$
and $\rho^n=n\alpha$.
We represent highest weights of representations of $M^n$ which are functionals
on ${\bf t}^n$ by elements of $(\tilde{\bf t}^n_{\bf C})^*$:
\begin{eqnarray*}
\hat M^n &\cong& \{\mu_{\sigma^n}=(m_1,\dots,m_n)\:|\:m_2\ge\dots
\ge m_n,\: m_i
\in{\bf Z} \}\\
&&\quad\mbox{ modulo translation by elements of the form }
(2\nu,\nu,\dots,\nu)\ .
\end{eqnarray*}
Then we can compute the scalar products with elements of $\Delta^n_r$ inside $(\tilde{\bf h}^n_{\bf C})^*$,
and the result will not depend on the chosen representative.
Since $\rho^n=n\alpha$ and $\beta_r=2\alpha$ we see that $\sigma^n$ belongs to the $\cot$-case iff $m_1\equiv n\:(2)$.
We obtain for $\lambda=z\alpha$
$$P_{\sigma^n}(\lambda)= C(\sigma^n)\cot(\frac{\pi}{2}z)2z\prod_{i=2}^{n}(z^2-(m_1-n-2(m_i+1-i))^2)\ .$$
We see that $P_{\sigma^n}(0)=0$ iff for one index $i_0\in\{2,\dots,n\}$ the
following equation holds:
$$\frac{m_1-n}{2}=m_{i_0}+1-i_0\ .$$
In this case set
$$\mu_{\sigma^{n+k}}:=(m_1,\dots,m_{i_0},\underbrace{m_{i_0},\dots,m_{i_0}}
_{\mbox{\scriptsize $k$ times}},m_{i_0+1},\dots,m_n)\ .$$
Then $\sigma^n\subset \sigma^{n+k}_{\ |M^n}$.
Depending on the parity of $k$ the representation $\sigma^{n+k}$ belongs to the
$\tan$-case or $\cot$-case, respectively. In any case, the function $f_{\sigma^{n+k}}$ has a first-order
pole at $\lambda=-\rho^{n+k}+\rho^n=-k\alpha$. But
in addition to the $i_0$-th factor $z^2-(m_1-n-k-2(m_{i_0}+1-i_0))^2$ of the polynomial part of $P_{\sigma^{n+k}}(z\alpha)$ also the $(i_0+k)$-th factor $z^2-(m_1-n-k-2(m_{i_0}+1-i_0-k))^2$ is equal to $z^2- k^2$. Thus
the polynomial part contributes a second order zero at $z=k$ and
$P_{\sigma^{n+k}}(-\rho^{n+k}+\rho^n)=P_{\sigma^{n+k}}(-k\alpha)=0$.
\begin{itemize}\item $G^n=Sp(1,n)$\end{itemize}
The group $M^n$ has the form
$$M^n= \left\{\left(\begin{array}{ccc}
q&0&0\\0&q&0\\0&0&B \end{array}\right)\:|\:\:
q\in Sp(1),B\in Sp(n-1)\right\}\ .$$
As a Cartan subalgebra of ${\bf g}^n$ we choose
$${\bf h}^n:=
\left\{ T_\nu:= \left(
{\scriptsize
\begin{array}{ccccc}
\imath \nu_1&\nu_0&&&\\
\nu_0&\imath \nu_1&&&\\
&&\imath\nu_2&&\\
&&&\ddots&\\
&&&&\imath\nu_n
\end{array}
}
\right)\:\Bigg|\:\nu_i\in{\bf R} \right\}\ \ ,$$
where
${\bf a}=\{T_\nu\:|\:\nu_i=0,\:i=1,\dots,n\}$ and ${\bf t}^n=\{T_\nu\:|\:\nu_0=0\}$.
We define elements $\alpha,\beta,e_i\in ({\bf h}_{\bf C}^n)^*$, $i=2,\dots,n$, by $\alpha(T_\nu):=\nu_0$, $\beta(T_\nu)=\imath\nu_1$ and
$e_i(T_\nu):=\imath \nu_i$. Then $\{\alpha,\beta,e_2,\dots,e_n\}$ becomes an orthogonal basis of $(\tilde{\bf h}_{\bf C}^n)^*$ and we normalize $\langle.,.\rangle$ such that
$|\alpha|^2=|\beta|^2=1$ and $|e_i|^2=2$.
Sometimes we will write elements of $\imath ({\bf t}^n)^*$ as $n$-tuples of reals with respect to the basis $\beta,e_2,\dots,e_n$. We choose the positive roots of ${\bf t}^n$ in ${\bf m}^n_{\bf C}$ as follows
$$\Delta_m^{n,+}:=\{2\beta,e_i\pm e_j, 2e_i \:|\: 2\le i<j\le n\}\ .$$
Then $\rho^n_m=(1,n-1,n-2,\dots,2,1)$ and the irreducible representations of
$M^n$ correspond to the highest weights
$$ \hat M^n \cong \{\mu_{\sigma^n}=(m_1,\dots,m_n)\:|\:m_1\ge 0,m_2\ge\dots\ge m_{n}\ge 0,\: m_i\in{\bf Z}\}\ .$$
Furthermore we have
$$\Delta^n_r=\{2\alpha,2(\alpha\pm\beta),\alpha\pm(\beta\pm e_i)\:|\:i=2,\dots,n\}$$
and $\rho^n=(2n+1)\alpha$.
Since $\beta_r=2\alpha$ we see that $\sigma^n$ goes with cot iff $m_1$ is odd.
We obtain for $\lambda=z\alpha$
\begin{eqnarray*}
P_{\sigma^n}(\lambda)&=& C(\sigma^n)\cot(\frac{\pi}{2}z)\:2z\:4(z^2-(m_1+1)^2)\\
&&\quad
\prod_{i=2}^{n}(z^2-(m_1+1+2(m_i+n+1-i))^2)(z^2-(m_1+1-2(m_i+n+1-i))^2)\ .
\end{eqnarray*}
We see that $P_{\sigma^n}(0)=0$ iff for one index $i_0\in\{2,\dots,n\}$ the
following equation holds:
$$\frac{m_1+1}{2}=m_{i_0}+n+1-i_0\ .$$
We are going to define $\sigma^{n+k}$ by an inductive procedure. It rests on
the following claim.
Let $l\in{\bf N}_0$ and $\mu_{\sigma^{n+l}}=(m^\prime_1,\dots,m^\prime_{n+l})$
be a highest weight of an irreducible representation of $M^{n+l}$ such that
$m_1^\prime=m_1$, $m_{i_0}^\prime=m_{i_0}$ and one of the following conditions
holds:
\begin{enumerate}
\item There exists an index $j_l\in\{i_0+l,\dots,n+l\}$ such that
$$ m_1+1-2(m^\prime_{j_l}+n+l+1-j_l)=2l\ .$$
\item $m_1+1=2l$ .
\item There exists an index $j_l\in\{i_0,\dots,n+l\}$ such that
$$ m_1+1+2(m^\prime_{j_l}+n+l+1-j_l)=2l\ .$$
\end{enumerate}
Then there exists $\sigma_{n+l+1}\in\hat M^{n+l+1}$ with the same properties
($l$ replaced by $l+1$) such that $\sigma^{n+l}\subset\sigma^{n+l+1}_{\ |M^{n+l}}$.
We now prove the claim. If $\sigma^{n+l}$ satisfies condition 1 and $m^\prime_{j_l}>
m^\prime_{j_l+1}$ (by convention $m^\prime_j:=0$ for $j>n+l$) we set
$$\mu_{\sigma^{n+l+1}}:=(m^\prime_1,\dots,m^\prime_{j_l},m^\prime_{j_l}-1, m^\prime_{j_l+1},\dots,m^\prime_{n+l})\:,\ j_{l+1}:=j_l+1\ .$$
Then $\sigma^{n+l+1}$ also satisfies condition 1.
If $\sigma^{n+l}$ satisfies condition 1 and $m^\prime_{j_l}=
m^\prime_{j_l+1}$ we set
$$\mu_{\sigma^{n+l+1}}:=(m^\prime_1,\dots,m^\prime_{j_l},m^\prime_{j_l}, m^\prime_{j_l}, m^\prime_{j_l+2},\dots,m^\prime_{n+l})\ .$$
If in addition $j_l<n+l$, set $j_{l+1}:=j_l+2$. Then again $\sigma^{n+l+1}$ satisfies condition 1. Otherwise we have $m^\prime_{n+l}=0$, hence
$m_1-1=2l$. It follows that $\sigma^{n+l+1}$ satisfies condition 2.
If $\sigma^{n+l}$ satisfies condition 2 or 3, then $\sigma^{n+l+1}$ defined by
$$\mu_{\sigma^{n+l+1}}:=(m^\prime_1,\dots,m^\prime_{n+l},0)$$
satisfies condition 3 with $j_{l+1}=n+l+1$ or $j_{l+1}=j_l$, respectively.
The branching rules for the restriction from $Sp(n+l+1)$ to $Sp(n+l)$
(see e.g. \cite{zelobenko73}, Ch. XVIII) show that in any case $\sigma^{n+l}\subset \sigma^{n+l+1}_{\ |M^{n+l}}$. The claim now follows.
Since $\sigma^n$ satisfies the induction hypothesis of the claim for $l=0$ and
$j_0=i_0$ we can define $\sigma^{n+k}$ inductively. We have to check that
$P_{\sigma^{n+k}}(-\rho^{n+k}+\rho^n)=P_{\sigma^{n+k}}(-2k\alpha)=0$. In fact, the claim ensures that $P_{\sigma^{n+k}}(z\alpha)$ containes in addition to
$z^2-(m_1+1-2(m_{i_0}+n+k+1-i_0))^2$ a second factor which contributes with
$z^2-(2k)^2$. Thus the pole originating from the $\cot$ factor cancels, and we have $P_{\sigma^{n+k}}(-2k\alpha)=0$. Now the lemma and, hence,
Proposition \ref{mystic} is proved.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\section{The essential spectrum}\label{esess}
In the present section we consider the spectral comparison between
$L^2(X,V(\gamma))$ and $L^2(Y,V_Y(\gamma))$.
Let ${\cal A}$ be any commutative algebra of invariant differential operators
on $V(\gamma)$ which is generated by selfadjoint elements and contains ${\cal Z}_\gamma$. As for ${\cal Z}_\gamma$ there are spectral decompositions
of $L^2(X,V(\gamma))$ and $L^2(Y,V_Y(\gamma))$ with respect to ${\cal A}$.
The main result is that the essential spectrum
of ${\cal A}$ on $L^2(X,V(\gamma))$ and $L^2(Y,V_Y(\gamma))$
coincides.
Recall the characterization of the essential spectrum in terms of Weyl sequences. Let $\{A_i\}$ be a finite set of generators of ${\cal A}$.
A character $\lambda$ of ${\cal A}$ belongs to the essential
spectrum of ${\cal A}$ iff there exists a Weyl sequence $\{\phi_\alpha\}\subset C^\infty_c$ (i.e. a sequence without accumulation points in $L^2$) such that
$\max_i \|A_i\phi_\alpha-\lambda(A_i)\phi_\alpha\|_{L^2}\rightarrow 0$ when
$\alpha\to \infty$.
\begin{prop}\label{esspec}
The essential spectrum of ${\cal A}$ on $L^2(X,V(\gamma))$ and on $L^2(Y,V_Y(\gamma))$
coincides.
\end{prop}
{\it Proof.$\:\:\:\:$}
The proof of the proposition relies on the transfer of Weyl sequences.
Let $\lambda$ be in the essential spectrum of ${\cal A}$ on $L^2(Y,V_Y(\gamma))$.
Then there is a Weyl sequence $\{\phi_\alpha\}$, satisfying
\begin{itemize}
\item $\|\phi_\alpha\|_{L^2(Y,V_Y(\gamma))}=1$, $\forall \alpha$
\item $\{\phi_\alpha\}$ has no accumulation point in $L^2(Y,V_Y(\gamma))$, and
\item $\max_i \| A_i\phi_\alpha-\lambda(A_i)\phi_\alpha\|_{L^2(Y,V_Y(\gamma))}\rightarrow 0$ as $\alpha\to \infty$.
\end{itemize}
Using a construction of Eichhorn \cite{eichhorn91}
we can modify this Weyl sequence such that it satisfies in addition
$\|\phi_\alpha\|_{L^2(K,V_Y(\gamma))}\rightarrow 0$ as $\alpha\to \infty$ for any compact $K\subset Y$.
Let $\{V_i\}$ be a finite number of open subsets covering $Y$ at infinity such that each
$V_i$ has a diffeomorphic lift $\tilde{V}_i\subset X$.
Using the method of Lemma \ref{lll} we can choose the $V_i$ such that there exists
a subordinated partition of unity $\{\chi_i\}$ (at infinity of $Y$)
such that for $A\in {\cal A}$
$$|[A,\chi_i](ka)| \le C(A) a^{-1},\quad \forall kaK\in \tilde{V}_i\ .$$
By taking a subsequence of the Weyl sequence and renumbering the $V_i$ we can assume that
$\|\chi_1 \phi_\alpha\|_{L^2(Y,V_Y(\gamma))}\ge c$ for some $c>0$ independent of $\alpha$.
We set $\psi_\alpha:=\chi_1\phi_\alpha /\|\chi_1 \phi_\alpha\|_{L^2(Y,V_Y(\gamma))}$.
We claim that $\psi_\alpha$ is again a Weyl sequence for $\lambda$.
In fact $\|\psi_\alpha\|_{L^2(Y,V_Y(\gamma))}=1$ by definition, $\| \psi_\alpha\|_{L^2(K,V_Y(\gamma))}\rightarrow 0$ as $\alpha\to \infty$ for any compact $K\subset Y$.
This implies that $\{\psi_\alpha\}$ has no accumulation points.
It remains to verify that for $A\in {\cal A}$
$$\|(A-\lambda(A))\psi_\alpha\|_{L^2(Y)}\rightarrow 0, \quad \alpha\to \infty\ .$$
We have
$$(A-\lambda(A))\psi_\alpha=\frac{\chi_1}{\|\chi_1 \phi_\alpha\|_{L^2(Y,V_Y(\gamma))}}(A-\lambda(A))\phi_\alpha
+\frac{[A,\chi_1]\phi_\alpha}{\|\chi_1 \phi_\alpha\|_{L^2(Y,V_Y(\gamma))}} \ .$$
Obviously we have
$$\|\frac{\chi_1}{\|\chi_1 \phi_\alpha\|_{L^2(Y,V_Y(\gamma))}}(A-\lambda(A))\phi_\alpha\|_{L^2(Y,V_Y(\gamma))}\rightarrow 0, \quad \alpha\to \infty\ .$$
For any $\epsilon>0$ we can choose $K\subset Y$ compact such that
$\sup_{x\in Y\setminus K} |[A,\chi_1](x)|<\epsilon$.
We then have
\begin{eqnarray*}
\lefteqn{\lim_{\alpha\to\infty}\|\frac{1}{\|\chi_1 \phi_\alpha\|_{L^2(Y,V_Y(\gamma))}}[A,\chi_1]\phi_\alpha\|_{L^2(Y,V_Y(\gamma))}}\hspace{1cm}\\
&\le &\lim_{\alpha\to\infty} \left( \frac{\sup_{x\in K}|[A,\chi_1](x)|}{\|\chi_1 \phi_\alpha\|_{L^2(Y,V_Y(\gamma))}} \|\phi_\alpha\|_{L^2(K,V_Y(\gamma))} + \frac{\epsilon}{\|\chi_1 \phi_\alpha\|_{L^2(Y,V_Y(\gamma))}}
\|\phi_\alpha\|_{L^2(Y\setminus K,V_Y(\gamma))}\right) \\
&\le& \epsilon/c\ .
\end{eqnarray*}
It follows that
$$\lim_{\alpha\to\infty}\|\frac{[A,\chi_1]\phi_\alpha}{\|\chi_1 \phi_\alpha\|_{L^2(Y,V_Y(\gamma))}} \|_{L^2(Y,V_Y(\gamma))}=0\ .$$
Thus $\{\psi_\alpha\}$ is a Weyl sequence of the algebra ${\cal A}$
with respect to the character $\lambda$ which is supported in $V_1$.
Lifting this sequence to $X$ we obtain a Weyl sequence of ${\cal A}$ to $\lambda$
in $L^2(X,V(\gamma))$.
Thus the essential spectrum of ${\cal A}$ on $L^2(Y,V_Y(\gamma))$ is contained in the essential spectrum of ${\cal A}$ on $L^2(X,V(\gamma))$.
In order to prove the opposite inclusion
we choose a finite cover of infinity of $X$ by sets $W_j$ such that the central projection of
$W_j$ is not surjective onto $\partial X$.
If the character $\lambda$ is in the essential spectrum of ${\cal A}$ on $L^2(X,V(\gamma))$, then repeating the above construction we can find a Weyl sequence $\{\psi_\alpha\}$ of ${\cal A}$ to $\lambda$ with
${\mbox{\rm supp}}(\psi_\alpha)\subset W_1$. There is a $g\in G$ such that $gW_1\subset V_1$.
Then $g^*\psi_\alpha$ can be pushed down to $Y$ and gives a Weyl sequence
for ${\cal A}$ to $\lambda$ on $L^2(Y,V_Y(\gamma))$.
Thus the essential spectrum of ${\cal A}$ on $L^2(X,V(\gamma))$ is contained in the essential spectrum of ${\cal A}$ on $ L^2(Y,V_Y(\gamma))$.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\section{Relevant generalized eigenfunctions}\label{relsec}
In principle a spectral decomposition of $L^2(Y,V_Y(\gamma))$ with respect to
${\cal Z}_\gamma$ is a way of expressing elements of $L^2(Y,V_Y(\gamma))$
in terms of generalized eigensection of ${\cal Z}$, i.e., sections
in $C^\infty(Y,V_Y(\gamma))$ on which ${\cal Z}$ acts by a character.
Its turns out that only a small portion of these eigenfunctions
is are needed for the spectral decomposition. We call them relevant.
Relevant eigenfunctions satisfy certain growth conditions.
In order to deal with these growth conditions properly we
introduce the Schwartz space $S(Y,V_Y(\gamma))$.
This Schwartz space is the immediate generalization of Harish-Chandra's Schwartz space $S(X,V(\gamma))$ to our locally symmetric situation.
The significance of the Schwartz space is the following.
If a generalized eigenfunction is relevant for the spectral decomposition
of $L^2(Y,V_Y(\gamma))$ with respect to ${\cal Z}_\gamma$ then
it defines a continuous functional on the Schwartz space.
The Schwartz space is defined as a Fr\'echet space which is contained in $L^2(Y,V_Y(\gamma))$. Then its (hermitian) dual $S(Y,V_Y(\gamma))^\prime$ contains the relevant generalized eigenfunctions.
We now turn to the definition of $S(Y,V_Y(\gamma))$.
The idea is to require the similar conditions as for $S(X,V(\gamma))$
locally at infinity.
Let $W\subset X\cup\Omega$ be any compact subset. For any $A\in{\cal U}({\bf g})$, $N\in {\bf N}$, we define the seminorm $q_{W,A,N}(f)\in[0,\infty]$
of $f\in C^\infty(X,V(\gamma))$ by
\begin{equation}\label{swart}q_{W,A,N}(f):=\int_{(W\cap X)K} \log (\|g\|^N ) |f(Ag)|^2 dg\ .\end{equation}
Here $\|g\|$ denotes the norm of ${\mbox{\rm Ad}}(g)$ on ${\bf g}$.
In the following definition we identifiy $C^\infty(Y,V_Y(\gamma))$ with the subspace of $\Gamma$-invariant sections in $C^\infty(X,V(\gamma))$.
\begin{ddd}
The Schwartz space is the space of sections $f\in C^\infty(Y,V_Y(\gamma))$ with $q_{W,A,N}(f)<\infty$ for all $W$, $A$ and $N$ as above.
The seminorms $q_{W,A,N}$ define the Fr\'echet
topology of $S(Y,V_Y(\gamma))$.
\end{ddd}
\begin{ddd}
An eigenvector of ${\cal Z}_\gamma$ belonging to
the dual of the Schwartz space $S(Y,V_Y(\gamma))^\prime$ is called
tempered.
\end{ddd}
The main goal of the present section is to provide a list of all
tempered eigenvectors of ${\cal Z}$. Unfortunately there exist exceptional
characters where such a description is difficult. Fortunately
this exceptional set is at most countable. We will cover the set
of all characters of ${\cal Z}$ which may provide difficulties
by a countable set $PS$ below.
The set of $\lambda\in{\bf h}_{\bf C}^*$ of parametrizing integral
characters of ${\cal Z}$ forms a lattice and is hence countable.
We denote that set of integral characters of ${\cal Z}$ by $PS_i$.
We choose a Cartan algebra ${\bf t}$ of ${\bf m}$ such that ${\bf a}\oplus{\bf t}={\bf h}$
and a positive root system of ${\bf t}$. Let $\rho_m$ denote half of the sum of the positive roots of $({\bf m},{\bf t})$.
For $\sigma\in \hat{M}$ let $\mu_\sigma\in{\bf t}^*$ be the highest weight.
Then the character $\chi_\lambda$ of ${\cal Z}$ on the principal series representation
$C^\infty(\partial, V(\sigma_\mu))$ is parametrized by
$\lambda:=\mu_\sigma+\rho_m-\mu\in{\bf h}_{\bf C}^*$.
We observe that if $\chi_\lambda\not\in PS_i$, then $\mu\in{\aca(\sigma)}$. Indeed, a
pole of $P_\sigma$ at $\mu$ implies the integrality $\chi_\lambda$ (compare
(\ref{blanche})), whereas a pole of $\hat J_{-\mu}$ for non-integral $\chi_\lambda$ cancels with a zero of $P_\sigma$ at $\mu$ because of (\ref{spex})
and the irreducibility of $C^{\infty}(\partial X,V(\sigma_{\mu}))$.
If $\chi\not\in PS_i$, then we have a complete understanding
of the corresponding eigenspace
$${\cal E}_\chi:= \{f\in C^\infty_{mg}(X,V(\gamma)), \quad (A-\chi(A))f=0 \quad \forall A\in{\cal Z} \}\ .$$
Let $(V_\gamma)_{|M}=\oplus_i V_\gamma(\sigma_i)$
denote the decomposition of $(V_\gamma)_{|M}$ into isotypic components.
Set $r_i=[(V_\gamma)_{|M}:V_{\sigma_i}]$ and
choose isomorphisms $T_i\in {\mbox{\rm Hom}}_M(\oplus_{j=1}^{r_i} V_{\sigma_i},V_\gamma(\sigma_i))$. Employing the fact that principal series representations with non-integral infinitesimal character are irreducible it is
a simple matter to deduce the following lemma from the results of \cite{olbrichdiss}.
\begin{lem}\label{oppp}
If $\chi\not\in PS_i$, then
the eigenspace ${\cal E}_\chi$ is the isomorphic
image of the Poisson transform $$\bigoplus_i P_{\mu_i}^{T_i}:\bigoplus_i \bigoplus_{j=1}^{r_i}C^{-\infty}(\partial X,V(\sigma_{i,\mu_i}))\rightarrow C^\infty_{mg}(X,V(\gamma))\ ,$$
where $\mu_i\in{\aaaa_\C^\ast}$ is uniquely characterized by $\chi_{\mu_{\sigma_i}+\rho_m-\mu_i}=\chi$, ${\rm Re }(\mu_i)>0$ or ${\rm Re }(\mu_i)=0$ and ${\rm Im}(\mu_i)\ge 0$.
\end{lem}
Let $PS_d$ denote the set of all characters $\chi$ of ${\cal Z}$
such that there exists $\sigma\subset\gamma_{|M}$ and $\mu\in{\aaaa_\C^\ast}$
with ${\rm Re }(\mu)\ge 0$, $\chi_{\mu_\sigma+\rho_m-\mu}=\chi$
and ${}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\mu))\not= 0$ or $\mu =0$.
By Lemma \ref{ghu} and the fact that $ext$ is meromorphic
on ${\aca(\sigma)}$ the set $PS_d$ is countable.
Let $PS=PS_i\cup PS_d$. Then $PS$ is countable, too.
We recall the asymptotic expansion of eigenfunctions
(see \cite{wallach88}, Ch. 4, \cite{wallach92}, Ch.11).
Since we want to avoid hyperfunction boundary values
we consider eigensections of moderate growth. A section
$f\in C^\infty(X,V(\gamma))$ is of moderate growth if
for any $A\in {\cal U}({\bf g})$ there exists $R\in{\bf R}$ such that
$$\sup_{g\in G} \|g\|^{-R}|f(Ag)| <\infty\ .$$
Let $C_{mg}^\infty(X,V(\gamma))$ be the space
of all sections of $V(\gamma)$ of moderate growth.
Let $L^+\subset {\bf a}^*_+$ be the semigroup generated by the
positive roots of $({\bf a},{\bf n})$ and $0$.
Let $f\in C^\infty(X,V(\gamma))$ be some ${\cal Z}$-finite section
which is $K$-finite with respect to the action of $K$ by left translations.
Then there is
a finite set of leading exponents $E(f)\subset {\aaaa_\C^\ast}$
such that
$$f(ka)\stackrel{a\to\infty}{\sim} \sum_{\mu\in E(f)} a^{\mu-\rho} \sum_{{\bf Q}\in L^+} a^{-Q} p(f,\mu,Q)(k)(\log(a))\ ,$$
where $p(f,\mu,Q)$ is a polynomial (required to be non-trivial for $Q=0$) on ${\bf a}$ with values in $C^\infty(K,V_\gamma)$.
To read this expansion properly consider $f$ as a function on $G$ with
values in $V_\gamma$.
The leading coefficient of the polynomial $p(f,\mu,0)$ has a continuous extension with respect to $f$ which is a $G$-equivariant continuous map
from the closed $G$-submodule of $C^\infty_{mg}(X,V(\gamma))$ generated by $f$ to $C^{-\infty}(\partial X,V(\gamma_{|M,\mu}))$.
If $f$ is an eigensection of ${\cal Z}$ and $\mu\not=0$, then $p(f,\mu,0)$ is in fact a constant polynomial
\cite{olbrichdiss}, Lemma 4.6.
If $f\in C^\infty(Y,V_Y(\gamma))$ is a tempered generalized eigensection of ${\cal Z}$, then its $\Gamma$-invariance implies that $f\in C_{mg}^\infty(X,V(\gamma))$.
It follows that $p(f,\mu,0)\in {}^\Gamma C^{-\infty}(\partial X,V(\gamma_{|M,\mu}))$ is a $\Gamma$-invariant distribution.
\begin{lem}\label{wo}
If $f$ is a tempered generalized eigensection of ${\cal Z}$, then
${\mbox{\rm supp}} (p(f,\mu,0))\subset\Lambda$ for all $\mu\in E(f)$ with ${\rm Re }(\mu)>0$.
\end{lem}
{\it Proof.$\:\:\:\:$}
We argue by contradiction.
Consider the exponent $\mu\in E(f)$ with the largest real part ${\rm Re }(\mu)>0$
such that ${\mbox{\rm supp}}(p(f,\mu,0))\cap\Omega \not= \emptyset$. We assume that such an exponent $\mu$ exists. Note that $p(f,\mu,0)$ is a constant polynomial on ${\bf a}$.
We study the support of of $p(f,\mu,0)$ by testing this distribution against
suitable test functions.
Let $F\subset X\cup\Omega$ be a fundamental domain of $\Gamma$
and $\partial F:=F\cap\Omega$.
Since $p(f,\mu,0)$ is $\Gamma$-invariant and since we have
freedom to choose $F$ it suffices to show that ${\mbox{\rm supp}}(p(f,\mu,0))\cap{\rm int}(\partial F)=\emptyset$.
Thus let $\phi\in C_c^\infty({\rm int}(\partial F),V(\gamma_{|M,-\bar{\mu}}))$ be a test function. The application of the distribution boundary value $p(f,\mu,0)$
to $\phi$ can be written as the limit
$$
(\phi,p(f,\mu,0)) = \lim_{a\to \infty} a^{\rho-\mu}
\int_K ( \phi(k), f(ka) )$$
(we write sesquilinear pairings as $(.,.)$).
Let $\chi\in C^\infty_c(0,1)$ be such that $\int_0^1\chi(t) dt=1$.
We extend $\chi$ to ${\bf R}$ by zero.
Then we define $\chi_n\in C_c^\infty(A_+)$ by
$\chi_n(a):=a^{-\rho-\bar{\mu}}\chi(|\log(a)|-n)$.
If we set $\phi_n(ka)= \chi_n(a)\phi(k)$,
then we can write
\begin{equation}\label{mnb}( \phi,p(f,\mu,0))
=\lim_{n\to \infty} (\phi_n,f) \ .\end{equation}
If $n$ is sufficiently large, then ${\mbox{\rm supp}}(\phi_n)\subset F$.
$\phi_n$ descends to a Schwartz space section $\tilde{\phi}_n\in S(Y,V_Y(\gamma))$. Using the fact that ${\rm Re }(\mu)>0$ one can easily check that $\lim_{n\to \infty}\tilde{\phi}_n=0$ in $S(Y,V_Y(\gamma))$.
The right-hand side of (\ref{mnb}) can be written as the application
of $f\in S(Y,V_Y(\gamma))^\prime$ to $\tilde{\phi}_n\in S(Y,V_Y(\gamma))$.
It follows that
$$( \phi,p(f,\mu,0))=\lim_{n\to 0}(\tilde{\phi}_n,f)=0\ .$$
Since $\phi$ was arbitrary we conclude that ${\mbox{\rm supp}}(p(f,\mu,0))\cap{\rm int}(\partial F)=\emptyset$.
As noted above it follows that ${\mbox{\rm supp}}(p(f,\mu,0))\cap\Omega=\emptyset$
and this contradicts our assumption.\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
The following proposition is a part of our description of the generalized eigenfunctions
of ${\cal Z}$ which are relevant for the spectral decomposition.
We adopt the notation of Lemma \ref{oppp}.
\begin{prop}\label{gener1}
If $\chi\not\in PS$ and $f\in {\cal E}_\chi\cap S(Y,V_Y(\gamma))^\prime$ is a tempered eigenfunction of ${\cal Z}$,
then $f=\bigoplus_i P_{\mu_i}^{T_i}(\phi_i)$, where
${\rm Re }(\mu_i)\ge 0$, $\chi_{\mu_{\sigma_i}+\rho_m-\mu_i}=\chi$.
Moreover, $\phi_i\not= 0$ implies that ${\rm Re }(\mu_i)=0$.
\end{prop}
{\it Proof.$\:\:\:\:$}
Assume that $\chi\not\in PS$. Let $0\not=f\in E_\chi$ be a tempered
eigenfunction of ${\cal Z}$. Then by Lemma \ref{oppp} we can represent $f$ as
$\sum_i P_{\mu_i}^{T_i}(\phi_i)$. Here $\phi_i\in \bigoplus_{j=1}^{r_i}{}^\Gamma C^{-\infty}(\partial X,V(\sigma_{i,\mu_i}))$ is uniquely characterized by $c_\gamma(\mu_i)T_i\phi_i=p(f,\mu_i,0)$.
If $\phi_i\not=0$, then by Lemma \ref{wo} and the definition of $PS_d\subset PS$ we have ${\rm Re }(\mu_i)=0$.\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\section{Wave packets and scalar products}\label{wxa}
In this section we introduce wave packets of Eisenstein series and
show that they belong to the Schwartz space.
Thus the scalar product of a wave packet with such a tempered generalized
eigenfunction of ${\cal Z}$ makes sense. If the corresponding character
does not belong to the exceptional set $PS$, then we obtain an explicit formula for this scalar product.
The subspace $L^2(Y,V_Y(\gamma))_c\subset L^2(Y,V_Y(\gamma))$ spanned by the wave packets is the absolute contiuous subspace.
We show that its complement is the discrete subspace $L^2(Y,V_Y(\gamma))_d$
and that there is no singular continuous subspace.
It turns out that the support of the continuous spectrum
of ${\cal Z}$ on $L^2(Y,V_Y(\gamma))$ concides with the support
of the continuous spectrum of ${\cal Z}$ on $L^2(X,V(\gamma))$
which is well known by the Harish-Chandra Plancherel theorem.
The main result of the present section is Theorem \ref{contsp}.
First we introduce the notion of an Eisenstein series.
Let $\sigma\in \hat{M}$,
$T\in{\mbox{\rm Hom}}_M(V_\sigma,V_\gamma)$, and let $P^T_\mu$, $\mu\in{\aaaa_\C^\ast}$,
be the associated Poisson transform (see Definition \ref{defofpoi}).
\begin{ddd}
The Eisenstein series associates to $\phi\in C^{-\infty}(B,V(\sigma_\mu))$
the eigensection of ${\cal Z}$
$$E(\mu,\phi,T):= P_\mu^T\circ ext(\phi)\in C^\infty(Y,V_Y(\gamma))$$
corresponding the character $\chi_{\mu_\sigma+\rho_m-\mu}$.
\end{ddd}
The meromorphic continuation of $ext$ immediately implies the following corollary.
\begin{kor} For $\mu\in{\aca(\sigma)}$ the Eisenstein series
$E(\mu,.,T):C^{-\infty}(B,V(\sigma_\mu))\rightarrow C^\infty(Y,V_Y(\gamma))$ is a meromorphic family of operators with finite-dimensional singularities.
The Eisenstein series is holomorphic on $\{{\rm Re }(\mu)=0,\mu\not=0\}$.
Moreover, $E(\mu,.,T)$ has at most first-order poles in the set $\{\lambda\in{\aca(\sigma)}\:|\:{\rm Re }(\mu)>0\}$.
\end{kor}
We now discuss the functional equations satisfied by the Eisenstein series.
This functional equation will be deduced from the corresponding functional equation of the Poisson transform.
Recall the definition (\ref{cgamma}) of $c_\gamma(\lambda)$.
Next we recall the functional equation satisfied by the Poisson transform
which is proved in \cite{olbrichdiss}:
\begin{equation}\label{mi9}c_{\sigma}(\lambda) P^T_\lambda \circ J_{-\lambda} = P^{\gamma(w)c_{ \gamma}(\lambda)T}_{-\lambda } \ ,\end{equation}
where $w\in N_K(M)$
represents the generator of the Weyl group of $({\bf g},{\bf a})$.
If we use $J_\lambda\circ ext = ext\circ S_\lambda$, then we obtain
the following corollary.
\begin{kor}\label{funeq}
The Eisenstein series satisfies the functional equation
$$E(\lambda,c_{\sigma}(\lambda)S_{-\lambda}\phi,T)=E(-\lambda ,\phi,\gamma(w)c_{ \gamma}(\lambda)T)\ .$$
(To be more precise, this is an identity of meromorphic quantities
valid for all $\lambda\in {\aaaa_\C^\ast}$ where all terms are meromorphic.)
\end{kor}
Now we turn to the definition of the wave packet transform.
Roughly speaking, a wave packet of Eisenstein series is
an average of the Eisenstein series over imaginary parameters
with a smooth, compactly supported weight function.
More precisely, the space of such weight functions
${\cal H}_0$ is the linear space of smooth families ${\bf a}^*_+\ni \mu\mapsto \phi_{\imath\mu}\in C^\infty(B,V_B(\sigma_{\imath\mu}))$ with compact support
with respect to $\mu$.
Because of the symmetry \ref{funeq} it will be sufficient to consider wave packets on the positive imaginary axis, only.
Next we fix a convenient choice of the endomorphisms $T$ entering the definition of the Eisenstein series.
Let $\tilde{\gamma}$ be the dual representation of $\gamma$ and let
$\tilde{T}\in {\mbox{\rm Hom}}_M(V_{\tilde{\sigma}},V_{\tilde{\gamma}})$ be such that $\tilde{T}^*T={\mbox{\rm id}}$.
We set $T(\lambda):=c_\sigma(\lambda)^{-1}T$ and $\tilde{T}(\lambda):=c_{\tilde{\sigma}}(\lambda)^{-1} \tilde{T}$.
There is a conjugate linear isomorphism of $\sigma$ and $\tilde{\sigma}$ and
hence $\bar{c}_{\sigma}(\bar{\lambda})=c_{\tilde{\sigma}}(\lambda)$.
Now we define the wave packet transform on ${\cal H}_0$.
Later we will extend it by continuity to a Hilbert space closure of ${\cal H}_0$.
\begin{ddd}
The wave packet transform is the map $E:{\cal H}_0\rightarrow C^\infty(Y,V(\gamma))$
given by
$$E(\phi) := \int_0^\infty E(\imath\mu,\phi_{\imath\mu},T(\imath\mu))\:d\mu\ .$$
The section $E(\phi)$, $\phi\in{\cal H}_0$, is called a wave packet (of Eisenstein series).
\end{ddd}
\begin{lem}\label{swa}
If $\phi\in {\cal H}_0$, then $E(\phi)\in S(Y,V_Y(\gamma))$.
\end{lem}
{\it Proof.$\:\:\:\:$}
We reduce the proof of this lemma to the case where $\Gamma$ is trivial.
In this case the assertion is well known \cite{arthur75}.
Let $W\subset X\cup\Omega$ be a compact subset such that $\Gamma W=X\cup\Omega$.
Let $\partial W:=W\cap\Omega$ and let $\chi\in C_c^\infty(\Omega)$
be a cut-off function such that $\chi_{|\partial W}=1$.
Let $\tilde{\phi}_{\imath\mu}=ext\:\phi_{\imath\mu}$ and set
$\psi_{\imath\mu}=\chi \tilde{\phi}_{\imath\mu}$,
$\xi_{\imath\mu}=(1-\chi)\tilde{\phi}_{\imath\mu}$.
Then ${\bf a}^*_+\ni\mu\mapsto \psi_{\imath\mu}\in C^\infty(\partial X,V(\sigma_{\imath\mu}))$ is a smooth family with compact support.
It was shown in \cite{arthur75} that
$$P(\psi):=\int P^T_{\imath\mu}(\psi_{\imath\mu}) d\mu\in S(X,V(\gamma))\ .$$
Since $E(\phi)=P(\psi)+P(\xi)$,
it remains to show that
$q_{W,A,N}(P(\xi))<\infty$ for all $A\in {\cal U}({\bf g})$, $N\in{\bf N}$, where $q_{W,A,N}$ is one of the seminorms (\ref{swart}) characterizing the Schwartz space.
By the $G$-equivariance of the Poisson transform we have
$$L_AP(\xi)=\int P^T_{\imath\mu}(\pi^{\sigma,\imath\mu}(A) \xi_{\imath\mu}) d\mu\ .$$
Note that $\xi_{|\partial W} =0$.
The expansion of the Poisson transform
obtained in Lemma \ref{eee}
leads to the following decomposition
into a leading and a remainder term.
For $k\in \partial WM$
\begin{eqnarray*}
P^T_{\imath\mu}(\pi^{\sigma,\imath\mu}(A) \xi_{\imath\mu})(ka)&=&
a^{-(\imath\mu+\rho)}\gamma(w)T (\hat{J}_{\imath\mu}(\pi^{\sigma,\imath\mu}(A) \xi_{\imath\mu}))(k)\\
&&+ a^{-(\imath\mu+\rho+\alpha_1)} p_1(a,k,\pi^{\sigma,\imath\mu}(A) \xi_{\imath\mu})\ ,
\end{eqnarray*}
where $p_1(a,k,f)$ is uniformly bounded as $ka\in W$
in terms of the distribution $f$.
Thus for all $N$ we have
$$\{W\ni ka\mapsto |\log(a)|^N \int a^{-(\imath\mu+\rho+\alpha_1)} p_1(a,k,\pi^{\sigma,\imath\mu}(A) \xi_{\imath\mu}) d\mu\}\in L^2(W,V(\gamma))\ .$$
This shows that the remainder term leads to something satisfying the
estimates of the Schwartz space.
We now analyse the leading term.
Since the family $\xi_{\imath\mu}$ has compact support
with respect to $\mu$ and $\hat{J}_{\imath\mu}(\pi^{\sigma,\imath\mu}(A) \xi_{\imath\mu})$ is smooth in $(\mu,k)$ by Lemma
\ref{off} we obtain for all $N\in{\bf N}_0$
$$|\int a^{-(\imath\mu+\rho)} \gamma(w) T (\hat{J}_{\imath\mu}(\pi^{\sigma,\imath\mu}(A) \xi_{\imath\mu}))(k) d\mu|\le C_N
(1+|\log(a)|)^{-N} a^{-\rho},\quad \forall ka\in W$$
for each $N\in{\bf N}$.
Thus the leading terms satisfies the Schwartz space estimates, too.
This proves the lemma.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
Let $\mu\not=0$, ${\rm Re }(\mu)=0$, and $\psi\in C^{-\infty}(B,V_B(\tilde{\sigma}_{-\imath\mu}))$. Set
$\tilde{\psi}:=E(-\imath\mu,\psi,\tilde{T}(-\imath\mu))$.
\begin{lem}\label{tttq}
$\tilde{\psi}\in S(Y,V_Y(\gamma))^\prime$.
\end{lem}
{\it Proof.$\:\:\:\:$}
We have $ext (\psi)\in C^{-\infty}(\partial X,V(\tilde{\sigma}_{-\imath\mu}))$.
It known by \cite{arthur75} that $P^{\tilde{T}}_{-\imath\mu}(ext(\psi))\in S(X,V(\gamma))^\prime$. This implies the lemma. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
By Lemmas \ref{swa} and \ref{tttq}
the pairing $\langle E(\phi),\tilde{\psi}\rangle$
between the wave packet $E(\phi)$ and the generalized eigensection $\tilde{\psi}$ is well defined.
The following proposition gives an explicit formula for this pairing.
\begin{prop}\label{scalar}
We have $\langle E(\phi),\tilde{\psi}\rangle=\pi\langle \phi_{\imath\mu},\psi\rangle$.
\end{prop}
{\it Proof.$\:\:\:\:$}
Let $\psi_n\in C^{\infty}(B,V_B(\tilde{\sigma}_{-\imath\mu}))$
be a sequence approximating the distribution $\psi$.
Then by \cite{arthur75} and the continuity of $ext$ we have
$$E(-\imath\mu,\psi_n,\tilde{T}(-\imath\mu))\to E(-\imath\mu,\psi,\tilde{T}(-\imath\mu))$$
in $S(Y,V_Y(\gamma))^\prime$ as $n\to \infty$.
Thus the proposition is a consequence of the following the special case.
\begin{lem}
Let $\psi\in C^{\infty}(B,V_B(\tilde{\sigma}_{-\imath\mu}))$, then
$\langle E(\phi),\tilde{\psi}\rangle=\pi\langle \phi_{\imath\mu},\psi\rangle$.
\end{lem}
{\it Proof.$\:\:\:\:$}
Let $W\subset\Omega$ be compact.
The following asymptotic expansions hold uniformly for $k\in WM$ and $a\in A_+$ large:
\begin{eqnarray}
E(\imath \mu,\phi_{\imath\mu},T(\imath\mu))(ka)& =& a^{\imath\mu -\rho}\frac{c_\gamma(\imath\mu)}{c_\sigma(\imath\mu)} T ext(\phi_{\imath\mu})(k)\nonumber\\
&&+a^{-\imath\mu-\rho}\gamma(w)T \frac{c_\sigma(-\imath\mu)}{c_\sigma(\imath\mu)} ext(S_{\imath\mu}\phi_{ \imath\mu})(k) + O(a^{-\rho-\epsilon})\label{w1w}\\
E(-\imath\mu,\psi_{-\imath\mu},\tilde{T}(-\imath\mu))(ka) & = & a^{-\imath\mu -\rho} \frac{c_{\tilde{\gamma}}(-\imath\mu)}{ c_{\tilde{\sigma}}
(-\imath\mu)} \tilde{T} ext(\psi_{-\imath\mu})(k)\nonumber \\
&&+a^{ \imath\mu-\rho}\tilde{\gamma}(w)\tilde{T} \frac{c_{\tilde{\sigma}}(\imath\mu)}{ c_{\tilde{\sigma}}
(-\imath\mu)} ext(S_{-\imath\mu}\psi_{ -\imath\mu})(k) + O(a^{-\rho-\epsilon})\ .\nonumber
\end{eqnarray}
These expansions are immediate consequences of the asymptotic
expansion of the Poisson transform of smooth sections and
can be differentiated with respect to $a$ and differentiated and integrated with respect to $\mu$.
In order to read these formulas appropriately identify sections of $V_Y(\gamma)$
with $\Gamma$-invariant functions on $G$ with values in $V_\gamma$,
sections of $V(\sigma_{\imath\mu})$ with functions on $G$ with values in $V_\sigma$, etc., as usual.
Let $\chi$ be a cut-off function as constructed in Lemma \ref{lll} and $B_R$ the ball of
radius $R$ around the origin of $X$.
If we define $A_{\imath\mu}:=-\Omega_G+\chi_{\mu_\sigma+\rho_m-\imath\mu}(\Omega_G)$, then
$A_{\imath\mu}E(\imath \mu,\phi_{\imath\mu},T(\imath\mu))=0$ and
$A_{-\imath\mu} E(-\imath\mu,\psi_{-\imath\mu},\tilde{T}(-\imath\mu))=0$.
We start with the following identity
\begin{eqnarray*}0&=& \langle\chi A_{ \imath\lambda}E(\imath\lambda,\phi_{\imath\lambda},T(\imath\lambda)), E(-\imath\mu,\psi_{-\imath\mu},\tilde{T}(-\imath\mu))\rangle_{L^2(B_R)}\\&& - \langle\chi E(\imath \lambda,\phi_{\imath\lambda},T(\imath\lambda)), A_{-\imath\mu} E(-\imath\mu,\psi_{-\imath\mu},\tilde{T}(-\imath\mu)) \rangle_{L^2(B_R)}\ .\end{eqnarray*}
By partial integration as in the proof of Proposition \ref{green}
we obtain
\begin{eqnarray}
\lefteqn{(\mu^2-\lambda^2)\langle \chi E(\imath\lambda,\phi_{\imath\lambda},T(\imath\lambda)), E(-\imath\mu,\psi_{-\imath\mu},\tilde{T}(-\imath\mu))\rangle_{L^2(B_R)}}\hspace{3cm}\\
&=&\langle \chi \nabla_n E(\imath\lambda,\phi_{\imath\lambda},T(\imath\lambda)), E(-\imath\mu,\psi_{-\imath\mu},\tilde{T}(-\imath\mu))\rangle_{L^2(\partial B_R)}\label{w2e}\\
&&-\langle \chi E(\imath\lambda,\phi_{\imath\lambda},T(\imath\lambda)), \nabla_n E(-\imath\mu,\psi_{-\imath\mu},\tilde{T}(-\imath\mu))\rangle_{L^2(\partial B_R)}\label{w3e}\\
&&-\langle [A_0,\chi] E(\imath\lambda,\phi_{\imath\lambda},T(\imath\lambda)), E(-\imath\mu,\psi_{-\imath\mu},\tilde{T}(-\imath\mu))\rangle_{L^2(B_R)}\ .\label{w4e}
\end{eqnarray}
We insert the asymptotic expansions (\ref{w1w}) which hold on the support of $\chi$.
We obtain with $a_R:={\rm e}^{R}$
\begin{eqnarray*}
(\ref{w2e})+(\ref{w3e}) &=& \imath(\lambda+\mu) a_R^{\imath (\lambda -\mu)}\langle\chi \frac{c_\gamma(\imath\lambda)}{c_\sigma(\imath\lambda)} T ext(\phi_{\imath\lambda}),\frac{c_{\tilde{\gamma}}(-\imath\mu)}{ c_{\tilde{\sigma}}
(-\imath\mu)} \tilde{T} ext(\psi_{-\imath\mu})\rangle \\
&&+\imath (\lambda-\mu) a_R^{\imath(\lambda+\mu)} \langle\chi \frac{c_\gamma(\imath\lambda)}{c_\sigma(\imath\lambda)} T ext(\phi_{\imath\lambda}), \tilde{\gamma}(w) \tilde{T} \frac{c_{\tilde{\sigma}}(\imath\mu) }{c_{\tilde{\sigma}}
(-\imath\mu)} ext(S_{-\imath\mu}\psi_{-\imath\mu})\rangle\\
&&+\imath(-\lambda+\mu) a_R^{\imath(-\lambda-\mu)} \langle\chi \gamma(w) T \frac{c_\sigma(-\imath\lambda)}{ c_\sigma(\imath\lambda) } ext(S_{\imath\lambda} \phi_{\imath\lambda}), \frac{c_{\tilde{\gamma}}(-\imath\mu)}{ c_{\tilde{\sigma}}
(-\imath\mu)}\tilde{T} ext(\psi_{-\imath\mu})\rangle\\
&&+\imath(-\lambda-\mu) a_R^{\imath(-\lambda+\mu)} \langle\chi \frac{c_\sigma(-\imath\lambda) c_{\tilde{\sigma}}(\imath\mu)}{c_\sigma(\imath\lambda )c_{\tilde{\sigma}}(-\imath\mu)} ext(S_{\imath\lambda}\phi_{\imath\lambda}), ext(S_{-\imath\mu}\psi_{-\imath\mu})\rangle\ .\\
&&o(1)\ .
\end{eqnarray*}
The pairings on the right-hand side are defined using the canonical $K$-equivariant
identification of the bundles $V(\sigma_\lambda)$ with $V(\sigma_0)$.
We combine the remainder $o(1)$ and the term (\ref{w4e}) to $F(\lambda,\mu,R)$.
Then we can write
\begin{eqnarray}
\lefteqn{ \langle \chi E(\imath\lambda,\phi_{\imath\lambda},T(\imath\lambda)), E(-\imath\mu,\psi_{-\imath\mu},\tilde{T}(-\imath\mu))\rangle_{L^2(B_R)}}\hspace{2cm}\label{e34r}\\
&=& \imath \frac{a_R^{\imath (\lambda -\mu)}}{ -\lambda+\mu } \langle\chi \frac{c_\gamma(\imath\lambda)}{c_\sigma(\imath\lambda)} T
ext(\phi_{\imath\lambda}),\frac{c_{\tilde{\gamma}}(-\imath\mu)}{ c_{\tilde{\sigma}}
(-\imath\mu)} \tilde{T} ext(\psi_{-\imath\mu})\rangle\label{sing1} \\
&&-\imath \frac{a_R^{\imath(\lambda+\mu)}}{ \lambda+\mu} \langle\chi ext(\phi_{\imath\lambda}), T^*\frac{c^*_\gamma(\imath\mu)}{c_\sigma(\imath\mu)}\tilde{\gamma}(w) \tilde{T} \frac{c_{\tilde{\sigma}}(\imath\mu)}{ c_{\tilde{\sigma}}
(-\imath\mu) } ext(S_{-\imath\mu}\psi_{-\imath\mu})\rangle\nonumber\\
&&+\imath \frac{a_R^{\imath(-\lambda-\mu)}}{\lambda+\mu} \langle\chi \tilde{T}^*\frac{c^*_{\tilde{\gamma}}(-\imath\mu)}{ c_{\tilde{\sigma}}
(-\imath\mu)}\gamma(w) T \frac{c_\sigma(-\imath\lambda)}{ c_\sigma(\imath\lambda) }ext(S_{\imath\lambda} \phi_{\imath\lambda}), ext(\psi_{-\imath\mu})\rangle\nonumber\\
&&+\imath \frac{a_R^{\imath(-\lambda+\mu)}}{\lambda-\mu} \langle\chi \frac{c_\sigma(-\imath\lambda)c_{\tilde{\sigma}}(\imath\mu)}{ c_\sigma(\imath\lambda )c_{\tilde{\sigma}}(-\imath\mu)} ext(S_{\imath\lambda}\phi_{\imath\lambda}), ext(S_{-\imath\mu}\psi_{-\imath\mu})\rangle\label{sing2}\\
&&+\frac{F(\lambda,\mu,R)}{\mu^2-\lambda^2}\ .
\end{eqnarray}
Since $S_{\imath\mu}$ is unitary,
$$\frac{c_\sigma(-\imath\lambda)c_{\tilde{\sigma}}(\imath\lambda)}{ c_\sigma(\imath\lambda )c_{\tilde{\sigma}}(-\imath\lambda)} =1\ ,$$
and
\begin{equation}\label{wonderid}P_{\sigma}(\imath\mu)^{-1}{\mbox{\rm id}}_{V_{\tilde{\gamma}}(\tilde{\sigma})}=
c_\sigma(\imath\mu)c_{\tilde{\sigma}}(-\imath\mu){\mbox{\rm id}}_{V_{\tilde{\gamma}}(\tilde{\sigma})}=T^* c_\gamma(\imath\mu)^*c_{\tilde{\gamma}}(-\imath\mu)\tilde{T}\ ,\end{equation}
the singularities of the terms (\ref{sing1}) and (\ref{sing2}) at $\mu=\lambda$ cancel and
$\frac{F(\lambda,\mu,R)}{\mu^2-\lambda^2}$
is smooth at $\mu=\lambda$. Moreover $F(\lambda,\mu,R)\to 0$ as $R\to \infty$
such that the $C^1$-norm with respect to $\lambda$ remains bounded.
By Lebesgue's theorem about dominated convergence we obtain
$$\lim_{R\to\infty} \int_0^\infty \frac{F(\lambda,\mu,R)}{\mu^2-\lambda^2} d\lambda = 0\ .$$
By the Lemma of Riemann-Lebesgue
\begin{eqnarray*}
\lim_{R\to\infty}\int_0^\infty \imath \frac{a_R^{\imath(-\lambda-\mu)}}{\lambda+\mu} \langle\chi \tilde{T}^*\frac{c^*_\gamma(\imath\mu)}{c_\sigma(\imath\mu)}\gamma(w) T \frac{c_\sigma(-\imath\lambda)}{ c_\sigma(\imath\lambda)} ext(S_{\imath\lambda} \phi_{\imath\lambda}), ext(\psi_{-\imath\mu})\rangle d\lambda &=& 0 \\
\lim_{ R\to\infty}\int_0^\infty \imath \frac{a_R^{\imath(\lambda+\mu)}}{ \lambda+\mu} \langle\chi ext(\phi_{\imath\lambda}), T^*\frac{c^*_{\tilde{\gamma}}(-\imath\mu)}{ c_{\tilde{\sigma}}
(-\imath\mu)}\tilde{\gamma}(w) \tilde{T}\frac{c_{\tilde{\sigma}}(\imath\mu)}{ c_{\tilde{\sigma}}
(-\imath\mu)} ext(S_{-\imath\mu}\psi_{-\imath\mu})\rangle d\lambda &=&0\ .
\end{eqnarray*}
We set $s:=\lambda-\mu$.
We regroup the remaining terms of (\ref{e34r}) to
\begin{eqnarray*}
\lefteqn{\frac{a_R^{\imath s}-a_R^{-\imath s}}{\imath s} \langle\chi \frac{c_\gamma(\imath\lambda)}{c_\sigma(\imath\lambda)} T ext(\phi_{\imath\lambda}),\frac{c_{\tilde{\gamma}}(-\imath\mu)}{ c_{\tilde{\sigma}}
(-\imath\mu)} \tilde{T} ext(\psi_{-\imath\mu})\rangle}\hspace{0cm}\\
&&+a_R^{-\imath s } \frac{\langle\chi \frac{c_\gamma(\imath\lambda)}{c_\sigma(\imath\lambda)} T ext(\phi_{\imath\lambda}),\frac{c_{\tilde{\gamma}}(-\imath\mu)}{ c_{\tilde{\sigma}}
(-\imath\mu)} \tilde{T} ext(\psi_{-\imath\mu})\rangle- \langle\chi \frac{c_\sigma(-\imath\lambda)c_{\tilde{\sigma}}(\imath\mu) }{c_\sigma(\imath\lambda )c_{\tilde{\sigma}}(-\imath\mu)} ext(S_{\imath\lambda}\phi_{\imath\lambda}), ext(S_{-\imath\mu}\psi_{-\imath\mu})\rangle}{\imath s}
\end{eqnarray*}
If we integrate the second term with respect to $s$ and perform
the limit $R\to\infty$, then the result vanishes by the Riemann-Lebesgue lemma.
Using the identity of distributions
$\lim_{r\to\infty}\frac{\sin(rs)}{s }=\pi\delta_0(s)$ and (\ref{wonderid}) the first term gives
$$
\lim_{R\to\infty} \int_{-\infty}^\infty \frac{a_R^{\imath s}-a_R^{-\imath s}}{2 \imath s} \langle\chi \frac{c_\gamma(\imath\lambda)}{c_\sigma(\imath\lambda)} T ext(\phi_{\imath\lambda}), \frac{c_{\tilde{\gamma}}(-\imath\mu)}{ c_{\tilde{\sigma}}
(-\imath\mu)} \tilde{T} ext(\psi_{-\imath\mu})\rangle ds
= \pi \langle \phi_{\imath\mu} , \psi \rangle\ .
$$
The limit as $R\to\infty$ of the integral of left-hand side of (\ref{e34r}) with respect to $\lambda$ is equal to
$\langle E(\phi),\tilde{\psi}\rangle$.
This proves the lemma and thus finishes the proof of the proposition. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
In order to deal with Hilbert spaces we go over to
employ sesquilinear pairings which will be denoted by $(.,.)$.
The unitary structure of $\sigma$ induces a conjugate
linear isomorphism of $\tilde{\sigma}_{-\imath\mu}$ with
$\sigma_{\imath\mu}$. Analogously the unitary structure of $\gamma$
induces a conjugate linear isomorphism of $\tilde{\gamma}$ with $\gamma$.
We choose $T$ to be unitary, then $\tilde{T}$ corresponds to $T$
under the above identifications.
Moreover, if $\psi\in C^{-\infty}(B,V_B(\tilde{\sigma}_{-\imath\mu}))$
corresponds to $\bar{\psi}\in C^{-\infty}(B,V_B(\sigma_{\imath\mu}))$,
then $\tilde{\psi}$ corresponds to $\tilde{\bar{\psi}}:= E(\imath\mu,\bar{\psi},T(\imath\mu))$ under the isomorphisms above.
Using the sesquilinear pairings on $\gamma$ and $\sigma_{\imath\mu}$ we can rewrite
the result of Proposition \ref{scalar} as
\begin{equation}\label{sscar}( E(\phi),\tilde{\bar{\psi}} )=\pi( \phi_{\imath\mu},\bar{\psi})\ .\end{equation}
We define a scalar product on ${\cal H}_0$ by
$$( \phi,\psi):=\pi \int_0^\infty ( \phi_{\imath\mu},\psi_{\imath\mu}) d\mu$$
and let ${\cal H}$ be the corresponding Hilbert space closure of ${\cal H}_0$.
The following corollary is a consequence of
Proposition \ref{scalar}.
\begin{kor}\label{hermit}
The wave packet transform extends by continuity to an
isometric embedding $E:{\cal H}\hookrightarrow L^2(Y,V_Y(\gamma))$.
\end{kor}
If $A\in{\cal Z}$ and $\phi\in{\cal H}_0$, then we have $AE(\phi)=E(\psi)$ with $\psi_{\imath\mu}=\chi_{\mu_\sigma+\rho_m-\imath\mu}(A)\phi_{\imath\mu}$.
Choose an orthogonal decomposition $(V_\gamma)_{|M}=\oplus_iV_{\sigma_i}$ and
let $T_i\in {\mbox{\rm Hom}}_M(V_{\sigma_i},V_\gamma)$ be the corresponding
unitary embeddings. Here some of the $\sigma_i$ may be equivalent.
Let ${\cal H}(i)$ be the Hilbert space corresponding to $\sigma_i$ and
$E_i$ the corresponding wave packet transform.
It is easy to modify the proof of Proposition \ref{scalar}
in order to show that the ranges
of the $E_i$ are pairwise orthogonal.
We define the unitary embedding $$E_\gamma:=\oplus_i E_i: {\cal H}(\gamma):=\bigoplus_i{\cal H}(i)\hookrightarrow L^2(Y,V_Y(\gamma))\ .$$
Then $E_\gamma$ represents an absolute-continuous subspace $L^2(Y,V_Y(\gamma))_c\subset L^2(Y,V_Y(\gamma))$ with respect to ${\cal Z}$.
We now prove that the orthogonal complement
of the range of $E_\gamma$ is the discrete subspace
and the corresponding characters belong to $PS$.
The abstract spectral decomposition of $L^2(Y,V_Y(\gamma))$
with respect to the commutative algebra ${\cal Z}$ provides an unitary equivalence
$$\alpha:L^2(Y,V_Y(\gamma))\cong H:= \int_{{\bf h}_{\bf C}^*/W} H_\lambda \kappa(d\lambda)\ ,$$
where the Hilbert space $H_\lambda$ is a ${\cal Z}$-module on which ${\cal Z}$ acts by $\chi_\lambda$.
A part of the structure of the direct integral is that $H$
is a space of sections $ {\bf h}_{\bf C}^*/W\ni\lambda\mapsto \psi_\lambda\in H_\lambda$ such that
${\rm clo}\{\psi_\lambda | \psi\in H\} = H_\lambda$ ($\kappa$-almost everywhere), and such that the scalar
products ${\bf h}_{\bf C}^*/W\ni \lambda\mapsto(\psi_\lambda,\psi_\lambda^\prime)\in{\bf C}$, $\psi,\psi^\prime\in H$,
are measurable functions.
Then scalar product on $H$ is given by
$$(\psi,\psi^\prime)=
\int_{{\bf h}_{\bf C}^*/W}(\psi_\lambda,\psi_\lambda^\prime)\kappa(d\lambda)\ .$$
Fix a base $\{X_i\}$ of ${\bf g}$ and let $I_N$, $N\in{\bf N}_0$, denote
the set of all multiindices $i=(i_1,\dots,i_{\dim({\bf g})})$, $|i|\le N$.
Let $\chi$ be the cut-off function constructed in Lemma \ref{lll}.
For $f\in S(Y,V_Y(\gamma))$ we define
$$\|f\|^2_N:=\sum_{i\in I_N}\int_G \chi(gk) |\log(a(g))|^N |f(X_i g)|^2 dg \ .$$
Note that if $f\in S(Y,V_Y(\gamma))$, then $\|f\|_N<\infty$.
By $S^N(Y,V_Y(\gamma))$ we denote the closure of the Schwartz space
$S(Y,V_Y(\gamma))$ with respect to $\|.\|_N$.
Then $S^N(Y,V_Y(\gamma))$ is a Hilbert space contained in $L^2(Y,V_Y(\gamma))$.
\begin{lem}\label{komppp}
If $N$ is sufficiently large, then
the inclusion
$$S^N(Y,V_Y(\gamma))\hookrightarrow L^2(Y,V_Y(\gamma))$$
is Hilbert-Schmidt.
\end{lem}
{\it Proof.$\:\:\:\:$}
This follows from the results of \cite{bernstein88}.
In order to provide some details we employ the
notions "space of polynomial growth" and "comparable scale functions"
introduced in \cite{bernstein88}.
Fix some base point $y\in Y$.
Let the scale function
$r:\Gamma\backslash G\rightarrow {\bf R}^+$ be given by
$r(\Gamma g)={\rm dist}_Y(y,\Gamma g K)$, where ${\rm dist}_Y$
denotes the Riemannian distance in $Y$.
Then $\Gamma\backslash G$ is a space of polynomial growth with respect to $r$,
i.e., if $A\subset G$ is a compact neighbourhood of the identity,
then there exist constants $d\ge 0$, $C>0$ such that for any $R>0$
there exists a set of $\le C(1+R)^d$ points $x_i$ of $\Gamma\backslash G$
such that $\{r(x)<R\}\subset \cup_i x_i A$. This follows essentially
from the fact that $G$ itself is of polynomial growth and
that ${\rm dist}_X(x,.)$ and ${\rm dist}_Y(y,.)$ are comparable
when restricted to a compact fundamental domain
$F\subset X\cup\Omega$ containing the lift $x$ of $y$.
Note that $g\mapsto r(\Gamma g)$ and $g\mapsto a(g)$ are comparable on this fundamental domain, too.
If we choose $N>\max\{d,\dim({\bf g})\}$, then the assertion of the lemma
is proved by the Proposition of \cite{bernstein88}, Sec. 3.4. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
In the following we choose $N$ suffiently large.
It follows by a theorem of Gelfand/Kostyuchenko (see \cite{bernstein88}) that
the composition
$$S^N(Y,V_Y(\gamma))\hookrightarrow L^2(Y,V_Y(\gamma))\rightarrow \int_{{\bf h}_{\bf C}^*/W} H_\lambda \kappa(d\lambda)$$
is pointwise defined, i.e., there exists a collection of continuous maps
$$\alpha_\lambda:S^N(Y,V_Y(\gamma))\rightarrow H_\lambda,\quad \lambda\in{\bf h}^*_{\bf C}/W$$ such that
for $\phi\in S^N(Y,V_Y(\gamma))$ we have $\alpha(\phi)_\lambda=\alpha_\lambda(\phi)$.
Let $S^N(Y,V_Y(\gamma))^*$ denote the Hermitean dual of $S^N(Y,V_Y(\gamma))$.
Then we have inclusions
$$S(Y,V_Y(\gamma))\hookrightarrow S^N(Y,V_Y(\gamma))\hookrightarrow L^2(Y,V_Y(\gamma))\hookrightarrow S^N(Y,V_Y(\gamma))^*\hookrightarrow S(Y,V_Y(\gamma))^*\ .$$
By changing $\alpha_\lambda$ on a set of $\lambda$'s of measure
zero (mod $\kappa$) we
can assume that for all $\lambda\in{\bf h}_{\bf C}^*/W$ the map
$$\alpha_\lambda:S(Y,V_Y(\gamma))\rightarrow H_\lambda$$
is a morphism of ${\cal Z}$-modules.
Let
$$\beta_\lambda:H_\lambda\rightarrow S^N(Y,V_Y(\gamma))^*$$ denote
the adjoint of $\alpha_\lambda$. Since
$$\beta_\lambda:H_\lambda\rightarrow S(Y,V_Y(\gamma))^*$$
is a morphism of ${\cal Z}$-modules we see that $\beta_\lambda(H_\lambda)$
consists of tempered eigensections of ${\cal Z}$ corresponding to the character
$\chi_\lambda$.
\begin{prop}\label{ortho}
Let $\psi\in L^2(Y,V_Y(\gamma))$ be represented by
$\alpha(\psi)$ such that $\alpha(\psi)_\lambda=0$ for all $\lambda\in PS$.
Assume further that $(E_\gamma(\phi),\psi)=0$ for all wave packets
$E_\gamma(\phi)$, $\phi=\oplus_i\phi_i\in \oplus_i{\cal H}_0(i) $.
Then $\psi=0$.
\end{prop}
{\it Proof.$\:\:\:\:$}
Let $\phi=\oplus\phi_i\in \oplus {\cal H}_0(i)$. Then we can write
\begin{eqnarray*}
0&=&( E_\gamma(\phi),\psi)\\
&=&\int_{{\bf h}_{\bf C}^*/W}(\alpha_\lambda(E_\gamma(\phi)),\alpha(\psi)_\lambda) \kappa(d\lambda)\\
&=&\int_{{\bf h}_{\bf C}^*/W} (E_\gamma(\phi),\beta_\lambda\alpha(\psi)_\lambda)\kappa(d\lambda)\ .
\end{eqnarray*}
We claim that for $\lambda\not\in PS$ we can write
$$\beta_\lambda\alpha(\psi)_\lambda=\sum_i E_\gamma(\imath\mu_i,\psi_{i,\imath\mu_i},T_i(\imath\mu_i))\ ,$$
where $\psi_{i,\imath\mu_i}\in C^{-\infty}(B,V_B(\sigma_{i,\imath\mu_i}))$,
${\rm Im}(\mu_i)=0$ if $\psi_{i,\imath\mu_i}\not=0$, and $\lambda=\mu_{\sigma_i}+\rho_m-\imath\mu_i$.
In fact, by Proposition \ref{gener1} we have
$\beta_\lambda\alpha(\psi)_\lambda=\sum_i P_{\mu_i}^{T_i}(\tilde{\psi}_{i,\imath\mu_i})$, where $\tilde{\psi}_{i,\imath\mu_i}\in {}^\Gamma C^{-\infty}(\partial X,V(\sigma_{i,\imath\mu_i}))$.
If ${\rm Im}(\mu_i)\not=0$ and $\tilde{\psi}_{i,\imath\mu_i}\not=0$, then ${\mbox{\rm supp}}(\tilde{\psi}_i)\in \Lambda$ by Lemma \ref{wo}. But then $\lambda \in PS_d$ and this case was excluded.
By the functional equation of the Eisenstein series
(Corollary \ref{funeq}) and since ${\bf a}^*\ni\mu_i\not=0$
(because of $\lambda\not\in PS$) we can assume that ${\rm Re }(\mu_i)>0$
for all relevant $i$.
By Proposition \ref{upperbound} we have
$\tilde{\psi}_{i,\imath\mu_i}=ext (\Psi_{i,\imath\mu_i})$
with $\Psi_{i,\imath\mu_i}=res (\tilde{\psi}_{i,\imath\mu_i})$.
Putting $\psi_{i,\imath\mu_i}=c_\sigma(\imath\mu)\Psi_{i,\imath\mu_i}$
we obtain the claim.
We consider $\mu_i$ as a function of $\lambda$. Then using (\ref{sscar})
we obtain
\begin{eqnarray*}
0&=&\sum_i\int_{{\bf h}_{\bf C}^*/W\setminus PS} ( E_\gamma(\phi),E(\imath\mu_i,\psi_{i,\imath\mu_i},T_i(\imath\mu_i))) \kappa(d\lambda)\\
&=&\pi\sum_i \int_{{\bf h}_{\bf C}^*/W\setminus PS} (\phi_{i,\imath\mu_i},\psi_{i,\imath\mu_i}) \kappa(d\lambda)\ .
\end{eqnarray*}
Let $f_i\in C_c^\infty({\bf h}_{\bf C}^*/W\setminus PS)$. Then $\{ (0,\infty) \ni \mu\rightarrow f_i(\lambda(\mu))\phi_{i,\imath\mu}\}\in{\cal H}_0(i)$ and thus
\begin{equation}\label{opop}0=\sum_i \int_0^\infty f_i(\mu_i) (\phi_{i,\imath\mu_i},\psi_{i,\imath\mu_i}) \kappa(d\lambda)\ .\end{equation}
We conclude that $( \phi_{i,\imath\mu_i},\psi_{i,\imath\mu_i})
=0$ for almost all $\lambda$ (mod $\kappa$).
We now trivialize the family of bundles
by identifying $V_B(\sigma_{i,\imath\mu})$ with $V_B(\sigma_{i,0})$
in some holomorphic manner. We choose a countable dense set $\{\phi_j\}\subset C^\infty(B,V_B(\sigma_{i,0}))$.
Viewing $\mu\mapsto \psi_{i,\imath\mu_i}$ as a family of sections
in $C^{-\infty}(B,V_B(\sigma_{i,0}))$ we form
$B_j:=\{\lambda|(\phi_j,\psi_{i,\imath\mu_i})\not=0\}\subset {\bf h}^*_{\bf C}/W$.
Then $\kappa(B_j)=0$. Moreover let $U:=\cup_j B_j$.
Then $\kappa(U)=0$ and we have
$(\phi_j,\psi_{i,\imath\mu_i})=0$ for all $\lambda\in {\bf h}_{\bf C}^*/W_+\setminus U$
and all $j$. Thus $\psi_{i,\imath\mu_i}=0$ for $\lambda\in {\bf h}_{\bf C}^*/W_+\setminus U$.
Hence $\alpha(\psi)_\lambda=0$ for almost all $\lambda$ (mod $\kappa$).
Hence $\psi=0$. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
The following theorem is the immediate consequence of Proposition \ref{ortho}.
\begin{theorem}\label{contsp}
The wave packet transform $E_\gamma$ is
an unitary equivalence of ${\cal H}$ with the absolute-continuous
subspace of $L^2(Y,V_Y(\gamma))$.
The orthogonal complement of the absolute-continuous
subspace is the discrete subspace $L^2(Y,V_Y(\gamma))$
and the corresponding eigencharacters belong to $PS$.
\end{theorem}
\section{The discrete spectrum}
In Section \ref{wxa} we obtained a complete description of the
continuous subspace $L^2(Y,V_Y(\gamma))$ in terms of
the wave packet transform. In the present section
we study the othogonal complement $L^2(Y,V_Y(\gamma))_d$ of the continuous subspace. The discrete subspace
decomposes further into a cuspidal, residual, and
scattering component (see Definition \ref{t6r}).
We show that the residual and the scattering
component are finite-dimensional and that the cuspidal
component is either trivial or infinite-dimensional.
The notions of the cuspidal and the residual components are similar
to the corresponding notions known in the finite volume
case. The appearence of the scattering component is a new phenomenon which does
not occur in the finite volume case. We
give examples where the scattering component is non-trivial.
In order to study the eigenspaces of ${\cal Z}$ on $L^2(Y,V_Y(\gamma))$
we decompose them further with respect to the full algebra
of invariant differential operators ${\cal D}_\gamma$ on $V(\gamma)$.
We first recall some facts concerning ${\cal D}_\gamma$
(see \cite{olbrichdiss}).
The algebra ${\cal D}_\gamma$ is in general a non-commutative
finite extension of ${\cal Z}_\gamma$.
The right action of ${\cal U}({\bf g})^K$ on $C^\infty(X,V(\gamma))$
induces a surjective homomorphism ${\cal U}({\bf g})^K\rightarrow {\cal D}_\gamma$.
Hence any representation of $D_\gamma$ can be lifted to
a representation of ${\cal U}({\bf g})^K$. For $\sigma\in\hat{M}$ and $\lambda\in{\aaaa_\C^\ast}$ there is a representation of $\chi_{\sigma,\lambda}$
of ${\cal U}({\bf g})^K$ into ${\mbox{\rm End}}_M(V_\gamma(\sigma))$ which descends
to ${\cal D}_\gamma$ such that its restriction to ${\cal Z}_\gamma$
induces $\chi_{\mu_\sigma+\rho_m-\lambda}$.
Here $V_\gamma(\sigma)$ denote the $\sigma$-isotypic component
of $V_\gamma$.
The representation
$\chi_{\sigma,\lambda}$ is characterized by
\begin{equation}\label{nearby}D\circ P^T_\lambda=P^{\chi_{\sigma,\lambda}(D)\circ T}_\lambda,\quad T\in {\mbox{\rm Hom}}_M(V_\sigma,V_\gamma),\:\: D\in D_\gamma\ ,\end{equation}
and where $P^T_\lambda$ denotes the Poisson transform.
To be more precise, the representations $\chi_{\sigma,\lambda}$ depend
holomorphically on $\lambda$ and (\ref{nearby}) is an identity
between holomorphic families of maps.
By ${\cal E}_{\sigma,\lambda}(X,V(\gamma))$ we denote the space
of all $f\in C^\infty(X,V(\gamma))$ which under ${\cal D}_\gamma$ generate a quotient of the representation
$\chi_{\sigma,\lambda}$.
Let ${\cal E}_{\sigma,\lambda}(Y,V_Y(\gamma))$ be the subspace of ${\cal E}_{\sigma,\lambda}(X,V(\gamma))$ of $\Gamma$-invariant sections.
Then ${\cal Z}$ acts on ${\cal E}_{\sigma,\lambda}(X,V(\gamma))$ and ${\cal E}_{\sigma,\lambda}(Y,V_Y(\gamma))$ by the character $\chi_{\mu_\sigma+\rho_m-\lambda}$.
Define $PS_{res}(\sigma)\subset {\aaaa_\C^\ast}$ by
$PS_{res}(\sigma):=\{\mu \:|\:{\rm Re }(\mu)>0,\: {}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\mu))\not= 0\}$.
By Proposition \ref{upperbound}, $3.$ we have $PS_{res}(\sigma)\subset (0,\rho)$.
\begin{prop}\label{ddsp}
\begin{enumerate}
\item $PS_{res}(\sigma)$ is a finite set.
\item $\dim {}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\mu)) <\infty$ for all $\mu\in PS_{res}(\sigma)$.
\item The singularities of
$ext:C^{-\infty}(B,V_B(\sigma_\mu))\rightarrow {}^\Gamma C^{-\infty}(\partial X,V(\sigma_\mu))$ are isolated in $\{\mu\in{\aaaa_\C^\ast}\:|\:{\rm Re }(\mu)\ge 0\}$.
\end{enumerate}
\end{prop}
{\it Proof.$\:\:\:\:$}
Let $L^2(X,V(\gamma))_d$ denote the discrete subspace of $L^2(X,V(\gamma))$.
\begin{lem}\label{mmm2}
There exist finitely many irreducible (hence finite-dimensional)
mutually inequivalent representations $(\chi_i,W_i)$ of $D_\gamma$
such that
\begin{equation}\label{mmm1}
L^2(X,V(\gamma))_d\cong\bigoplus_{i}V_{\pi_i}\otimes W_i
\end{equation}
as a $G\times D_\gamma$-module. Here $V_{\pi_i}$ are representations
of the discrete series $\hat{G}_d$ of $G$. Given $\sigma\in\hat{M}$ and $\lambda\in{\aaaa_\C^\ast}$, ${\rm Re }(\lambda)>0$, let $\gamma$ be a minimal $K$-type of the principal series
representation $C^\infty(\partial X,V(\sigma_\lambda))$ of $G$.
Then $\chi_i\not\cong \chi_{\sigma,\lambda}$, $\forall i$.
\end{lem}
{\it Proof.$\:\:\:\:$}
The Harish-Chandra Plancherel Theorem for $L^2(G)$ implies
that
$$L^2(X,V(\gamma))_d=\bigoplus_{\pi\in \hat{G}_d} V_\pi\otimes {\mbox{\rm Hom}}_K(V_\pi,V_\gamma)\ .$$
Since the representations $V_\pi$ are irreducible and mutually non-equivalent
the same is true for the ${\cal U}({\bf g})^K$-modules ${\mbox{\rm Hom}}_K(V_\pi,V_\gamma)$
(see \cite{wallach88}, 3.5.4.). Since there is only a finite number
of $\pi\in \hat{G}_d$ with ${\mbox{\rm Hom}}_K(V_\pi,V_\gamma)\not=0$, equation
(\ref{mmm1}) follows.
Let $\gamma$ now be a minimal $K$-type of
$C^\infty(\partial X,V(\sigma_\lambda))$.
Argueing by contradiction we assume that $\chi_{\sigma,\lambda}\cong\chi_i$
for some $i$. Then there is an embedding
$$V_{\pi_i}\otimes W_i\hookrightarrow {\cal E}_{\sigma,\lambda}(X,V(\gamma))\cap L^2(X,V(\gamma))=:L^2_{\sigma,\lambda}\ .$$
There exists $f\in L^2_{\sigma,\lambda}$ such that $f(1)\not=0$.
Let $f_\gamma$ be its projection onto the (left) $K$-type $\gamma$.
Then $f_\gamma\not=0$. But $f_\gamma$ is the Poisson transform
of some $\phi\in C^\infty(\partial X,V(\sigma_\lambda))(\gamma)$
(see \cite{olbrichdiss}, Thm. 3.6, and \cite{minemura92}).
Thus $f_\gamma=P^T_\lambda(\phi)$ and asymptotically
$$f_\gamma(ka)\stackrel{a\to\infty}{\sim} c_\sigma(\lambda) T\phi(k) a^{\lambda-\rho}\ .$$
Since ${\rm Re }(\lambda)>0$ we have $c_\sigma(\lambda)\not=0$. We conclude that $f_\gamma\not\in L^2$. This is a contradiction to $f_\gamma\in L^2_{\sigma,\lambda}$. \hspace*{\fill}$\Box$ \\[0.5cm]\noindent
For the following two lemmas let $\gamma$ be a minimal $K$-type of
$C^\infty(\partial X,V(\sigma_\lambda))$.
Then its
multiplicity is one.
By Frobenius reciprocity $[\gamma:\sigma]=1$ and
$\chi_{\sigma,\lambda}$ is a one-dimensional representation.
\begin{lem}\label{mko1}
Let ${\rm Re }(\lambda)>0$. If there exist $\sigma^\prime\in\hat{M}$, $\lambda^\prime\in{\aaaa_\C^\ast}$, ${\rm Re }(\lambda^\prime)\ge 0$, such that
${\cal E}_{\sigma,\lambda}(X,V(\gamma))\cap {\cal E}_{\sigma^\prime,\lambda^\prime}(X,V(\gamma))\not= 0$, then ${\rm Re }(\lambda^\prime)\ge{\rm Re }(\lambda)$.
\end{lem}
{\it Proof.$\:\:\:\:$}
As in the proof of Lemma \ref{mmm2} there exists $$0\not=f_\gamma\in \left({\cal E}_{\sigma^\prime,\lambda^\prime}(X,V(\gamma))\cap {\cal E}_{\sigma^\prime,\lambda^\prime}(X,V(\gamma))\right)(\gamma)\ .$$
Again $f_\gamma=P^T_\lambda(\phi)=P^{T^\prime}_{\lambda^\prime}(\phi^\prime)$
for $\phi\in C^\infty(\partial X,V(\sigma_\lambda))(\gamma)$,
$\phi^\prime\in C^\infty(\partial X,V({\sigma}^\prime_{\lambda^\prime}))(\gamma)$.
Thus on the one hand we have
$$f_\gamma(ka)\stackrel{a\to\infty}{\sim} c_\sigma(\lambda) T\phi(k) a^{\lambda-\rho}\ ,$$
and on the other hand for any $\epsilon>0$
$$f_\gamma(ka)\stackrel{a\to\infty}{\sim} o(a^{\lambda^\prime-\rho+\epsilon})\ .$$
This implies ${\rm Re }(\lambda^\prime)\ge{\rm Re }(\lambda)$.\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
\begin{lem}\label{o9o9o}
If $\lambda\in{\bf a}^*$, $\lambda>0$, then there exists a commutative algebra extension ${\cal A}\subset{\cal D}_\gamma$ of ${\cal Z}_\gamma$ which is generated by selfadjoint elements such that the character $(\chi_{\sigma,\lambda})_{|{\cal A}}$
does not belong to the spectrum of ${\cal A}$ on $L^2(X,V(\gamma))$.
\end{lem}
{\it Proof.$\:\:\:\:$}
Consider the finite set
$B_{\sigma,\lambda}:=\{(\sigma^\prime,\lambda^\prime)\:|\: \sigma^\prime
\subset \gamma_{|M}, \lambda^\prime\in{\aaaa_\C^\ast}, (\chi_{\sigma^\prime,\lambda^\prime})_{|{\cal Z}}=(\chi_{\sigma,\lambda})_{|{\cal Z}}\}$.
If $(\sigma^\prime,\lambda^\prime)\in B_{\sigma,\lambda}$, then
$\lambda^\prime\in{\bf a}^*$ since $(\chi_{\sigma,\lambda})_{|{\cal Z}}$
is a real character.
We define
$B_{\sigma,\lambda}^+:=\{(\sigma^\prime,\lambda^\prime)\in B_{\sigma,\lambda}\:|\: {\rm Re }(\lambda^\prime)=0\}$.
Note that the $\chi_i$ (the $\chi_i$ have been introduced in Lemma \ref{mmm2}) and the $\chi_{\sigma^\prime,\lambda^\prime}$, $(\sigma^\prime,\lambda^\prime)\in B_{\sigma,\lambda}^+ $ are $*$-representations of ${\cal D}_\gamma$. While the $\chi_i$ were already
irreducible the $\chi_{\sigma^\prime,\lambda^\prime}$, $(\sigma^\prime,\lambda^\prime)\in B_{\sigma,\lambda}^+$,
can be completely decomposed into irreducible components.
Let $\chi^+$ denote the representation of ${\cal D}_\gamma$,
which is obtained by taking the direct sum of the $\chi_i$
and mutually inequivalent representatives of the irreducible components
of $\chi_{\sigma^\prime,\lambda^\prime}$, $(\sigma^\prime,\lambda^\prime)\in B_{\sigma,\lambda}^+$.
Set $I_{\sigma,\lambda}:=\ker\chi_{\sigma,\lambda}\subset {\cal D}_\gamma$ and
$I^+:=\ker\chi^+\subset {\cal D}_\gamma$.
We claim that $I^+\not\subset I_{\sigma,\lambda}$.
Argueing by contradiction we assume that $I^+\subset I_{\sigma,\lambda}$.
Let $R^+$ denote the range of
the representation $\chi^+$.
Since $\chi^+$ is the direct sum of mutually inequivalent representations
of ${\cal D}_\gamma$ the commutant of $R^+$ is generated by the projections
onto these components. Thus $R^+$ is a finite
direct sum $\oplus_j {\rm Mat}(l_j,{\bf C})$ of matrix algebras.
$R^+$ admits a character $\kappa:R^+\rightarrow {\bf C}$ such that the composition
${\cal D}_\gamma\rightarrow R^+\stackrel{\kappa}{\rightarrow} {\bf C}$
coincides with $\chi_{\sigma,\lambda}$.
If $l_j>1$, then the restriction of $\kappa$ to the summand ${\rm Mat}(l_j)$
vanishes. Thus the representation $\chi_{\sigma,\lambda}$
of ${\cal D}_\gamma$ must be one of the one-dimensional components defining $\chi^+$.
By Lemma \ref{mmm2} we have $\chi_{\sigma,\lambda}\not=\chi_i$
for all $i$. Hence there exists $(\sigma^\prime,\lambda^\prime)\in B_{\sigma,\lambda}^+$ such that $\chi_{\sigma^\prime,\lambda^\prime}$
contains $\chi_{\sigma,\lambda}$ as an irreducible component.
But then ${\cal E}_{\sigma,\lambda}(X,V(\gamma))\subset {\cal E}_{\sigma^\prime,\lambda^\prime}(X,V(\gamma))$.
By Lemma \ref{mko1} we conclude ${\rm Re }(\lambda^\prime)\ge \lambda>0$.
This is in conflict with the definition of
$B_{\sigma,\lambda}^+$.
Since the ideal $I^+$
is a $*$-ideal there exists a selfadjoint $A\in {\cal D}_\gamma$ with
$A\in I^+\setminus I_{\sigma,\lambda}$.
Let ${\cal A}$ be the algebra generated by $A$ and ${\cal Z}_\gamma$.
Let $f\in S(X,V(\gamma))$ be an eigenfunction of ${\cal A}$
corresponding to $(\chi_{\sigma,\lambda})_{|{\cal A}}$. The Harish-Chandra
Plancherel theorem for the Schwartz space $S(X,V(\gamma))$
(see \cite{arthur75})
implies that $f=f_c+f_d$, where $f_d\in L^2(X,V(\gamma))_d$
and $$f_c\in\sum_{(\sigma^\prime,\lambda^\prime)\in B_{\sigma,\lambda}^+}
{\cal E}_{\sigma^\prime,\lambda^\prime}(X,V(\gamma))\ .$$
It follows that $$f=\frac{1}{\chi_{\sigma,\lambda}(A)} A f = \frac{1}{\chi_{\sigma,\lambda}(A)} (Af_c + A f_d)=0\ .$$
We conclude that there are no tempered eigenfunctions of ${\cal A}$
for the character $\chi_{\sigma,\lambda}$. Thus $\chi_{\sigma,\lambda}$
is not on the spectrum of ${\cal A}$ on $L^2(X,V(\gamma))$.
This finishes the proof of the lemma.\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
Now we finish the proof of Proposition \ref{ddsp}.
Fix $\sigma\in \hat{M}$. Let $\gamma$ be a minimal $K$-type
of the principal series representation associated to $\sigma$.
We choose an $M$-equivariant embedding $T:V_\sigma\rightarrow V_\gamma$.
If $\mu,\lambda\in{\bf a}^+$, $\lambda>\mu>0$, and if
$\phi\in {}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))$, $\psi\in
{}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\mu))$, then
$P^T_\lambda(\phi)\perp P^T_\mu(\psi)$ in $L^2(Y,V_Y(\gamma))$.
If ${\rm Re }(\mu)>0$, then $c_\sigma(\mu)\not=0$ and the Poisson transform
$P^T_\mu$ is injective.
By Corollary \ref{nahenull} there exists $\epsilon>0$ such that
$PS_{res}(\sigma)\cap(0,\epsilon)=\emptyset$.
In the following we argue by contradiction.
Assume that
$\oplus_{\lambda\in [\epsilon,\rho]} {}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))$ is infinite-dimensional.
Since $[\epsilon,\rho]$ is compact this would imply
that there exists a sequence $\mu_i$ and $\psi_i\in {}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\mu))$, such that
$\mu_i\to \mu\in [\epsilon,\rho]$ and the $\psi$
are pairwise orthonormal. Let $\chi:=\chi_{\sigma,\mu}$ and
let ${\cal A}$ be the algebra constructed for $\chi$ in Lemma \ref{o9o9o}
such that $\chi$ does not belong to the essential
spectrum of ${\cal A}$ on $L^2(X,V(\gamma))$. But our assumption
implies that $\chi$ belongs to the essential spectrum of ${\cal A}$ on
$L^2(Y,V_Y(\gamma))$. In fact for any $A\in{\cal A}$ we have
$$\lim_{i\to\infty}\|(A-\chi(A))\psi_i\|\le \lim \lim_{i\to\infty}\|(A-\chi_{\sigma,\mu_i}(A))\psi_i\|+\lim_{i\to\infty}\|(\chi_{\sigma,\lambda}(A)-\chi(A))\psi_i\|=0\ .$$
By Proposition \ref{esspec} we conclude that $\chi$ belongs to the essential spectrum of ${\cal A}$ on $L^2(Y,V_Y(\gamma))$. But this contradicts our construction of ${\cal A}$.
Thus $\oplus_{\lambda\in [\epsilon,\rho]} {}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))$ is finite-dimensional.
This shows $1.$ and $2.$ of Proposition \ref{ddsp}.
Assertion $3.$ follows from $1.$ and Lemma \ref{lead}.\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
We now investigate the fine structure of the discrete subspace
$L^2(Y,V_Y(\gamma))_d$ for any $\gamma\in\hat{K}$.
By definition $L^2(Y,V_Y(\gamma))_d$ is the closure
of the subspace $L^2(Y,V_Y(\gamma))_{\cal Z}$ of ${\cal Z}$-finite
vectors. As explained in Section \ref{relsec},
if $f\in L^2(Y,V_Y(\gamma))_{\cal Z}$, then it has an asymptotic expansion at infinity. Since by Theorem \ref{contsp} the eigencharacters of ${\cal Z}$ on $L^2(Y,V_Y(\gamma))_d$ belong to $PS$ they are real.
This implies (see \cite{knapp86}, Ch.8) that the set of leading exponents
$E(f)$ is contained in ${\bf a}^*$.
To be precise with the zero exponent we distinguish two types of leading exponents $\mu=0$ which we
write as $0_0,0_1$.
We say that $f$ has the leading exponent $\mu=0_1$ ($\mu=0_0$)
if $p(f,0,0)$ is a non-constant (constant) polynomial on ${\bf a}$.
We use the leading exponents in order to define a filtration $F_*$ of $L^2(Y,V(\gamma))_{{\cal Z}}$.
For any exponent $\mu$ we set
$$F_\mu L^2(Y,V(\gamma))_{{\cal Z}}:=\{f\in L^2(Y,V(\gamma))_{{\cal Z}}\:|\:\mu \ge\lambda \quad \forall \lambda \in E(f)\}\ .$$
Then $F_\mu L^2(Y,V(\gamma))_{{\cal Z}}$ is the subspace of $L^2(Y,V(\gamma))_{{\cal Z}}$
on which the boundary value map $p(.,\mu,0)$ is well-defined
(if $\mu=0_1$, then we consider the leading coefficient $p_1(.,\mu,0)=:p(.,0_1,0)$ of $p(.,\mu,0)$). We define for any leading exponent
$$L^2(Y,V_Y(\gamma))_{{\cal Z}}^\mu:=F_\mu L^2(Y,V(\gamma))_{{\cal Z}}\cap \ker(p(.,\mu,0))^\perp\ .$$
\begin{ddd}\label{t6r}
We define
\begin{eqnarray*}
L^2(Y,V_Y(\gamma))_{res}&:=&\bigoplus_{\mu>0} L^2(Y,V_Y(\gamma))_{{\cal Z}}^\mu\\
L^2(Y,V_Y(\gamma))_{cusp}&:=&\bigoplus_{\mu<0} L^2(Y,V_Y(\gamma))_{{\cal Z}}^\mu\\
L^2(Y,V_Y(\gamma))_{scat}&:=&\bigoplus_{\mu=0_0,0_1} L^2(Y,V_Y(\gamma))_{{\cal Z}}^\mu\ .
\end{eqnarray*}
\end{ddd}
We apriori have $$L^2(Y,V_Y(\gamma))_d=\overline{L^2(Y,V_Y(\gamma))_{res}\oplus L^2(Y,V_Y(\gamma))_{cusp}\oplus L^2(Y,V_Y(\gamma))_{scat}}\ ,$$
but by $1.$ of Theorem \ref{poinye} the subspaces defined above are already closed.
We now describe these spaces in detail.
\begin{theorem}\label{poinye}
\begin{enumerate}
\item
The spectrum of ${\cal Z}$ on $L^2(Y,V_Y(\gamma))_d$ is finite.
In particular $L^2(Y,V_Y(\gamma))_{*}$, $*\in\{res,cusp,scat\}$,
are closed subspaces.
\item
The space $L^2(Y,V_Y(\gamma))_{res}$ is finite-dimensional.
There is an embedding $$L^2(Y,V_Y(\gamma))_{res}\hookrightarrow \bigoplus_{\sigma\subset \gamma_{|M}} \bigoplus_{\mu \in PS(\sigma)} {}^\Gamma C^{-\infty}(\Lambda,V(\sigma_{\mu}))\otimes {\mbox{\rm Hom}}_M(V_\sigma,V_\gamma)\ .$$
\item
The space $L^2(Y,V_Y(\gamma))_{cusp}$ is trivial or infinite-dimensional.
More precisely, for any discrete series representation $\pi\in\hat{G}_d$ there exists
an infinite-dimensional subspace $V_{\pi,\Gamma}\subset{}^\Gamma V_{\pi,-\infty}$
($V_{\pi,-\infty}$ denotes the distribution vector globalization) such that
for any $\gamma\in\hat{K}$
\begin{equation}\label{y7y7}L^2(Y,V_Y(\gamma))_{cusp}\cong\bigoplus_{\pi\in\hat{G}_d} V_{\pi,\Gamma}\otimes {\mbox{\rm Hom}}_K(V_\pi,V_\gamma)\ .\end{equation}
\item
There exists an embedding
$$L^2(Y,V_Y(\gamma))_{scat}\hookrightarrow
\bigoplus_{\sigma\subset \gamma_{|M}} \bigoplus_{0_0,0_1} {}^\Gamma C^{-\infty}(\Lambda,V(\sigma_0))\otimes {\mbox{\rm Hom}}_M(V_\sigma,V_\gamma)\ .$$
The space $L^2(Y,V_Y(\gamma))_{scat}$ is finite-dimensional.
\end{enumerate}
\end{theorem}
{\it Proof.$\:\:\:\:$}
The assertion $1.$ follows from $2.$ and (\ref{y7y7}).
Assertion $2.$ follows from Lemma \ref{wo} and Proposition \ref{ddsp}.
In fact, $\oplus_{\mu>0} p(.,\mu,0)$ defines the embedding
$$L^2(Y,V_Y(\gamma))_{res}\hookrightarrow \bigoplus_{\sigma\subset \gamma_{|M}} \bigoplus_{\mu \in PS(\sigma)} {}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\mu))\otimes {\mbox{\rm Hom}}_M(V_\sigma,V_\gamma)\ .$$
We now prove $3$.
For $\pi\in \hat{G}_d$ set $V_{\pi,\Gamma}:={\mbox{\rm Hom}}_G(V_\pi^*,L^2(\Gamma\backslash G))$.
For any Banach representation of $G$ on $V$ let $V_\infty$ denote the space of smooth vectors.
We have
\begin{eqnarray*}
{\mbox{\rm Hom}}_G(V_\pi^*,L^2(\Gamma\backslash G))&=&{\mbox{\rm Hom}}_G((V_\pi)^*_\infty,L^2(\Gamma\backslash G)_\infty)\\
&\subset&{\mbox{\rm Hom}}_G((V_\pi)^*_\infty,C^\infty(\Gamma\backslash G))\\
&=&{\mbox{\rm Hom}}_\Gamma((V_\pi)^*_\infty,{\bf C})\\
&=&{}^\Gamma V_{\pi,-\infty}\ .
\end{eqnarray*}
Let $I_\pi$ be the natural injection
$$I_\pi:V_{\pi,\Gamma}\otimes {\mbox{\rm Hom}}_K(V_\pi,V_\gamma)\hookrightarrow (L^2(\Gamma\backslash G)\otimes V_\gamma)^K=L^2(Y,V_Y(\gamma))\ .$$
The range of $I_\pi$ consists of eigenfunctions of ${\cal Z}$ which are matrix coefficients
of the discrete series representation $V_\pi$.
Matrix coefficients of discrete series representations
are characterizwed by the fact that all their leading exponents are negative.
We conclude that $I_\pi$ injects into $L^2(Y,V(\gamma))_{cusp}$.
Conversely, if $f\in L^2(Y,V(\gamma))_{cusp}$, then all its leading
exponents are negative and hence $f$ is a finite sum of matrix coefficients of discrete series
representations. Given $\gamma$ the set $\{\pi_i\}$ of discrete series representations of $G$ satisfying $I_\pi\not=0$
is finite.
Thus
\begin{equation}\label{scd}L^2(Y,V(\gamma))_{cusp}=\bigoplus_i {\mbox{\rm im}}(I_{\pi_i})\end{equation} and the spectrum of ${\cal Z}$ on
$L^2(Y,V(\gamma))_{cusp}$ is finite.
This finishes the proof of $1$.
To prove $3.$ it remains to show that
$V_{\pi,\Gamma}$ is infinite-dimensional.
Let $\gamma$ be a minimal $K$-type of the discrete series representation
$V_\pi$ ( or more general a $K$-type occuring with multiplicity one in $V_\pi$).
Let $\chi$ denote the corresponding character of ${\cal D}_\gamma$
induced by the representation of ${\cal U}({\bf g})^K$ on ${\mbox{\rm Hom}}_K(V_\pi,V_\gamma)$.
Furthermore, let $\chi_i$ denote the irreducible representations of ${\cal D}_\gamma$
introduced in Lemma \ref{mmm2}.
Without loss of generality we can assume that $\chi=\chi_1$.
Let $\{\chi_1,\dots,\chi_r\}$ denote the subset of these representations satisfying $(\chi_i)_{|{\cal Z}}=\chi_{|{\cal Z}}$.
We claim that there exists an abelian extension ${\cal A}\subset{\cal D}_\gamma$
of ${\cal Z}_\gamma$ which separates $\chi$ from all the characters occcuring in $(\chi_i)_{|{\cal A}}$, $\forall i>1$. Let $\chi^+$ denote the sum of all $\chi_i$ with $i>1$. Since the $\chi_i$ are all mutually inequivalent, the range of $\chi^+$ is a finite sum of matrix algebras $\oplus_{i>1}{\rm Mat}(l_i,{\bf C})$.
Let $I^+\subset {\cal D}_\gamma$, $I\subset {\cal D}_\gamma$ denote the kernels of $\chi^+$, $\chi$. We claim that $I^+\not\subset I$.
Assuming the contrary $R^+$ would admit an character $\kappa:R^+\rightarrow {\bf C}$
such that $\chi$ is given by $\chi:{\cal D}_\gamma\rightarrow R^+\stackrel{\kappa}{\rightarrow} {\bf C}$. But this is impossible by the definition of $\chi^+$. Choose a selfadjoint $A\in I^+\setminus I$ and let ${\cal A}={\cal Z}_\gamma[A]$.
Then $\chi_i(A)=0$ for all $i>1$ but $\chi(A)\not=0$.
Now $\chi_{|{\cal A}}$ belongs to the essential spectrum of ${\cal A}$ on $L^2(X,V(\gamma))$.
By Proposition \ref{esspec} the character $\chi_{|{\cal A}}$ belongs to the essential spectrum of ${\cal A}$ on $L^2(Y,V_Y(\gamma))$.
The characters $\chi_{|{\cal Z}}$ and $\chi_{|{\cal A}}$ are separated from
the continuous spectrum on $L^2(X,V(\gamma))$.
By Theorem \ref{contsp} the character $\chi_{|{\cal A}}$
is also separated from the continuous spectrum of ${\cal A}$ on $L^2(Y,V_Y(\gamma))_c$ and from the spectrum on $L^2(Y,V_Y(\gamma))_{scat}$.
Since the discrete spectrum of ${\cal Z}$ and of ${\cal A}$ on $L^2(Y,V_Y(\gamma))$
is finite the eigenspace of ${\cal A}$ in $L^2(Y,V_Y(\gamma))$ according to
$\chi_{|{\cal A}}$ must be infinite-dimensional.
Since $L^2(Y,V_Y(\gamma))_{res}$ can only contribute an finite-dimensional
subspace to this eigenspace the eigenspace of ${\cal A}$ to $\chi_{|{\cal A}}$
in $L^2(Y,V_Y(\gamma))_{cusp}$ is infinite-dimensional. By (\ref{scd})
this eigenspace is just given by $I_\pi (V_{\pi,\Gamma}\otimes {\mbox{\rm Hom}}_K(V_\pi,V_\gamma))$. It follows that $\dim\:V_{\pi,\Gamma}=\infty$.
This finishes the proof of $3.$.
We now prove $4$.
There is an embedding
$$p(.,0_0,0)\oplus p(.,0_1,0):L^2(Y,V_Y(\gamma))_{scat}\hookrightarrow
{}^\Gamma C^{-\infty}(\partial X,V(\gamma_{|M,0}))\oplus {}^\Gamma C^{-\infty}(\partial X,V(\gamma_{|M,0}))\ .$$
We prove the assertion about the support.
We show that if $f\in L^2(Y,V_Y(\gamma))_{{\cal Z}}^{0_1}$,
then ${\mbox{\rm supp}}(p(f,0_1,0))\subset \Lambda$.
The argument is similar to the one used in the proof of Lemma
\ref{wo}.
Let $U\subset \bar{U}\subset \Omega$
be open. If $\phi\in C_c^\infty(U,V(\gamma_{|M,0}))$, we have
$$(\phi,p(f,0_1,0))=\lim_{a\to\infty} a^{\rho-\mu}|\log(a)|^{-1} \int_K (\phi(k),f(ka)) dk\ .$$
Constructing the sequence $\phi_n$ with ${\mbox{\rm supp}}(\phi_n)\subset UMA^+K$ as in the proof of Lemma \ref{wo} but using $\chi_n(a):=|\log(a)|^{-1}a^{-\rho-\bar{\mu}}\chi(|\log(a)|-n)$
we can write
$$(\phi,p(f,0_1,0))=\lim_{n\to\infty} (\phi_n,f)\ .$$
By construction $\phi_n\to 0$ weakly in $L^2(UMA^+K,V(\gamma))$
and $f_{|UMA^+K}\in L^2(UMA^+K,V(\gamma))$.
We obtain $(\phi,p(f,0_1,0))=0$.
Since $U$ and $\phi$ were arbitrary,
this proves that ${\mbox{\rm supp}}(p(f,0_1,0))\subset \Lambda$.
The same argument works in the case $\mu=0_0$.
The following lemma (for $\lambda=0$) implies that $L^2(Y,V_Y(\gamma))_{scat}$ is finite-dimensional. For $\lambda>0$ it provides an alternative proof of Proposition \ref{ddsp}, $2$.
\begin{lem}
If $\lambda\ge 0$, then
${}^\Gamma C^{-\infty}(\Lambda,V)(\sigma_\lambda))$ is finite-dimensional.
\end{lem}
{\it Proof.$\:\:\:\:$}
We first prove an apriori estimate of the order of elements
of ${}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))$, i.e.
we show that there exists a $k\in{\bf N}$ such that $f\in {}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))$ implies
$f\in C^k(\partial X,V(\tilde{\sigma}_{-\lambda}))^\prime$.
The inclusion ${}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))\hookrightarrow
C^k(\partial X,V(\tilde{\sigma}_{-\lambda}))^\prime$ induces on ${}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))$ the structure of a Banach space.
Since ${}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))$ is a closed
subspace of the Montel space $C^{-\infty}(\partial X,V(\sigma_\lambda))$
it must be finite-dimensional. It remains to prove the apriori estimate
of the order.
If $\lambda\not=0$ or if $\lambda=0$ and the principal series
representation $\pi^{\sigma,\lambda}$ is irreducible,
then let $\gamma$ be a minimal $K$-type of the principal series representation
$\pi^{\sigma,\lambda}$ and
fix an unitary $T\in {\mbox{\rm Hom}}_M(V_\sigma,V_\gamma)$.
If $\lambda=0$
and $\sigma=\sigma^\prime\oplus\sigma^{\prime w}$ is reducible, then let
$\gamma$ be the sum of two copies of a minimal $K$-type of the principal
series representation $\pi^{0,\sigma^\prime}$.
In this case we let $T:=T^\prime_0\oplus T^\prime_1$, where
$T^\prime_0\in {\mbox{\rm Hom}}_M(V_{\sigma^\prime},V_\gamma)$ and $T^\prime_1\in {\mbox{\rm Hom}}_M(V_{\sigma^{w \prime}},V_\gamma)$ are unitary.
In the remaining case where $\lambda=0$, $\sigma$ is irreducible and $\pi^{\sigma,\lambda}$ splits as a sum of two irreducible representations
we let $\gamma$ be the sum of minimal $K$-types of these representations.
In this case we let $T\in {\mbox{\rm Hom}}_M(V_\sigma,V_\gamma)$ denote
the "diagonal" embedding of $\sigma$ into $\gamma$.
In any case let
$P:=P^T_\lambda$ denote the associated injective Poisson transform.
Consider $f\in {}^\Gamma C^{-\infty}(\Lambda,V(\sigma_\lambda))$.
The asymptotic expansion (\ref{epan}) shows that $Pf$
is bounded along $\Omega$. Using the $\Gamma$-invariance we conclude
that $Pf$ is a uniformly bounded section of $V(\gamma)$.
Let $\chi$ be the infinitesimal character of ${\cal Z}$ on the
principal series representation $\pi^{\sigma,\lambda}$.
Let $C^\infty_{mg}(X,V(\gamma))_\chi$ denote the corresponding eigenspace.
As a topological vector space $C^\infty_{mg}(X,V(\gamma))_\chi$ is a direct
limit of Banach spaces
$$C^\infty_{R}(X,V(\gamma))_\chi:=\{f\in C^\infty_{mg}(X,V(\gamma))_\chi\:|\:
\sup_{g\in G}\|g\|^{-R} |f(g)| <\infty\}\ .$$
In particular, $Pf\in C^\infty_{0}(X,V(\gamma))_\chi$.
The range of the Poisson transform $P$ is a closed $G$-submodule ${\cal M}$
of $C^\infty_{mg}(X,V(\gamma))_\chi$.
We claim that there is a boundary value map $\beta$
defined on ${\cal M}$ which is continuous and inverts $P$.
Before proving the claim we finish the proof of the apriori estimate assuming the claim. On the one hand the topological vector space ${\cal M}$ is the direct limit of the Banach spaces
${\cal M}_R$, where ${\cal M}_R:={\cal M}\cap C^\infty_{R}(X,V(\gamma))_\chi$.
On the other hand $C^{-\infty}(\partial X, V(\sigma_\lambda))$
is the direct limit of Banach spaces $C^k(\partial X,V(\tilde{\sigma}_{-\lambda}))^\prime$.
Since $\beta$ is continuous for any $R\ge 0$ there exists a $k\in{\bf N}$ such that
$\beta({\cal M}_R)\subset C^k(\partial X,V(\tilde{\sigma}_{-\lambda}))^\prime$.
Since $Pf\in {\cal M}_0$ this yields the apriori estimate we looked for.
We now show the existence of the boundary value $\beta$. It is intimately related with the leading asymptotic coefficient $p(\phi,\lambda,0)$, $\phi\in {\cal M}$.
Let first $\lambda>0$.
Then we can define $\beta(\phi)$, $\phi\in {\cal M}$, by
$$( \beta(\phi),\psi) := c_\sigma(\lambda)^{-1}\lim_{a\to\infty} a^{-\lambda+\rho} \int_K (\phi(ka),T\psi(k)) dk, \quad \psi\in C^\infty(\partial X,V(\sigma_{-\lambda}))\ .$$
Let now $\lambda=0$.
If $c_\sigma(\mu)$ has a pole at $\mu=0$, then
$\pi^{\sigma,\lambda}$ is irreducible an we define
$\beta(\phi)$, $\phi\in {\cal M}$, by
$$(\beta(\phi),\psi) := \frac{1}{2{\rm res}_{\mu=0}c_\sigma(\mu)} \lim_{a\to\infty}\log(a)^{-1} a^{\rho} \int_K (\phi(ka),T\psi(k) ) dk, \quad \psi\in C^\infty(\partial X,V(\sigma_{0})) .$$
If $c_\sigma(\mu)$ is regular at $\mu=0$ and
$\sigma=\sigma^\prime\oplus\sigma^{\prime w}$, then we define $\beta$ by
$$(\beta(\phi),\psi) := \frac{1}{c_\sigma(0)} \lim_{a\to\infty}
a^{\rho} \int_K (\phi(ka),T\psi(k) ) dk, \quad \psi\in C^\infty(\partial X,V(\sigma_{0})) .$$
In the remaining case $c_\sigma(\mu)$ is regular at $\mu=0$
and $\pi^{\sigma,0}$ is reducible. Let $\gamma=\gamma_1\oplus\gamma_2$ and
$t_i\in{\mbox{\rm Hom}}_M(V_\sigma,V_{\gamma_i})$, $i=1,2$, be such that $T=t_1\oplus t_2$.
Let $c_{\gamma_i,\sigma}(\mu)$ denote the value of $c_{\gamma_i}(\mu)$
on the range of $t_i$. Note that $c_{\gamma_i,\sigma}(0)\not=0$.
We define $\beta$ by
$$(\beta(\phi),\psi) := \lim_{a\to\infty}
a^{\rho} \int_K (\phi(ka),(t_1 c_{\gamma_1,\sigma}(0)^{-1}\oplus t_2 c_{\gamma_2,\sigma}(0)^{-1}) \psi(k) ) dk, \quad \psi\in C^\infty(\partial X,V(\sigma_{0})) .$$
One can check in each case that $\beta$ inverts the Poisson transform $P$
(e.g. use the method of the proof of \cite{olbrichdiss}, Lemma 4.31).
The fact that $\beta$ is continuous follows from the globalization
theory for Harish-Chandra modules.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
This finishes the proof of $4.$ and hence of the theorem.
\hspace*{\fill}$\Box$ \\[0.5cm]\noindent
It is clear by Theorem \ref{contsp}
that $L^2(Y,V_Y(\gamma))_c$ is always non-trivial.
By Theorem \ref{poinye} there are examples
with $L^2(Y,V_Y(\gamma))_{cusp}$ non-trivial.
If $\delta_\Gamma>0$, then the Patterson-Sullivan
measure leads to a non-trivial element in $L^2(Y)_{res}$.
Thus examples with non-trivial $L^2(Y,V_Y(\gamma))_{res}$ exist.
We now give an example with $L^2(Y,V_Y(\gamma))_{scat}\not=0$.
Let $\Gamma\subset SO(1,2)$ be a cocompact Fuchsian group.
Consider $\Gamma\subset SO(1,3)$ in the standard way.
Then $Y$ is a $3$-dimensional hyperbolic manifold of the type considered
in the present paper. $Y$ has two ends, i.e., $B$ has two
components. It was shown by Mazzeo-Phillips \cite{mazzeophillips90},
Corollary 3.20, that the dimension of the space of harmonic, square-integrable one-forms is $\ge \sharp\{\mbox{ends of $Y$}\}-1=1$. The character of ${\cal Z}$ corresponding to harmonic one-forms is the boundary of the continuous
spectrum of ${\cal Z}$ on one-forms. Thus square-integrable harmonic one-forms
are elements of $L^2(Y,V_Y(\gamma))_{scat}$.
As indicated after the proof of Proposition \ref{upperbound}
one can obtain vanishing results for the discrete spectrum.
One can bound the residual spectrum in terms of $\delta_\Gamma$.
For certain $K$-types (e.g. for the trivial one) one
can show that $L^2(Y,V_Y(\gamma))_{scat}$ vanishes (see the remark following the proof of Lemma \ref{th43}).
\bibliographystyle{plain}
| proofpile-arXiv_065-434 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}\setcounter{equation}{0}
Systems of two-dimensional electrons in a perpendicular strong magnetic field
are giving many exciting physics these days\cite{r1}.
Electrons in these systems have discrete energies with finite maximum
degeneracy\cite{r2}.
Among infinitely many representations, an invariant representation under two
dimensional translation is convenient for many purposes.
Invariant representation
under continuous translations does not exist due to the magnetic
field but lattice translational invariant one does exist.
von Neumann lattice coherent state representation\cite{r3}is such
representation and we have used it to verify the integer quantum Hall
effect.
We give a new representation of von Neumann lattice in this paper, and
propose a new mean field theory of the quantum Hall liquid
based on flux condensed state.
The fractional Hall state\cite{r4} is regarded as a kind of integer Hall
effect due to the condensed flux, and has a large energy gap in our theory.
It is convenient to use relative coordinates\cite{r5} which
are perpendicular
to velocity operators and guiding center coordinates for describing
two-dimensional electrons in the perpendicular magnetic field.
Coherent states in guiding center coordinates have minimum spatial
extensions and its suitable complete subset whose element has
discrete eigenvalues defined on lattice site are used.
Field operators in the present representation have two-dimensional lattice
coordinates in
addition to Landau level index, and the many-body theory is described by a
lattice field theory of having internal freedom.
Exact identities such as current conservation and Ward-Takahashi
identity\cite{r3} derived from current conservation are written in a
transparent way and play the important roles for establishing
an exact low-energy theorem of the
quantum Hall effect.
We are able to expess flux state in a symmetric way, also.
One of the hardest and most important thing of the fractional Hall
effect is to find a mechanism of generating energy gap.
Without interactions there is no
energy gap at any fractional filling state. Hence interaction generates gap.
Once the gap is generated, fluctuations are suppressed.
Higher order corrections
are expected to be small if starting approximate ground state has
the energy gap and close to eigenstate.
Flux state has a modified symmetry under translation and can have energy
gap at certain filling states.
We study flux states of the fractional Hall system.
In our mean field theory of the fractional quantum Hall effect,
an order parameter is a magnitude of the flux condensation.
Because the system
changes its property drastically with the change of flux it is quite natural
to treat the flux as an order parameter. Due to the dynamical fux, our mean
field Hamiltonian is close to that
of Hofstadter\cite{r6} and new integer Hall effect occurs within the lowest
Landau level space of the external magnetic field.
$ $From Hofstadter's analysis and others\cite{r7} ground state has
the lowest energy and the largest energy gap,
if its flux is proportional to filling factor.
We regard, in fact, these states as the fractional Hall states.
Especially the principal series at $\nu=p/(2p\pm1)$
satisfy the self-consistency condition and have large energy gap.
They can be observed even in the Hofstadter's spectrum of butterfly shape.
Our mean field theory has similarities also with a mean field theory of
composite fermion\cite{r8} in regarding the fractional
Hall effect as a kind of integer Hall effect.
However, ours includes interaction effects in the space of the lowest
Landau level at the mean field level and so the effective mass,
energy gap, and other quantities of the fractional quantum Hall states
are close to the observed value even in the lowest order.
These features are seen in ours but are not seen in the
composite fermion mean field theory.
We, also, found our ground
state energy has slightly higher energy than that of Laughlin for
$\nu=1/3$ case.
Due to the energy gap, fluctuations are small in the fractional quantum Hall
state.
Higher order corrections are small and the perturbative expansions
converge well, generally.
Exact half-filling is, however, exceptional and there is no energy gap.
Fermi surface is composed of isolated points in the lowest order, but the
fluctuations are extremely large.
So, the structure around the Fermi energy may be changed drastically
from that of the lowest order by interactions at the half-filling.
Thus our mean field may not be good at the exact half-filling.
The paper is organized in the following way.
In section 2, we formulate von Neumann lattice representation and
verify the integer Hall effect.
In section 3 flux state mean field theory on the von Neumann lattice is
formulated and is compared with the existing experiments in Section 4.
Conclusions are given in section 5.
\section{Quantum Hall dynamics on von Neumann lattice}\setcounter{equation}{0}
Quantum Hall system is described by the following Hamiltonian,
\begin{eqnarray}
&H=\int d{\bf x}[\Psi^\dagger(x){({\bf p}+e{\bf A})^2\over 2m}\Psi(x)+
\rho(x){V(x-y)\over2}\rho(y)], \\
&\partial_1 A_2-\partial_2 A_1=B,\ \rho(x)=\Psi^\dagger(x)\Psi(x),
\ V(x)={e^2\over\kappa}{1\over\vert x\vert}.
\nonumber
\end{eqnarray}
We ignore the disorders in this paper.
It is convenient to use the following two sets of variables,
\begin{eqnarray}
&\xi={1\over eB}(p_y+eA_y),\ X=x-\xi,\\
&\eta=-{1\over eB}(p_x+eA_x),\ Y=y-\eta,\nonumber
\end{eqnarray}
which satisfy,
\begin{eqnarray}
&[\xi,\eta]=-[X,Y]=-i{\hbar\over eB},\nonumber \\
&[\xi,X]=[\xi,Y]=0,\\
&[\eta,X]=[\eta,Y]=0.\nonumber
\end{eqnarray}
We expand the electron field operators with a complete set of base
functions, $f_l(\xi,\eta)\otimes\vert R_{m,n}\rangle$, which are
defined by,
\begin{eqnarray}
&{({\bf p}+{\bf A})^2\over 2m}f_l={e^2 B^2\over 2m}(\xi^2+\eta^2)f_l=
E_l f_l,\ E_l={\hbar eB\over 2m}(2l+1)\\
&(X+iY)\vert R_{m,n}\rangle=a(m+in)\vert R_{m,n}\rangle,
\ a=\sqrt{2\pi\hbar\over eB}.
\nonumber
\end{eqnarray}
The coherent states on von Neumann lattice are constructed as,
\begin{eqnarray}
&\vert R_{m,n}\rangle=(-1)^{mn+m+n}e^{A^\dagger\sqrt{\pi}(m+in)-
A\sqrt{\pi}(m-in)}\vert 0\rangle,\\
&A=\sqrt{eB\over 2\hbar}(X+iY),\ [A,A^\dagger]=1.
\nonumber
\end{eqnarray}
The expressions of electron field are given by,
\begin{eqnarray}
&\Psi(x)=\sum a_l(m,n)f_l(\xi,\eta)\otimes\vert R_{m,n}\rangle,\\
&\Psi^\dagger(x)=\sum a^\dagger_l(m,n)
f_l(\xi,\eta)\otimes\langle R_{m,n}\vert,
\nonumber
\end{eqnarray}
and they are substituted into the action integral,
\begin{eqnarray}
S&=&\int d{\bf x}\Psi^\dagger(x)i\hbar{\partial\over\partial t}\Psi(x)
-H \\
&=&\sum\langle R_{m_1,n_1}\vert R_{m_2,n_2}\rangle a^\dagger_l(R_1)(
i\hbar{\partial\over\partial t}-E_l)a_l(R_2)+
\int d{\bf k}\rho({\bf k}){V({\bf k})\over 2}\rho(-{\bf k}).
\nonumber
\end{eqnarray}
By using the conjugate momentum to the lattice site, ${\bf R}_{m,n}$,
the action is written as,
\begin{eqnarray}
S=&\sum\sum a^\dagger_l({\bf p}_1)e^{i\phi({\bf p},{\bf N})}
\beta^*({\bf p}_1)\beta({\bf p}_2)(
i\hbar{\partial\over\partial t}-E_l)a_l({\bf p}_2)
\delta({\bf p}_1-{\bf p}_2+{2\pi\over a}{\bf N}) \nonumber\\
&+\int d{\bf k}\rho({\bf k}){V({\bf k})\over 2}\rho(-{\bf k}),
\end{eqnarray}
\begin{eqnarray*}
\beta^*({\bf p})&=&2^{1\over4}e^{-{a^2\over 4\pi}p^2_x}
\Theta_1({ia\over 2\pi}(p_x+ip_y),i),\nonumber\\
\beta^*({\bf p}+{2\pi\over a}{\bf N})&=&e^{i\phi({\bf p},{\bf N})}
\beta^*({\bf p}),\nonumber\\
\phi({\bf p},{\bf N})&=&N_x(ap_y+\pi)+N_y\pi,\nonumber\\
&&-\pi/a\leq p_i\leq\pi/a, \nonumber
\end{eqnarray*}
where $\Theta_1(x,i)$ is the elliptic theta function of
the first kind\cite{r9}
and ${\bf N}$ is a two-dimensional vector which has integers as components.
The momentum is conserved and its eigenvectors are orthogonal
if their eigenvalues are different but they are not normalized.
We introduce normalized operators,
\begin{eqnarray}
b_l({\bf p})&=&\beta({\bf p})a_l({\bf p}),\\
b^\dagger_l({\bf p})&=&\beta^*({\bf p})a^\dagger_l({\bf p}).
\nonumber
\end{eqnarray}
$ $From the definition, $\beta({\bf p})$ vanishes at ${\bf p}=0$.
This reflects a constraint of the coherent states
on von Neumann lattice\cite{r10},
but it causes no difficulty in our method.
They satisfy,
\begin{eqnarray}
&\{b_{l_1}({\bf p}_1),b^\dagger_{l_2}({\bf p}_2)\}=
\sum_{{\bf N}}e^{-i\phi({\bf p}_1,{\bf N})}\delta({\bf p}_1-{\bf p}_2
+{2\pi\over a}{\bf N}),\\
&b_l({\bf p}+{2\pi\over a}{\bf N})=e^{-i\phi({\bf p}_1,{\bf N})}b_l(
{\bf p}).\nonumber
\end{eqnarray}
The action is written as,
\begin{eqnarray}
S=\sum_l\sum_{{\bf N},{\bf p}_1,{\bf p}_2}
&b^\dagger_l({\bf p}_1)e^{i\phi({\bf p}_1,{\bf N})}(
i\hbar{\partial\over\partial t}-E_l)b_l({\bf p}_2)
\delta({\bf p}_1-{\bf p}_2+{2\pi\over a}{\bf N}) \nonumber\\
&+\int d{\bf k}\rho({\bf k}){V({\bf k})\over 2}\rho(-{\bf k}),
\end{eqnarray}
\begin{equation}
\rho({\bf k})=\sum b^\dagger_{l_1}({\bf p}_1)b_{l_2}({\bf p}_2)
\delta({\bf p}_1-{\bf p}_2-{\bf k}+{2\pi\over a}{\bf N})
(l_1\vert e^{i{\bf k}\cdot\hbox{$\xi$\kern-.5em\hbox{$\xi$}}}\vert l_2)e^{i\phi({{\bf p}_1+
{\bf p}_2\over2},{\bf N})+i{a^2\over 4\pi}k^y(p_1^x+p_2^x)}.
\end{equation}
The current operator is also written as,
\begin{eqnarray}
&j_i({\bf k})=\sum b^\dagger_{l_1}({\bf p}_1)b_{l_2}({\bf p}_2)\delta(
{\bf p}_1-{\bf p}_2-{\bf k}+{2\pi\over a}{\bf N})
(l_1\vert v_i e^{i{\bf k}\cdot\hbox{$\xi$\kern-.5em\hbox{$\xi$}}}\vert l_2)e^{i\phi({{\bf p}_1+
{\bf p}_2\over2},{\bf N})
+i{a^2\over 4\pi}k^y(p_1^x+p_2^x)},
\nonumber\\
&{\bf v}={eB\over m}(-\eta,\xi).
\end{eqnarray}
The commutation relations (2.10), and the action (2.11), have no
singularity in ${\bf p}$.
The zero of $\beta({\bf p})$ in Eq.(2.9) does not cause any problem.
Hence meaningful theory is defined in this way contrary to naive
expectations.
To make sure this point further, we solve the one impurity problem
by the present representation. We find an agreement with
previous results. Namely eigenvectors are localized around the
impurity position if corresponding eigenvalues are isolated and
located between Landau levels.
Their energies are moved with the impurity, also.
Appendix A is devoted for the short range impurity problem.
$ $From Eqs.(2.10) and (2.12),
we have commutation relations between charge density and field operators,
\begin{eqnarray}
&[\rho({\bf k}),b({\bf p})]=-(l_1\vert e^{i{\bf k}\cdot\hbox{$\xi$\kern-.5em\hbox{$\xi$}}}
\vert l_2)
e^{i{a^2\over 4\pi}k_y(2p_x-k_x)}b_{l_2}({\bf p}-{\bf k})
e^{i\pi N_x N_y+iak_y N_x},\nonumber\\
&\vert {\bf p}-{\bf k}+{2\pi\over a}{\bf N}\vert\leq {\pi\over a},\\
&[\rho({\bf k}),b^\dagger({\bf p})]=
(l_1\vert e^{i{\bf k}\cdot\hbox{$\xi$\kern-.5em\hbox{$\xi$}}}\vert l_2)
e^{i{a^2\over 4\pi}k_y(2p_x+k_x)}b^\dagger_{l_1}({\bf p}+{\bf k})
e^{-i\pi N_x N_y-iak_y N_x},\nonumber\\
&\vert {\bf p}+{\bf k}-{2\pi\over a}{\bf N}
\vert\leq {\pi\over a}.\nonumber
\end{eqnarray}
For the momentum $\bf p$ in the fundamental region and infinitesimal
$\bf k$, $\bf N$ vanishes.
The right-hand sides have linear terms in $\bf k$.
Hence Ward-Takahashi identity between the vertex part and the propagator
is modified from that of the naive one.
We introduce the unitary operator $U({\bf p})$ which satisfies
\begin{equation}
\delta_{l_1,l_2}{\partial\over\partial p_i}U({\bf p})+
\{i(l_1\vert\xi_i\vert l_2)+i{a^2\over 2\pi}p_x\delta_{l_1,l_2}\}U(p)=0,
\end{equation}
and make transformation of the propagator and the vertex part as,
\begin{eqnarray}
&\tilde S({\bf p})=U({\bf p})S^{(0)}({\bf p})U^\dagger({\bf p}),\\
&\tilde \Gamma_\mu({\bf p}_1,{\bf p}_2)=U({\bf p}_1)
\Gamma_\mu({\bf p}_1,{\bf p}_2)U^\dagger({\bf p}_2).\nonumber
\end{eqnarray}
They satisfy, then,
\begin{equation}
\tilde \Gamma_\mu(p,p)={\partial\tilde
S^{-1}(p)\over\partial p_\mu}.
\end{equation}
The current correlation function in the momentum representation is
written as,
\begin{eqnarray}
\pi_{\mu\nu}(q)&=&\langle{\rm T}(j_\mu(q_1)j_\nu(q_2))\rangle \\
&=&\sum S^{(0)}_{l_1,l_4}(p_1)S^{(0)}_{l_3,l_2}(p_3)
(l_1\vert\Gamma_\mu e^{i{\bf q}\cdot\hbox{$\xi$\kern-.5em\hbox{$\xi$}}}\vert l_2)
(l_3\vert\Gamma_\nu e^{-i{\bf q}\cdot\hbox{$\xi$\kern-.5em\hbox{$\xi$}}}\vert l_4)
\times \nonumber \\
&&\delta({\bf p}_1-{\bf p}_3-{\bf p}_1+{2\pi\over a}{\bf N})
\delta(-{\bf q}_1-{\bf p}_2+{2\pi\over a}({\bf M}+{\bf N})),
\nonumber\\
S^{(0)}_{l_1,l_4}(p_1)&=&{1\over p_1^0-E_{l_1}}\delta_{l_1,l_4},
\nonumber
\end{eqnarray}
in the lowest order of the interaction.
The Hall conductance is the slope of $\pi_{\mu\nu}(q_1)$ at the origin
and is expressed, under the use of the transformed propagator and
the Ward-Takahashi identity (2.17), as,
\begin{equation}
\sigma_{xy}={e^2\over h}{1\over 24\pi^2}\int d^3 p\epsilon_{\mu\nu\rho}
{\rm Tr}[{\partial\tilde S^{-1}(p)\over\partial p_\mu}\tilde S(p)
{\partial\tilde S^{-1}(p)\over\partial p_\nu}\tilde S(p)
{\partial\tilde S^{-1}(p)\over\partial p_\rho}\tilde S(p)].
\end{equation}
The right-hand side is a three-dimensional winding number of the mapping
defined by the propagator $\tilde S(p)$.
The space of the propagator is decomposed into SU(2) subspace,
a space spanned by $l$-th Landau level and $(l+1)$-th Landau level
from Eq.(2.18).
Note also that $S^{(0)}(p)$ is diagonal in $l_i$.
In this subspace, $\tilde S(p)$ is defined on the torus and the Hall
conductance becomes integer multiple of $e^2\over h$ in the
quantum Hall regime where there is no two-dimensionally extended
states around the Fermi energy.
The value is stable under perturbation effects such as interactions,
disorders, and others as far as the series converge, as were shown by
Coleman, Hill\cite{r11}, and others\cite{r12}\cite{r3}.
The value of $\sigma_{xy}$ stays constant while the Fermi energy is
moved if there is no singularity involved.
This occurs actually at the quantum Hall regime where there is
no two-dimensionally extended states but there are only
localized states with discrete energies or one-dimensionally extended
states.
In this situation, the value is computed by the lowest order calculation
and has no correction from the higher order corrections.
\section{Flux state mean field theory}\setcounter{equation}{0}
We propose a new mean field theory based on flux state on von Neumann
lattice in this section.
The dynamical flux is generated by interactions
and plays the important role in our mean field theory.
It is described by a lattice Hamiltonian, which is due to
the external magnetic field, and by the dynamical magnetic flux due to
interaction, although the original electrons are defined
on the continuum space. Consequently,
our mean field Hamiltonian is close to Hofstadter Hamiltonian and
hence there are similarities between their solutions.
Hofstadter Hamiltonian shows remarkable structures.
As is seen in Fig.1 the largest gap exists along a line
$\Phi=\nu\Phi_0$ with a unit of flux $\Phi_0$.
Ground state energy becomes minimum also with this flux.
These facts may suggest that Hofstadter problem has some connection
with the fractional Hall effect.
We pursue a mean field theory of the condensed flux states in the
quantum Hall system and point out that the Hofstadter problem is
actually connected with the fractional Hall effect.
$ $From the dynamical magnetic field, are defined new Landau levels.
If integer number of these Landau levels are filled completely,
the integer quantum Hall effect occurs.
The ground state has a large energy gap and is stable against
perturbations, just like ordinary integer quantum Hall effect.
We study these states and will identify them as fractional quantum Hall
states.
We postulate, in the quantum Hall system of the filling factor $\nu$,
the dynamical flux per plaquette and dynamical magnetic field of
the following magnitudes,
\begin{eqnarray}
&\Phi_{\rm ind}=\nu\Phi_0,\ \Phi_0=\Phi_{\rm external\ flux},\\
&B_{\rm ind}=\nu B_0,\ B_0=B_{\rm external\ magnetic\ field},
\nonumber
\end{eqnarray}
where $\nu$ is the filling factor measured with the external magnetic
field.
We obtain a self-consistent solution with this flux.
Then the integer quantum Hall effect due to induced magnetic field
could occur just at filling factor $\nu$, because the density
satisfies the integer Hall effect condition,
\begin{eqnarray}
&{eB_{\rm ind}\over 2\pi}N={eB_0\over 2\pi}\nu,\\
&N=1.\nonumber
\end{eqnarray}
The ground state has a large energy gap, generally.
At the half-filling $\nu=1/2$, half-flux $\Phi_0/2$ is induced.
This situation has been studied in detail by Lieb\cite{r13} and
others\cite{r14} in
connection with Hubbard model or t-J model.
Lieb gave a quite general proof that the energy is optimal with
half-flux at the half-filling case.
We study first the state of $\nu=1/2$ and the states of
$\nu=p/(2p\pm1)$, next.
At $\Phi=\Phi_0/2$, band structure is that of massless Dirac field and has
doubling symmetry.
When even number of Landau levels of the effective magnetic field,
$B_{\rm ind}-B_0/2$, are filled, ground states have large energy gaps.
This occurs if the condition of the density,
\begin{eqnarray}
{e\over 2\pi}\vert\nu-{1\over2}\vert B_0\cdot 2p={eB_0\over2\pi}\nu,\\
\nu={p\over 2p\pm1}\ ;\ p,\ {\rm integer},\nonumber
\end{eqnarray}
is satisfied. A factor 2 in the left-hand side is due to doubling of
states and will be discussed later.
We study these states in detail based on von Neumann lattice representation.
Action, Eq.(2.11), and density operator, Eq.(2.12), show that
there is an effective magnetic field in the momentum space.
Area of the momentum space is given by a finite value, $(2\pi/a)^2$,
and the total flux is hence finite.
The total flux in the momentum space is in fact unit flux.
In the thermodynamic limit, in which the density in space is finite,
the density in momentum space is infinite.
Consequently, it is possible to make this phase factor disappear by
a singular gauge transformation in the momentum space with
infinitesimally small coupling.
We make a singular gauge transformation of the field in the momentum
space,
\begin{eqnarray}
&c_l({\bf p})=e^{i\tilde e\lambda({\bf p})}b_l({\bf p}),
\nonumber\\
&\lambda({\bf p})={1\over 2\pi}\int\theta({\bf p}-{\bf p}')\rho({\bf p}')
d{\bf p}',\\
&\tan\theta={(p-p')_y\over(p-p')_x},\nonumber
\end{eqnarray}
where $\rho({\bf p})$ is the density operator in the momentum space and
$\tilde e$ is determined from Eqs.(B.3) and (B.4) in Appendix B.
With the transformed field, the commutation relation and the
charge density are expressed as,
\begin{eqnarray}
&\{c_{l_1}({\bf p}_1),c^\dagger_{l_2}({\bf p}_2)\}=
\sum\delta_{l_1,l_2}\delta({\bf p}_1-{\bf p}_2+{2\pi\over a}{\bf N}),\\
&\rho({\bf k})=\sum c^\dagger_{l_1}({\bf p}_1)c_{l_2}
({\bf p}_2)\delta({\bf p}_1-{\bf p}_2-{\bf k}+{2\pi\over a}{\bf N})
(l_1\vert e^{i{\bf k}\cdot\hbox{$\xi$\kern-.5em\hbox{$\xi$}}}\vert l_2).\nonumber
\end{eqnarray}
By a Chern-Simons gauge theory in the momentum space, the gauge
transformation, Eq.(3.4), is realized as is expressed in Appendix B.
Here, the coupling constant $\tilde e$ is infinitesimally
small, hence fluctuations of the Chern-Simons gauge field have small
effect and we ignore the fluctuations.
The action in the coordinate representation is given by,
\begin{eqnarray}
&S=\sum c^\dagger_l({\bf R})(i\hbar{\partial\over\partial t}-E_l)
c_l({\bf R})-{1\over 2}\sum v_{l_1,l_2;l_3,l_4}({\bf R}_2-{\bf R}_1)
c^\dagger_{l_1}({\bf R}_1)c^\dagger_{l_2}({\bf R}_1)
c_{l_3}({\bf R}_2)c_{l_4}({\bf R}_2),\nonumber\\
&v_{l_1,l_2;l_3,l_4}({\bf R}_2-{\bf R}_1)=\int_{{\bf k}\neq0}
d{\bf k} V({\bf k})
(l_1\vert e^{i{\bf k}\cdot\hbox{$\xi$\kern-.5em\hbox{$\xi$}}}\vert l_2)
(l_3\vert e^{-i{\bf k}\cdot\hbox{$\xi$\kern-.5em\hbox{$\xi$}}}\vert l_4)
e^{i{\bf k}\cdot({\bf R}_2-{\bf R}_1)}.
\end{eqnarray}
Hence the Hamiltonian in the lowest Landau level space is given by,
\begin{eqnarray}
&H=-{1\over 2}\sum v({\bf R}_2-{\bf R}_1) c_0^\dagger({\bf R}_1)
c_0({\bf R}_2) c_0^\dagger({\bf R}_2) c_0({\bf R}_1),
\\
&v({\bf R})=
{\pi\over a}e^{-{\pi\over2}{\bf R}^2}I_0({\pi\over2}{\bf R}^2),
\nonumber
\end{eqnarray}
where $I_0$ is zero-th order modified Bessel function.
We study a mean field solution of this Hamiltonian.
We have an expectation value and a mean field Hamiltonian,
\begin{eqnarray}
&\langle c_0^\dagger({\bf R}_1) c_0({\bf R}_2)\rangle=
U_0({\bf R}_1-{\bf R}_2)e^{i\int_{{\bf R}_2}^{{\bf R}_1}{\bf A}_{\rm ind}
\cdot d{\bf x}},\\
&H_{\rm mean\ field}=-\sum v({\bf R}_2-{\bf R}_1)U_0({\bf R}_1-{\bf R}_2)
e^{i\int_{{\bf R}_2}^{{\bf R}_1}{\bf A}_{\rm ind}\cdot d{\bf x}}
c_0^\dagger({\bf R}_2)c_0({\bf R}_1),
\nonumber
\end{eqnarray}
and solve the equations self-consistently.
The mean field Hamiltonian coincides to that of Hofstadter if the
potential is of short range of nearest neighbor type.
The spectrum obtained by Hofstadter shows characteristic structures,
and has a deep connection with the structure of the fractional quantum
Hall effect.
\
(i) Half-filled case, $\nu=1/2$.
At half-filling $\nu=1/2$, the system has a half flux
$\Phi=\Phi_0/2$.
The system, then, is described equivalently with the two-component
Dirac field by combining the field at even sites with that at
odd sites.
In the gauge, $A_x=0,\ A_y=Bx$, the mean field Hamiltonian reads,
\begin{eqnarray}
&H_{\rm M.F.}=
\sum\Psi^\dagger(X')
\left( \begin{array}{cc}
a_{ee}(X'-X) & a_{eo}(X'-X)\\
a_{oe}(X'-X) & a_{oo}(X'-X)
\end{array}\right)\Psi(X),
\nonumber\\
&\Psi(X)=
\left(\begin{array}{c}
c(2m,n)\\
c(2m+1,n)
\end{array}\right),
\end{eqnarray}
\begin{eqnarray}
&a_{ee}(m'-m,n'-n)=\langle c^\dagger(2m',n')c(2m,n)\rangle
v(2m'-2m,n'-n),\nonumber\\
&a_{oo}(m'-m,n'-n)=\langle c^\dagger(2m+1',n')c(2m+1,n)\rangle
v(2m'-2m,n'-n),\nonumber\\
&a_{eo}(m'-m,n'-n)=\langle c^\dagger(2m',n')c(2m+1,n)\rangle
v(2m'-2m-1,n'-n),\nonumber\\
&a_{oe}(m'-m,n'-n)=\langle c^\dagger(2m'+1,n')c(2m,n)\rangle
v(2m'-2m+1,n'-n),\nonumber\\
&a_{eo}=a_{oe},\\
&a_{ee}+a_{oo}=0.\nonumber
\end{eqnarray}
We obtain the self-consistent solutions numerically.
Fig.2 shows the spectrum.
As is expected, spectrum has two minima and two zeros corresponding
to doubling.
We have the momentum space expression of the mean field Hamiltonian,
\begin{eqnarray}
&H_{\rm M.F.}&=\sum c^\dagger_\xi({\bf p})F_{\xi\eta}({\bf p})c_\eta
({\bf p})\\
&&=\sum\epsilon_\alpha({\bf p})c^\dagger_\xi({\bf p})U^\dagger
_{\xi\xi'}({\bf p})U_{\xi'\eta}({\bf p})c_\eta({\bf p}),\nonumber\\
&&\epsilon_\pm({\bf p})=\pm\sqrt{a_{ee}^2({\bf p})+a_{eo}^2({\bf p})},
\nonumber\\
&&U_{\xi\eta}({\bf p})=
\left(\begin{array}{cc}
a_{eo}/N_+ & (\epsilon_+ -a_{ee})/N_+\\
a_{eo}/N_- & (\epsilon_- -a_{ee})/N_-
\end{array}\right),\nonumber\\
&&N^2_{\pm}=2(a^2_{ee}+a^2_{eo}-a_{ee}\epsilon_{\pm}),
\nonumber
\end{eqnarray}
$$
a_{ee}(p_x,p_y+\pi/a)=-a_{ee}(p_x,p_y),\
a_{ee}(p_y,p_x)=a_{eo}(p_x,p_y).\nonumber
$$
The matrix $F_{\xi\eta}({\bf p})$ is approximated well with a nearest
neighbor form,
\begin{equation}
F({\bf p})=2K\left(\begin{array}{cc}
\cos p_y&\cos p_x\\
\cos p_x&-\cos p_y
\end{array}\right),
\end{equation}
$$
K=0.107{e^2\over\kappa l_B},\ l_B=\sqrt{\hbar\over eB}.
$$
Around minima, the energy eigenvalue of Eq.(3.11) is approximated as,
\begin{equation}
E(p)=E_0+{({\bf p}-{\bf p}_0)^2\over 2m^*},\
{\bf p}_0=(0,0),(0,\pi/a),
\end{equation}
and they are approximated around zeros as,
\begin{eqnarray}
&E(p)=\gamma{\hbox{$\alpha$\kern-.7em\hbox{$\alpha$}}}\cdot{\bf p},\\
&\alpha_x=\left(\begin{array}{cc}
0&1\\
1&0
\end{array}\right),\
\alpha_y=\left(\begin{array}{cc}
1&0\\
0&-1
\end{array}\right),
\nonumber
\end{eqnarray}
in $2\times2$ expression.
The $m^*$ in Eq.(3.13) is the effective mass and $\gamma$ is the effective
velocity.
They are computed numerically.
Its value is,
\begin{eqnarray}
&m^*=0.225\sqrt{B\over B_0}m_e,\ B_0=20{\rm Tesla},\\
&\gamma=0.914{e^2\over\kappa}.\nonumber
\end{eqnarray}
The effective mass is proportional to the square root of
$B$ in our method.
$\nu=1/2$ mean field Hamiltonian is invariant under a kind of
Parity, $P$, and
anti-commutes with a chiral transformation, $\alpha_5$, which are
defined by,
\begin{eqnarray}
P:\ &\Psi(p_x,p_y)&\rightarrow \alpha_x\Psi(p_x,p_y+\pi/a),\\
\alpha_5:\ &\Psi(p_x,p_y)&\rightarrow\alpha_5\Psi(p_x,p_y),\nonumber\\
&&\alpha_5=\alpha_x\alpha_y.\nonumber
\end{eqnarray}
If the parity is not broken spontaneously, there is a degeneracy
due to parity doublet.
Doubling of the states\cite{r15} appears also at $\nu\neq1/2$ and plays
important role,
when we discussed the states away from $\nu=1/2$ in the next part.
When an additional vector potential with the same gauge,
$A_x=0,\ A_y=Bx$, is added, the Hamiltonian satisfies the properties
under the above transformations and
the doubling due to parity doublet also appears. Thus,
the factor 2 is necessary in Eq.(3.3) and leads the principal
series at $\nu=p/(2p\pm1)$ to have maximum energy gap.
\
(ii) $\nu={p\over 2p\pm1}$
If the filling factor, $\nu$, is slightly away from 1/2, total system
can be regarded as a system with a small magnetic field
$(\nu-1/2)B_0$.
A band structure may be slightly modified.
It is worthwhile to start from the band of $\nu=1/2$ as a first
approximation and to make iteration in order to obtain self-consistent
solutions at arbitrary $\nu=p/(2p\pm1)$.
We solve the following mean field Hamiltonian under the self-consistency
condition at $\nu=1/2+\delta$,
\begin{eqnarray}
&H_{\rm M}=\sum U^{({1\over 2}+\delta)}_0({\bf R}_1-{\bf R}_2)
e^{i\int({{\bf A}}^{({1\over 2})}+
\delta{\bf A})d{\bf x}}v({\bf R}_1-{\bf R}_2)
c^\dagger({\bf R}_1)c({\bf R}_2),\\
&\langle c^\dagger({\bf R}_1)c({\bf R}_2)\rangle_{1/2+\delta}=
U^{({1\over 2}+\delta)}_0 e^{i\int({{\bf A}}^{({1\over 2})}
+\delta{\bf A})}.\nonumber
\end{eqnarray}
Here we solve, instead, an Hamiltonian which has the
phase of Eq.(3.17) but has the magnitude of the $\nu=1/2$ state.
Namely we study a series of states defined at $\nu=1/2$,
\begin{eqnarray}
H_{\rm M}^{(2)}&=\sum U^{({1\over2})}_0({\bf R}_1-{\bf R}_2)
e^{i\int{\bf A}^{({1\over2})}d{\bf x}}v({\bf R}_1-{\bf R}_2)
e^{i\int\delta{\bf A}d{\bf x}}c^\dagger({\bf R}_1)c({\bf R}_2)
\nonumber\\
&=\sum F^{({1\over2})}({\bf R}_1,{\bf R}_2;\delta{\bf A})
c^\dagger({\bf R}_1)c({\bf R}_2).
\end{eqnarray}
Integer quantum Hall state has an energy gap
of the Landau levels due to $\delta{\bf A}$.
This occurs when the integer number of Landau levels are filled
completely.
Landau level structure is determined by the phase factor and
Eq.(3.17) and Eq.(3.18) have the same Landau level structure.
Magnitudes of the physical quantities of Eq.(3.17) may be modified,
nevertheless.
$F^{({1\over2})}({\bf R}_1,{\bf R}_2;0)$ was obtained in the previous part,
and is approximated with either the effective mass formula (3.13) or
with the nearest neighbor form (3.12).
$ $From Eqs.(3.13) and (3.18),
the Hamiltonian $H^{(2)}_{\rm M}$ is written in the former method as
\begin{eqnarray}
H_{\rm M}^{(2)}&=\sum c^\dagger({\bf R}_1)U^\dagger({\bf R}_1,
{\bf R}'_1,\delta{\bf A})
\{E_0+{({\bf p}+e\delta{\bf A})^2\over2m^*}\}
U({\bf R}'_2,{\bf R}_2,\delta{\bf A})c({\bf R}_2)\nonumber\\
&=\sum\tilde c^\dagger(R_1)
\{E_0+{({\bf p}+e\delta{\bf A})^2\over2m^*}\}\tilde c({\bf R}_1),\\
&\tilde c({\bf R}_1)=\sum U({\bf R}_1,{\bf R}'_1)c({\bf R}'_1),\\
&U_{\xi\eta}({\bf R}_1,{\bf R}'_1)=\int d{\bf p} e^{i{\bf p}\cdot({\bf R}_1-
{\bf R}'_1)}U_{\xi\eta}({\bf p}).
\end{eqnarray}
Integer Hall states of Eq.(3.19) satisfy,
\begin{equation}
\langle{\tilde c}^\dagger_\xi({\bf R}_1)
{\tilde c}_\eta({\bf R}_2)\rangle=
\tilde u_{\xi\eta}({\bf R}_1-{\bf R}_2)
e^{i\int^{{\bf R}_1}_{{\bf R}_2}\delta{\bf A}d{\bf x}},
\end{equation}
and leads the expectation value,
\begin{eqnarray}
\langle c^\dagger({\bf R}_1)c({\bf R}_2)\rangle&=&U^\dagger(
{\bf R}_1,{\bf R}'_1)
\langle{\tilde c}^\dagger({\bf R}'_1){\tilde c}({\bf R}'_2)\rangle
U({\bf R}'_2,{\bf R}_2)\\
&=&U^\dagger({\bf R}_1,{\bf R}'_1)\tilde u_{\xi\eta}({\bf R}'_1-{\bf R}'_2)
e^{i\int^{{\bf R}'_1}_{{\bf R}'_2}\delta{\bf A}d{\bf x}}
U({\bf R}'_2,{\bf R}_2).\nonumber
\end{eqnarray}
Here, we solve the equations obtained from Eq.(3.19) with continuum
approximation, first.
We have,
\begin{eqnarray}
&[E_0+{({\bf p}-{\bf p}_0^{(i)}+\delta{\bf A}_{\rm ind})^2\over 2m^*}]
u^{(i)}_p=E_p u_p^{(i)},\\
&i=1,2,\nonumber\\
&E_p=E_0+{e\delta B_{\rm eff}\over 2m^*}(2p+1)\\
&\delta B_{\rm eff}=\vert \nu-{1\over2}\vert B_0.\nonumber
\end{eqnarray}
$ $From Eq.(3.25),
$p$ Landau levels are completely filled and the integer
quantum Hall effect occurs at $\nu=p/(2p\pm1)$.
The energy gap is given by Landau level spacing,
\begin{equation}
\Delta E_{\rm gap}={e\delta B_{\rm eff}\over m^*}={eB_0\over m^*}
\vert\nu-{1\over2}\vert.
\end{equation}
Hamiltonian with the nearest neighbor form is a $2\times2$ matrix of the
tight-binding type.
Based on this Hamiltonian, we solve Landau level equation of
the tight-binding form with the vector potential
$\delta{\bf A}$,
\begin{eqnarray}
&K\{c_1({\bf n}+{\bf\Delta}_x)e^{i\int_{\bf n}^{{\bf n}+{\bf\Delta}_x}
\delta{\bf A}\cdot d{\bf x}}+
c_2({\bf n}+{\bf \Delta}_y)e^{i\int_{\bf n}^{{\bf n}+{\bf\Delta}_y}
\delta{\bf A}\cdot d{\bf x}}+{\rm h.c.}\}=Ec_1({\bf n}),\nonumber\\
&K\{c_1({\bf n}+{\bf\Delta}_x)e^{i\int_{{\bf n}}^{{\bf n}+{\bf\Delta}_x}
\delta{\bf A}\cdot d{\bf x}}-
c_2({\bf n}+{\bf\Delta}_y)e^{i\int_{\bf n}^{{\bf n}+{\bf\Delta}_y}
\delta{\bf A}\cdot d{\bf x}}+{\rm h.c.}\}=Ec_2({\bf n}).\nonumber\\
&
\end{eqnarray}
The equation are solved numerically and the energy gaps and the
widths of excited bands are given in Fig.3.
Some bands are narrow and some bands are wide.
Near $\nu=1/2$, the effective magnetic field approaches to zero and
Landau level wave functions have large spatial extensions.
Lattice structure becomes negligible and spectrum shows simple
Landau levels of the continuum equation in these regions.
Near $\nu=1/3$, lattice structure is not negligible and bands have
finite widths.
There are non-negligible corrections from those of continuum
calculations, Eq.(3.24).
Due to energy gap of the integer Hall effect caused by the induced
dynamical magnetic field, the states at $\nu=p/(2p\pm1)$ are stable
and fluctuations are weak.
Invariance under $P$, moreover, ensures these states to
have uniform density.
In systems with impurities, localized states with isolated discrete
energies are generated by impurities and have energies in the gap regions.
They contribute to the density but do not contribute to the conductance.
If the Fermi energy is in one of these gap regions,
the Hall conductance $\sigma_{xy}$ is given by a topological formula,
Eq.(2.19), and stays constant, at
${e^2\over h}\cdot{p\over 2p\pm1}$. The fractional Hall effect is realized.
At a value of $\nu$ smaller than 1/3,
Hofstadter butterfly shows other kind of
structures hence suggests other structures of the fractional Hall effect do
exist.
For instance at $\nu=$1/5, 1/7, 1/9,$\dots$the energy gap is large.
In low density, it may be complicated, in fact.
There could exist completely
different kind of phase, such as Wigner crystal phase\cite{r16}.
Competition between two phases may be important.
They will not be presented in
the present paper but it will be presented in a later work.
\
(iii) Fluctuations of FQHE at $\nu={p\over 2p\pm1}$.
Ground states have energy gap, hence the fluctuations are small, just as
in the integer Hall effect.
Density and phase fluctuations are described by the massive
Chern-Simons gauge theory\cite{r17}with a mass of order energy gap.
\
(iv) Fluctuations at half-filling, $\nu={1\over2}$.
Ground state has no energy gap at $\nu=1/2$.
Fluctuations are described by the action,
\begin{equation}
S=\int d^3x\Psi^\dagger(i\hbar{\partial\over\partial t}+a_0)\Psi+
\gamma\Psi^\dagger\hbox{$\alpha$\kern-.7em\hbox{$\alpha$}}\cdot({\bf p}+{\bf a})\Psi
\end{equation}
The fermion field integration leads severe infra-red divergence.
If the energy dispersion is changed to
\begin{equation}
E_0=\tilde\gamma\vert p\vert^{1+\delta},
\end{equation}
by interaction, physics at $\nu=1/2$ is completely different from that
of the mean field.
We will not discuss physical properties of $\nu=1/2$ state in this paper.
\section{Comparison with experiments}\setcounter{equation}{0}
In the previous section we presented our mean field theory based on
flux condensation, where lattice structure generated by the external magnetic
field and condensed flux due to interaction
are important ingredient.
Consequently, our mean field Hamiltonian becomes very similar to
that of Hofstadter which is known to show actually large energy gap
zone along $\Phi=\nu\Phi_0$ line.
The line $\Phi=\nu\Phi_0$ is special in Hofstadter problem
and hence in our mean field Hamiltonian, too.
This explains why the experiments of the fractional quantum Hall effects
shows characteristic behavior at $\nu=p/(2p\pm1)$.
The ground states at $\nu=p/(2p\pm1)$ have lowest energy
and largest energy gap, hence these states are
stable.
In this section we compare the energy gaps of the principal series
with the experiments in the lowest order.
and the ground state energy of $\nu=1/3$ state with the Laughlin
variatioal wave function\cite{r18}.
The effective mass $m^*$ of Eq.(3.14) was obtained from the curvature
of the energy dispersion and should show a characteristic mass
scale of the fractional Hall effect.
Eq.(3.24) gives Landau level energy in the lowest approximation and
the gap energy is given in Eq.(3.25).
The gap energy from the nearest neighbor approximation, Eq.(3.12) is
given by solving Eq.(3.27).
They are compared with the experimental values\cite{r19}
in Fig.4.
The agreement is not perfect but should be regarded good as the
lowest mean field approximation.
Near $\nu=1/3$, the bands have finite widths and near $\nu=1/2$, the
widths are infinitesimal.
The dependence of the width upon the filling factor, $\nu$,
and the whole structure of the bands expressed in Fig.1 are
characteristic features of the present mean field and
should be tested experimentally.
Finally we compute the ground state energy of the $\nu=1/2$ state and
the $\nu=1/3$ state.
Using the fact that our mean field Hamiltonian of $\nu=1/3$ is very
close to the short-range tight-binding Hamiltonian,
we compute the ground state energy of $\nu=1/3$ with the tight-binding
model wave function.
The wave function is obtained numerically and is substituted
to the total energy per particle, Eq.(3.7), as
\begin{eqnarray}
E_{1/3}&=&{3\over N}\langle\Psi\vert H\vert\Psi\rangle \\
&=&-{3\over2}\sum_{\bf X}v({\bf X})\vert U_0({\bf X})\vert^2
.\nonumber
\end{eqnarray}
where $N$ is the number of sites.
The result is,
\begin{equation}
E_{1/3}=-0.340{e^2\over\kappa l_B}.
\end{equation}
where $l_B=\sqrt{\hbar/eB}$.
This value should be compared with that of Laughlin wave function,
\begin{equation}
E_{1/3}=-0.416{e^2\over\kappa l_B}.
\end{equation}
The value of Eq.(4.2) is higher than Eq.(4.3), but the difference is
not large.
This may suggest that mean field flux state is infact close to
Laughlin wave function and to the exact solution.
For the state at $\nu=1/2$, we use our self-consistent solution
for computing the ground state energy per particle,
\begin{equation}
E_{1/2}=-0.347{e^2\over\kappa l_B}.
\end{equation}
As is mentioned in Section.3, the fluctuations are extremely large and
the mean field value get large corrections from the higher order
effects.
So the results about the $\nu=1/2$ state should not be
taken seriously.
\section{Summary}\setcounter{equation}{0}
We formulated the quantum Hall effects, integer Hall effect and
fractional Hall effect with von Neumann lattice representation of
two-dimensional electrons in a strong magnetic field.
von Neumann lattice is a subset of coherent state.
The overlapp of the states is expressed with theta function.
They give a systematic way of expressing quantum Hall dynamics.
Topological invariant expression of the Hall conductance was
obtained in which compactness of the momentum space is
ensured by the lattice of the coordinate space.
Because the lattice has an origin in the external magnetic field,
topological character of the Hall conductance is ensured by
the external magnetic field.
The conductance is quantized exactly as $(e^2/h)\cdot N$ at the
quantum Hall regime.
The integer $N$ increases with chemical potential.
A new mean field theory of the fractional Hall effect that has
dynamical flux condensation is studied.
If the filling $\nu$ is less than one, many particle states
of Landau levels have no energy gap unless interaction switchs on.
In the tight-binding model, the situation is very different.
The spectrum that was found by Hofstadter first is changed drastically
when the flux is changed and it has a
large energy gap in some regions.
In our mean field theory, lattice structure is introduced from von
Neumann lattice and flux is introduced dynamically.
The mean field Hamiltonian becomes a kind of tight-binding model and
rich structure of the tight-binding model is seen, in fact, as
characteristic features of the fractional Hall effect.
Our mean field flux states have liquid property of uniform density with
energy gap.
They are defined as special integer quantum Hall states
in the lowest Landau level space hence the band structures shown in
Figs.2 and 3 are very different from those of normal integer Hall states.
We gave a dynamical reason why the principal series at $\nu=p/(2p\pm1)$ are
observed dominantly.
These states satisfy the self-consistency condition of having
the lowest energy and the largest energy gap.
The physical quantities of our mean field theory are close to the
experimental values in the lowest order at $\nu=p/(2p\pm1)$.
At the exact half-filling $\nu=1/2$, fluctuations are very large and
corrections from mean field value may be large, too.
\section*{Acknowledgements}
We are indebted to Professors P. Wiegmann and H. Suzuki for their
useful comments in the early stage of the present work.
One of the authors(K.I.) thanks Professors A. Luther and
H. Nielsen for useful discussions.
The present work was partially supported by the special Grant-in-Aid
for Promotion of Education and Science in Hokkaido University
Provided by the Ministry of Education, Science, Sports, and Culture,
a Grant-in-Aid for Scientific Research(07640522), and
Grant-in-Aid for International Scientific Research(Joint Research
07044048) the Ministry of Education, Science, Sports, and Culture,
Japan.
\newpage
\renewcommand{\theequation}{A.\arabic{equation}}
\noindent {\Large \bf APPENDIX A}
\setcounter{equation}{0}
\bigskip
In the present representation, a short-range impurity term can be
expressed as,
\begin{eqnarray}
&H_{\rm impurity}=a^4\int^{\pi/a}_{-\pi/a}{d^2p_1\over (2\pi)^2}
{d^2p_2\over (2\pi)^2}b^\dagger(p_1)\sum_{{\bf N}}({2\pi\over a})^2
e^{ia\int^{p_2^N}_{p_1}{\bf A}\cdot d{\bf p}}\tilde V(p_1-p_2^N)e^{
-{a^2(p_1-p_2^N)^2\over8\pi}}b(p_2^N)\nonumber\\
&=a^2\int^{\pi/a}_{-\pi/a}{d^2p_1\over (2\pi)^2}b^\dagger(p_1)\int^\infty
_{-\infty}d^2 k\tilde V(-k)e^{-{a^2k^2\over8\pi}}e^{iak^i D_i}b(p_1),
\nonumber\\
&D_i={1\over i}{\partial\over a\partial p_i}+A_i,\
{\bf A}=({ap_y\over2\pi},0),
\\
&{\bf p}_2^N={\bf p}_2+2\pi{\bf N},\
V(x)=\int^\infty_{-\infty}d^2k\tilde V(k)e^{i{\bf k}\cdot{\bf x}}.
\nonumber
\end{eqnarray}
With the creation and annihilation operator defined by,
\begin{eqnarray}
&A=D_x-iD_y,\\
&A^\dagger=D_x+iD_y,\nonumber\\
&[A,A^\dagger]={1\over\pi},\nonumber
\end{eqnarray}
the above Hamiltonian can be written as,
\begin{equation}
H_{\rm impurity}=a^2\int^{\pi/a}_{-\pi/a}{d^2p\over (2\pi)^2}
b^\dagger(p)\int^\infty_{-\infty}d^2 k\tilde V(-k)e^{{a\over2}(ik_x-k_y)A}
e^{{a\over2}(ik_x+k_y)A^\dagger}b(p).
\end{equation}
We represent $H_{\rm impurity}$ in Landau level representation of
the momentum space defined by,
\begin{eqnarray}
b(p)&=&\sum_l b_l\tilde\psi_l(p),\
\vert l\rangle=\pi^{l\over2}{1\over \sqrt{l!}}
(A^\dagger)^l\vert 0\rangle,\\
A\vert 0\rangle&=&0,\nonumber\\
\tilde\psi_l(p)&=&\langle p\vert l\rangle
=C\sum_N e^{-i\pi N}e^{ik_N a p_x}H_l({a(p_y+{2\pi\over a}
k_N)\over\sqrt{\pi}})e^{-{a^2\over4\pi}(p_y+{2\pi\over a} k_N)^2},
\nonumber
\end{eqnarray}
where $k_N=N+{1\over2}$ and $C$ is the normalization constant.
The normalized lowest Landau level wave function is,
\begin{equation}
\tilde\Psi_0=2^{1/4}e^{-{a^2 p_y^2\over4\pi}}
\Theta_1({a(p_x+ip_y)\over2\pi},i).
\end{equation}
$H_{\rm impurity}$ is reduced to,
\begin{eqnarray}
&H_{\rm impurity}=\sum_{l_1,l_2}b^\dagger_{l_1}\tilde V_{l_1,l_2}
b_{l_2},\\
&\tilde V_{l+n,l}=4\pi\sqrt{l!\over (l+n)!}\int d^2 k
\tilde V(-2\sqrt{\pi} k)a^n (ik_x+k_y)^n L_l^{(n)}
(k^2 a^2)e^{-a^2 k^2},\nonumber\\
&\tilde V_{l,l+n}=4\pi\sqrt{l!\over (l+n)!}\int d^2 k
\tilde V(-2\sqrt{\pi} k)a^n (ik_x-k_y)^n L_l^{(n)}
(k^2 a^2)e^{-a^2 k^2}.\nonumber
\end{eqnarray}
For a short range potential, we have
\begin{eqnarray}
&V(x)=g\delta(x),\
\tilde V(-k)={g\over (2\pi)^2},\nonumber\\
&\tilde V_{l+n,l}=\tilde V_{l,l+n}=0,\ {\rm for}\
n\neq0,\\
&\tilde V_{l,l}=0,\ {\rm for}\ l\neq0,\nonumber\\
&\tilde V_{0,0}={g\over (2\pi)^2}4\pi\int d^2 k
L_0^{(0)}(a^2k^2)e^{-a^2k^2}={g\over a^2},\nonumber\\
&H_{\rm impurity}=\tilde V_{0,0}b^\dagger_0 b_0.
\end{eqnarray}
Hence the one Landau level has the energy shift $\tilde V_{00}$, and
all the other Landau levels have no effect from the impurity.
The state $\vert l=0\rangle$ has an isolated energy and its wave function
is square-integrable.
This state corresponds to localized state.
\newpage
\renewcommand{\theequation}{B.\arabic{equation}}
\noindent {\Large \bf APPENDIX B}
\setcounter{equation}{0}
\bigskip
In Chern-Simons gauge theory in momentum space, the action is
\begin{equation}
\int dtd{\bf p}
{1\over2}\epsilon_{\mu\nu\rho}a^\mu{\partial\over\partial p_\nu}
a^\rho+a_0 j^0({\bf p})+
F(c^\dagger(p_1)e^{i\tilde e\int^{p_1}_{p_2}a_i dp_i}c
(p_2)),
\end{equation}
where in the last term operators $b^\dagger(p_1)b(p_2)$ are replaced with
$c^\dagger(p_1)e^{i\tilde e\int^{p_1}_{p_2}a_i dp_i}c
(p_2)$, and the theory is defined on the torus\cite{r20}.
The vector potential $a_i$ satisfy,
\begin{equation}
\epsilon_{0ij}{\partial\over\partial p_i} a^j+j^0({\bf p})=0.
\end{equation}
Its solution is substituted into the first term of (B.1).
By choosing the coupling strength $\tilde e$ from the condition,
\begin{eqnarray}
\tilde e j^0({\bf p})+{a^2\over 2\pi}=0,\\
j^0({\bf p})=({a\over 2\pi})^2 N_{\rm total}.
\end{eqnarray}
the phase factor which expresses the magnetic field in the momentum space
in Eqs.(2.11), (2.12), (2.13), and (2.14) are cancelled.
In thermodynamic limit, $j^0({\bf p})$ diverges,
hence $\tilde e$ in infinitesimally small.
The fluctuation of Chern-Simons gauge field can be ignored.
The first two terms in Eq.(B.1) cancell if Eq.(B.2) is satisfied and
do not contribute to the energy of the system.
\newpage
| proofpile-arXiv_065-435 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
During its 17 plus years of operation, the {\it International Ultraviolet
Explorer\/} ({\it IUE\/}) satellite (\cite{bog78}) acquired a large number
($> 10^5$) of UV spectra from all classes of astronomical objects. Cataclysmic
variables have been studied intensively with {\it IUE\/}, and these spectra
have revealed a wide variety of phenomena which were previously unknown or
poorly studied. These include mass loss via high-velocity winds, the heating
and cooling of the white dwarf, the spectrum of the accretion disk, and the
response of the disk to dwarf nova outbursts. As a result, we now have a much
more complete understanding of the nature of CVs: the geometry, velocity law,
mass-loss rate, and ionization structure of the wind; the disk instability
mechanism and the heating and cooling waves which drive the disk to and from
outburst; the relative energetics of the accretion disk and boundary layer;
subclasses; and even evolutionary scenarios. These advances are due in part
to the fact that CVs emit a large fraction of their luminosity in the UV.
With few exceptions, the strategy which has prevailed to date in the study
of CVs with {\it IUE\/} and other instruments is to make an observation or
set of observations of a single object and to analyze and interpret those
data separately from the data of other similar objects. Only when this step
is complete are the data compared with those from other similar objects. An
alternative strategy is to examine the data from all the objects of a given
class simultaneously. This is practical only in situations where a large body
of data exists, of fairly uniform quality, and when the various members of
a given class form a relatively homogeneous group. A few such studies of CVs
using various manifestations of the {\it IUE\/} archive have appeared in
the literature. Verbunt (1987) studied the absolute continuum spectral flux
distributions of disk-fed CVs. la Dous (1991) studied the relative continuum
spectral flux distributions of all classes of CVs, but concentrated on disk-fed
systems. For the discrete spectral features, only the inclination dependence of
the equivalent widths was discussed. Deng et~al.\ (1994) studied the dependence
on the orbital period and inclination of the relative continuum spectral
flux distributions and the equivalent widths of the lines of dwarf novae in
quiescence.
It is unfortunate that so little quantitative work has been performed to date
on the emission lines of CVs. Emission lines are powerful diagnostics of the
physical conditions in all types of objects including these interacting
binaries. The emission line fluxes or relative fluxes can constrain the
temperature, ionization state, density, elemental abundance, and geometrical
distribution of the gas. However, the dependence of measurable quantities on
these parameters is often complicated, necessitating numerical model fitting
in many situations. The range of possible assumptions regarding the physical
mechanisms of heating and ionization, together with the technical challenges
of constructing models, add to the difficulty of performing such studies. As a
result, emission lines have received less attention as diagnostics of CVs than
have continuum properties.
In the present paper, we begin to remedy this situation with an analysis of
the flux ratios of the emission lines of all classes of CVs using spectra
extracted from the {\it IUE\/} Uniform Low Dispersion Archive (ULDA). We begin
by describing the data and our analysis procedures (\S 2). We then present the
results of the data analysis (\S 3), followed by a discussion of models for
the line emission (\S 4). In \S 5 we compare data and models and in \S 6 close
with a summary of our conclusions.
\section{Data Analysis}
The {\it IUE\/} ULDA (\cite{wam89}) provides a unique source of archival data
which satisfies the requirements for a comprehensive study of a large sample of
objects owing to its uniform quality and the large number of spectra accumulated
over the long life of {\it IUE\/}. Although the {\it IUE\/} final archive
(\cite{nic94}) has since superseded the ULDA, the reprocessing effort of the
final archive had not begun when this study was begun in 1992. To build the
data sample, the {\it IUE\/} Merged Log was searched using software at the
{\it IUE\/} Regional Analysis Facility, then at the University of Colorado.
The search criteria were the camera (SWP), dispersion (low), and {\it IUE\/}
object class (dwarf novae, classic novae, irregular variables, and nova-like
variables). All spectra meeting these criteria were extracted from the ULDA
resident at Goddard Space Flight Center, which at that time contained (nearly)
all low-dispersion {\it IUE\/} spectra through 1988 (SWP sequence numbers
through $\approx 35190$). The spectra were then written to disk with header
information extracted from the {\it IUE\/} Merged Log. Excluded from further
consideration were 13 very early (prior to 1978 July) spectra of AM~Her and
SS~Cyg lacking data quality flags. The final dataset consisted of $\approx
1300$ spectra of $\approx 100$ CVs.
From this dataset, we selected all spectra exhibiting pure emission lines.
Sources which met this criteria include AM~Her stars, DQ~Her stars, dwarf
novae in quiescence, and eclipsing nova-like variables and dwarf novae in
outburst. To this dataset we added the {\it IUE\/} spectra of the eclipsing
nova-like variable V374~Pup listed in Table~1 of Mauche et~al.\ (1994), as
we were working on that source at the time. Excluded from analysis were
non-eclipsing nova-like variables and dwarf novae in outburst, since the UV
resonance lines of such systems are either in absorption or are P~Cygni
profiles.
For these spectra, we measured the integrated fluxes above the continuum of
the strongest emission lines in the SWP bandpass: N~V $\lambda 1240$, Si~IV
$\lambda 1400$, C~IV $\lambda 1550$, and He~II $\lambda 1640$. To determine
the flux of these lines, the continuum to the left and right of each line
(e.g., from 1500--1530 \AA \ and from 1570--1600 \AA \ for C~IV) was fitted
with a linear function by the method of least squares. This method works well
with all of the emission lines except N~V, because the left side of this line
falls on the wing of the geocoronal Lyman $\alpha $ emission line. To overcome
this problem, a continuum point to the left of the N~V line was specified
interactively, and the fit included the region to the right of the line
(1255--1270 \AA ) plus this continuum point to the left. Care was taken to
insure that the continuum over this small wavelength region was a reasonable
extrapolation both in normalization and slope to the continuum at longer
wavelengths. In all cases the least-squares fit was made under the assumption
that the errors $\sigma _i$ in the flux density $f_i$ at each wavelength point
$\lambda _i$ were the same. After the normalization $a$ and slope $b$ of the
continuum was determined, the size of the errors $\sigma $ were estimated by
assuming that the reduced $\chi ^2$ of the fit was equal to 1: $\sigma ^2 =
\sum _{i=1}^N (f_i-a-b\lambda _i)^2/(N-2)$. This procedure is necessary
because the ULDA (unlike the {\it IUE\/} final archive) does not include the
error associated with each flux point. The integrated fluxes of the emission
lines were then determined by summing the flux minus the fitted continuum
over the region of the line (e.g., 1530--1570 \AA \ for C~IV). The error on
the integrated flux was determined by propagating the error $\sigma $ through
the calculation. Finally, careful track was kept of the number of pixels
labeled by the data quality flags as either saturated or nearly saturated
(extrapolated ITF). Such pixels compromise the quality of the flux density
measurements and consequently the integrated line fluxes; saturated pixels
are particularly bad in this regard, because they systematically produce
only lower limits to the flux density. In the final cut to the data, flux
measurements for a given line were retained only if the total number of
saturated and extrapolated pixels included in the summation was less than
or equal to some number $n$. While $n=1$ would have been the ideal choice,
such a stringent selection criterion would cull too many measurements. As
a reasonable compromise, we settled on $n=3$ for this selection criterion,
allowing a few mild sinners to pass with the saints. The roster at this stage
of the analysis consisted of $\approx 700$ spectra of $\approx 60$ CVs. Of
this number, we present results based on $\approx 430$ spectra of the 20
systems with the largest number ($> 10$) of flux measurements.
\section{Observational Results}
Just as color-color diagrams are a useful way to discuss the continuum
spectral flux distributions of CVs (e.g., \cite{wad88}; \cite{den94}), flux
ratio diagrams are a useful way to display and discuss the fluxes of their
emission lines. Flux ratios remove the variables of luminosity and distance
from the comparison of different sources; flux ratios are unaffected by
aperture losses, which occur for {\it IUE\/} spectra obtained through the
small aperture; and flux ratios are much less sensitive to reddening than the
fluxes themselves. Due to the increase of the extinction efficiency at short
wavelengths, reddening has the largest differential effect on the fluxes of
the He~II and N~V lines. Luckily, for a typical extinction of $E_{B-V}\leq
0.1$ (\cite{ver87}), the ratio of the measured fluxes of these lines is only
$\leq 18$\% higher than intrinsic ratio. As we shall see, this correction,
while systematic, is much smaller than either the typical measurement errors,
or the typical dispersion in the flux ratios of any given source.
Of the many possibilities, we choose to plot the ratios N~V $\lambda
1240$/C~IV $\lambda 1550$ and He~II $\lambda 1640$/C~IV $\lambda 1550$ vs.\
Si~IV $\lambda 1400$/C~IV $\lambda 1550$ (Figs.~1--8). In what follows, we
often refer to these ratios simply as N~V/C~IV and He~II/C~IV vs.\ Si~IV/C~IV.
It should be emphasized that the identifications of these lines are not unique;
at $\Delta\lambda\approx 5$~\AA , the {\it IUE\/} resolution is not sufficient
to exclude several other possible lines in the vicinity of these wavelengths.
We defer a discussion of such potential confusion, and of the physical
motivation for our choice of lines until the following section, and begin by
presenting the phenomenological behavior of these ratios as derived from the
data. We begin by discussing the various members of the various CV subclasses
and then discuss the sources which do not appear to fit the pattern established
by the other members of their subclass (sources we enjoy referring to as
``weird'').
\subsection{Results Arranged by Subclass}
\subsubsection{AM~Her stars: AM~Her, V834~Cen, and QQ~Vul}
Figure~1 shows that AM~Her, V834~Cen, and QQ~Vul form a sequence of increasing
N~V/C~IV with increasing Si~IV/C~IV. He~II/C~IV is reasonable constant at
$\approx 0.2\pm 0.1$.
\subsubsection{DQ~Her stars: EX~Hya, TV~Col, FO~Aqr, and DQ~Her}
Among these DQ~Her stars, TV~Col displays the greatest amount of variability
in its continuum and line fluxes, and, as is evident in Figure~2, in its UV
line ratios. The upper right portion of the observed range of line ratios
is populated by the observations obtained during the optical and UV flare
observed by Szkody \& Mateo (1984); during the flare, both N~V and He~II
increased relative to C~IV. The relatively tight phase space occupied by the
line ratios of FO~Aqr is approximately the same as that of TV~Col in outburst.
The ionization state of the line-emitting gas in EX~Hya and DQ~Her is lower
than the two other DQ~Hers, as evidenced by their higher Si~IV/C~IV line
ratios and the lower He~II/C~IV line ratios.
\subsubsection{Dwarf Novae in Quiescence: SS~Cyg, SU~UMa, RU~Peg, and WX~Hya}
As shown in Figure~3, WX~Hyi has the largest ratios of N~V/C~IV and Si~IV/C~IV
among all the AM~Her stars, DQ~Her stars, and other dwarf novae in quiescence.
SU~UMa and RU~Peg have N~V/C~IV ratios of $\approx 0.3$, but the value for
SS~Cyg is $\approx 30\%$ lower at $\approx 0.2$.
\subsubsection{Eclipsing Dwarf Novae: Z~Cha and OY~Car}
As shown in Figure~4, relative to all other ``normal'' CVs, Z~Cha and OY~Car
both have strong N~V relative to C~IV. Almost all of the spectra of Z~Cha
were obtained during the peak and decline of the superoutburst of 1987 April
(\cite{har2a}) and of a normal outburst of 1988 January (\cite{har2b}).
The evolution of the spectrum during these observations is manifest most
clearly in the Si~IV/C~IV ratio, which was highest during the decline of the
superoutburst ($V\approx 12.3$--12.7) and lowest during the decline of the
normal outburst ($V\approx 13.3$--14.0). During these intervals, both the
N~V/C~IV and He~II/C~IV ratios remained roughly constant, though there is
some indication that the He~II/C~IV ratio increased as the Si~IV/C~IV ratio
decreased. The spectra of OY~Car were obtained during the decline of the
superoutburst of 1985 May (\cite{nay88}) and produce flux ratios which
roughly equal those of Z~Cha in superoutburst.
\subsubsection{Eclipsing Nova-like Variables: UX~UMa, V347~Pup, and RW~Tri}
As shown in Figure~5, the eclipsing nova-like variables UX~UMa, V347~Pup,
and RW~Tri have moderately strong N~V relative to C~IV, with N~V/C~IV ratios
intermediate between those of dwarf novae in quiescence and eclipsing dwarf
novae. The phase space occupied by V347~Pup is amazingly tight, and is disturbed
only by eclipse effects: the single discrepant Si~IV/C~IV ratio was obtained
from the eclipse spectrum of this source, and is due to that fact that the
C~IV emission line is eclipsed less than the other emission lines. The cause
of the large spread in line ratios of UX~UMa appears to due to a different
mechanism: there are epochs when Si~IV is weak relative to C~IV, and other
epochs when it is reasonably strong. These epochs do not seem to correlate
with the continuum flux. The He~II/Si~IV ratio of UX~UMa is typically less
than that of V347~Pup.
\subsubsection{Intercomparison of Various Subclasses}
Figure~6 combines the line ratios of the objects from the previous figures
grouped together into magnetic (upper panels) and non-magnetic (lower panels)
systems. It is apparent from this figure and the previous discussion that the
line ratios of the various subclasses are rather homogeneous, with a dispersion
of $\sim 1$ decade. This dispersion to be due to almost equal contributions
from the dispersion of values from the various observations of individual
objects ($\sim 0.5$ decade) and from the dispersion from one object to another
($\sim 0.5$ decade).
There are clear and significant differences between these two subclasses: (i)
He~II/C~IV is $\sim 0.25$ decades larger in magnetic systems; (ii) Si~IV/C~IV
is slightly larger in the non-magnetic systems, but the difference is small
compared to the dispersion; (iii) The upper limit on the N~V/C~IV distribution
is greater in the non-magnetic systems, while the lower limit of the
distribution is similar in the two cases.
It is also instructive to consider the differences within these two subclasses.
The greater range of the N~V/C~IV distribution is due to the eclipsing dwarf
novae Z~Cha and OY~Car (squares and circles, respectively, in the lower panels
of Fig.~6); otherwise the N~V/C~IV distributions of magnetic and non-magnetic
systems are similar. Among the magnetic systems, TV Col (triangles in the upper
panels of Fig.~6) has significantly lower Si~IV/C~IV than any other object; the
Si~IV/C~IV distribution would be much tighter if this object were excluded from
the sample.
\subsection{Systems Showing Anomalous Behavior: ``Weird'' CVs}
\subsubsection{GK Per}
GK~Per is unusual among CVs in a number of respects. It has an evolved secondary
and hence a very long orbital period and a large accretion disk. Unlike most
CVs, the UV continuum of this DQ~Her-type system peaks in the {\it IUE\/}
bandpass. The cause of this anomaly is thought to be due to the disruption
of the inner disk by the magnetic field of the white dwarf (\cite{bia83};
\cite{wu89}; \cite{mau90}; \cite{kim92}). As shown in Figure~7, GK~Per also
distinguishes itself from other CVs in its anomalously large He~II/C~IV ratio.
In outburst, both He~II and N~V are stronger than C~IV, whereas the opposite is
true in quiescence. While the Si~IV/C~IV ratio spans a rather broad range of
$\approx 0.2$--1, this range is present in both outburst and quiescence.
\subsubsection{V~Sge}
V~Sge is an unusual nova-like variable which occasionally manifests
brightenings of as much as 3 mag (\cite{her65}). The model for this system is
highly uncertain; both white dwarf and neutron star binary models have been
considered (\cite{koc86}; \cite{wil86}). The {\it IUE\/} spectra of V~Sge
obtained during 1978 and 1979 are characterized by strong He~II and N~V emission
lines relative to C~IV (\cite{koc86}): N~V/C~IV $\approx 2.5$ and He~II/C~IV
$\approx 3$. These distinguishing line ratios are suppressed in outburst. In
1985 August, the UV continuum was enhanced by factor of $\approx 2$ relative
to the earlier spectra, while the N~V/C~IV ratio fell to $\approx 0.4$ and the
He~II/C~IV ratio fell to $\approx 1$. In 1985 April, the UV continuum was
enhanced by factor of $\approx 4$, while the N~V/C~IV ratio fell to $\approx
0.15$; measurements of the He~II/C~IV ratio are not possible during this epoch
because the He~II emission line was overexposed. This behavior is opposite to
that of GK~Per, whose N~V/C~IV and He~II/C~IV ratios increased mildly in
outburst.
\subsubsection{BY~Cam}
BY~Cam (H~0538+608) is one of only three AM~Her stars above the period gap
(V1500 Cyg and AM~Her being the other two) and one of only two which rotates
asynchronously (V1500 Cyg being the other one). As pointed out by
Bonnet-Bidaud \& Mouchet (1987), BY~Cam is unusual in having weak C~IV and
strong N~V. Figure~7 shows that all the line ratios are large because C~IV
is weak.
\subsubsection{AE~Aqr}
AE~Aqr exhibits a host of bizarre phenomena including variable flare-like
radio and optical emission; rapid, coherent oscillations; and QPOs. Its UV
spectrum has been discussed by Jameson et~al.\ (1980) and is distinguished
by the near-absence of C~IV. Unfortunately, the flux measurements of the C~IV
emission line are very uncertain because this line is so weak and because the
measurements are possibly contaminated by the Si~II 3p--4s multiplet at 1526.71,
1533.43~\AA \ (e.g., \cite{kel87}). Nevertheless, it is clear that N~V and
Si~IV have similar strengths and are $\approx 10$ times stronger than He~II
and C~IV, which have similar strengths.
\section{Line Ratio Modeling}
We now present some simple models for the systematic behavior we expect from CV
line ratios. Although CVs as a class share a number of fundamental properties,
they also differ between subclasses in the likely properties of the gas
responsible for line emission. For example, in disk-fed systems, such as
nova-like variables and dwarf novae, a possible site for line emission is the
disk atmosphere. In this case, the atmosphere may be heated either by viscosity
(e.g., \cite{sha91}) or by photons from other parts of the disk or from the
vicinity of the white dwarf (e.g., \cite{ko96}). A wind is also present in
nova-like variables and dwarf novae in outburst (see \cite{mau6b} for a recent
review of CV winds). In AM~Her stars, the line emission may come from the stream
of material being transferred from the secondary to the white dwarf which is
heated either mechanically or by photons from the shock at the white dwarf
surface. DQ~Her stars may have emission from either or both of these sites.
Beyond these simple considerations, there is considerable uncertainty about the
nature of the emission region: the geometrical arrangement of the gas relative
to the source of ionizing photons (if present), the density and optical depth
of the gas, the elemental abundances of the gas, the spectrum and flux of the
ionizing radiation, and the presence and distribution of any other sources of
heating. These quantities are not only uncertain, they may differ from one
object to another in a subclass. Furthermore, all subclasses contain anomalous
members. We therefore consider models for CV emission regions which employ the
very simplest assumptions, namely, slab models each with a single ionizing
spectrum, gas composition, and density. We then explore plausible choices for
photon flux, column density, and mechanical heating rate (if any). The results
serve not only as a test of the validity of the models when compared to the
observations, but also provide insight into the physics of the line emission
which will hopefully remain useful when more detailed models are considered.
Further support for this strategy comes from the results of the previous
section: the objects of a given subclass show significant clustering in the
two line ratio diagrams, and there are clear differences from one subclass
to another. Therefore, we can define our modeling goals as follows: to test
whether one or more sets of models can reproduce the mean values of the observed
line ratios for the various subclasses of objects, and to test whether any
models can reproduce the dispersion or systematic variability of the observed
line ratios. Given the uncertainties of the models and the observed dispersion
in line ratios, we will consider agreement between the models and observations
to be adequate if the two agree to within the dispersion of the observed values,
i.e., approximately one decade. We will show that although this is clearly a
very crude criterion, it turns out to be very constraining for the models.
\subsection{Model Ingredients}
Although the mechanism of line emission is uncertain, we favor photoionization
as the source of ionization, excitation, and heating of the gas. This is partly
because the observed continuum spectra, and their extrapolation into the
unobservable spectral regions, provide a convenient and plausible energy source.
Further support for this idea comes from Jameson et~al.\ (1980) and King et~al.\
(1983) who compared the observed line strengths from AE Aqr and UX UMa to
simple predictions of both photoionized and coronal models and found the coronal
models inconsistent with the observed lines. Nevertheless, for the sake of
completeness, we examine both photoionized and coronal models.
The models are calculated using the XSTAR v1.19 photoionization code
(\cite{kal82}; \cite{kal93}). The models consist of a spherical shell of gas
with a point source of continuum radiation at the center; this may be used to
represent a slab in the limit that the shell thickness is small compared with
the radius. In what follows we will use the line fluxes emergent from the
illuminated face of the slab, which is equivalent to those emitted into the
interior of the spherical shell. The input parameters include the source
spectrum, the gas composition and density, the initial ionization parameter
(determining the initial radius, see below for a definition), and the column
density of the shell (determining the outer radius). Construction of a model
consists of the simultaneous determination of the state of the gas and the
radiation field as a function of distance from the source. The state of the
gas at each radius follows from the assumption of a stationary local balance
between heating and cooling and between ionization and recombination.
When the gas is optically thin, the radiation field at each radius is determined
simply by geometrical dilution of the given source spectrum. Then, as shown
by Tarter et~al.\ (1969), the state of the gas depends only on the ionization
parameter $\xi$, which is proportional to the ratio of the radiation flux to the
gas density. We adopt the definition of the ionization parameter used by Tarter
et~al.: $\xi=L/(nr^2)$, where $L$ is the ionizing energy luminosity of the
central source (between 1 and 1000 Ry), $n$ is the gas density, and $r$ is the
distance from the source. This scaling law allows the results of one model
calculation to be applied to a wide variety of situations. For a given choice of
spectral shape, this parameter is proportional to the various other customary
ionization parameter definitions: $U_H=F_H/n$, where $F_H$ is the incident
photon number flux above the hydrogen Lyman limit; to $\Gamma=
F_\nu(\nu_L)/(2hcn)$, where $F_\nu(\nu_L)$ is incident energy flux at the Lyman
limit; and to $\Xi=L/(4\pi R^2 cnkT)$. This simple picture breaks down when the
cloud optical depth is non-negligible, since the source spectrum then depends
on position, and the escape of cooling radiation in lines and recombination
continua depends on the total column density of each ion species and hence on
the ionization state of the gas throughout the cloud. In addition, the rates for
cooling and line emission can depend on gas density owing to the density
dependence of line collisional deexcitation and dielectronic recombination.
However, even in this case the ionization parameter remains a convenient means
of characterizing the results.
The state of the gas is defined by its temperature and by the ion abundances.
All ions are predominantly in the ground state, and except for hydrogen and
helium the populations of excited levels may be neglected. The relative
abundances of the ions of a given element are found by solving the ionization
equilibrium equations under the assumption of local balance, subject to the
constraint of particle number conservation for each element. Ionization balance
is affected by a variety of physical processes, most notably photoionization
and radiative and dielectronic recombination. The temperature is found by
solving the equation of thermal equilibrium, by equating the net heating of the
gas due to absorption of incident radiation with cooling due to emission by
the gas. These rates are derived from integrals over the absorbed and emitted
radiation spectra. Although Compton scattering is not explicitly included as a
source or sink of radiation, its effect is included in the calculation of the
thermal balance.
The emitted spectrum includes continuum emission by bremsstrahlung and
recombination and line emission by a variety of processes including
recombination, collisions, and fluorescence following inner shell
photoionization. Line transfer is treated using an escape probability
formalism and includes the effects of line destruction by collisions and
continuum absorption. Transfer of the continuum is calculated using a
single stream approximation, as described in Kallman \& McCray (1982).
Rates for atomic processes involving electron collisions have been modified
since the publication of Kallman \& McCray (1982) to be consistent with
those used by Raymond \& Smith (1977). Recombination and ionization rates for
Fe have been updated to those of Arnaud \& Raymond (1992). In addition, we
have added many optical and UV lines from ions of medium-Z elements (C, N, O,
Ne, Si, and S) using collisional and radiative rates from Mendoza (1983). The
elements Mg, Ar, Ca, and Ni have also been added. The models have a total of
168 ions, producing 1715 lines, of which 665 have energies greater than 120 eV
(10 \AA ), and approximately 800 are resonance lines. For each ion we also
calculate the emission from radiative recombination onto all the excited levels
which produce resonance lines. The number of such continua is equal to the
number of resonance lines in the calculation. Some of the results of the models
are sensitive to the rates for dielectronic recombination. We use the high
temperature rates given by Aldrovandi \& Pequignot (1973), together with the
low temperature rates from Nussbaumer \& Storey (1983). These rates differ
significantly for several relevant ions, e.g., Si~III, relative to those of
Shull \& Van Steenberg (1983), which were used in most earlier versions of
XSTAR. Finally, all models assume element abundances which are close to solar:
H:He:C:N:O:Ne:Mg:Si:S:Ar:Ca:Fe:Ni =
1:0.1:3.7E-4:1.1E-4:6.8E-4:2.8E-5:3.5E-5:3.5E-5:1.6E-5:4.5E-06:2.1E-6:2.5E-5:2.0E-6
(\cite{wit71}).
An emission line may be produced by two types of physical processes. First is
what we will call ``thermal emission,'' which is recombination or collisional
excitation by thermal electrons. Second is excitation by photons from the
incident continuum. Since this is an elastic scattering process, we will refer
to it as ``scattering.'' This process will produce an apparent emission feature
if our line of sight to the continuum source is at least partially blocked, and
the scattering region must be concentrated on the plane of the sky (see
\cite{kro95} for more discussion of these issues). If the scattering region has
a bulk velocity greater than the thermal line width then a P~Cygni or inverse
P~Cygni profile will form even if the scattering region is spherical. Scattering
fails to account for the presence of He~II $\lambda$1640, since this is a
subordinate line and the population of the lower level will be negligibly small
under conditions appropriate to photoionized gas. Significant opacity in this
line would require a temperature greater than $\sim 3\times 10^5$ K and
level populations which are close to LTE values. In spite of this, we have
investigated the possibility that this mechanism can explain the ratios of the
other lines we consider. We assume that the fluxes in scattered lines are
proportional to the opacities in the lines. This is likely to be justified if
the line optical depths are less than unity, and if the wind ionization balance
is approximately uniform. If so, the line flux ratios will equal the opacity
ratios.
\subsection{Input Parameters}
The models we explore fall into two categories: photoionized models and
collisional (coronal) models. For each we present the line ratios for both
thermal emission and scattering emission mechanisms. The photoionization model
parameters are chosen in an attempt to crudely represent the range of observed
ionizing spectra from CVs. These typically consist of a soft component which is
consistent with a 10--50 eV blackbody, together with a hard X-ray component such
as a 10 keV bremsstrahlung (e.g., \cite{cor2a}). The ratio of strengths of these
two components varies from one object to another, and with outburst state and
subclass. However, a typical ratio is 100:1 (soft:hard) for non-magnetic systems,
and 1:1 for magnetic systems (\cite{cor2b}). Beyond such simple considerations,
the detailed shape of the ionizing radiation field is difficult to determine
accurately. This is due to the strong influence of photoelectric absorption by
interstellar and circumstellar gas, and to the limited bandpass and spectral
resolution of most past observations in the soft X-ray band (e.g., \cite{ram94};
\cite{mau6a}). Owing to such uncertainties we choose a few very simple ionizing
spectra for consideration in our photoionization models: Model A: 30 eV
blackbody, Model B: 10 keV brems, Model C: 50 eV blackbody, Model D: 10 eV
blackbody. Model G is the mechanically heated (coronal) ionization model. All
these models have a total (neutral + ionized) hydrogen column density of $N_{\rm
H}=10^{19}~\rm cm^{-2}$, chosen to make them optically thin to the continuum and
effectively thin to the escaping resonance lines. In addition, we present two
photoionized models which are close to being optically thick, both of which have
of column density $N_{\rm H}=10^{23}~\rm cm^{-2}$. Model E has a 30 eV blackbody
ionizing spectrum, and Model F has a 10 keV brems ionizing spectrum. For each
model we determine the net line flux contained in the wavelength intervals
1237--1243~\AA , 1390--1403~\AA , 1547--1551~\AA , and 1639--1641~\AA , which we
refer to as N~V, Si~IV, C~IV, and He~II, respectively. As we will show, these
wavelength intervals also contain other lines which can mimic these strong
lines, and these other lines are likely to affect the interpretation of the {\it
IUE\/} low resolution data as well. The results of our models---the Si~IV/C~IV,
N~V/C~IV, and He~II/C~IV line ratios---are summarized in Tables 1--3. For the
photoionization models, we consider various values of the ionization parameter
for six choices of ionizing continuum shape. For the collisional models, we
consider ten values of the gas temperature. We also tested for the dependence on
gas density and found that it is negligible for the conditions we consider; all
the models presented here have density $n=10^9~\rm cm^{-3}$.
\subsection{Model Results}
When the line ratios shown in Tables 1--3 are plotted in two diagrams in the
same way as the {\it IUE\/} data, several common properties emerge. These are
shown in Figures 9 and 10, with various symbols denoting the ionizing spectra:
Model A \vrule height6pt width5pt depth-1pt, Model B = $\bullet $, Model C =
$\times $, Model D = +, Model E = $\Box $, Model F = $\circ $. We have also
tried a model consisting of a 30 eV blackbody together with a 10 keV brems
spectrum in a ratio of 99:1; the results are so similar to Model A as to be
indistinguishable, i.e., the 30 eV blackbody has far more influence on the
ionization balance than does the 10 keV brems in these ratios. Model G, the
coronal case, produces line ratio combinations which are almost entirely outside
the range spanned by these figures. For the other spectra, as expected, there is
little or no dependence on model density for optically thin photoionized models.
The shape of the trajectory of the model results in these planes is similar
for most of the models. They resemble a {\sf U} shape, although in some cases
the right upright of the {\sf U} is missing, and in others it is tipped nearly
45 degrees to the vertical. We can understand the results better if we label
the points along the {\sf U} as follows: A = upper left extreme of trajectory;
B = bottom of steepest part of left upright; C = lowest point of trajectory; D =
bottom of right upright; E = upper right extreme of trajectory. These are shown
schematically in Figure~11. In general, the trajectory is traversed from point A
to point E as the ionization parameter decreases, and may be understood in terms
of the relative ease of ionization of the various ions responsible for line
emission. The ionization parameter at which the abundance of a given ion peaks,
relative to its parent element, can be derived crudely from the ionization
potential. Thus, the four ions responsible for the strongest observed lines may
be ordered in terms of decreasing ease of ionization according to: Si~IV, He~II,
C~IV, N~V. For the metal ions the ionization parameter at which the emissivities
of the lines peak is approximately the same as the ionization parameter at which
the ion abundances peak. Thus, at the highest ionization parameters we consider,
nitrogen is ionized to or beyond N~V, carbon is ionized beyond C~IV, and silicon
is ionized beyond Si~IV. However, the abundance of N~V, and hence its line
emissivity, is greater than the corresponding quantities for C~IV, which in
turn are greater than for Si~IV. Thus, at high ionization parameter, N~V/C~IV
is relatively large and Si~IV/C~IV is small, corresponding to point A of the
trajectory in the N~V/C~IV vs.\ Si~IV/C~IV plane. At intermediate ionization
parameter, carbon recombines to C~IV and nitrogen recombines below N~V, so that
N V/C~IV is small and Si~IV/C~IV is also small (point B). At lower ionization
parameter, silicon recombines to Si~IV and carbon may recombine to below C~IV,
resulting in an intermediate value of Si~IV/C~IV (point C). At very low
ionization parameter, Si~IV/C~IV is at a maximum, and there is an apparent
increase in N~V/C~IV (points D, E). This cannot be understood in terms of
conventional ionization balance, since the C~IV abundance always exceeds the N~V
abundance at low ionization parameter. Rather, it is due to confusion between
the N~V 2s--2p doublet at 1238.82, 1242.80~\AA \ and the Mg~II 3s--4p doublet at
1239.93, 1240.39~\AA \ (e.g., \cite{kel87}). Since Mg~II has a lower ionization
potential than any of the other ions in question, Mg~II/C~IV increases at low
ionization parameter, thus explaining the apparent increase in N~V/C~IV .
The behavior of He~II $\lambda$1640 differs from the other lines owing to
the fact that it is emitted by recombination, while the others are emitted by
collisional excitation. Thus, the emissivity of He~II $\lambda$1640 remains
nearly constant at high ionization parameter, while the C~IV $\lambda$1550 line
emissivity decreases with increasing ionization parameter. This is in spite of
the fact that He~II is more easily ionized than C~IV. This explains the AB
part of the trajectory in the He~II/C~IV vs.\ Si~IV/C~IV plane. As the
ionization parameter decreases, C~IV recombines to C~III and below, while He~II
and He~III are still abundant, thus explaining the DE part of the trajectory.
The model behavior under the scattering scenario is qualitatively similar to
the thermal excitation scenario, except for the weakness of He~II. The line
strengths in the scattering case are less dependent on the gas temperature
than in the thermal excitation case, and the oscillator strength for the Mg~II
3s--4p transition is small, so that the trajectory lacks the DE segment in the
N~V/C~IV vs.\ Si~IV/C~IV plane.
The fact that the coronal models fail almost completely to produce line ratios
within the range of our diagrams indicates that it is unlikely that this
process dominates in CV line-emitting gas. This result is not surprising, owing
to the well-known fact that coronal equilibrium produces ion abundances that
have less overlap in parameter space (e.g., temperature) between adjacent ion
stages than does photoionization.
\subsection{Spectral Dependence}
The shape of the ionizing spectrum influences the location of the various
points along the trajectory in the two line ratio diagrams, and also the
existence of part of the trajectory, most notably the segments between points
C and E.
For soft spectra, such as the 30 eV blackbody, recombination to species below
Si~IV does not occur for the parameter range considered ($\log\xi =-1.5$ to
+1.0), so the CDE part of the trajectory is absent in the N~V/C~IV vs.\
Si~IV/C~IV plane. Also, when the ionization parameter is suitable for producing
large N~V/C~IV, the Si~IV/C~IV ratio is so small as to be off the scale of
Figure~9.
For very soft spectra, such as the 10 eV blackbody, there are insufficient hard
photons to produce N~V. Thus, points A and B are missing in the N~V/C~IV vs.\
Si~IV/C~IV plane. The DE part of the trajectory is entirely due to Mg~II
$\lambda1240$.
For hard spectra, such as the 10 keV brems spectrum, X-rays can make Si~IV via
Auger ionization even at very low ionization parameter. Thus, the range of
Si~IV/C~IV is greatly expanded, and can reach 10 at point D in the N~V/C~IV
vs.\ Si~IV/C~IV plane. Other models, such as a 30 eV blackbody, can make large
Si~IV/C~IV at low ionization parameter, but they also have very weak N~V line
emission, so that the 1240~\AA \ feature is dominated by Mg~II and they are on
DE segment of the trajectory.
At high ionization parameter He~II is a ``bolometer'' of the ionizing spectrum,
since it is dominated by recombination. That is, the He~II strength depends only
on the number of photons in the He~II Lyman continuum ($\varepsilon\geq 54.4$
eV). So, harder spectra make stronger He~II, and conversely. This behavior may
still hold at point C. At small ionization parameter, He~II is likely to be more
abundant than C~IV or Si~IV. This explains the CDE segment of the trajectory
in the He~II/C~IV vs.\ Si~IV/C~IV plane. This fact does not seem to depend
strongly on spectral shape, although the value of Si~IV/C~IV at points C, D,
and E does depend on the spectrum; this ratio increases at all these points for
harder spectra, and conversely.
The 10 eV blackbody models show a ``hook'' in their trajectory in which the
Si~IV/C~IV appears to increase with increasing ionization parameter at the upper
end of the range of values we consider. This is counter to the expected behavior
of Si~IV at high ionization parameter, since we expect silicon to become ionized
beyond Si~IV and the Si~IV abundance to decrease at high ionization parameter.
The reason for the model behavior is the confusion between the Si~IV doublet
at 1393.76, 1402.77~\AA \ and the O~IV $\rm 2s^22p$--$\rm 2s2p^2$ multiplet
at 1397--1407~\AA \ (e.g., \cite{kel87}). The unique behavior of the 10 eV
blackbody models is due to the fact that the O~IV lines increase at high
ionization parameter only for the softest spectra; harder spectra ionize oxygen
past O~IV when other ion abundances are at similar values. Like the N~V and
Mg~II lines near 1240~\AA , the Si~IV and O~IV lines near 1400~\AA \ can be
confused in {\it IUE\/} low resolution data.
The overlap in ionization parameter space of the regions where N~V, Si~IV,
C~IV, and He~II predominate does not differ greatly between models with hard
X-ray spectra (e.g., 10 keV brems) and those with blackbodies with $kT>30$ eV.
A more pronounced difference is due to the fact that the latter spectra have
all their ionizing photons crammed into a smaller energy range. Therefore, the
ionization parameter scale, which simply counts the energy in ionizing photons,
and the distribution of ionization, which really depends on the photon density
in the EUV/soft X-ray region, are very different in the two cases. For example,
in the 10 keV brems case, C~IV predominates at $\log\xi =0$, while in the 30 eV
blackbody case it predominates near $\log\xi=-1$. Thus, our model grid, which
spans the range $-3< \log\xi < +1.5$, does not include the region where the gas
has recombined below C~IV, etc., in the blackbody case. This accounts for the
absence of the CDE part of the line ratio trajectory for the blackbody models.
\section{Comparison with Observations}
The models presented in the previous section provide a useful context in which
to examine the likely physical conditions in CV line-emitting regions. As was
discussed earlier, we do not expect these simple models to account for the
details of the observed spectra, but we do hope that they will at least crudely
reproduce some of the features of the observations. These might include: the
range of the observed ratios to within the dispersion of the observed values,
i.e., approximately one decade, and possibly the differential behavior of a
given source as it varies in time. In fact, we find little evidence for
agreement between the observations and any of the models beyond the simplest
measures of consistency for some of the line ratios. We begin the comparison of
model results with observations by considering the ``normal'' CVs.
\subsection{``Normal'' CVs}
The observed ratios of ``normal'' CVs lumped together regardless of class
(Fig.~6) are clustered within a range of $\sim 1$ decade for log Si~IV/C~IV
$\approx -0.5$ and log He~II/C~IV $\approx -1.0$ and $\sim 1.5$ decades for log
N~V/C~IV $\approx -0.25$. The larger range of the N~V/C~IV ratio is due largely
to the large line ratios of the eclipsing dwarf novae Z~Cha and OY~Car (squares
and circles, respectively, in Fig.~6). Otherwise, the range is $\sim 1$ decade
centered on $\approx -0.5$. This same general range is spanned by the models,
but there is little detailed agreement.
One notable failure of the models is the behavior in the He~II/C~IV vs.\
Si~IV/C~IV plane. The photoionization models always produce He~II/C~IV $\gax$
Si~IV/C~IV. This is because, although the He~II line is emitted following
radiative recombination and the C~IV and Si~IV lines are formed by collisional
excitation, photons at energies greater than 54.4 eV which are responsible
for ionizing He~II are also responsible for heating the gas. So, models which
efficiently heat the gas and emit Si~IV and C~IV also have efficient ionization
of He~II and hence efficient production of the 1640~\AA \ line. In contrast,
``normal'' CVs show He~II/C~IV values which are less than Si~IV/C~IV by $\sim
0.5$ decades. Magnetic systems have He~II/C~IV line ratios which are
systematically higher (and Si~IV/C~IV line ratios which are systematically
lower) than those of the non-magnetic systems by $\sim 0.25$ decades. Only the
DQ~Her stars TV~Col and FO~Aqr (triangles and pentagons, respectively, in
Fig.~6) have He~II/C~IV $\geq$ Si~IV/C~IV.
It is interesting to note that the ``hook'' in the trajectory of the 10 eV
blackbody models referred to in the previous section occurs near the values of
these line ratios where most observed objects cluster. Furthermore, the 10 eV
models come closest to reproducing the observed He~II/C~IV ratios; they are
the only ones for which He~II/C~IV falls below Si~IV/C~IV.
The coronal models produce He~II/C~IV values of order 1\% of Si~IV/C~IV when
this latter ratio is in the observed range; even at this relatively favorable
point, the N~V/C~IV ratio is much smaller than observed. This suggests that
the coronal emission mechanism is less likely than photoionization for all CVs.
In the N~V/C~IV vs.\ Si~IV/C~IV plane, the models and the data span a similar
range of ratios, so there is less indication of the failure of any of the
models. The 50 eV blackbody models are least successful at reproducing the
most commonly observed values of these ratios simultaneously. The 30 eV
blackbody and 10 keV brems models both span the observed range, as do the
optically thick models (which also use these ionizing spectra). The ``hook''
in the trajectory of the 10 eV blackbody models causes the Si~IV/C~IV ratio to
lie almost entirely in the range $-1.5\leq $ log Si~IV/C~IV $\leq -0.5$, which
is close to that spanned by the observed ratios of most ``normal'' CVs.
In addition to asking whether there exists a model which can reproduce a given
ratio, we can ask whether the distribution of observed ratios is consistent with
the analogous model quantity. For example, if the ionizing spectrum and emission
mechanism are independent of time, but the luminosity and hence the ionization
parameter varies with time, we expect that the ratios of a given object will
lie along a {\sf U}-shaped trajectory in the line ratio diagram. In contrast,
there appears no clear pattern in the observed ratios for objects with many
observations, other than a clustering in a well-defined region of the diagram.
\subsection{``Weird'' CVs}
Consider next the ``weird'' CVs. Figures~7 and 8 demonstrate that V~Sge, GK~Per,
BY~Cam, and AE~Aqr form a sequence of dramatically increasing Si~IV/C~IV and
N~V/C~IV at nearly constant He~II/C~IV. In V~Sge and GK~Per, He~II/C~IV $>$
Si~IV/C~IV, unlike most of the ``normal'' CVs, but consistent with the
photoionization models. The extreme line ratios of BY~Cam and AE~Aqr are harder
to understand. Bonnet-Bidaud \& Mouchet (1987) have suggested a depletion of
carbon by a factor of $\sim 10$ to explain the anomalous line ratios of BY~Cam.
If a similar deficiency applies to AE~Aqr, depletion by a factor of $\sim 60$
is required. Although such abundance anomalies are possible, they are unlikely
to explain the positive correlation between N~V/C~IV and Si~IV/C~IV observed
in AE~Aqr and possibly BY~Cam (see Figs.~7 and 8). The observed correlation
between these ratios lends support to the hypothesis that confusion between
the N~V 2s--2p doublet and the Mg~II 3s--4p doublet, together with a low value
of the ionization parameter and hence a large value of the Si~IV/C~IV ratio,
is responsible for the apparent anomalous line ratios of BY~Cam and AE~Aqr.
However, none of the models reproduce the nearly perfect linear proportionality
between N~V/C~IV and Si~IV/C~IV observed in AE~Aqr.
There are several possibilities why the models and the observed line ratios
are discrepant. First, it is possible that we have failed to consider ionizing
spectra of the right type. Since the 10 eV blackbody appears to come closest to
providing agreement with the He~II/C~IV and Si~IV/C~IV ratios simultaneously,
it is possible that there are confusing lines which we have not included in
our models, or which become important under other conditions, which affect the
results. Alternatively, the emission region may consist of multiple components
with differing physical conditions. If so, the various components must have line
ratios which bracket the observed values. This is ruled out by our models: no
single set of models brackets the observed ratio of He~II/C~IV, for example. The
observed values of this ratio are bracketed by the photoionization models on
the high side and the coronal models on the low side, so that a superposition
of these models might provide consistent line ratios. However, we consider this
possibility to be somewhat contrived, and a more detailed exploration is needed
to test whether it can account for all the ratios simultaneously. Another
possible explanation for the He~II/C~IV ratio is that our assumption of a
stationary steady state is invalid for the line-emitting region. If, for
example, the heating and ionization of the gas occurs as the result of many
impulsive events, then the time-average value of the ratios could differ
significantly from the steady-state model predictions owing to the differing
timescales for relaxation of the upper levels of the He~II line from that of
C~IV. Such a scenario has been suggested to account for the strength of the
He~II lines from the Sun (\cite{ray90}). The differential behavior of the line
ratios from a given object could be due at least in part to changes in the
ionizing spectrum or ionization mechanism, rather than simply due to changes
solely in ionization parameter. This could account for the departures from the
variability behavior predicted by the models. In spite of the difficulty in
reproducing the observed line ratios, steady-state photoionization models are
capable of fitting the absolute strengths of the observed lines (Ko et~al.\
1996).
\section{Summary}
We have presented a statistical analysis of the Si~IV/C~IV, N~V/C~IV, and
He~II/C~IV emission line ratios of 20 CVs based on $\approx 430$ UV spectra
extracted from the {\it IUE\/} ULDA. We find for most systems that these ratios
are clustered within a range of $\sim 1$ decade for log Si~IV/C~IV $\approx
-0.5$ and log He~II/C~IV $\approx -1.0$ and $\sim 1.5$ decades for log N~V/C~IV
$\approx -0.25$. The larger range of the N~V/C~IV ratio is due largely to the
large line ratios of the eclipsing dwarf novae Z~Cha and OY~Car; otherwise, the
range of log N~V/C~IV is $\sim 1$ decade centered on $\approx -0.5$. The
clearest difference between magnetic and non-magnetic CVs is the He~II/C~IV
ratio, which is $\sim 0.25$ decades larger in magnetic systems.
To place constraints on the excitation mechanism and the physical conditions
of the line-emitting gas of CVs, we have investigated the theoretical line
ratios of gas in either photoionization and collisional ionization equilibrium.
Given the uncertain and variable geometry, density, optical depth of the
line-emitting gas and the shape and luminosity of the ionizing spectrum, we
have restricted ourselves to consideration of simple slab models each with fixed
gas composition, density, and column density. The variables have been the shape
and ionization parameter of the ionizing spectrum and the density and column
density of the slab; for the collisional models, the temperature was varied.
Line emission is produced in these models by recombination or collisional
excitation by thermal electrons or by excitation by the ionizing continuum
(``scattering'').
Within the confines of these simple models, we find little agreement between the
observations and any of the models. Specifically, the observed Si~IV/C~IV line
ratios are reproduced by many of the models, but the predicted N~V/C~IV line
ratios are simultaneously too low by typically $\sim 0.5$ decades. Worse, for no
parameters are any of the models able to reproduce the observed He~II/C~IV line
ratios; this ratio is far too small in the collisional and scattering models and
too large by typically $\sim 0.5$ decades in the photoionization models. Among
the latter, the 10 eV blackbody models do the best job of reproducing the
three line ratios simultaneously, but the match with the N~V/C~IV line ratio is
accomplished only if the observed emission feature near 1240~\AA \ is due to the
Mg~II 3s--4p doublet at 1239.93, 1240.39~\AA \ instead of the N~V 2s--2p doublet
at 1238.82, 1242.80~\AA .
Despite the above generally unfavorable comparisons between observations and
simple photoionization and collisional models, our investigation has proven
useful in revealing both the problems and promises of understanding the UV line
ratios of CVs. Future detailed work could be profitably performed on any and
all of the above CV subclasses with more detail in the shape of the ionizing
spectrum and the geometrical distribution, density, and column density of the
emission region(s). Where the distance is well known, not only line ratios but
absolute line strengths can be fit. With larger effective area, weaker lines,
less subject to optical depth effects, can be included. With higher spectral
resolution, the UV lines be can uniquely identified, thus removing the annoying
ambiguity of some of the line identifications. Additional constraints on the
physical conditions and optical depth of the line-emitting gas is possible if
the UV doublets are resolved. At comparable or slightly higher spectral higher
resolution, the velocity field of the line-emitting gas can be constrained to
constrain the ionization parameter and hence the density. With realistic photon
transport in the models, the line shapes further constrain the models. By
extending the bandpass into the far-UV, lines from species with both lower
and higher ionization potentials (e.g., C~III, N~III, O~VI, P~V, S~IV, S~VI)
provide additional diagnostics. The UV data can and is being obtained with
{\it HST\/}, but to obtain the far-UV data, we require the likes of {\it HUT\/}
(e.g., \cite{lon96}), {\it ORFEUS\/} (e.g., \cite{ray95}), and {\it FUSE\/}.
\acknowledgments
We thank John Raymond for useful insights and suggestions and the referee for
helpful comments which significantly improved the original manuscript. Work at
Lawrence Livermore National Laboratory was performed under the auspices of the
U.S.\ Department of Energy under contract No.~W-7405-Eng-48.
\clearpage
\begin{center}
\begin{tabular}{cccc}
\multicolumn{4}{c}{\bf TABLE 1}\\
\multicolumn{4}{c}{PHOTOIONIZATION MODEL LINE RATIOS}\\
\tableline
\tableline
$\log\xi $& log Si~IV/C~IV& log N~V/C~IV& log He~II/C~IV\\
\tableline
\multicolumn{4}{l}{Model A:
30 eV Blackbody Spectrum:
\vrule height6pt width5pt depth-1pt}\\
\tableline
$-1.5$& $?0.00$& $-2.02$& $?0.68$\\
$-1.0$& $-0.55$& $-1.40$& $-0.44$\\
$-0.5$& $-1.12$& $-0.97$& $-0.90$\\
$?0.0$& $-1.56$& $-0.59$& $-0.76$\\
$?0.5$& $-2.20$& $-0.37$& $-0.58$\\
$?1.0$& $-2.92$& $-0.24$& $-0.41$\\
\tableline
\multicolumn{4}{l}{Model B:
10 keV Bremsstrahlung Spectrum:
$\bullet $}\\
\tableline
$-1.5$& $?0.78$& $?0.91$& $?3.65$\\
$-1.0$& $?0.73$& $-0.15$& $?2.99$\\
$-0.5$& $?0.62$& $-1.38$& $?1.87$\\
$?0.0$& $?0.19$& $-1.39$& $?0.52$\\
$?0.5$& $-0.76$& $-1.15$& $-0.64$\\
$?1.0$& $-1.05$& $-0.68$& $-0.84$\\
\tableline
\multicolumn{4}{l}{Model C:
50 eV Blackbody Spectrum:
$\times $}\\
\tableline
$-3.0$& $?1.54$& $?2.18$& $?5.06$\\
$-2.5$& $?1.26$& $?1.04$& $?4.19$\\
$-2.0$& $?0.97$& $-0.25$& $?3.16$\\
$-1.5$& $?0.67$& $-1.66$& $?1.90$\\
$-1.0$& $?0.11$& $-1.69$& $?0.47$\\
$-0.5$& $-1.00$& $-1.17$& $-0.71$\\
$?0.0$& $-2.01$& $-0.97$& $-0.90$\\
$?0.5$& $-2.69$& $-0.62$& $-0.67$\\
$?1.0$& $-3.20$& $-0.26$& $-0.32$\\
$?1.5$& $-3.56$& $?0.22$& $?0.31$\\
\tableline
\end{tabular}
\end{center}
\clearpage
\begin{center}
\begin{tabular}{cccc}
\multicolumn{4}{c}{\bf TABLE 1 --- continued}\\
\multicolumn{4}{c}{PHOTOIONIZATION MODEL LINE RATIOS}\\
\tableline
\tableline
$\log\xi $& log Si~IV/C~IV& log N~V/C~IV& log He~II/C~IV\\
\tableline
\multicolumn{4}{l}{Model D:
10 eV Blackbody Spectrum:
+}\\
\tableline
$-3.0$& $-1.86$& $?0.35$& $?3.50$\\
$-2.5$& $-1.50$& $-0.90$& $?2.30$\\
$-2.0$& $-0.86$& $-2.23$& $?1.06$\\
$-1.5$& $-0.48$& $-2.90$& $?0.15$\\
$-1.0$& $-0.71$& $-2.52$& $-0.48$\\
$-0.5$& $-0.90$& $-1.88$& $-0.75$\\
$?0.0$& $-0.80$& $-1.08$& $-0.79$\\
$?0.5$& $-0.58$& $-0.31$& $-0.69$\\
$?1.0$& $-0.45$& $?0.25$& $-0.48$\\
$?1.5$& $-0.45$& $?0.57$& $-0.16$\\
\tableline
\multicolumn{4}{l}{Model E:
30 eV Blackbody Spectrum, $\log N_{\rm H}=23$:
$\Box $}\\
\tableline
$-1.5$& $?0.15$& $-1.83$& $?1.46$\\
$-1.0$& $-0.31$& $-1.47$& $?0.18$\\
$-0.5$& $-0.74$& $-1.13$& $-0.49$\\
$?0.0$& $-0.90$& $-0.96$& $-0.59$\\
$?0.5$& $-0.93$& $-0.92$& $-0.58$\\
$?1.0$& $-0.93$& $-0.91$& $-0.58$\\
$?1.5$& $-0.93$& $-0.89$& $-0.58$\\
\tableline
\multicolumn{4}{l}{Model F:
10 keV Bremsstrahlung Spectrum, $\log N_{\rm H}=23$:
$\circ $}\\
\tableline
$-1.5$& $?0.86$& $?1.33$& $?4.20$\\
$-1.0$& $?0.77$& $?0.44$& $?3.49$\\
$-0.5$& $?0.67$& $-0.81$& $?2.42$\\
$?0.0$& $?0.30$& $-1.36$& $?1.05$\\
$?0.5$& $-0.37$& $-1.15$& $-0.07$\\
$?1.0$& $-0.69$& $-0.91$& $-0.44$\\
$?1.5$& $-0.73$& $-0.79$& $-0.48$\\
\tableline
\end{tabular}
\end{center}
\clearpage
\begin{center}
\begin{tabular}{cccc}
\multicolumn{4}{c}{\bf TABLE 2}\\
\multicolumn{4}{c}{SCATTERING MODEL LINE RATIOS}\\
\tableline
\tableline
$\log\xi $& log Si~IV/C~IV& log N~V/C~IV& log He~II/C~IV\\
\tableline
\multicolumn{4}{l}{Model A$^\prime $:
30 eV Blackbody Spectrum:
\vrule height6pt width5pt depth-1pt}\\
\tableline
$-1.5$& $?0.36$& $-1.44$& $*-9.44$\\
$-1.0$& $-0.26$& $-0.98$& $*-8.84$\\
$-0.5$& $-1.06$& $-0.65$& $*-8.04$\\
$?0.0$& $-1.86$& $-0.33$& $*-6.40$\\
$?0.5$& $-3.09$& $-0.19$& $*-4.86$\\
$?1.0$& $-4.65$& $-0.13$& $*-3.35$\\
$?1.5$& $-6.36$& $-0.10$& $*-2.23$\\
\tableline
\multicolumn{4}{l}{Model B$^\prime $:
10 keV Bremsstrahlung Spectrum:
$\bullet $}\\
\tableline
$-1.5$& $?1.26$& $?0.68$& $-12.02$\\
$-1.0$& $?1.20$& $?0.08$& $-10.71$\\
$-0.5$& $?1.07$& $-0.71$& $*-9.49$\\
$?0.0$& $?0.68$& $-0.86$& $*-8.10$\\
$?0.5$& $-0.59$& $-0.81$& $*-7.14$\\
$?1.0$& $-2.25$& $-0.47$& $*-4.57$\\
$?1.5$& $-5.26$& $-0.06$& $*-0.79$\\
\tableline
\multicolumn{4}{l}{Model C$^\prime $:
50 eV Blackbody Spectrum:
$\times $}\\
\tableline
$-3.0$& $?2.01$& $?1.86$& $-11.36$\\
$-2.5$& $?1.70$& $?1.02$& $-10.81$\\
$-2.0$& $?1.41$& $?0.07$& $-10.32$\\
$-1.5$& $?1.10$& $-0.97$& $*-9.69$\\
$-1.0$& $?0.53$& $-1.24$& $*-8.85$\\
$-0.5$& $-0.75$& $-0.79$& $*-8.45$\\
$?0.0$& $-2.23$& $-0.68$& $*-7.13$\\
$?0.5$& $-4.35$& $-0.42$& $*-5.09$\\
$?1.0$& $-6.98$& $-0.14$& $*-3.19$\\
$?1.5$& $-\infty$& $?0.26$& $*-1.42$\\
\tableline
\end{tabular}
\end{center}
\clearpage
\begin{center}
\begin{tabular}{cccc}
\multicolumn{4}{c}{\bf TABLE 2 --- continued}\\
\multicolumn{4}{c}{SCATTERING MODEL LINE RATIOS}\\
\tableline
\tableline
$\log\xi $& log Si~IV/C~IV& log N~V/C~IV& log He~II/C~IV\\
\tableline
\multicolumn{4}{l}{Model D$^\prime $:
10 eV Blackbody Spectrum:
+}\\
\tableline
$-3.0$& $-1.28$& $?0.35$& $-11.04$\\
$-2.5$& $-1.24$& $-0.48$& $-10.32$\\
$-2.0$& $-0.61$& $-1.41$& $*-9.50$\\
$-1.5$& $-0.13$& $-2.22$& $*-9.62$\\
$-1.0$& $-0.45$& $-2.06$& $*-9.77$\\
$-0.5$& $-0.85$& $-1.48$& $*-9.57$\\
$?0.0$& $-1.08$& $-0.74$& $*-8.62$\\
$?0.5$& $-1.16$& $-0.02$& $*-7.52$\\
$?1.0$& $-1.19$& $?0.51$& $*-6.85$\\
$?1.5$& $-1.20$& $?0.81$& $*-6.52$\\
\tableline
\multicolumn{4}{l}{Model E$^\prime $:
30 eV Blackbody Spectrum, $\log N_{\rm H}=23$:
$\Box $}\\
\tableline
$-1.5$& $?2.01$& $?0.92$& $*-6.19$\\
$-1.0$& $?1.85$& $?0.27$& $*-5.06$\\
$-0.5$& $?1.85$& $-0.03$& $*-3.90$\\
$?0.0$& $?1.90$& $-0.01$& $*-2.24$\\
$?0.5$& $?1.90$& $-0.02$& $*-0.77$\\
$?1.0$& $?1.92$& $-0.18$& $*?0.58$\\
$?1.5$& $?1.96$& $?0.09$& $*?1.91$\\
\tableline
\multicolumn{4}{l}{Model F$^\prime $:
10 keV Bremsstrahlung Spectrum, $\log N_{\rm H}=23$:
$\circ $}\\
\tableline
$-1.5$& $?1.23$& $?1.21$& $-11.89$\\
$-1.0$& $?1.19$& $?0.77$& $-10.29$\\
$-0.5$& $?1.19$& $?0.28$& $*-8.47$\\
$?0.0$& $?1.19$& $-0.21$& $*-6.36$\\
$?0.5$& $?1.14$& $-0.62$& $*-4.50$\\
$?1.0$& $?1.08$& $-0.79$& $*-1.93$\\
$?1.5$& $?1.05$& $-0.81$& $*?1.19$\\
\tableline
\end{tabular}
\end{center}
\clearpage
\begin{center}
\begin{tabular}{cccc}
\multicolumn{4}{c}{\bf TABLE 3}\\
\multicolumn{4}{c}{COLLISIONAL IONIZATION MODEL LINE RATIOS}\\
\tableline
\tableline
$T$ (10{,}000 K)& log Si~IV/C~IV& log N~V/C~IV& log He~II/C~IV\\
\tableline
\multicolumn{4}{l}{Model G:
Constant Temperature}\\
\tableline
*3& $-0.12$& $-2.63$& $-1.67$\\
*5& $-0.46$& $-3.00$& $-2.61$\\
*7& $-1.83$& $-2.91$& $-3.54$\\
*9& $-2.65$& $-2.79$& $-4.04$\\
11& $-2.80$& $-2.16$& $-3.99$\\
13& $-2.65$& $-1.29$& $-3.58$\\
15& $-2.44$& $-0.52$& $-3.13$\\
17& $-2.29$& $?0.04$& $-2.74$\\
19& $-2.17$& $?0.38$& $-2.42$\\
21& $-2.09$& $?0.52$& $-2.16$\\
\tableline
\end{tabular}
\end{center}
\clearpage
| proofpile-arXiv_065-436 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
The Mott transition is one of the most fascinating phenomenons arising
from electron-electron interactions, and occurs in a wide range of
materials \cite{mott_metal_insulator}.
In fact two different Mott transitions exist: one can
either stay at a given (commensurate) filling and vary the strength of
the interactions (I call this transition Mott-U and it occurs in e.g.
vanadium oxides or
in organic (quasi-)one dimensional systems), one can also keep the
strength of the interaction constant and dope the system to move away
from the commensurate density (a Mott-$\delta$ transition, a situation
realized in High Tc superconductors or in quantum wires).
Although the basic underlying physics behind these transitions is by
now well understood it has proved incredibly difficult to tackle it in
either $d=2$ or $d=3$ due to our lack of tools to treat strongly
interacting systems \cite{mott_mean_fields}. In fact nearly all the fine
points of the transition, such as the critical properties or the
transport properties remain unknown.
One dimension constitute a special case where a rather complete study of
the Mott transition can be done. This offers special
interest both for theoretical and experimental reasons. From a
theoretical point of view, the effect of e-e interactions is
particularly strong and leads to a non-fermi
liquid state (the so called Luttinger liquid (LL)).
One can therefore expect drastic effects on the transport properties of
the system. From the experimental point of view, both transitions at
constant doping and by varying the doping can be realized, e.g. in
organic conductors \cite{jerome_revue_1d} and quantum wires
\cite{tarucha_quant_cond} or Josephson junction networks
\cite{vanoudenaarden_josephson_mott}.
Although the thermodynamic properties of the Mott transition
were understood a long time ago for the Hubbard model, which was shown
to be a Mott insulator at half
filling \cite{lieb_hubbard_exact,emery_revue_1d,solyom_revue_1d},
very little was
known of the Mott-$\delta$ transition and of the transport
properties: a parquet treatment gave the effective scattering
\cite{gorkov_pinning_parquet} but was limited to half filling and small
(perturbative) interactions, and only the zero frequency
conductivity (i.e.
the Drude weight) could be computed by Bethe-Ansatz for the particular
case of the Hubbard model
\cite{shastry_twist_1d,schulz_conductivite_1d}).
Recently a complete picture of both
Mott transitions as well as a full description of the transport
properties $\sigma(\omega,T,\delta)$ for
any commensurate filling, both for bosons or fermions,
was obtained
\cite{giamarchi_umklapp_1d,giamarchi_attract_1d,giamarchi_curvature}.
In these proceedings I will review some of these results.
No derivation will be given and
the reader is referred to
\cite{giamarchi_umklapp_1d,giamarchi_attract_1d,giamarchi_curvature}
for derivation as well as the complete
analytical expressions, for the figures presented here. Such a
presentation is done in section~\ref{section2} where umklapp effects are
presented and in section~\ref{section3} where both the critical
properties of the Mott transition(s) and the transport properties are
examined. In section~\ref{section4} application of these results to the
physics of the organic materials is done.
\section{Lattice effects and umklapp terms} \label{section2}
In the continuum e-e interactions conserve momentum, and thus
current, and cannot lead to any finite conductivity as
a consequence of Galilean invariance. In the presence of
a lattice however the momentum need only to be conserved modulo one
vector of the reciprocal lattice, and such interaction process (named
umklapp process) can lead to finite resistivity. In a fermi liquid,
umklapps are responsible for the intrinsic resistivity $\rho(T)\sim T^2$.
In one dimension it was rapidly
realized \cite{emery_revue_1d,solyom_revue_1d}
that umklapps are also responsible
for the Mott-U transition at half filling. Away from half filling they
are ``frozen'' due to the mismatch in momentum, and are usually
discarded as irrelevant: the system becomes then a perfect metal.
However both for the transport properties and
to study the Mott-$\delta$ transition it is necessary to
have a description of the umklapp processes even for finite doping.
This can be achieved using the so called bosonization representation,
that uses that {\bf all} excitations of a one dimensional
system can be described in term of density oscillations
\cite{emery_revue_1d,solyom_revue_1d,haldane_bosonisation}.
The charge properties of a
{\bf full} interacting one dimensional system (excluding umklapp terms)
is therefore described by
\begin{equation} \label{quadra}
H_0 = \frac1{2\pi} \int dx \; u_\rho K_\rho (\pi\Pi_\rho)^2 +
\frac{u_\rho}{K_\rho} (\nabla \phi)^2
\end{equation}
where $\nabla \phi = \rho(x)$, $\rho(x)$ is the charge density and $\Pi$
is the conjugate momentum to $\phi$. All the interaction effects are
hidden in the parameters $u_\rho$ (the velocity of charge excitations)
and $K_\rho$ (the Luttinger liquid exponent controlling the decay of
all correlation functions). This description (\ref{quadra}) is
valid for an arbitrary
one-dimensional interacting system, {\bf provided} one uses the proper
$u$ and $K$ (in the following I will drop the $\rho$ index).
In a general way $K =1$ is the
noninteracting point, $K>1$ means attraction whereas $K<1$ means
repulsion.
The umklapp process can also be given in terms of boson operators
\cite{emery_revue_1d,solyom_revue_1d}. In fact umklapps exist not only
at half filling but for higher
commensurabilities as well by transferring more particles across the fermi
surface (such processes are generated in higher order in perturbation
theory) \cite{giamarchi_curvature,schulz_mott_revue}.
For even commensurabilities the
Hamiltonian corresponding to the umklapp process is
\begin{equation} \label{um1n}
H_{\frac1{2n}} = g_{\frac1{2n}} \int dx \cos(n \sqrt8 \phi_\rho(x) +
\delta x)
\end{equation}
where $n$ is the order of the commensurability ($n=1$ for half filling -
one particle per site; $n=2$ for quarter filling - one particle every
two sites and so on). The coupling constant $g_{1/2n}$ is the umklapp
process corresponding to the commensurability $n$ and $\delta$ the
deviation (doping) from the commensurate filling.
$g_{\frac12}$ corresponds to one particle per site (half filling)
For simple models such as the Hubbard model $g_{\frac12}=U$, but
this does not need to be the case for more general models (see in
particular section~\ref{section4}). For $1/4$ filling and a typical
interaction $U$ one has $g_{1/4}\sim U (U/E_F)^3$.
Odd commensurability involves spin \cite{schulz_mott_revue} but can be
treated similarly.
Similar expressions can be derived for the case of bosons
\cite{giamarchi_attract_1d}.
It is therefore remarkable that in one dimension $H_0+H_{\frac1{2n}}$
provides the solution to {\bf all} Mott transitions,
for {\bf all} systems and {\bf all} (for particles with spin: even)
commensurabilities.
\section{Mott transition(s)} \label{section3}
Let us now examine the physical properties of the Mott transitions close
to a commensurability of order $n$ described by $H_0+H_{\frac1{2n}}$.
\subsection{Phase diagram}
The Mott-U and Mott-$\delta$ transitions are radically different and
lead to the phase diagram shown in Figure~\ref{phasediag}.
\begin{figure}
\centerline{\epsfig{file=figure1.eps,angle=-90,width=7cm}}
\caption{Phase diagram close to a commensurability of order $n$ ($n=1$
for half filling and $n=2$ for quarter filling). $U$ denotes a general
(i.e. not necessarily local) repulsion. $\mu$ is the chemical potential
and $\delta$ the doping. MI means Mott insulator and LL Luttinger
liquid (metallic) phase. The critical exponent $K_c$ and velocity $u$
depends on whether it is a Mott-U or Mott-$\delta$ transition.}
\label{phasediag}
\end{figure}
The Mott-U is of the Kosterlitz Thouless type
\cite{emery_revue_1d,solyom_revue_1d} and
occurs for a critical value of $K$, $K_c=1/n^2$
\cite{giamarchi_curvature,schulz_mott_revue}. For
half filling the transition point is the noninteracting one ($K_c=1$)
but for higher commensurability one reaches the Mott insulator only for
very repulsive interactions (for example for quarter filling
$K_c^U=1/4$). In the metallic phase the system is a LL, with finite
compressibility and Drude weight. The Mott insulator has a gap in the
charge excitations (thus zero compressibility). At the transition there
is a finite jump both in the compressibility and Drude weight.
The dynamical exponent is $z=1$.
To study the Mott-$\delta$ transition it is useful
\cite{giamarchi_umklapp_1d} to map the
sine-gordon Hamiltonian $H_0+H_{\frac1{2n}}$ to a spinless fermion model
(known as massive Thiring model \cite{emery_revue_1d,solyom_revue_1d}),
describing the charge excitations
(solitons) of the sine-gordon model. The remarkable fact is that {\bf
close} to the Mott-$\delta$ transition the solitons become
non-interacting, and one is simply led to a simple semi-conductor
picture of two bands separated by a gap (see figure~\ref{Thiring}).
\begin{figure}
\centerline{\epsfig{file=figure2.eps,angle=-90,width=6cm}}
\caption{Lower Hubbard band and Upper Hubbard band. This concept can be
made rigorous in one dimension by mapping the full interacting system to
a massive Thiring model. Optical transitions can be made either within
or between the two ``bands''.}
\label{Thiring}
\end{figure}
This image has to be used with caution since the solitons are only
non-interacting for infinitesimal doping (or for a very special value
of the initial interaction) and has to be supplemented by other
techniques \cite{giamarchi_umklapp_1d}. Nevertheless it provides a very
appealing description of the LHB and UHB and a good guide to understand
the phase diagram and transport properties.
The Mott-$\delta$ transition is of
the commensurate-incommensurate type. The {\bf universal}
(independent of the interactions) value of the
exponents $K_c^\delta = 1/(2n^2)$ is half of the one of Mott-U
transition. Since at the Mott-$\delta$ transition the
chemical potential is at the bottom of a band the velocity goes to zero
with doping. This leads to a continuous vanishing of the Drude weight
and compressibility. The dynamical
exponent is now $z=2$. For more details see
\cite{giamarchi_umklapp_1d,giamarchi_curvature,schulz_mott_revue,mori_mott_1d}.
For bosons, the phase diagram of
figure~\ref{phasediag} is well compatible with numerical results
\cite{batrouni_bosons_numerique} and higher dimensional proposals
\cite{fisher_boson_loc}.
\subsection{Transport properties}
Let us now look at the transport properties. The full conductivity
(real and imaginary part) $\sigma(\omega,T,\delta)$ can be found in
\cite{giamarchi_umklapp_1d,giamarchi_attract_1d,giamarchi_curvature} and
we just examine here simple limits.
The ac conductivity (at $T=0$) for
$\delta=0$ is shown in figure~\ref{sigacdel0}.
In the Mott insulator
$\sigma$ is zero until $\omega$ can make transitions between the LHB and
UHB. At the threshold one has the standard square root singularity
coming from the density of states (see figure~\ref{Thiring}). For higher
frequencies interactions dress the umklapps and give a
nonuniversal (i.e. interaction-dependent) power law-like decay. Such
a power law is beyond the reach of the simple noninteracting description
of Figure~\ref{Thiring}.
\begin{figure}
\centerline{\epsfig{file=figure3.eps,angle=-90,width=7cm}}
\caption{ac conductivity for $\delta=0$ for a commensurability of order
$n$. $\Delta$ is the Mott gap. The full line is the conductivity in the
Mott insulator. The dashed one is $\sigma$ in the metallic
regime. It contains both a Drude peak of weight $D$ and a regular part.}
\label{sigacdel0}
\end{figure}
Away from commensurate filling ($\delta\ne 0$) the conductivity is shown
in Figure~\ref{sigacdeln0} (only the case where the half filled system
is a MI is shown. For the other case see \cite{giamarchi_attract_1d}).
\begin{figure}
\centerline{\epsfig{file=figure4.eps,angle=-90,width=7cm}}
\caption{ac conductivity for $\delta\ne 0$ for a commensurability of
order $n$. $\Delta$ is the Mott gap. In addition to the Drude peak of
weight $D$ the regular part has two distinct regimes. }
\label{sigacdeln0}
\end{figure}
Features above the Mott gap are unchanged (the system has no way to
know it is or not at half filling at high frequencies). The two new
features are a Drude peak with a weight proportional to $\delta/\Delta$,
and an $\omega^3$ absorption \cite{giamarchi_curvature} at small
frequency. Features above the Mott
gap come from inter (hubbard)-band transitions whereas they come from
intra-band processes below the Mott gap (see figure~\ref{Thiring}).
The dc conductivity can be computed by the same methods and is shown in
figure~\ref{dcsig}.
\begin{figure}
\centerline{\epsfig{file=figure5.eps,angle=-90,width=7cm}}
\caption{dc conductivity as a function of $T$. Full line is for the
Mott insulator, dashed line is in the metallic regime. $\Delta$ is the
Mott gap.}
\label{dcsig}
\end{figure}
Here again the dressing of umklapps by the other interactions
results in a nonuniversal power law dependence. If the interactions are
repulsive enough the resistivity can even {\bf increase} as a function
of temperature well above the Mott gap.
Two universal behavior are expected: at the Mott-U transition one has
$\rho(T)\sim T$ and $\sigma(\omega)\sim 1/(\omega\ln(\omega)^2)$,
whereas at the Mott-$\delta$ transition due to the different $K_c$ one
expects $\rho(T)\sim 1/T$.
All this results are completely general and apply to any
one-dimensional systems for which $\Delta$ is smaller than the scale
above which {\bf all} interactions can be treated perturbatively
(typically $U$), a situation that covers most of
the experimentally relevant cases (see section~\ref{section4}).
It is noteworthy that the above results are also valid in the
presence not of umklapp processes but of a simple periodic potential
(the lattice corresponds itself to a $4k_F$ periodic potential). For a
$2k_F$ periodic potential transport properties are similar to the one
above with the replacement of $4n^2 K_\rho$ by $1+K_\rho$.
\section{Organic compounds} \label{section4}
The above results have a direct application to
organic conductors. These compounds are $1/4$ filled by chemistry but
due to a slight dimerization of the chain an half
filled umklapp $g_{1/2}\sim U (D/E_F)$ also exists where $D$ is the
dimerization gap, and $U$ a typical strength of the interactions
\cite{jerome_revue_1d}. Since $D/E_F$ is quite small the
umklapp term is much smaller than the other interactions leading to a
quite small Mott gap (see e.g. \cite{penc_numerics} for a numerical
estimation of the parameters). There is also a
$1/4$ filled umklapp
$g_{1/4}\sim U (U/E_F)^3$, which is as we saw less relevant but can be
depending on the typical interaction $U$ much larger in magnitude than
$g_{1/2}$.
Since the organic conductors are only quasi-one dimensional systems with
a perpendicular hopping integral $t_\perp$ between the chains one can
distinguish various domains in energy scale (temperature or
frequency) as shown in figure~\ref{scales}
\begin{figure}
\centerline{\epsfig{file=figure6.eps,angle=-90,width=7cm}}
\caption{Four important energy regimes for quasi-one dimensional
systems. In ``pert.'' everything can be treated perturbatively. In
``1D'' the
interactions lead to the one dimensional physics, and hopping from chain
to chain is incoherent. In ``2D-3D'' the hoping between chains is
coherent. The system orders in ``Ordered''.}
\label{scales}
\end{figure}
The most relevant questions being of course: what is the strength of the
interactions in these systems, what is the scale for $T_{\rm cr}$ (the
bare $t_\perp$ or lower \cite{hopping_general}), and what is the physics
below $T_{\rm cr}$.
In the absence of $t_\perp$ one expects therefore these compounds to be
Mott insulators. This is the case for the TMTTF family that has indeed
a conductivity \cite{creuzet_tmttf} similar to the one of
figure~\ref{dcsig} (full line).
Indeed if $\Delta > T_{\rm cr}$, one expects the Mott gap to render the
single particle hopping $t_\perp$ irrelevant ($E_{\rm cr}$ would thus
not exist). This family should
be described by one-dimensional physics. Further check of this can be
provided by examination of the optical (ac) conductivity, and comparing
it to figure~\ref{sigacdel0}. Measurements of the transverse
conductivity would also give information on the relevance of the
transverse hopping. Note that the temperature dependence of
the dc resistivity and the frequency dependence of the optical
conductivity provide a {\bf direct} measure of the $K_\rho$ exponent of
the Luttinger Liquid and give therefore crucial information on the
importance of interactions in such systems (the optical conductivity
has the advantage to be free from thermal expansion problems).
A naive fit in TMTTF would give a value of $K_\rho\sim 0.8$, widely
different from the one of $K_\rho=0.3$ extracted from the NMR
\cite{wzietek_tmtsf_nmr}. A way
to get out of this predicament could be that the conductivity is in fact
dominated by $1/4$ filling umklapp processes till very close to the Mott
gap giving
\begin{equation}
\rho(T) \sim g_{1/2}^2 T^{4K-3} + g_{1/4}^2 T^{16K-3} \sim g_{1/4}^2 T^{16K-3}
\end{equation}
but this point clearly deserves further investigation.
On the other hand, the TMTSF family shows
a rather good metallic behavior with a $T^2$ resistivity, indicating the
importance of transverse hopping. This is to be expected if
$\Delta > E_{\rm cr}$. There is important controversy on the value of
$E_{\rm cr}$ \cite{behnia_transport_magnetic,gorkov_sdw_tmtsf}.
Regardless of the value of $E_{\rm cr}$ the physics for
$(\omega, T ) > E_{\rm cr}$ will still
be controlled by one dimensional
effects. Indeed For the TMTSF family the optical conductivity
\cite{dressel_optical_tmtsf} is
very well compatible with the figure~\ref{sigacdeln0}.
In particular the optical peak can easily be interpreted in term of the
Mott insulator described here. Of course more detailed comparison of the
structure above the gap an in particular a check for the power law decay
of figure~\ref{sigacdeln0} would be worthy to do. The low energy
features and in particular the metallic behavior are closer to the
{\bf doped} system rather than the commensurate one.
``doping'' is not
too surprising since if one particle hopping between chains is relevant, one
expect small deviations to the commensurate filling due to the warping
of the Fermi surface. One therefore expect a very small spectral weight
in the $\delta(\omega)$ part.
Since one has a clear idea of the (purely)
one-dimensional conductivity (figure~\ref{sigacdeln0}), a detailed
comparison with experimental data should provide an indication on the
value of $E_{\rm cr}$. The question on whether the physics below
$E_{\rm cr}$ is simply ``fermi liquid'' like
\cite{gorkov_sdw_tmtsf} or still retains some
features of one-dimensionality and interactions is still open.
One way to settle this issue is a detailed examination of the
low frequency part of the optical conductivity and measurements of the
transverse dc conductivity in this regime.
Another way would
be to examine the effects of impurities on the dc conductivity. Indeed
one expect drastic localization in a one dimensional regime and very
weak effects for a FL \cite{giamarchi_loc}.
{\bf Acknowledgments:}
It is a pleasure to thank L. Degiorgi, L.P. Gor'kov, G. Gr\"uner,
D. J\'erome, A.J. Millis, H.J. Schulz and B.S. Shastry
for many interesting discussions.
| proofpile-arXiv_065-437 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
\vskip 10pt
The precise measurement of the $W$-boson mass $M_W$ constitutes a
primary task
of the forthcoming experiments at the high energy electron--positron
collider LEP2 ($2 M_W \leq \sqrt{s} \leq 210$~GeV).
A meaningful comparison between theory and
experiment requires an accurate description of
the fully exclusive processes $e^+ e^- \to 4f$, including the
main effects of radiative corrections, with the final goal of
providing predictions for the distributions
measured by the experiments. A large effort in the direction of developing
tools dedicated to the investigation of
this item has been spent within the Workshop
``Physics at LEP2'', held at CERN during 1995. Such an effort has led to the
development of several independent four-fermion codes, both semianalytical and
Monte Carlo, extensively documented in~\cite{wweg}. {\tt WWGENPV} is one of
these codes, and the aim of the present paper is to describe in some detail
the developments performed with respect to the original version~\cite{cpcww},
where a description of the
formalism adopted and the physical ideas behind it can be found.
As discussed in~\cite{wmass}, the most promising methods for measuring the
$W$-boson mass at LEP2 are the so
called ``threshold'' and ``direct reconstruction'' methods.
For the first one,
a precise evaluation of the threshold cross section is required. For the
second one, a precise description of the invariant-mass shape of the hadronic
system in semileptonic and hadronic decays is mandatory. In order to meet these
requirements, the previous version of the program has been improved, both from
the technical and physical point of view.
On the technical side, in addition
to the ``weighted event integration'' and
``unweighted event generation'' branches, the present version can
also be run as an ``adaptive Monte Carlo'' integrator, in order to obtain
high numerical precision results for cross sections and other relevant
observables. In the ``weighted event integration'' branch, a ``canonical''
output can be selected, in which several observables are processed in parallel
together with their most relevant moments~\cite{wweg}.
Moreover, the program offers the possibility of generating events
according to a specific flavour quantum number assignment for the final-state
fermions, or of generating ``mixed samples'', namely a fully leptonic, fully
hadronic or semileptonic sample.
On the physical side, the class of tree-level EW diagrams taken into
account has been extended to include all the single resonant diagrams
({\tt CC11/CC20}), in such a way that all the
charged current processes are covered.
Motivated by the physical relevance of keeping under control the effects of
the transverse degrees of freedom of photonic radiation, both for the $W$ mass
measurement and for the detection of anomalous couplings,
the contribution of QED radiation has been fully developed in the leading
logarithmic approximation, going beyond the initial-state, strictly collinear
approximation, to include $p_T / p_L $ effects both for initial- and
final-state photons. Last, an hadronic interface to {\tt JETSET}
in the generation branch has been added.
In the present version the neutral current backgrounds are neglected in the
fully hadronic and leptonic decay channels, but this is not a severe
limitation of
the program since, at least in the LEP2 energy range, these backgrounds
can be suppressed by means of proper invariant mass cuts. On the other hand,
the semileptonic decay channels are complete at the level of the Born
approximation EW diagrams ({\tt CC11/CC20} diagrams),
and this feature allows to treat at best those
channels that are expected to be the most promising for the direct
reconstruction of the $W$ mass, because free of systematics such as the
``color reconnection'' and the ``Bose-Einstein correlation'' problems.
In the development of the code, particular attention has been paid to the
possibility of obtaining precise results in relatively short CPU time. As shown
in~\cite{wweg}, {\tt WWGENPV} is one of the most precise four fermion Monte
Carlo's from the numerical point of view. This feature allows the use of the
code also for fitting purposes.
\noindent
\section{The most important new features}
\vskip 10pt
In the following we list and briefly describe the most important
technical and physical developments implemented in the new version
of {\tt WWGENPV}.
\vskip 24pt \noindent
TECHNICAL IMPROVEMENTS
The present version of the program consists of three branches, two
of them already present in the original version but upgraded in some
respect, the third one completely new.
\noindent
\begin{itemize}
\item Unweighted event generation branch.
This branch, meant for simulation purposes, has been
improved by supplying an option for an hadronisation interface
(see more details later on).
\item Weighted event integration branch.
This branch, intended for
computation only, includes as a new feature an option for
selecting a ``canonical'' output containing predictions for several
observables and their most relevant moments together with a
Monte Carlo estimate of the errors. According to the strategy
adopted in~\cite{wweg}, the first four Chebyshev/power
moments of the following quantities are computed:
the production angle of the $W^+$ with respect to the positron beam
({\tt TNCTHW, N=1,2,3,4}), the production angle $\vartheta_{d}$
of the down fermion with respect to the positron beam ({\tt TNCTHD}),
the decay angle $\vartheta_{d}^*$ of the down fermion with respect
to the direction of the decaying $W^-$ measured in its rest frame
({\tt TNCTHSR}), the energy $E_{d}$ of the down fermion
normalized to the beam energy ({\tt XDN}),
the sum of the energies of all radiated photons ({\tt XGN})
normalized to the beam energy,
the lost and visible photon energies normalized to the beam energy
({\tt XGNL} and {\tt XGNV}), respectively, and, finally
\begin{eqnarray*}
<x_m> \, = \, {1 \over \sigma} \, \int \left( { { \sqrt{s_+} +
\sqrt{s_-} -2 M_W} \over {2 E_b} } \right) \, d \sigma
\end{eqnarray*}
where $s_+$ and $s_-$ are the invariant masses of the $W^+$ and $W^-$
decay products, respectively ({\tt XMN}).
\item Adaptive integration branch.
This new branch is intended for computation
only, but offers high precision performances. On top of the importance
sampling, an adaptive Monte Carlo integration algorithm is used. The
code returns the value of the cross section together with a Monte Carlo
estimate of the error. Moreover, if QED corrections are taken into
account, also the
average energy and invariant mass losses are printed. The program must
be linked to NAG library for the Monte Carlo adaptive routine. Full
consistency between non-adaptive and adaptive integrations has been
explicitly proven.
\end{itemize}
In each of these three branches the user is asked to specify the
four-fermion final state which is required. The final states at present
available are those containing {\tt CC03} diagrams as a subset.
Their list, as appears
when running the code, is the following:
\begin{verbatim}
PURELY LEPTONIC PROCESSES
[0] ---> E+ NU_E E- BAR NU_E
[1] ---> E+ NU_E MU- BAR NU_MU
[2] ---> E- BAR NU_E MU+ NU_MU
[3] ---> E+ NU_E TAU- BAR NU_TAU
[4] ---> E- BAR NU_E TAU+ NU_TAU
[5] ---> MU+ NU_MU MU- BAR NU_MU
[6] ---> MU+ NU_MU TAU- BAR NU_TAU
[7] ---> MU- BAR NU_MU TAU+ NU_TAU
[8] ---> TAU+ NU_TAU TAU- BAR NU_TAU
SEMILEPTONIC PROCESSES
[9] ---> E+ NU_E D BAR U
[10] ---> E- BAR NU_E BAR D U
[11] ---> E+ NU_E S BAR C
[12] ---> E- BAR NU_E BAR S C
[13] ---> MU+ NU_MU D BAR U
[14] ---> MU- BAR NU_MU BAR D U
[15] ---> MU+ NU_MU S BAR C
[16] ---> MU- BAR NU_MU BAR S C
[17] ---> TAU+ NU_TAU D BAR U
[18] ---> TAU- BAR NU_TAU BAR D U
[19] ---> TAU+ NU_TAU S BAR C
[20] ---> TAU- BAR NU_TAU BAR S C
HADRONIC PROCESSES
[21] ---> D BAR U BAR D U
[22] ---> D BAR U BAR S C
[23] ---> S BAR C BAR D U
[24] ---> S BAR C BAR S C
MIXED SAMPLES
[25] ---> LEPTONIC SAMPLE
[26] ---> SEMILEPTONIC SAMPLE
[27] ---> HADRONIC SAMPLE
\end{verbatim}
It is worth noting that in the weighted event integration and unweighted
event
generation branches, besides the possibility of selecting a specific
four-fermion final state, an option is present for considering three
realistic mixed samples corresponding to
the fully leptonic, fully hadronic or semileptonic decay channel,
respectively.
When the generation of a mixed sample is required,
as a a first step the cross section for
each contributing channel is calculated; as a second step, the unweighted
events are generated for each contributing channel with a frequency
given by the weight of that particular channel with respect to the total.
In the generation branch, if the hadronisation interface is not enabled,
an $n$-tuple is created with the following structure:
\begin{verbatim}
'X_1','X_2','EB' ! x_{1,2} represent the
! energy fractions of incoming e^- and
! e^+ after ISR; EB is the beam energy;
'Q1X','Q1Y','Q1Z','Q1LB' ! x,y,z components of the momentum
! of particle 1 and the particle label
! according to PDG; the final-state
! fermions are assumed to be massless;
'Q2X','Q2Y','Q2Z','Q2LB' ! as above, particle 2
'Q3X','Q3Y','Q3Z','Q3LB' ! " " " 3
'Q4X','Q4Y','Q4Z','Q4LB' ! " " " 4
'AK1X','AK1Y','AK1Z' ! x,y,z components of the momentum of
! the photon from particle 1;
! they are 0 if no FSR has been chosen
! and/or if particle 1 is a neutrino;
'AK2X','AK2Y','AK2Z' ! as above, particle 2
'AK3X','AK3Y','AK3Z' ! " " " 3
'AK4X','AK4Y','AK4Z' ! " " " 4
'AKEX','AKEY','AKEZ' ! x,y,z components of the momentum of
! the photon from the initial-state
! electron; they are 0 if no ISR has
! been chosen;
'AKPX','AKPY','AKPZ' ! as above, initial-state positron;
\end{verbatim}
If the hadronisation interface is enabled, fully hadronised events are
instead made available to a user routine in the {\tt /HEPEVT/} format
(see below).
\vskip 24pt \noindent
PHYSICAL IMPROVEMENTS \\
The main theoretical developments with respect to the original version
concern the inclusion of additional matrix elements to
the tree-level kernel and a more sophisticated treatment of the photonic
radiation, beyond the initial-state, strictly collinear approximation.
Moreover, an hadronisation interface to { \tt JETSET} has been also provided.
{\it Tree-level EW four-fermion diagrams} -- In addition to double-resonant
charged-current diagrams {\tt CC03}
already present in the previous version,
the matrix element includes also the
single-resonant charged-current diagrams {\tt CC11} for $\mu $ and $\tau$'s in
the final state,
and {\tt CC20} for final states containing electrons.
This allows a complete treatment at the level of four-fermion
EW diagrams of the semileptonic sample, which appears the most
promising and cleanest for the direct mass reconstruction
method due to the absence of potentially large ``interconnection''
effects~\cite{wmass}.
Concerning {\tt CC20} diagrams,
the importance sampling technique has been extended to
take care of the peaking behaviour of the matrix element
when small momenta of the virtual photon are involved. As a consequence
of the fact that the tree-level matrix-element is
computed in the massless limit, a cut on the minimum electron (positron)
scattering angle must be imposed. The inclusion of such a cut
eliminates the problems connected with gauge-invariance in the case of
{\tt CC20} processes, for which the present version does not
include any so-called ``reparation'' scheme~\cite{bhf,wwcd}. Anyway, when
for instance a set of ``canonical cuts''~\cite{wweg} is used,
the numerical relevance of
such gauge-invariance restoring schemes has been shown~\cite{bhf,wwcd} to be
negligible compared with the expected experimental accuracy.
{\it Photonic corrections} -- As far as photonic effects are concerned,
the original version, as
stated above, included only leading logarithmic initial-state corrections
in the collinear approximation within the SF formalism. The
treatment has been extended in a two-fold way: the contribution of
final-state radiation has been included and the $p_T / p_L$ effects
have been implemented both for initial- and final-state radiation. The
inclusion of the transverse degrees of freedom has been achieved by generating
the fractional energy $x_{\gamma}$ of the radiated photons by means
of resummed electron structure
functions $D(x; s)$~\cite{sf} ($x_{\gamma} = 1 - x$) and the angles
using an angular factor inspired by the pole behaviour
$1 / (p \cdot k)$ for each charged emitting fermion. This allows
to incorporate leading QED radiative corrections originating from
infrared and collinear singularities, taking into account at the same
time the dominant kinematic effects due to non-strictly collinear
photon emission, in such a way that the universal factorized photonic spectrum
is recovered. According to this procedure, the
leading logarithmic corrections from initial- and final-state radiation
are isolated as a gauge-invariant
subset of the full calculation (not yet available) of the
electromagnetic corrections to $e^+ e^- \to 4f$.
Due to the inclusion of $p_T$-carrying photons at
the level of initial-state radiation, the Lorentz boost allowing the
reconstruction of the hard-scattering event from the c.m. system to
the laboratory one has been generalized to keep under control the
$p_T$ effects on the beam particles.
Final state radiation and $p_T / p_L$ effects are not taken into account
in the adaptive integration branch.
{\it Non-QED corrections} -- Coulomb correction is treated as in
the original version
on double-resonant {\tt CC03} diagrams. QCD corrections are
implemented in the present version in the naive form
according to the recipe described in~\cite{wweg,wwcd}. The
treatment of the leading EW contributions is unchanged with respect to the
original version.
{\it Hadronisation} -- Final-state quarks
issuing from the electroweak 4-fermion scattering are
not experimentally observable.
An hadronisation interface is provided to the {\tt JETSET} package~\cite{jetset}
to allow events to be extrapolated to the hadron level, for example for
input to a detector simulation program.
Specifically, the 4-fermion event structure is converted to the
{\tt /HEPEVT/} convention, then {\tt JETSET} is called to simulate QCD
partonic evolution (via routines {\tt LUJOIN} and {\tt LUSHOW}) and
hadronisation (routine {\tt LUEXEC}).
In making this conversion, masses must be added to the outgoing fermions,
considered massless in the hard scattering process.
This is done by rescaling the fermion momenta by a single scale factor,
keeping the flight directions fixed in the rest frame of the four fermion
system.
In the QCD evolution phase, strings join quarks coming from the same W decay.
The virtuality scale of the QCD evolution is taken to be the invariant
mass-squared of each evolving fermion pair.
No colour reconnection is included by default, although it could be implemented
by appropriate modification of routine {\tt WWGJIF} if required.
Bose - Einstein correlations are neglected in the present version.
The resulting event structure is then made available to the user
in the {\tt /HEPEVT/} common block via a routine
{\tt WWUSER} for further analysis, such as writing out for later
input to a detector simulation program.
The {\tt WWUSER} routine is also called at program initialisation time to
allow the user to set any non-standard {\tt JETSET} program options, for
example, and at termination time to allow any necessary clean-up.
A dummy {\tt WWUSER} routine is supplied with the program.
The only {\tt JETSET} option which is changed by
{\tt WWGENPV} from its default value
controls emission of gluons and photons by final-state
partons\footnote{\footnotesize {\tt JETSET} parameter {\tt MSTJ(41)}},
turning off final-state photon emission
simulation from {\tt JETSET} if activated in {\tt WWGENPV},
to avoid double counting.
\\
All the new features of the program can be switched on/off by means of
separate flags, as described in the following.
\section{Input}
\vskip 10pt
Here we give a short explanation of the input parameters and flags
required when running the program.
\noindent
{\bf \begin{verbatim}
OGEN(CHARACTER*1)
\end{verbatim}}
\noindent
It controls the use of the program as a Monte Carlo
event generator of unweighted events ({\tt OGEN = G})
or as a Monte Carlo/adaptive integrator
for weighted events ({\tt OGEN = I}).
\vskip 5pt
\noindent
{\bf \begin{verbatim}
RS(REAL*8)
\end{verbatim}}
\noindent
The centre-of-mass energy (in GeV).
\vskip 5pt
\noindent
{\bf \begin{verbatim}
OFAST(CHARACTER*1)
\end{verbatim}}
\noindent
It selects ({\tt OFAST = Y}) the adaptive integration branch, when
{\tt OGEN = I}. When this choice is done, the required relative accuracy
of the numerical integration has to be supplied by means of the {\tt REAL*8}
variable {\tt EPS}.
\noindent
{\bf \begin{verbatim}
NHITWMAX(INTEGER)
\end{verbatim}}
\noindent
Required by the Monte Carlo integration branch. It is the maximum number of
calls for the Monte Carlo loop.
\vskip 5pt
\noindent
{\bf \begin{verbatim}
NHITMAX(INTEGER)
\end{verbatim}}
\noindent
Required by the event-generation branch. It is the maximum number of
hits for the hit-or-miss procedure.
\vskip 5pt
\noindent
{\bf \begin{verbatim}
IQED(INTEGER)
\end{verbatim}}
\noindent
This flag allows the user to switch on/off the contribution of the
initial-state radiation. If {\tt IQED = 0} the distributions are
computed in lowest-order approximation, while for {\tt IQED = 1}
the QED corrections are included in the calculation.
\vskip 5pt
\noindent
{\bf \begin{verbatim}
OPT(CHARACTER*1)
\end{verbatim}}
\noindent
This flag controls the inclusion of $p_T / p_L$ effects for
the initial-state radiation. It is ignored in the adaptive integration
branch where only initial-state strictly collinear radiation is allowed.
\vskip 5pt
\noindent
{\bf \begin{verbatim}
OFS(CHARACTER*1)
\end{verbatim}}
\noindent
It is the option for including final-state radiation. It is assumed
that final-state radiation can be switched on only if initial-state
radiation including $p_T / p_L$ effects is on, in which case
final-state radiation includes $p_T / p_L$ effects as well.
Ignored in the adaptive integration branch.
\vskip 5pt
\noindent
{\bf \begin{verbatim}
ODIS(CHARACTER*1)
\end{verbatim}}
\noindent
Required by the integration branch.
It selects the kind of experimental distribution. For {\tt ODIS = T}
the program computes the total cross section
(in pb) of the process;
for {\tt ODIS = W} the value of the invariant-mass distribution
$d \sigma / d M$ of the system $d \bar u$ ({\tt IWCH = 1}) or of the system
${\bar d} u$ ({\tt IWCH = 2}) is returned (in pb/GeV).
\vskip 5pt
\noindent
{\bf \begin{verbatim}
OWIDTH(CHARACTER*1)
\end{verbatim}}
\noindent
It allows a different choice of the value of the $W$-width.
{\tt OWIDTH = Y} means that the tree-level Standard Model
formula for the $W$-width is used; {\tt OWIDTH = N} requires that
the $W$-width is supplied by the user in GeV.
\vskip 15pt
\vfil\eject
\noindent
{\bf \begin{verbatim}
NSCH(INTEGER)
\end{verbatim}}
\noindent
The value of
{\tt NSCH} allows the user to choose the calculational scheme for the
weak mixing angle and the gauge coupling. Three choices are available. If
{\tt NSCH=1}, the input parameters
used are $G_F, M_W, M_Z$ and the calculation is
performed at tree level. If {\tt NSCH = 2} or
{\tt 3}, the input parameters used are
$\alpha(Q^2), G_F, M_W$ or $\alpha(Q^2), G_F, M_Z$, respectively,
and the calculation is performed using the QED coupling constant at a
proper scale $Q^2$, which is requested as further input. The
recommended choice is {\tt NSCH = 2}, consistently with~\cite{wweg}.
\noindent
{\bf \begin{verbatim}
OCOUL(CHARACTER*1)
\end{verbatim}}
\noindent
This flag allows the user to switch on/off the contribution of the
Coulomb correction. Unchanged with respect to the old version of
the program.
\noindent
{\bf \begin{verbatim}
OQCD(CHARACTER*1)
\end{verbatim}}
\noindent
This flag allows the user to switch on/off the contribution of the naive
QCD correction.
\noindent
{\bf \begin{verbatim}
ICHANNEL(INTEGER)
\end{verbatim}}
\noindent
A channel corresponding to a specific flavour quantum number assignment
can be chosen.
\noindent
{\bf \begin{verbatim}
ANGLMIN(REAL*8)
\end{verbatim}}
\noindent
The minimum electron (positron) scattering angle (deg.) in the laboratory
frame. It is ignored when {\tt CC20} graphs are not selected.
\noindent
{\bf \begin{verbatim}
SRES(CHARACTER*1)
\end{verbatim}}
\noindent
Option for switching on/off single-resonant diagrams ({\tt CC11}).
\noindent
{\bf \begin{verbatim}
OCC20(CHARACTER*1)
\end{verbatim}}
\noindent
Option for switching on/off single-resonant diagrams when
electrons (positrons) occur in the final state ({\tt CC20}).
\noindent
{\bf \begin{verbatim}
OOUT(CHARACTER*1)
\end{verbatim}}
\noindent
Option for ``canonical'' output containing results for several observables
and their most important moments. It is active only in the Monte Carlo
integration branch.
\noindent
{\bf \begin{verbatim}
OHAD(CHARACTER*1)
\end{verbatim}}
\noindent
Option for switching on/off hadronisation interface in the
unweighted event generation branch.
\section{Test run output}
\vskip 10pt
The typical new calculations that can be performed with the
updated version of the
program are illustrated in the following examples.
\vskip 8pt\noindent
\leftline{\bf Sample 1} An example of adaptive integration is
provided. The process considered is $e^+ e^- \to e^+ \nu_e d \bar u$
({\tt CC20}).
The output gives the cross section, together with the energy and
invariant-mass losses from initial-state radiation. ``Canonical'' cuts
are imposed as in~\cite{wweg}. The input card is as follows:
\begin{verbatim}
OGEN = I
RS = 190.D0
OFAST = Y
EPS = 1.D-2
IQED = 1
OPT = N
OFS = N
ODIS = T
OWIDTH = Y
NSCH = 2
ALPHM1 = 128.07D0
OCOUL = N
OQCD = N
ICHANNEL = 9
ANGLMIN = 10.D0
SRES = Y
OCC20 = Y
OOUT = N
\end{verbatim}
\vskip 8pt\noindent
\leftline{\bf Sample 2} An example of weighted event integration
is provided. Here the process considered is $e^+ e^- \to \mu^+ \nu_{\mu}
d \bar u$ ({\tt CC11}). ``Canonical'' cuts
are imposed as before. The ``canonical'' output is provided. The input card
differs from the previous one as follows:
\begin{verbatim}
RS = 175.D0
OFAST = N
NHITWMAX = 100000
OPT = Y
OFS = Y
OCOUL = Y
OQCD = Y
ICHANNEL = 13
OCC20 = N
OOUT = Y
\end{verbatim}
\vskip 8pt\noindent
\leftline{\bf Sample 3} An example of unweighted event generation including
hadronisation is provided. A sample of 100 events
corresponding to the full semileptonic channel is generated. The detailed
list of an hadronised event is given. The input card
differs from the first one as follows:
\begin{verbatim}
OGEN = G
RS = 175.D0
NHITMAX = 100
OPT = Y
OFS = Y
OCOUL = Y
OQCD = Y
ICHANNEL = 26
ANGLMIN = 5.D0
OHAD = Y
\end{verbatim}
\vskip 10pt
\vskip 15pt
\section{Conclusions}
The program {\tt WWGENPV 2.0} has been described. In its present version it
allows the treatment of all the four-fermion reactions including the {\tt CC03}
class of diagrams as a subset. This means that all the semileptonic channels
are complete from the tree-level diagrams point of view, whereas fully leptonic
and fully hadronic channels are treated in the CC approximation. Since the most
promising channels for the $W$ mass reconstruction are the semileptonic ones,
the present version of the code allows a precise analysis of such data.
Moreover, NC backgrounds can be suppressed by proper invariant mass cuts, so
that the code is also usable for fully hadronic and leptonic events analysis,
with no substantial loss of reliability. Initial- and final-state QED radiation
is taken into account within the SF formalism, including finite $p_T / p_L$
effects in the leading logarithmic approximation. The Coulomb correction is
included for the {\tt CC03 } graphs. Naive QCD and leading EW corrections
are implemented as well. An hadronic interface to
{\tt JETSET} is also provided.
The code as it stands is a valuable tool for the analysis of LEP2 data, with
particular emphasis to the threshold and direct reconstruction methods for the
measurement of the $W$-boson mass. Speed and high numerical accuracy allow the
use of the program also for fitting purposes.
The code is supported. Future releases of {\tt WWGENPV} will include:
\begin{itemize}
\item an interface to the code {\tt HIGGSPV}~\cite{wweg,egdp} in
order to treat all the possible
four-fermion processes in the massless limit, including Higgs-boson signals;
\item implementation of anomalous couplings;
\item implementation of CKM effects;
\item the extension of the hadronic interface to {\tt HERWIG}~\cite{herwig}.
\end{itemize}
| proofpile-arXiv_065-438 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
The formulation of quantum field theory in terms of
Haag Kastler nets of local observable algebras
(``local quantum physics" \cite{Haa}) has
turned out to be well suited for
the investigation of general structures.
Discussion of concrete models, however, is mostly done
in terms of pointlike localized fields.
In order to be in a precise mathematical framework,
these fields might be assumed to obey the Wightman axioms \cite{StW}.
Even then, the interrelation between both concepts is not
yet completely understood (see \cite{BaW,BoY} for the present stage).
Heuristically, Wightman fields are constructed out
of Haag-Kastler nets by some scaling limit which, however, is
difficult to formulate in an intrinsic way \cite{Buc2}.
In a dilation invariant
theory scaling is well defined,
and in the presence of massless particles the construction
of a pointlike field was performed in \cite{BuF}.
Here, we study the possibly simplest
situation: Haag-Kastler nets in 2 dimensional Min\-kows\-ki
space with trivial translations in one light cone
direction (``chirality") and covariant under
the real M\"obius group which acts on the other lightlike
direction.
In \cite{FrJ}, it has been shown that in the vacuum representation
pointlike localized fields can be constructed. Their
smeared linear combinations are affiliated to the original
net and generate it. We do not know at the moment whether
they satisfy all Wightman axioms,
since we have not yet found
an invariant domain of definition.
In \cite{Joe3}, we have generalized this to the charged sectors of a
theory. We have constructed pointlike localized fields carrying
arbitrary charge with finite statistics
and therefore intertwining between the different
superselection sectors of the theory.
(In Conformal Field Theory
these objects are known as ``Vertex Operators".)
We have obtained
the unbounded field
operators as limits of elements of the reduced field bundle
\cite{FRS1,FRS2}
associated
to the net of observables of the theory.
In this paper, we start again from chiral conformal Haag-Kastler nets
and present an canonical construction of
N-point-functions that can be shown to fulfill the Wightman axioms.
We proceed by generalizing the conformal cluster theorem \cite{FrJ} to
higher N-point-functions and by examining the momentum space limit of
the algebraic N-point-functions at $p=0$.
We are not able to prove that these Wightman fields can be identified
with the pointlike localized fields constructed in \cite{FrJ} and \cite{Joe3}.
\section{First Steps}
In this section, we give an explicit formulation
of the setting frow which this work starts. We then present the proof
of the conformal cluster theorem and the results on the construction
of pointlike localized fields in \cite{FrJ} and \cite{Joe3}.
\subsection{Assumptions}
Let $\A=(\A(I))_{I\in\KKK}$
be a family of von Neumann algebras on some
separable Hilbert space \H. \k\ denotes the set of nonempty bounded
open intervals on \R.
$\A$ is assumed to satisfy the following conditions.
\begin{enumerate}
\def\roman{enumi}){\roman{enumi})}
\def\roman{enumi}{\roman{enumi}}
\item Isotony:
\be \A(I_1)\subset\A(I_2)\;\;\;\;\;\mb{for}\;\;\;\;I_1\subset I_2,
\;\;\;\;I_1, I_2\in\k.\ee
\item Locality:
\be \A(I_1)\subset\A(I_2)'\;\;\;\;\;\mb{for}\;\;\;\;I_1\cap I_2=\{\},
\;\;\;\;I_1, I_2\in\k\ee
($\A(I_2)'$ is the commutant of $\A(I_2)$).
\item
There exists a strongly continuous unitary representation $U$ of
$G=SL(2,\R)$ in $H$ with $U(-1)=1$ and
\be U(g)\,\A(I)\,U(g)^{-1}=\A(gI),\;\;\;\;\;I,gI\in\k
\ee
($SL(2,\R)\ni g=\left(\begin{array}{cc}a&b\\c&d\end{array}\right)$
acts
on $\R\cup\{\infty\}$ by
$x\mapsto
\frac{ax+b}{cx+d}$ with the appropriate interpretation for
$x, gx=\infty$).
\item
The conformal Hamiltonian \HH, which generates the
restriction of $U$ to $SO(2)$, has a nonnegative spectrum.
\item
There is a unique (up to a phase) $U$-invariant unit vector
$\OM\in\H$.
\item
\H\ is the smallest closed subspace containing the vacuum
$\OM$ which
is invariant under $U(g),$ $g\in SL(2,\R),$ and $A\in\A(I),I\in\k$
(``cyclicity").
\footnote{This assumption is seemingly weaker than cyclicity of
$\OM$ w.r.t.\,the algebra of local observables on $\R$.}
\end{enumerate}
It is convenient to extend the net to intervals $I$ on the circle
$S^1=\R\cup\{\infty\}$ by setting
\be
\A(I)=U(g)\;\A(g^{-1}I)\;U(g)^{-1},\;\;\;\;\;\;g^{-1}I\in\k,\;g\in
SL(2,\R).
\ee
The covariance property guarantees that $\A(I)$ is well defined
for all intervals $I$ of the form $I=gI_0,\;I_0\in\k,\;g\in SL(2,\R),$
i.e.\ for all nonempty nondense open intervals on $S^1$ (we denote
the set of these intervals by \K).
\subsection{Conformal Cluster Theorem}
In this subsection, we derive a bound on conformal two-point-functions
in algebraic quantum field theory (see \cite{FrJ}). This bound specifies the
decrease properties of conformal two-point-functions in
the algebraic framework to be exactly those known from theories
with pointlike localization. The Conformal Cluster Theorem plays a central role in this work.
\medskip
{\bf Conformal Cluster Theorem (see \cite{FrJ}):}
Let $(\A(I))_{I\in\KKK}$ be a conformally covariant local net on $\R$.
Let $a,b,c,d\in\R$ and
$a<b<c<d$.
Let $A\in\A(\,(a,b)\,)$, $B\in\A(\,(c,d)\,)$, $n\in\N$ and
$P_k A\OM=P_k A^*\OM=0,\;k<n$. $P_k$ here denotes the
projection on the subrepresentation of $U(G)$ with conformal dimension $k$.
We then have
\be
|(\,\OM,BA\OM\,)|\leq \left(\frac{(b-a)\,(d-c)}{(c-a)\,(d-b)}\right)^n
\;\|A\|\,\|B\|.
\ee
\medskip
{\bf Proof:} Choose $R>0$. We consider the following 1-parameter
subgroup of $G=SL(2,\R)$\,:
\be
g_t\,:\,x\longmapsto\frac{x\,
\mb{cos}\frac{t}{2}+R\,\mb{sin}\frac{t}{2}}
{-\frac{x}{R}\,\mb{sin}\frac{t}{2}+\mb{cos}\frac{t}{2}}\,.
\ee
Its generator ${\rm \bf H}_R$ is within each subrepresentation
of $U(G)$
unitarily equivalent to the
conformal Hamiltonian ${\rm \bf H}$. Therefore, the spectrum of $A\OM$
and $A^*\OM$ w.r.t.\ ${\rm \bf H}_R$ is bounded below by $n$.
Let $0<t_0<t_1<2\pi$ such that $g_{t_0}(b)=c$ and $g_{t_1}(a)=d$.
We now define
\be
F(z)=\left\{\begin{array}{ll}
(\,\OM,\,B\,z^{-{\rm \bf H}_R}\,A\OM\,)
&|z|>1\\
(\,\OM,\,A\,z^{{\rm \bf H}_R}\,B\OM\,)&|z|<1\\
(\,\OM,\,A\,\alpha_{g_t}(B)\,\OM\,)&z=e^{it},\,t\not\in
[t_0,t_1]
\end{array},
\right.
\ee
a function analytic in its domain of definition, and then
\be
G(z)=\,(z-z_0)^n\,(z^{-1}-z_0^{-1})^n\,F(z),\;\;
z_0=e^{\frac{i}{2}(t_0+t_1)}\,.
\ee
(Confer the idea in \cite{Fre}.)
At $z=0$ and $z=\infty$ the function
$G(\cdot)$ is bounded
because of the bound on the spectrum of ${\rm \bf H}_R$ and
can therefore be analytically continued.
As an analytic function it reaches its maximum at the boundary of its
domain of definition, which is the interval
$[e^{it_0},e^{it_1}]$
on the unit circle:
\be
\mb{sup}|G(z)|\,\leq\,\|A\|\,\|B\|\,|e^{it_0}-e^{\frac{i}{2}
(t_0+t_1)}|^{2n}\,=\,\|A\|\,\|B\|\,(2\,
\mb{sin}\frac{t_0-t_1}{4})^{2n}
\,.
\ee
This leads to
\beam
|(\,\OM,\,BA\OM\,)|&=&|F(1)|\,=\,|G(1)|\ |1-e^{\frac{i}{2}(t_0+t_1)}
|^{
-2n}\,=\,|G(1)|\ (2\,\mb{sin}\frac{t_0+t_1}{4})^{-2n}\nn\\
&\leq&\mb{sup}|G|\,(2\,\mb{sin}\frac{t_0+t_1}{4})^{-2n}\,\leq\,
\|A\|\,\|B\|\,\left(\frac{\mb{sin}\frac{t_0-t_1}{4}}{\mb{sin}
\frac{t_0+t_1}{4}}\right)^{2n}\,.
\eeam
Determining $t_0$ and $t_1$ we obtain
\be
\lim_{R\rightarrow\infty}R\,t_0=2(c-b)\;\;\;\mb{and}\;\;\;
\lim_{R\rightarrow\infty}R\,t_1=2(d-a)\,.
\ee
We now assume $a-b=c-d$ and find $\left(\frac{t_0-t_1}{t_0+t_1}
\right)^2=\frac{(a-b)\,(c-d)}{(a-c)\,(b-d)}=:x\,.$
Since the bound on $|(\,\OM,\,BA\OM\,)|$ can only depend
on the conformal cross ratio $x$,
we can drop the assumption and the theorem is
proven.\,\hfill $\Box$
\subsection{The Construction of Pointlike Localized
Fields from Conformal Haag-Kastler Nets}
This subsection presents
the argumentation and results of \cite{FrJ} and \cite{Joe3}:\\
The idea for the definition of conformal fields is the following:
Let $A$ be a local observable,
\be
A\in\bigcup_{I\in\KKK}\A(I),
\ee
and $P_\tau$ the projection onto an irreducible subrepresentation
$\tau$ of $U$. The vector $P_\tau A\OM$ may then be thought of as
$\varphi_\tau(h)\,\OM$ where $\varphi_\tau$ is a conformal field of
dimension $n_\tau=:n$ and $h$ is an appropriate
function on $\R$.
The relation between
$A$ and $h$, however, is unknown at the moment, up to the known
transformation properties under $G$,
\be
U(g)\,P_\tau A\OM=\varphi_\tau(h_g^{(n)})\,\OM
\ee
with $h_g^{(n)}(x)=(cx-a)^{2n-2}\,h(\frac{dx-b}{-cx+a})$,
$g=\left(\begin{array}{cc}a&b\\c&d\end{array}\right)\in G$.
We may now scale the vector $P_\tau A\OM$ by dilations $D(\l)=
U\left(\begin{array}{cc}\l^{\frac{1}{2}}&0\\0&\l^{-\frac{1}{2}}
\end{array}\right)$ and find
\be
D(\l)\,P_\tau A\OM=\l^n\,\varphi_\tau(h_{\l})\,\OM
\ee
where $h_{\l}(x)=\l^{-1}\,h(\frac{x}{\lambda
})$. Hence, we obtain formally
for $\lambda\downarrow 0$
\be
\l^{-n}\,D(\l)\,P_\tau A\OM\longrightarrow
\int dx\,h(x)\;\varphi_\tau(0)
\,\OM.
\ee
In order to obtain a Hilbert space vector in the limit, we smear over
the group of
translations $T(a)=U\left(\begin{array}{cc}1&a\\0&1\end{array}
\right)$ with some test function $f$ and obtain formally
\be
\label{a}
\lim_{\lambda
\downarrow 0}\l^{-n}\int da\,f(a)\;T(a)\,D(\l)\,P_\tau A\OM=
\int dx\,h(x)\;\varphi_\tau(f)\,\OM.\label{f}
\ee
We now interpret the left-hand side as a definition of a conformal
field $\varphi_\tau$ on the vacuum, and try to obtain densely defined
operators with the correct localization by defining
\be
\varphi_\tau^I(f)\,A'\OM=A'\varphi_\tau^I(f)\,\OM,\;\;
f\in\D(I),\,A'\in\A(I)',\,I\in\K.
\ee
In order to make this formal construction meaningful, there are two problems to overcome.
The first one is the fact that the limit on the left-hand side of
(\ref{a})
does not exist in general if $A\OM$ is replaced by an arbitrary
vector in $H$. This corresponds to the possibility that the function
$h$ on the right-hand side might not be integrable.
We will show that after smearing the operator $A$
with a smooth function on $G$,
the limit is well defined. Such operators will be called regularized.
The second problem is to show that the smeared field operators
$\varphi_\tau^I(f)$ are closable, in spite of the nonlocal
nature of the projections $P_\tau$.
We omit the technical parts of \cite{FrJ} and \cite{Joe3} and
summarize the results in a compact form and as general as possible.
Due to the positivity condition, the representation
$U(\tilde{G})$ is completely
reducible into irreducible subrepresentations
and the irreducible components $\tau$ are up to equivalence
uniquely characterized by the conformal dimension $n_\tau\in\Rp$
($n_\tau$ is the lower bound of the spectrum of the conformal
Hamiltonian \HH\ in the representation $\tau$).
Associated with each irreducible subrepresentation $\tau$ of $U$
we find
for each $I\in\k$ a
densely defined operator-valued distribution $\varphi_\tau^I$
on the space $\D(I)$
of Schwartz functions with support in $I$
such that the following statements hold for all $f\in\D(I).$
\begin{enumerate}
\def\roman{enumi}){\roman{enumi})}
\def\roman{enumi}{\roman{enumi}}
\item
The domain of definition of
$\varphi_\tau^I(f)$ is given by $\A(I')\,\OM$.
\item
\be
\varphi_\tau^I(f)\,\OM\in P_\tau\H_{red}
\ee
with $P_\tau$ denoting the projector on the module of $\tau$.
\item
\be
U(\tilde{g})\;
\varphi_\tau^I(x)\;
U(\tilde{g})^{-1}=(cx+d)^{-2n_\tau}\varphi_\tau^{gI}
(\tilde{g}x)
\ee
with the covering projection $\tilde{g}\mapsto g$ and
$g=\left(\begin{array}{cc}a&b\\c&d\end{array}\right)\in
SL(2,\R),\;I, gI\in\k$.
\item
$\varphi_\tau^I(f)$ is closable.
\item
The closure of $\varphi_{\tau}^I(f),\,f\in\D(I),$ is affiliated to
$\A(I)$.
\item
$\A(I)$ is the smallest von Neumann algebra to which all operators
$\varphi_{\tau}^I(f),\,f\in\D(I),$ are affiliated.
\item The exchange algebra of the reduced field bundle \cite{FRS2}
and the existence of the closed field operators
$\varphi_\tau^I(f)$, mapping a dense set of the vacuum Hilbert space
into some charged sector with finite statistics,
suffice to construct closed field operators
$\varphi_{\tau,\alpha}^I(f)$,
mapping a dense set of an arbitrary charged
sector $\alpha$ with finite statistics
into
some (other) charged sector with finite statistics.
Here,
the irreducible
module $\tau$ of
$U(\tilde{G})$ labels orthogonal irreducible fields defined
in the same sector $\alpha$\,.
\item
The closure of any $\varphi_{\tau,\alpha}^I(f),\,f\in\D(I),$ is affiliated to
$\F_{red}(I)$.
\item
$\F_{red}(I)$ is the smallest von Neumann algebra to which all operators
$\varphi_{\tau,\alpha}^I(f),\,f\in\D(I),$ are affiliated.
\end{enumerate}
With the existence of pointlike localized fields we are able to
give a proof of a generalized Bisognano-Wichmann property. We can
identify the conformal group and the reflections as generalized
modular structures in the reduced field bundle. Especially, we obtain
a PCT operator on $\H_{red}$ proving the PCT theorem for the full
theory.
Moreover, the existence of pointlike localized fields gives a
proof of the hitherto unproven Spin-Statistics theorem for conformal
Haag-Kastler nets in 1+1 dimensions.
It was also possible to prove an operator product expansions
for arbitrary local observables: \\
For each $I\in\k$ and each $A\in \A(I)$
there is a local expansion
\be
A\,=\,\sum_{\tau}\varphi_{\tau}^I(f_{\tau,A})\,
\ee
into a sum over all irreducible modules
$\tau$ of $U(G)$
with \be
\mb{supp}f_{\tau,A}\subset I\,,\ee
which converges on $\A(I')\OM$ $*$-strongly (cf.\ the definition in
\cite{BrR}). Here, $I'$ denotes the complement of $I$ in $\k$.
\section{Canonical Construction of Wightman Fields}
Starting from a chiral
conformal Haag-Kastler net, pointlike localized fields have been
constructed in \cite{FrJ,Joe3}. Their smeared linear combinations are affiliated to the
original net and generate it. We do not know at the moment whether
these fields satisfy all Wightman axioms, since we have not found an
invariant domain of definition.
In this section, we construct in a canonical manner a complete set of
pointlike localized correlation functions out of
the net of algebras we have been starting from.
We proceed by generalizing the conformal cluster theorem to
higher N-point-functions and by examining the momentum space limit of
the algebraic N-point-functions at $p=0$. This canonically constructed
set of correlation
functions can be shown to fulfill the conditions for Wightman functions (cf.\ \cite{StW} and
\cite{Jos}). Hence, we can construct an associated
field theory fulfilling the Wightman axioms.
We are not able to prove that these Wightman fields can be identified with the pointlike
localized fields constructed in \cite{FrJ} and \cite{Joe3}. We do not know
either how the Haag-Kastler theory, we have been starting
from, can be reconstructed from the Wightman theory.
Such phenomena
have been investigated by Borchers and Yngvason \cite{BoY}. Starting
from a Wightman theory, they could not rule out in general the possibility that
the associated local net has to be defined in an enlarged Hilbert space.
\subsection{Conformal Two-Point-Functions}
First, we will determine the general form of conformal
two-point-functions of local observables:\\
It has been shown (cf.\ e.g.\ \cite{Joe1}) that a two-point-function
$(\,\OM,\,B\,U(x)\,A\OM\,)$
of a chiral local net with translation covariance is of
Lebesgue class $L^p$ for any $p>1$. The
Fourier
transform of this two-point-function is a measure concentrated on
the positive half line. Therefore, it is - with the possible
exception of a trivial
delta function at zero - fully determined by the Fourier transform
of the commutator function $(\,\OM,\,
[B,\,U(x)\,A\,U(x)^{-1}]\,\OM\,)\,.$ Since $A$ and $B$ are local
observables, the commutator function has compact support and an
analytic Fourier transform $G(p)$.
The restriction $\Theta(p)\,G(p)$
of this analytic
function to the positive half line is then the Fourier transform
of $(\,\OM,\,B\,U(x)\,A\OM\,)\,.$
In the conformally covariant case with
$P_kA\OM=P_kA^*\OM=0,\,k<n$, the
conformal cluster theorem implies that the
two-point-function
$(\,\OM,\,B\,U(x)\,A\OM\,)$
decreases as $x^{-2n}$. Therefore,
its Fourier transform is $2n\!-\!2\,$times continuously differentiable
and can be written as $\Theta(p)\,p^{2n-1}\,H(p)$ with an
appropriate analytic function $H(p)$.
Using this result, we are able to present a sequence of canonically
scaled two-point-functions of local observables
converging as distributions to the two-point-function known from
conventional conformal field theory (cf.\ \cite{Joe1,Reh}):
\be
\lim_{\lambda\downarrow 0}\l^{-2n}\ (\,\OM,\,B\,U(
\lambda^{-1}x)\,A\OM\,)\ =\
\lim_{\lambda\downarrow 0}\l^{-2n}\ {\cal F}_{p\rightarrow x}\
\Theta(p)\,(\l p)^{2n-1}\,H(\l p)\,\l\,dp\ =\
H(0)\ (x+i\varepsilon)^{-2n}\,.
\ee
\subsection{Conformal Three-Point Functions}
We consider the properties of chiral algebraic three-point functions
\be
(\,\OM,\,A_1\,U(x_1-x_2)\,A_2\,U(x_2-x_3)\,A_3\,\OM\,)
\ee
of local observables
$A_i$\,, $ i=1,2,3$\,.
The general form of a
(truncated) chiral three-point function of local observables
is restricted by locality and by the condition of positive energy. The
Fourier transform of an algebraic three-point function
can be shown to be the sum of the restrictions of analytic functions to
disjoint open wedges in the domain of positive energy:
\\ If $F$ now denotes the Fourier
transform of $(\,\OM,\,A_1\,U(\cdot)\,A_2\,U(\cdot)\,A_3\,\OM\,)$,
we get by straightforward calculations as a first result (cf.\ \cite{Joe4})
\be
F(p,q)\,=\,\Theta(p)\,\Theta(q-p)\,G^+(p,q)\,+\,
\Theta(q)\,\Theta(p-q)\,G^-(p,q)
\ee
with appropriate analytic functions
$G^+$ and $G^-$.
In the case of conformal covariance the general
form of these algebraic three-point functions is even more restricted by the
following generalization of the conformal cluster theorem \cite{FrJ}:
\medskip
{\bf Theorem:} Let $(\A(I))_{I\in\KKK}$ be a conformally covariant local net on $\R$\,.
Let $a_i,b_i\in\R\,,\ i=1,2,3$\,, and
$a_1<b_1<a_2<b_2<a_3<b_3$\,.
Let $A_i\in\A(\,(a_i,b_i)\,)$\,, $n_i\in\N$\,, $i=1,2,3$\,, and
\be
P_k\,A_i\,\OM=P_k\,A_i^*\,\OM=0\,,\;k<n_i\,.
\ee
$P_k$ here denotes the
projection on the subrepresentation of $U(SL(2,R))$ with conformal dimension $k$\,.
We then have the following bound:
\beam
|(\,\OM,A_1A_2A_3\OM\,)|&\leq&
\left|\frac{(a_1-b_1)+(a_2-b_2)}{(a_2-a_1)+(b_2-b_1)}\right|^{(n_1+n_2-n_3)}\\
&&
\left|\frac{(a_1-b_1)+(a_3-b_3)}{(a_3-a_1)+(b_3-b_1)}\right|^{(n_1+n_3-n_2)}
\;\;\nn\\&&
\left|\frac{(a_2-b_2)+(a_3-b_3)}{(a_3-a_2)+(b_3-b_2)}\right|^{(n_2+n_3-n_1)}
\;\|A_1\|\,\|A_2\|\,\|A_3\|\,.\nn
\eeam
If we additionally assume
\be
a_1-b_1=a_2-b_2=a_3-b_3\,,
\ee
we get
\be
|(\,\OM,A_1A_2A_3\OM\,)|\ \leq\
r_{12}^{(n_1+n_2-n_3)/2}\,r_{23}^{(n_2+n_3-n_1)/2}\,
r_{13}^{(n_1+n_3-n_2)/2}\ \|A_1\|\,\|A_2\|\,\|A_3\|\,,
\ee
with the conformal
cross ratios
\be
\frac{(a_i-b_i)\,(a_j-b_j)}{(a_i-a_j)\,(b_i-b_j)}=:r_{ij}\,,\,i,j=1,2,3\,.
\ee
\medskip
{\bf Proof:} This proof follows, wherever possible, the line of
argument in the proof of the conformal cluster theorem for
two-point functions (cf.\ \cite{FrJ}). \\
Choose $R>0$\,. Let us consider the following one-parameter
subgroup of $SL(2,\R)$\,:
\be
g_t\,:\,x\longmapsto\frac{x\,
\mb{cos}\frac{t}{2}+R\,\mb{sin}\frac{t}{2}}
{-\frac{x}{R}\,\mb{sin}\frac{t}{2}+\mb{cos}\frac{t}{2}}\,.
\ee
Its generator ${\rm \bf H}_R$ is within each subrepresentation
of $U(SL(2,R))$
unitarily equivalent to the
conformal Hamiltonian ${\rm \bf H}$\,. Therefore, the spectrum of $A_i\,\OM$
and $A^*_i\,\OM$ with respect to ${\rm \bf H}_R$ is bounded from below by $n_i$\,, $i=1,2,3$\,.
Let $0<t^-_{ij}<t^+_{ij}<2\pi$ such that
\be
g_{t^-_{ij}}(b_i)=a_j
\ee
and
\be
g_{t^+_{ij}}(a_i)=b_j\,
\ee
for $i,j=1,2,3$\,, $i<j$\,.
We now define
\be
F(z_1,z_2,z_3):=(\,\OM,\,A_{i_1}\,(\frac{z_{i_1}}{z_{i_2}})^{{\rm \bf H}_R}\,A_{i_2}\,(\frac{z_{i_2}}{z_{i_3}})^{{\rm \bf H}_R}\,A_{i_3}\OM\,)
\ee
in a domain of definition given by
\be
|z_{i_1}|<|z_{i_2}|<|z_{i_3}|
\ee
with permutations $(i_1,i_2,i_3)$
of $(1,2,3)$\,. This definition can uniquely be extended
to certain boundary values with $|z_{j}|=|z_{k}|$\,, $j,k=1,2,3$\,, $j\neq
k$\,:\\
$F$ shall be continued to this boundary of its domain of definition
if
\be
t_{jk}:=-i\log\frac{z_{j}}{z_{k}}\notin[t^-_{jk},t^+_{jk}]+2\pi{\bf
Z}
\ee
or equivalently if
\be
g_{t_k}([a_k,b_k])\cap g_{t_j}([a_j,b_j])\neq\emptyset\,,
\ee
using the notation
\be
t_i:=-i\log z_i\,,\ i=1,2,3\,.
\ee
Thereby, boundary points with coinciding absolute values are included in the
domain of definition. The definition of $F$ is chosen in
analogy to the analytic continuation of general Wightman functions
(cf., e.g.,
\cite{StW,Jos})
such that the edge-of-the-wedge theorem for distributions with
several variables \cite{StW} proves $F$ to be an analytic function:\\
Permuting the local observables $A_i$\,, $i=1,2,3$\,, we have six
three-point functions
\be
(\,\OM,\,A_{i_1}\,U(x_{i_1}-x_{i_2})\,A_{i_2}\,U(x_{i_2}-x_{i_3})\,A_{i_3}\OM\,)\,.
\ee
These six functions have by locality identical values on a domain
\be
E:=\{(y_1,y_2)\in {\bf
R}^2\,|\,|y_1|>c_1,\,|y_2|>c_2,\,|y_1+y_2|>c_3\}
\ee
with appropriate $c_1,c_2,c_3\in\Rp$\,. Each single function can be continued analytically by the condition of positive energy
to one of the six disjoint subsets in
\be
U:={\bf R}^2+iV:=\{(z_1,z_2)\in
{\bf C}^2\,
|\,\mb{Im}z_1\neq 0 \neq
\mb{Im}z_2\,,\,\mb{Im}z_1
+\mb{Im}z_2\neq 0\}\,.
\ee
In this geometrical situation, the edge-of-the-wedge theorem (cf.\
\cite{StW}, theorem 2.14) proves the assumed analyticity of $F$.\\
With the abbreviation
\be
z_{ij}^0:=e^{i(t^-_{ij}+t^+_{ij})/2}\,,\ i,j=1,2,3\,,
\ee
we then define
\beam
\lefteqn{
G(z_1,z_2,z_3)}\\
&:=&F(z_1,z_2,z_3)\,\prod_{(i,j,k)\in T(1,2,3)}\,(\frac{z_i}{z_j}-z^0_{ij})^{(n_i+n_j-n_k)/2}\,(\frac{z_j}{z_i}-z^0_{ji})^{(n_i+n_j-n_k)/2}\,,\nn
\eeam
where $T(1,2,3)$ denotes the set $\{(1,2,3),\,(1,3,2),\,(2,3,1)\}$\,. The added polynomial in
$z_i$\,, $i=1,2,3$\,, is constructed such that the degree of
the leading terms are restricted by the assumption on the conformal
dimensions of the
three-point function $F$. Also,
using the binomial formula, it can be controlled by straightforward
calculations that no half odd integer exponents
appear after multiplication of the product. Hence,
at $z_i=0$ and $z_i=\infty$\,, $i=1,2,3$\,, the function
$G$ is bounded
because of the bound on the spectrum of ${\rm \bf H}_R$ and
can therefore be analytically continued. We can find estimates on $G$ by the maximum
principle for analytic functions. In order to get the estimate needed in
this proof,
we do not use the maximum principle for several complex variables
\cite{BoM}. Instead, we present an iteration
of the maximum principle argument used in
the proof of the conformal cluster theorem \cite{FrJ} for the single
variables $z_i$\,, $i=1,2,3$\,, of $G(\cdot,\cdot,\cdot)$ and derive a bound on $G(1,1,1)$\,:\\
Applying the line of argument known from the case of the two-point functions
now to $G(\cdot,1,1)$\,, we get the estimate
\beam
|G(1,1,1)|&\leq& \mbox{sup}_{z_1}\,|G(z_1,1,1)|\nn\\
&=& \mbox{sup}_{z_1\in
B_{\cdot,1,1}}\,|G(z_1,1,1)|\,.
\eeam
The boundary of the domain of definition of the maximal analytical
continuation of $G(\cdot,1,1)$ is here denoted by
\be
B_{\cdot,1,1}:=\{e^{it}\,|\,t\notin [t_{12}^-,t_{12}^+]\cup
[t_{13}^-,t_{13}^+]+2\pi{\bf Z}\}\,.
\ee
Applying this argument to $G(z_1,\cdot,1)$, we analogously get the estimate
\beam
|G(z_1,1,1)|&\leq& \mbox{sup}_{z_2}\,|G(z_1,z_2,1)|\nn\\
&=& \mbox{sup}_{z_2\in
B_{z_1,\cdot,1}}\,|G(z_1,z_2,1)|\,
\eeam
with $B_{z_1,\cdot,1}$ denoting the boundary of the domain of definition of the maximal analytical
continuation of $G(z_1,\cdot,1)$\,.
Applying this argument finally to $G(z_1,z_2,\cdot)$\,, we analogously get the estimate
\beam
|G(z_1,z_2,1)|&\leq& \mbox{sup}_{z_3}\,|G(z_1,z_2,z_3)|\nn\\
&=& \mbox{sup}_{z_3\in
B_{z_1,z_2,\cdot}}\,|G(z_1,z_2,z_3)|\,
\eeam
with $B_{z_1,z_2,\cdot}$ denoting the boundary of the domain of definition of the maximal analytical
continuation of $G(z_1,z_2,\cdot)$\,.
Having iterated this maximum principle argument for the single
variables $z_i$\,, $i=1,2,3$\,, we can combine the derived estimates and get
\be
|G(1,1,1)|\ \leq\ \mbox{sup}_{t_{jk}=-i\log\frac{z_j}{z_k}\notin[t^-_{jk},t^+_{jk}]+2\pi{\bf
Z}\,,\ j\neq k}\,|G(z_1,z_2,z_3)|\,.
\ee
Hence, the boundary values of
$G$ have to be evaluated on the domain described by
\be
g_{t_k}([a_k,b_k])\cap g_{t_j}([a_j,b_j])\neq\emptyset\,
\ee
with
$
t_i=-i\log z_i\,,\ i=1,2,3\,.
$
We find the supremum with the same
calculation as in the proof of the conformal cluster theorem above
(cf.\ \cite{FrJ}):
\beam
|G(1,1,1)|\,&\leq&\,\|A_1\|\,\|A_2\|\,\|A_3\|\,\prod_{(i,j,k)\in T(1,2,3)}\,|e^{it^-_{ij}}-e^{i(t^-_{ij}+t^+_{ij})/2}|^{n_i+n_j-n_k}\nn\\&=&\,\|A_1\|\,\|A_2\|\,\|A_3\|\,\prod_{(i,j,k)\in T(1,2,3)}\,|2\,
\mb{sin}\frac{t^-_{ij}-t^+_{ij}}{4}|^{n_i+n_j-n_k}
\eeam
This leads to another estimate:
\beam
|(\,\OM,A_1A_2A_3\OM\,)|&=&\,|F(1,1,1)|\nn\\
&=&\,|G(1,1,1)|\,\prod_{(i,j,k)\in T(1,2,3)}\,|1-e^{i(t^-_{ij}+t^+_{ij})/2}|^{n_i+n_j-n_k}\nn\\
&=&\,|G(1,1,1)|\,\prod_{(i,j,k)\in T(1,2,3)}\,|2\,\mb{sin}\frac{t^-_{ij}+t^+_{ij}}{4}|^{n_i+n_j-n_k}\nn\\
&\leq&\,\|A_1\|\,\|A_2\|\,\|A_3\|\,\prod_{(i,j,k)\in T(1,2,3)}\,\left|\frac{\mb{sin}\frac{t^-_{ij}-t^+_{ij}}{4}}{\mb{sin}
\frac{t^-_{ij}+t^+_{ij}}{4}}\right|^{n_i+n_j-n_k}
\eeam
Determining $t^-_{ij}$ and $t^+_{ij}$\,, we obtain for $i,j=1,2,3$
\be
\lim_{R\rightarrow\infty}R\,t^-_{ij}=2(a_j-b_i)
\ee
and
\be
\lim_{R\rightarrow\infty}R\,t^+_{ij}=2(b_j-a_i)
\ee
and the first bound in the theorem is proven. If we now assume
\be
a_1-b_1=a_2-b_2=a_3-b_3\,,
\ee
we find
\be
\left(\frac{t^-_{ij}-t^+_{ij}}{t^-_{ij}+t^+_{ij}}
\right)^2=\frac{(a_i-b_i)\,(a_j-b_j)}{(a_i-a_j)\,(b_i-b_j)}=r_{ij}\,,\;\;\;\,i,j=1,2,3\,,
\ee
and the theorem is
proven.\hfill $\Box$
\medskip
This theorem can be used to get deeper
insight in the form of the Fourier transforms of algebraic
three-point functions.
As in the case of the two-point functions, we proceed by transferring the decrease properties of the function in
position space into regularity properties of the
Fourier transform in momentum space.
In conventional
conformal field theory, the three-point function with
conformal dimensions $n_i\,,\ i=1,2,3$\,, is known up to multiplicities as
\beam
f_{n_1n_2n_3}(x_1,x_2,x_3)
&=&(x_1-x_2+i\varepsilon)^{-(n_1+n_2-n_3)}\nn\\
&&(x_2-x_3+i\varepsilon)^{-(n_2+n_3-n_1)}\nn\\
&&(x_1-x_3+i\varepsilon)^{-(n_1+n_3-n_2)}
\eeam
(cf.\ \cite{ChH,Reh}).
Its Fourier transform
\be
\tilde{f}_{n_1n_2n_3}(p,q)\,=:\,\Theta(p)\,\Theta(q)\,Q_{n_1n_2n_3}(p,q)
\ee
can be calculated to be a sum of the restrictions of homogeneous polynomials
$Q^+_{n_1n_2n_3}$ and $Q^-_{n_1n_2n_3}$ of degree
$n_1+n_2+n_3-2$
to disjoint open wedges $W_+$ and $W_-$ in the domain of
positive energy (cf.\ \cite{Reh}).
By the bound in the cluster theorem above, we know that
a conformally covariant algebraic three-point function
$(\,\OM,\,A_1\,U(x_1-x_2)\,A_2\,U(x_2-x_3)\,A_3\OM\,)$
of local observables $A_i$ with minimal conformal dimensions
$n_i$\,,
$i=1,2,3\,,$
decreases in position space at
least as fast
as the associated pointlike three-point function $f_{n_1n_2n_3}(x_1,x_2,x_3)$ known from conventional conformal field theory. Hence,
the Fourier transform
$F_{A_1A_2A_3}(p,q)$
of this algebraic
three-point function has to be at least as regular in momentum space
as the Fourier transform $\tilde{f}_{n_1n_2n_3}(p,q)$ of the associated pointlike
three-point function
known from
conventional conformal field theory: \\
Technically, we use a well-known formula
from the theory of Fourier transforms,
\be
{\cal F}(\mbox{Pol}(X)S)=\mbox{Pol}(\frac{\partial}{\partial Y}){\cal F}S\,,
\ee
for arbitrary temperate distributions $S$ and polynomials Pol$(\cdot)$
with a (multi-dimensional)
variable $X$ in position space and an appropriate associated differential operator
$\frac{\partial}{\partial Y}$ in momentum space.
$\F$ denotes the Fourier transformation from position space to
momentum space.
Let now $S$ be the conformally covariant algebraic three-point function
of local observables $A_i$ with minimal conformal dimensions
$n_i$\,,
$i=1,2,3\,$:
\be
S\,:=\,(\,\OM,\,A_1\,U(x_1-x_2)\,A_2\,U(x_2-x_3)\,A_3\,\OM\,)\,
\ee
and $X$ be a pair of two difference variables out of $x_i-x_j$\,,
$i,j=1,2,3\,.$ By the
cluster theorem proved above, we can now choose
an appropriate homogeneous polynomial $\mbox{Pol}(X)$ of degree
$n_1+n_2+n_3-4$ such that the product $\mbox{Pol}(X)\,S$ is still
absolutely integrable in position space.
Using the formula given above, we see that
$\mbox{Pol}(\frac{\partial}{\partial Y}){\cal F}S$ is continuous and bounded in
momentum space.
Furthermore, we have already derived the form of the Fourier transform $F$
of an arbitrary (truncated) algebraic three-point function in a chiral
theory to be
\be
F(p,q)\,=\,\Theta(p)\,\Theta(q-p)\,G^+(p,q)\,+\,
\Theta(q)\,\Theta(p-q)\,G^-(p,q)
\ee
with appropriate analytic functions
$G^+$ and $G^-$.
Thereby, we see that
in the case of conformal covariance with minimal conformal dimensions
$n_i$\,, $i=1,2,3$\,,
the analytic function $G^+$ ($G^-$)
can be expressed as the product of an appropriate
homogeneous polynomial $P^+$ ($P^-$) of degree $n_1+n_2+n_3-2$ restricted to the wedge $W_+$
($W_-$) and an appropriate
analytic function $H^+$ ($H^-$)\,.
Hence, we have proved that the Fourier transform $F_{A_1A_2A_3}$ of the algebraic
three-point function
$(\,\OM,\,A_1\,U(x_1-x_2)\,A_2\,U(x_2-x_3)\,A_3\,\OM\,)$
can be written as
\be
F_{A_1A_2A_3}(p,q)\ =\ \Theta(p)\,\Theta(q)\,P_{A_1A_2A_3}(p,q)\,H_{A_1A_2A_3}(p,q)
\ee
with an appropriate homogeneous function $P_{A_1A_2A_3}(p,q)$ of degree $n_1+n_2+n_3-2$ and
an appropriate continuous and bounded function $H_{A_1A_2A_3}(p,q)$\,.
These results suffice to control the pointlike limit of the
considered correlation functions.
Scaling an algebraic
three-point function in a canonical manner, we construct a sequence
of distributions that converges to the three-point function of
conventional conformal field theory:
\beam
&&\nn\\
\lefteqn{ \lim_{\lambda\downarrow 0}\l^{-(n_1+n_2+n_3)}\ (\,\OM,\,A_1\,U(
\frac{x_1-x_2}{\lambda})\,A_2\,U(
\frac{x_2-x_3}{\lambda})\,A_3\,\OM\,)}\nn\\
&&\nn\\
&=&\lim_{\lambda\downarrow 0}\l^{-(n_1+n_2+n_3)}
\ {\cal F}_{p\rightarrow x_1-x_2\atop q\rightarrow x_2-x_3}\
F_{A_1A_2A_3}(\l p,\l q)\,\l^2\,dp\,dq\nn\\
&&\nn\\
&=&\lim_{\lambda\downarrow 0}\l^{-(n_1+n_2+n_3)}\nn\\
&&
\ {\cal F}_{p\rightarrow x_1-x_2\atop q\rightarrow x_2-x_3}\ \Theta(p)\,\Theta(q)\,
\l^{n_1+n_2+n_3-2}\,P_{A_1A_2A_3}(p,q)\,H_{A_1A_2A_3}(\l p,\l
q)\,\l^2\,dp\,dq\nn\\
&&\nn\\
&=&(x_1-x_2+i\varepsilon)^{-(n_1+n_2-n_3)}\nn\\
&&(x_2-x_3+i\varepsilon)^{-(n_2+n_3-n_1)}\nn\\
&&(x_1-x_3+i\varepsilon)^{-(n_1+n_3-n_2)}\ H_{A_1A_2A_3}(0,0)\,.\\
&&\nn
\eeam
\subsection{Conformal N-Point Functions}
Since the notational expenditure increases strongly as we come to the
construction of higher N-point functions, we concentrate on
qualitatively new aspects not occurring in the case of
two-point functions and three-point functions. These qualitatively new
aspects in the construction of higher N-point functions are related to
the fact that in conventional field theory the form of higher
N-point functions is not fully determined by conformal covariance.
In conventional conformal field theory conformal covariance restricts
the form of correlation functions of field operators
$\varphi_i(x_i)\,,\ i=1,2,...\,,N\,,$ with
conformal dimension $n_i$ in the following manner (cf.\ \cite{ChH,Reh}):
\be
(\OM,\,\left(\prod_{1\leq i\leq N}\,\varphi_i(x_i)\right)\,\OM)\
=\ \left(\prod_{1\leq i<j\leq N}\frac{1}{(x_j-x_i+i\e)^{c_{ij}}}\right)\;f(r_{t_1u_1}^{v_1s_1},...\,,r_{t_{N-3}u_{N-3}}^{v_{N-3}s_{N-3}})\,.\ee
Here, $f(\cdot,...\,,\cdot)$
denotes an appropriate function
depending on $N\!-\!3$ algebraicly independent conformal cross ratios
\be r_{tu}^{vs}:=\frac{(x_v-x_s)}{(x_v-x_t)}\frac{(x_t-x_u)}{(x_s-x_u)}\,.
\ee
The exponents $c_{ij}$ must fulfill the consistency conditions
\be
\sum_{j=1\atop j\neq i}^N c_{ij}=2n_i\,,\ c_{ij}=c_{ji}\,,\ 1\leq i\leq N\,.
\ee
These conditions do not fully determine the exponents $c_{ij}$ in the
case of $N\!\geq\!4$\,.
Hence, in conventional conformal field theory four-point functions and
higher
N-point functions are not fully determined by conformal covariance.
In the case of conformal two-point functions and conformal three-point functions, our
strategy to construct pointlike localized correlation functions was
the following: First, we proved that the algebraic
correlation functions decrease in position space as fast as
the associated correlation functions in conventional field theory,
which are uniquely determined by conformal covariance.
Then, we
transferred this property by Fourier transformation into regularity
properties in momentum space. Finally, we were able to prove that
the
limit $\lambda\!\downarrow\!0$ of canonically scaled algebraic
correlation functions converges to (a multiple of) the associated
pointlike
localized correlation functions in conventional conformal field theory.
In the case of four-point functions and higher N-point functions, the situation has changed and
we cannot expect to be able to fully determine the form of
the pointlike localized limit in this construction, since for $N>4$
the correlation functions in conventional
conventional field theory are not any longer uniquely determined by conformal
covariance.
Beginning with the discussion of the general case with $N\!\geq\!4$\,,
we consider
algebraic N-point functions
\be
(\,\OM,\,\left(\prod_{1\leq i\leq
N}\,U(-x_i)\,A_i\,U(x_i)\right)\,\OM\,)
\ee
of local observables $A_i$ with minimal conformal dimensions $n_i$\,,
$i=1,2,...\,,N$\,, in a chiral theory with conformal covariance. We
want to examine the pointlike limit of canonically scaled correlation functions
\be
\lim_{\lambda\downarrow 0}\l^{-\left(\sum_{1\leq i\leq N}n_i\right)}\
(\,\OM,\,\left(\prod_{1\leq i\leq N}\,U(-
\frac{x_i}{\lambda})\,A_i\,U(
\frac{x_i}{\lambda})\right)\,\OM\,)\,.
\label{i}
\ee
Our procedure in the construction of pointlike localized
N-point functions for $N\!\geq\!4$ will be the following: We consider all possibilities to
form a set of exponents $c_{ij}$ fulfilling the
consistency conditions
\be
\sum_{j=1\atop j\neq i}^N c_{ij}=2n_i\,,\ c_{ij}=c_{ji}\,,\ i=1,2,3,...\,,N\,.
\ee
For each consistent set of exponents a bound on algebraic N-point functions in
position space can be proved. Each single bound on algebraic
N-point functions in position space can be transferred
into a regularity property of algebraic N-point functions in momentum
space.
We can use the same techniques as in the case of three-point functions.
Finally, we will
control the canonical scaling limit in (\ref{i}) and construct pointlike localized
conformal N-point functions.
We present the following generalization of the conformal
cluster theorem proved above (cf.\ \cite{FrJ}) to algebraic
N-point functions of local observables:
\medskip
{\bf Theorem:} Let $(\A(I))_{I\in\KKK}$ be a conformally covariant local net on $\R$\,.
Let $a_i,b_i\in\R\,,\ i=1,2,3,...\,,N$\,, and
$a_i<b_i<a_{i+1}<b_{i+1}$ for $i=1,2,3,...\,,N\!-\!1$\,.
Let $A_i\in\A(\,(a_i,b_i)\,)$\,, $n_i\in\N$\,, and
\be
P_k\,A_i\,\OM=P_k\,A_i^*\,\OM=0\,,\;k<n_i\,,\;i=1,2,3,...\,,N\,.
\ee
$P_k$ here denotes the
projection on the subrepresentation of $U(SL(2,R))$ with conformal dimension $k$\,.
We then have for each set of exponents $c_{ij}$ fulfilling
the consistency conditions
\be
\sum_{j=1\atop j\neq i}^N c_{ij}=2n_i\,,\ c_{ij}=c_{ji}\,,\ i=1,2,3,...\,,N\,,
\ee
the following bound:
\beam
\lefteqn{
|(\,\OM,\,\left(\prod_{1\leq i\leq N}A_i\right)\OM\,)|}\nn\\
&\leq&
\left(\prod_{1\leq i<j\leq N}\left|\frac{(a_i-b_i)+(a_j-b_j)}{(a_j-a_i)+(b_j-b_i)}\right|^{c_{ij}}\right)
\;\prod_{1\leq i\leq N}\|A_i\|\,.
\eeam
If we additionally assume
\be
a_1-b_1=a_2-b_2=...=a_N-b_N\,,
\ee
we can introduce
conformal cross ratios and get
\beam
\lefteqn{
|(\,\OM,\,\left(\prod_{1\leq i\leq N}A_i\right)\OM\,)|}\nn\\
&\leq&
\left(\prod_{1\leq i<j\leq N}\left(\frac{(a_i-b_i)\,(a_j-b_j)}{(a_i-a_j)\,(b_i-b_j)}\right)^{c_{ij}/2}\right)
\;\prod_{1\leq i\leq N}\|A_i\|\,.
\eeam
\medskip
{\bf Proof:} If we pay attention to the obvious modifications needed for the
additional variables, we can use in this proof the assumptions,
the notation, and the line of argument introduced in the
proof of the cluster theorem in the case of three-point functions. \\
We choose an arbitrary set of exponents $c_{ij}$ fulfilling
the consistency conditions
\be
\sum_{j=1\atop j\neq i}^N c_{ij}=2n_i\,,\ c_{ij}=c_{ji}\,,\ i=1,2,3,...\,,N\,.
\ee
Let $R>0$\,. We consider the generator ${\rm \bf H}_R$ of the
following one-parameter
subgroup of $SL(2,\R)$\,:
\be
g_t\,:\,x\longmapsto\frac{x\,
\mb{cos}\frac{t}{2}+R\,\mb{sin}\frac{t}{2}}
{-\frac{x}{R}\,\mb{sin}\frac{t}{2}+\mb{cos}\frac{t}{2}}\,.
\ee
We know that ${\rm \bf H}_R$ is within each subrepresentation
of $U(SL(2,R))$
unitarily equivalent to the
conformal Hamiltonian ${\rm \bf H}$. Therefore, the spectrum of $A_i\,\OM$
and $A^*_i\,\OM$ with respect to ${\rm \bf H}_R$ is bounded from below by $n_i$\,, $i=1,2,...\,,N$\,.
Let $0<t^-_{ij}<t^+_{ij}<2\pi$ such that
\be
g_{t^-_{ij}}(b_i)=a_j
\ee
and
\be
g_{t^+_{ij}}(a_i)=b_j\,,
\ee
for $i,j=1,2,...\,,N$\,, $i<j$\,.
We introduce
\be
F(z_1,...\,,z_N)\ :=\ (\,\OM,\,\left(\prod_{i=1}^N z_{p(i)}^{-{\rm \bf
H}_R}\,A_{p(i)}\,
z_{p(i)}^{{\rm \bf H}_R}\right)\,
\OM\,)
\ee
in a domain of definition given by
\be
|z_{p(1)}|<|z_{p(2)}|<...<|z_{p(N)}|
\ee
with permutations
$(\,p(1),p(2),...\,,p(N)\,)$ of $(\,1,2,...\,,N\,)$\,.
This definition can uniquely be extended
in analogy to the case of three-point functions to boundary points
with $|z_{j}|=|z_{k}|$\,, $j,k=1,2,...\,,N$\,, $j\neq k$\,,
if
\be
g_{t_k}([a_k,b_k])\cap g_{t_j}([a_j,b_j])\neq\emptyset\,,
\ee
thereby introducing
\be
t_i:=-i\log z_i\,,\ i=1,2,...\,,N\,.
\ee
The line of argument presented above in the case of three-point functions
and developed for general Wightman functions in \cite{StW,Jos} proves that
this continuation is
still an analytic function. We then define
\be
G(z_1,...\,,z_N)\ :=\ F(z_1,...\,,z_N)\,\prod_{1\leq i<j\leq N}\,(\frac{z_i}{z_j}-z^0_{ij})^{c_{ij}/2}\,(\frac{z_j}{z_i}-z^0_{ji})^{c_{ji}/2}\,,
\ee
using the abbreviation
\be
z_{ij}^0:=e^{i(t^-_{ij}+t^+_{ij})/2}\,,\ i,j=1,2,...\,,N\,.
\ee
This function is constructed such that with the consistency conditions
for $c_{ij}$ and with the bound on the spectrum of
${\rm \bf H}_R$ we get the following result in analogy to the cluster theorem for
three-point functions:
At the boundary points $z_i=0$ and $z_i=\infty$\,, $i=1,2,...\,,N$\,, the function
$G$ is bounded
and
can therefore be analytically continued.
As in the case of three-point functions, we get with the maximum principle for
analytic functions further estimates
on $G$\,: Iterating the
well-known maximum
principle argument for the single variables, one obtains
\be
|G(1,...\,,1)|\ \leq\ \mbox{sup}_B\
|G(z_1,...\,,z_N)|\,,
\ee
where $B$ denotes the set of boundary points
\be
B\ :=\
\{\,|z_{j}|=|z_{k}|\ |\ g_{t_k}([a_k,b_k])\,\cap\,
g_{t_j}([a_j,b_j])
\neq\emptyset\,,\ j\neq k
\}
\ee
with
$t_i=-i\log z_i$\,, $i=1,2,...\,,N$\,.
The supremum of the boundary values of
$G$ can be calculated in full analogy to the
case of the three-point functions and to the proof of the conformal
cluster theorem (cf.\ \cite{FrJ}). We obtain straightforward:
\beam
|(\,\OM,\,\left(\prod_{1\leq i\leq N}A_i\right)\OM\,)|
&\leq&
\left(\prod_{1\leq i\leq N}\|A_i\|\right)\,
\prod_{1\leq i<j\leq N}\,\left|\frac{\mb{sin}\frac{t^-_{ij}-t^+_{ij}}{4}}{\mb{sin}
\frac{t^-_{ij}+t^+_{ij}}{4}}\right|^{c_{ij}}.\;\;
\eeam
This estimate
converges in the limit $R\downarrow 0$ with
\be
\lim_{R\rightarrow\infty}R\,t^-_{ij}=2(a_j-b_i)
\ee
and
\be
\lim_{R\rightarrow\infty}R\,t^+_{ij}=2(b_j-a_i)
\ee
for $i,j=1,2,...\,,N$
to the first bound asserted
in the theorem. If we assume
\be
a_1-b_1=a_2-b_2=...=a_N-b_N\,,
\ee
we
find
\be
\left(\frac{t^-_{ij}-t^+_{ij}}{t^-_{ij}+t^+_{ij}}
\right)^2=\frac{(a_i-b_i)\,(a_j-b_j)}{(a_i-a_j)\,(b_i-b_j)}=r_{ij}\,,\;\;\;\,
i,j=1,2,...\,,N\,,
\ee
and
get the second bound. Hence, the theorem is proven.\hfill $\Box$
\medskip
For each consistent set of exponents
$c_{ij}$\,, $i,j=1,2,3,...\,,N$\,, we have proved a different bound on conformal
four-point functions of chiral local observables. Hence, we know that
the algebraic N-point function
\be
(\,\OM,\,\left(\prod_{1\leq i\leq
N}\,U(-x_i)\,A_i\,U(x_i)\right)\,\OM\,)
\ee
decreases in position space at
least as fast
as the set of associated pointlike N-point functions known from conventional conformal field theory. Therefore,
the Fourier transform
of the algebraic
N-point function has to be
at least as regular in momentum space
as the Fourier transforms of the associated pointlike
N-point functions
known from
conventional conformal field theory.
Technically, we follow the line of argument in the case of
three-point functions and use the formula
\be
{\cal F}(\mbox{Pol}(X)S)=\mbox{Pol}\left(\frac{\partial}{\partial Y}\right){\cal F}S
\ee
for arbitrary temperate distributions $S$ and polynomials Pol$(\cdot)$
with a (multi-dimensional)
variable $X$ in position space and an appropriate associated differential operator
$\frac{\partial}{\partial Y}$ in momentum space.
$\F$ denotes the Fourier transformation from position space to
momentum space.
Now, we choose $S$ to be an algebraic N-point function
\be
(\,\OM,\,\left(\prod_{1\leq i\leq
N}\,U(-x_i)\,A_i\,U(x_i)\right)\,\OM\,)
\ee
of local observables $A_i$ with minimal conformal dimensions
$n_i\,,\ i=1,2,...\,,N\,,$
and $X$ to be a tuple of $N-1$ algebraicly independent difference
variables
out of $x_i-x_j$\,, $i,j=1,2,...\,,N\,.$
The estimates in the
cluster theorem proved above imply, that appropriate homogeneous polynomials $\mbox{Pol}(X)$ of degree
\be
\mbox{deg}(\mbox{Pol})\ =\ \left(\sum_{i=1}^N n_i\right)
-2N+2
\ee can be found such that the product $\mbox{Pol}(X)\,S$ is
still absolutely integrable in position space.
We then see that
$\mbox{Pol}(\frac{\partial}{\partial Y}){\cal F}S$ is continuous and bounded in
momentum space.
By locality and the condition of positive energy, the Fourier transform $F$
of an arbitrary (truncated) algebraic N-point function is known to be of the form
\be
F(p_1,...\,,p_{N-1})\ =\ G(p_1,...\,,p_{N-1})\ \prod_{i=1}^{N-1}\,\Theta(p_i)\,,
\ee
where $G$ denotes a sum of restrictions
of appropriate analytic functions to subsets of momentum space (cf.\
the case of three-point functions in the section above).
One can now proceed in analogy to the argumentation in the case of
three-point functions: In a situation with conformal covariance
and minimal conformal dimensions $n_i$\,, $i=1,2,...\,,N$\,,
the function $G$
can be expressed as the product of an appropriate
homogeneous polynomial $P$ of degree
\be
\mbox{deg}(P)\ =\ \left(\sum_{i=1}^N n_i\right)
-N+1
\ee
and an appropriate function $H$\,, where $H$ denotes another sum of
restrictions of analytic functions to subsets of momentum space.
Hence, we have proved that the Fourier transform of the algebraic
N-point function
\be
(\,\OM,\,\left(\prod_{1\leq i\leq
N}\,U(-x_i)\,A_i\,U(x_i)\right)\,\OM\,)\,
\ee
can be written as
\be
F(p_1,...\,,p_{N-1})\ =\ P(p_1,...\,,p_{N-1})\ H(p_1,...\,,p_{N-1})\
\prod_{i=1}^{N-1}\,\Theta(p_i)
\ee
with an appropriate homogeneous function $P$ of degree
\be
\mbox{deg}(P)\ =\ \left(\sum_{i=1}^N n_i\right)
-N+1
\ee
and
an appropriate continuous and bounded function $H$\,.
Using this result, we can now show in full analogy to the procedure in the last
section
that by
canonically scaling an algebraic N-point function we construct a
sequence of distributions that converges to an appropriate pointlike
localized N-point function of conventional conformal field theory:
\beam
&&\nn\\
\lefteqn{ \lim_{\lambda\downarrow 0}\l^{-\left(\sum_{1\leq i\leq
N}n_i\right)}
\
(\,\OM,\,\left(\prod_{1\leq i\leq N}\,U(-
\frac{x_i}{\lambda})\,A_i\,U(
\frac{x_i}{\lambda})\right)\,\OM\,)}
\nn\\
&&\nn\\
&=&\lim_{\lambda\downarrow 0}\l^{-\left(\sum_{1\leq i\leq
N}n_i\right)}
\ {\cal F}_{p_i\rightarrow x_i-x_i+1}\
F(\l p_1,...\,,\l p_{N-1})\,\l^{N-1}\,\prod_{1\leq i\leq N-1}dp_i\nn\\
&&\nn\\
&=&\lim_{\lambda\downarrow 0}
\ {\cal F}_{p_i\rightarrow x_i-x_i+1}\
P(p_1,...\,,p_{N-1})\
H(\l p_1,...\,,\l p_{N-1})\
\prod_{1\leq i\leq N-1}\ \Theta(p_i)\ dp_i\nn\\
&&\nn\\
&=&\left(\prod_{1\leq i<j\leq N}\frac{1}{(x_j-x_i+i\e)^{c_{ij}}}\right)\;
f(r_{t_1u_1}^{v_1s_1},...\,,r_{t_{N-3}u_{N-3}}^{v_{N-3}s_{N-3}})\,.\\
&&\nn
\eeam
Again, $f(\cdot,...\,,\cdot)$
denotes an appropriate function
depending on $N\!-\!3$ algebraicly independent conformal cross ratios
\be r_{tu}^{vs}:=\frac{(x_v-x_s)}{(x_v-x_t)}\frac{(x_t-x_u)}{(x_s-x_u)}\,.
\ee
The exponents $c_{ij}$ must fulfill the consistency conditions
\be
\sum_{j=1\atop j\neq i}^N c_{ij}=2n_i\,,\ c_{ij}=c_{ji}\,,\ 1\leq i\leq N\,,
\ee
which
do not fully determine the exponents.
Hence, the general form of the pointlike localized conformal correlation
functions constructed from algebraic quantum field theory has been
determined to be exactly the general form of the N-point functions known from
conventional conformal field theory. In both approaches conformal
covariance
does not fully determine the form of N-point functions for $N>4$\,.
\subsection{Wightman Axioms and Reconstruction Theorem}
The most common axiomatic system for pointlike localized quantum
fields is the formulation of Wightman axioms given in \cite{StW} and
\cite{Jos}. (If braid group
statistics has to be considered and the Bose-Fermi alternative does not hold in general, the classical formulation of
\cite{StW} and \cite{Jos} has to be modified for the charged case by introducing the axiom of weak locality
instead of locality \cite{FRS1,FRS2}.)
The
construction of pointlike localized correlation functions in this paper uses sequences of algebraic correlation functions of local
observables.
The algebraic correlation functions obviously fulfill positive
definiteness,
conformal covariance, locality, and the spectrum
condition.
Hence,
if the sequences converge, the set of pointlike limits of
algebraic correlation functions fulfills the Wightman axioms (see
\cite{StW}) by
construction.
By the reconstruction theorem in \cite{StW} and \cite{Jos}, the
existence of Wightman fields associated with the Wightman functions is
guaranteed and this Wightman field theory is unique up to unitary
equivalence.
We do not know at the moment whether the Wightman fields
can be identified with the pointlike localized field operators
constructed in \cite{FrJ} from the
Haag-Kastler theory. We do
not know either whether the Wightman fields are affiliated to the
associated von Neumann algebras of local observables and how the Haag-Kastler net we
have been starting from can be reconstructed from the Wightman
fields. Possibly, the Wightman fields cannot even be realized in the
same Hilbert space as the Haag-Kastler net of local observables.
We do know, however, that the Wightman theory associated with
the Haag-Kastler theory is non-trivial: The two-point functions of this
Wightman fields are, by construction,
identical with the two-point functions of the pointlike localized
field operators constructed in \cite{FrJ}.
And
we have already proved that those pointlike field vectors can be chosen to
be non-vanishing and that the vacuum vector is cyclic for a set of all
field operators localized in an arbitrary interval.
It shall be
pointed out again that those pointlike fields constructed in
\cite{FrJ,Joe3} could not be proved to fulfill the Wightman axioms, since we were not
able to find a domain of definition that is stable under the
action of the field operators.
To summarize this paper, we state that starting from a chiral conformal
Haag-Kastler theory we have found a canonical construction of
non-trivial Wightman fields. The reconstruction of
the original net of von Neumann algebras of local observables from the Wightman fields
could not explicitly be presented, since we do not know whether the
Wightman fields can be realized in the same Hilbert space as the
Haag-Kastler net.
Actually, Borchers and Yngvason \cite{BoY} have investigated similar
situations and have shown that such problems can occur in quantum field theory. In \cite{BoY} the question is discussed under which
conditions a Haag-Kastler net can be associated with a Wightman
theory.
The condition for the locality of the associated algebra net turned out be
a property of the Wightman fields called ``central positivity". Central positivity is
fulfilled for Haag-Kastler nets and is stable under pointlike limits \cite{BoY}.
Hence, the Wightman fields constructed in this thesis fulfill
central positivity.
The
possibility, however, that the local net has to be defined in an enlarged
Hilbert space could not be ruled out in general by
\cite{BoY}.
Furthermore, it has been proved in \cite{BoY} that Wightman fields
fulfilling generalized H-bounds (cf.\ \cite{DSW}) have associated
local nets of von Neumann algebras
that can be defined in the same Hilbert space. The closures of the
Wightman field operators are then affiliated to the associated local
algebras.
We could not prove generalized H-bounds for the Wightman fields
constructed in this thesis. Actually, we suppose that the
criterion of generalized H-bounds is too strict for general conformal --
and therefore massless -- quantum field theories. (Generalized) H-bounds have been proved,
however,
for massive theories, i.e.\ for models in quantum field theory with
massive
particles (cf.\ also \cite{DrF,FrH,Sum,Buc1}).
\paragraph{Acknowledgements}\nichts\\
This paper is one part of the author's dissertation.
We would like to thank Prof.\,Dr.\,K.\,Fre\-den\-ha\-gen for his
confidence, constant encouragement, and the numerous inspiring
discussions over the whole period of the work. I am indebted to him
for many important insights I received by his guidance. His
cooperation was crucial and fruitful for this work.
The financial support given by the Friedrich-Ebert-Stiftung is
gratefully acknowledged.
{\small
| proofpile-arXiv_065-439 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
A goal of the future heavy ion collision experiments at the
relativistic heavy ion collider (RHIC) at Brookhaven and at
the large hadron collider (LHC) at CERN is to find the
quark-gluon plasma. The primary aim is of course to show
that quarks and gluons can indeed be freed from their hadronic
``prison'' and exist as individual entities in a hot plasma.
Once this is realized, one can then turn to the diverse
physics of such a new state of matter. One of these is
the relation of the various thermodynamic variables to
each other or in other words, the equation of state \cite{shury1}.
In order to probe this in experiments, an equilibrated quark-gluon
plasma is required. In this work, we look at how far can one
expect to have such a plasma in equilibrium. Because of the
importance of this question, various different approaches have
already been taken to address this issue. In particular, Shuryak
\cite{shury2} argued that equilibration of the plasma
proceeds via two stages in the ``hot gluon scenario''. First the
equilibration of the gluons and then that of the quarks follows
with a certain time delay. Thermal equilibration is quite short
for gluon $\le$ 1 fm with high initial temperature of 440 MeV at
LHC and 340 MeV at RHIC. However, these estimates are based
on thermal reaction rates for large and small angle scatterings
and on the assumption that one scattering is sufficient to
achieve isotropy of momentum distribution. As has been shown
in \cite{heis&wang1} using a family of different power behaviours
for the time-dependence of the collision time, the assumption of
one scattering is sufficient is a serious underestimate. With
a larger number of scatterings, using the same arguments as
in \cite{shury2}, the initial temperature will be lowered and
the thermalization time will be increased. Also, we argue that
estimates based on using the scattering rate alone is incorrect,
since in a medium, one must consider the difference of the
scattering going forward and backward both weighed with
suitable factors of particle distribution functions. Hence, the
process with the largest cross-section is not necessarily the
more important. However, we will show the two-stage
equilibration scenario or in other words, gluons equilibrate
much faster than quarks and antiquarks.
The other approach is the semi-classical parton cascade model
(PCM) \cite{geig&mull,geig,geig&kap}, which is based on solving a
set of relativistic transport equations in full six-dimensional
phase space using perturbative QCD calculation for the interactions,
predicts an equilibration time of 2.4 fm/c for Au+Au collision
at 200 GeV/nucleon. This approach, which uses a spatial and
momentum distribution obtained from the measured nuclear structure
functions for the partons as initial state, is very complicated.
Due to the finite size of the colliding nuclei, it is hard
to clearly identify thermalization in terms of the expected
time-dependent behaviours of the various collective variables
\cite{geig}. But by fitting the total particle rapidity and
transverse momentum distributions of the defined central volume,
roughly identical temperatures are obtained \cite{geig} and hence
the claim of thermalization. However, in terms of the same
distributions of the individual parton components, this becomes
less obvious to be the case \cite{geig&kap}. As was stated in
\cite{geig&kap}, the momentum distributions are not perfect
exponentials and therefore there is no complete thermalization
in any case.
We will look at this problem of equilibration using a much
simpler approach which is based on the Boltzmann equation and
the relaxation time approximation for the collision terms.
Initially used by Baym \cite{baym} to study thermal equilibration
and has subsequently been used in the study
of various related problems \cite{gavin,kaj&mat,heis&wang2,wong}.
The conclusion of these works is, in general, if the collision time
$\theta$ which enters in the relaxation approximation, grows less fast
than the expansion time $\tau$, then thermal equilibration can be
achieved eventually. In the case of the quark-gluon plasma, it is
not sufficient to know that equilibration will be achieved
eventually because the plasma has not an infinite lifetime in
which to equilibrate. We would like to know how far can it
equilibrate before the phase transition. To answer such a
question, we will use both the relaxation time approximation
and the interactions obtained from perturbative QCD for
the collision terms to determine $\theta$. This approach has been
used previously to study both thermal and chemical equilibration
in a gluon plasma \cite{wong} where it was found that with the
initial conditions obtained from HIJING results, the gluon plasma
had not quite enough time to completely equilibrate. In the
present case of a quark and gluon parton plasma, quarks and gluons
are treated as different particle species rather than as generic
partons and so they have different time-dependent collision times.
As a result, they approach equilibrium at different rates and
towards different target temperatures. The latters will converge only
at large times. It follows that the system can only equilibrate as one
single system at large times. This lends support to the two-stage
equilibration scenario \cite{shury2}.
In an expanding system, particles are not in equilibrium early on
because interactions are not fast enough to maintain this so they
are most likely to start off free streaming in the beam direction
\cite{heis&wang2,dan&gyul}. Thermalization will be seen as the
gradual reduction of this free streaming effect as interactions gain
pace and momentum transfer processes are put into action to bring
the particle momenta into an isotropic distribution. The present
approach takes into account of these effects.
As in the previous work \cite{wong}, isotropic momentaneously
thermalized initial conditions are used at both RHIC and LHC
energies. These are obtained from HIJING results after allowing
the partons to free stream until the momentum distribution
becomes isotropic for the first time \cite{biro&etal1,lev&etal,wang}.
From then on, interactions are turned on but the distribution
becomes anisotropic again due to the tendency of the particles
to continue to free stream. It is the role of interactions to
reduce this and to progressively bring the distributions into
the equilibrium forms. We have shown that, surprisingly,
kinetic equilibration in a pure gluon plasma is driven
mainly by gluon multiplication and not gluon-gluon elastic
scattering. In this paper, we include quarks and antiquarks
and consider the equilibration of a proper QCD plasma. We
explicitly break down the equilibration process into each of
its contributing elements and show which interactions are more
important and hence uncover the dominant processes for
equilibration. In fact, our result is {\em inelastic}
interactions are most important for this purpose both for
quarks and for gluons.
Our paper is organized as follows. In Sect. \ref{sec:relax}, we
describe the Boltzmann equations with the relaxation time approximation
for two particle species. In Sect. \ref{sec:therm}, the time-dependent
behaviour of the collision times, $\theta$'s, necessary for equilibration
will be analysed and extracted. The particle interactions entering
into the collision terms and details of their calculations will be
explained in Sect. \ref{sec:cal}. Initial conditions used will be
given in Sect. \ref{sec:ic} and lastly the results of the evolution
of the plasma will be shown and discussed in Sect. \ref{sec:result}.
We finish with a brief discussion of the differences with the
results of PCM.
\section{Relaxation Time Approximation for Two Particle Species}
\label{sec:relax}
In the absence of relativistic quantum transport theory derived from
first principle of QCD
\cite{heinz1,elze&etal,elze&heinz,elze,heinz2,heinz3},
we base our approach on Boltzmann equation with both the
relaxation time approximation for the collision terms
and the real collision terms obtained from perturbative QCD.
Treating quarks and gluons on different footings, we
write down the Boltzmann equations
\begin{equation} {{\mbox{$\partial$} f_i} \over {\mbox{$\partial$} t}}+{\vo v}_{{\mathbf p}\; i} \!\cdot\!
{{\mbox{$\partial$} f_i} \over {\mbox{$\partial$} {\mathbf r}}} = C_i ({\mathbf p},{\mathbf r},t)
\end{equation}
where $f_i$ is the one-particle distribution and $C_i$ stands
for the collision terms and includes all the relevant
interactions for particle species $i$ and $i=g, q,\bar q$.
Concentrating in the central region of
the collision where we assumed to be spatially homogeneous,
baryon free and boost invariant in the z-direction (beam direction)
so that $f_q=f_{\bar q}$ and $f_i=f_i({\mathbf p}_\perp,{\mathbf p}'_z,\tau)$ where
$p'_z =\gamma (p_z-u p)$ with $\gamma=1/\sqrt {1-u^2}$ and $u=z/t$
is the boosted particle z-momentum component and $\tau =\sqrt {t^2-z^2}$
is the proper time. Following Baym \cite{baym}, the Boltzmann equation
can be rewritten as
\begin{equation} {{\mbox{$\partial$} f_i} \over {\mbox{$\partial$} \tau}} \Big |_{p_z \tau}
=C_i(p_\perp,p_z,\tau)
\label{eq:baymeq}
\end{equation}
in the central region. Using the relaxation time approximation
\begin{equation} C_i(p_\perp,p_z,\tau)=-
{{f_i(p_\perp,p_z,\tau)-f_{eq \; i}(p_\perp,p_z,\tau)}
\over \theta_i(\tau)} \;
\label{eq:relaxapp}
\end{equation}
where $f_{eq \; i}$ is the equilibrium distribution
and $\theta_i$ is the collision time for species $i$,
this allows us to write down a solution to \eref{eq:baymeq}.
\begin{equation} f_i({\mathbf p},\tau)=f_{0\; i}(p_\perp,p_z \tau/\tau_0) e^{-x_i}
+\int^{x_i}_0 dx'_i e^{x'_i-x_i}
f_{eq\; i}(\sqrt{p^2_\perp+(p_z \tau/\tau')^2},T_{eq\; i}(\tau')) \; ,
\label{eq:baymeqsol}
\end{equation}
where
\begin{equation} f_{0\; i}(p_\perp, p_z \tau/\tau_0) = \Big ( \mbox{\rm exp}
(\sqrt {p^2_\perp + (p_z \tau/\tau_0)^2}/T_0)/l_{0\; i}
\mp 1 \Big )^{-1} \; ,
\end{equation}
is the solution to \eref{eq:baymeq} when $C=0$ which is also the
distribution function at the initial isotropic time $\tau_0$,
with initial fugacities $l_{0\; i}$ and temperature $T_0$.
It is of such a form because of the assumption of momentaneously
thermalized initial condition. The functions $x_i(\tau)$'s, given by
\begin{equation} x_i(\tau)=\int^\tau_{\tau_0} d\tau'/ \theta_i(\tau') \; ,
\end{equation}
play the same role as $\theta_i$'s in the sense that their
time-dependent behaviours control thermalization.
$T_{eq\; i}$, that appears in $f_{eq\; i}$, is the
time-dependent momentaneous target equilibrium temperature
for the $i$ particle species. The two terms of equation
\eref{eq:baymeqsol} can be thought of, up to exponential
factor, as the free streaming (first term) and equilibrium
term (second term). Whether species $i$ equilibrates or not
depends on which of the two terms dominates.
In the present case of two species, the energy conservation
equations are, in terms of the equilibrium ideal gas energy densities
$\epsilon_{eq\; g}=a_2 \Tg^4$, $\epsilon_{eq\; q}=n_f b_2 \Tq^4$, $a_2=8 \pi^2/15$,
$b_2=7\pi^2/40$ and $n_f$ is the number of quark flavours,
\begin{equation} {{d \epsilon_i} \over {d \tau}}+{{\epsilon_i+p_{L\; i}} \over \tau}
=-{{\epsilon_i-\epsilon_{eq\; i}} \over \theta_i}
\end{equation}
and
\begin{equation} {{d \epsilon_{tot}} \over {d \tau}}+{{\epsilon_{tot}+p_{L\; tot}} \over \tau} =0
\; ,
\end{equation}
where $\epsilon_{tot}=\sum_i \epsilon_i$ and $p_{L\; tot}=\sum_i p_{L\; i}$,
or in other words
\begin{equation} \sum_i {{\epsilon_i-\epsilon_{eq\; i}} \over \theta_i} =0 \; .
\label{eq:e_cons}
\end{equation}
The above equation only expresses the fact that energy loss of one
species must be the gain of the other. The transport equations
of the different particle species are therefore coupled as they
should be. The longitudinal and transverse pressures are defined
as before
\begin{equation} p_{L,T\; i}(\tau)=\nu_i \int \frac{d^3 \P}{(2\pi)^3} {{p_{z,x}^2} \over p}
f_i (p_\perp,p_z,\tau) \; ,
\label{eq:pres}
\end{equation}
with $\nu_g=2\times 8=16$ and $\nu_q=2\times 3\times n_f=6\, n_f$,
the multiplicities of gluons and quarks respectively.
Here the equilibrium target temperatures $\Tg$ and $\Tq$
cannot be the same in general since, as we will see in Sect.
\ref{sec:result}, $\qg \ne \qq =\theta_{\bar q}$. Therefore gluons
and quarks will approach equilibrium at different rates. Note
that energy conservation here {\em does not} mean
\begin{equation} \epsilon_g+2 \epsilon_q = \epsilon_{eq\; g} + 2 \epsilon_{eq\; q}
\label{eq:eeqe}
\end{equation}
since $\qg < \qq$ always, at least at small times, so gluon
energy density $\epsilon_g$ will approach $\epsilon_{eq\; g}$ faster than
$\epsilon_q$ approaches $\epsilon_{eq\; q}$ so the two equilibrium energy
densities should not be considered to be those which can coexist
at the same moment. This can only be true at large $\tau$
when $\Tg \simeq \Tq$ and $\qg \simeq \qq$.
If \eref{eq:eeqe} were true, the condition for energy
conservation \eref{eq:e_cons} could not hold when
$\qg \neq \qq$. Since our QCD plasma is a dynamical
system under one-dimensional expansion as well as particle
production, the target temperatures $\Tg$ and $\Tq$ must be
changing continuously and must approach each other at large times
before the gluon and quark (antiquark) subsystems can merge
into one system and exist at one single temperature.
Likewise, we believe $\qg$ and $\qq$ should also converge
to a single value at large times, unfortunately, this will
take too long to happen in the evolution of our
plasma although we can be sure that both $\qg$ and $\qq$
increase less fast than the expansion time $\tau$ near the end
of the evolution, a condition which, as has already been stated in
the introduction and we will see again in Sect. \ref{sec:therm},
is necessary for thermalization.
\section{Conditions on $\qg$ and $\qq$ for Thermalization}
\label{sec:therm}
Before considering the evolution of the QCD plasma
under real interactions, we can deduce analytically, using
\eref{eq:baymeq} and \eref{eq:baymeqsol}, the conditions
on the $\theta_i$'s under which the plasma will come
to kinetic equilibrium. Multiplying \eref{eq:baymeqsol} by
particle energy and integrating over momentum, we have the
equations for the $\epsilon_i$'s. Further manipulating these gives,
\begin{equation} \int^{x_i}_0 d x'_i \; e^{x'_i} \Big \{
\tau' h(\tau'/\tau) \Big (\epsilon_{eq\; i}(\tau')-\epsilon_i(\tau') \Big )
-{d \over {d x'_i}} \Big (\tau' h(\tau'/\tau) \epsilon_i(\tau') \Big )
\Big \} =0 \; ,
\label{eq:cond_theta}
\end{equation}
where
\begin{equation} h(r)=\int^1_0 dy \sqrt {1-y^2 (1-r^2)}
=\half \bigg (r+{\sin^{-1} {\sqrt {1-r^2}} \over {\sqrt {1-r^2}}}
\bigg )
\end{equation}
and $x'_i=x_i(\tau')$. Supposing as $\tau \rightarrow \infty$, $x_g \rightarrow \infty$
and $x_q \rightarrow \infty$ then the integrand in \eref{eq:cond_theta}
will be weighed by the $\tau' \rightarrow \infty$ or large $x'_i$
limit. It follows that the term within braces in \eref{eq:cond_theta}
must be zero at large $\tau'$ so using $h'(r)|_{r=1}=1/3$, we have
\begin{equation} {{d \epsilon_i} \over {d \tau}}+{4 \over 3} {\epsilon_i \over \tau}
= -{{\epsilon_i-\epsilon_{eq\; i}} \over \theta_i} \; .
\label{eq:hydro_i}
\end{equation}
This means each species will undergo near hydrodynamic expansion
at large $\tau$ modified by energy lost to or energy gained from
the other species. The latter should be small at such times.
Summing \eref{eq:hydro_i} over species, we obtain the energy
conservation equation for a system undergoing hydrodynamic
expansion
\begin{equation} {{d \epsilon_{tot}} \over {d \tau}}+{4 \over 3} {\epsilon_{tot} \over \tau}
= 0 \; ,
\label{eq:hydro}
\end{equation}
with $p_{L\; tot}=\epsilon_{tot}/3$.
If one $\theta_i$ is such that the corresponding
$x_i \rightarrow x_{i\; \infty} < \infty$ as $\tau \rightarrow \infty$
then hydrodynamic expansion does not apply to that species since
we have
\begin{equation} {{d (\epsilon_i \tau)} \over {d \tau}} =-{{(\epsilon_i-\epsilon_{eq\; i}) \tau} \over \theta_i}
-p_{L\; i} \; ,
\label{eq:x->x<inf}
\end{equation}
where now $p_{L\; i} \neq \epsilon_i/3$, so kinetic equilibrium is
not established. The r.h.s. of \eref{eq:x->x<inf} is negative if
these particles are losing energy or gaining energy at a rate
less than $p_{L\; i}/\tau$ at large $\tau$. Therefore $\epsilon_i \tau$
must decrease towards a non-zero asymptotic value $(\epsilon_i \tau)_\infty$,
since $x_{i\; \infty} < \infty \Longrightarrow \epsilon_i \tau >0$
always, which results in a free streaming final state for
these particles
\begin{equation} \epsilon_i(\tau \rightarrow \infty) \sim (\epsilon_i \tau)_\infty /\tau \; .
\label{eq:asymp_e}
\end{equation}
A similar free streaming final state will be reached if the rate
of gaining energy is larger than $p_{L\; i}/\tau$ at large $\tau$.
In this case, although $\epsilon_i \tau$ is increasing, the $\epsilon_j$ of
the other particle species with $x_j \rightarrow \infty$ as
$\tau \rightarrow \infty$ will be close to $\epsilon_{eq\; j}$ and
so the energy transfer will be very small. One can deduce
that as $\tau \rightarrow \infty$
\begin{equation} 1 \gg {{\epsilon_{eq\; i}-\epsilon_i} \over \theta_i} \rightarrow 0
\; > {p_{L\; i} \over \tau} \; \Longrightarrow \;
{{d (\epsilon_i \tau)} \over {d \tau}} \rightarrow 0 \; ,
\label{eq:onlot}
\end{equation}
hence $\epsilon_i \tau\rightarrow (\epsilon_i \tau)_\infty$. That is
$\epsilon_i \tau$ now increases towards some asymptotic value instead
of decreasing towards one as in the previous case.
But it ends up with a free streaming final state nevertheless.
We do not consider the case where the relative rate
$(\epsilon_{eq\; i}-\epsilon_i)\tau/\theta_i p_{L\; i}$ oscillates about one at
large $\tau$ except to say that on the average
$d (\epsilon_i \tau)/d \tau \sim 0$ and so an average free streaming final
state is likely.
The last possibility where
$x_i \rightarrow x_{i\; \infty} <\infty$ as $\tau \rightarrow \infty$
for both particle species, \eref{eq:x->x<inf} applies to both.
Barring the case of the oscillating relative rate, one particle
species must lose energy and so by the above argument, a free
streaming final state results. For the remaining particle species,
it does not matter whether $d (\epsilon_i \tau)/d\tau$ is or is
not positive at large $\tau$, these particles will also be in
a free streaming final state. If the rate is negative, then the
same argument that leads to \eref{eq:asymp_e} applies. If it is
positive, since the species that is losing energy is approaching
free streaming so the energy transfer must go to zero. Then
we are back to \eref{eq:onlot}.
The conclusions are therefore, depending on the time-dependent
behaviours of $\qg$ and $\qq$,
\begin{enumerate}
\item $x_g \rightarrow \infty$ and $x_q \rightarrow \infty$ as
$\tau \rightarrow \infty$ are required for the whole system to
completely thermalize.
\item $x_g \rightarrow \infty$ and $x_q \rightarrow x_{q\; \infty}
< \infty$ or $x_q \rightarrow \infty$ and $x_g \rightarrow
x_{g\; \infty} < \infty$ as $\tau \rightarrow \infty$
imply that only the species with $x_i \rightarrow \infty$ will
thermalize, the other species will not equilibrate but free
streams at the end. The system will end up somewhere between
free streaming and hydrodynamic expansion.
\item Both $x_g \rightarrow x_{g\; \infty} <\infty$ and
$x_q \rightarrow x_{q\; \infty} <\infty$ as $\tau \rightarrow \infty$
then the whole system will end up in a free streaming final state.
\end{enumerate}
One can understand these $x_i$ behaviours in terms of $\theta_i$'s
by assuming simple power $\tau$-dependence for the latters.
One finds that $\theta_i$'s must all grow slower than $\tau$ for
the whole system to achieve thermalization. If either one
or more grow faster then a mixed or a complete free
streaming final state results.
\section{Particle Interactions --- Collision Terms}
\label{sec:cal}
To investigate the evolution of a proper QCD plasma, we consider
the following simplest interactions at the tree level
\begin{equation} gg \longleftrightarrow ggg \; \; , \; \; \; gg \longleftrightarrow gg \; ,
\label{eq:ggi}
\end{equation}
\begin{equation} gg \longleftrightarrow q\bar q \; \; , \; \; \; g q \longleftrightarrow g q \; \; ,
\; \; \; g\bar q \longleftrightarrow g\bar q \; ,
\label{eq:gqi}
\end{equation}
\begin{equation} q\bar q \longleftrightarrow q\bar q \; \; , \; \; \; qq \longleftrightarrow qq \; \; ,
\; \; \; \bar q \bar q \longleftrightarrow \bar q \bar q \; .
\label{eq:qqi}
\end{equation}
As in \cite{biro&etal1,lev&etal,wang}, we include only the
leading inelastic processes i.e. the first interaction of
\eref{eq:ggi} and \eref{eq:gqi}\footnote{The first one of
\eref{eq:qqi} could also be inelastic but here we give the same
chemical potential to all the fermions so we do not consider
quark-antiquark annihilations into different flavours as
inelastic for our purpose.}. We will return to this point
later on in Sect. \ref{sec:result}.
In the solutions \eref{eq:baymeqsol} to the Boltzmann equations
\eref{eq:baymeq}, there are two time-dependent unknown parameters
$\theta_i$ and $T_{eq\;i}$ for each species which very much control
the particle distributions. To determine them, we need two
equations each for gluons and for quarks. In order to show the
relative importance of the various interactions
\etrref{eq:ggi}{eq:gqi}{eq:qqi} in equilibration, we find
these time-dependent parameters by constructing equations from
the rates of energy density transfer between quarks (antiquarks)
and gluons and the collision entropy density rates.
From \etrref{eq:baymeq}{eq:relaxapp}{eq:baymeqsol},
the energy density transfer rates are
\begin{equation} {{d \epsilon_i} \over {d \tau}}+{{\epsilon_i + p_{L\; i}} \over \tau}
= -{{\epsilon_i-\epsilon_{eq\; i}} \over \theta_i}
= \nu_i \int \frac{d^3 \P}{(2\pi)^3} \; p \; C_i (p_\perp,p_z,\tau) = {\cal E}_i \; ,
\label{eq:e_trans}
\end{equation}
where ${\cal E}_i$ is the energy gain or loss of species $i$ per unit
time per unit volume. As stated in Sect. \ref{sec:relax}, ${\cal E}_i$'s
must obey $\sum_i {\cal E}_i =0$ for energy conservation.
The other equations, the collision entropy rates can be deduced
from the explicit expression of the entropy density in terms
of particle distribution function \cite{groot}
\begin{equation} s_i(\tau)=-\nu_i \int \frac{d^3 \P}{(2\pi)^3} \Big \{f_i({\mathbf p},\tau) \ln f_i({\mathbf p},\tau)
\mp (1 \pm f_i({\mathbf p},\tau)) \ln (1 \pm f_i({\mathbf p},\tau)) \Big \} \; ,
\end{equation}
where the different signs are for bosons and fermions respectively.
They are, using again \etrref{eq:baymeq}{eq:relaxapp}{eq:baymeqsol},
\begin{eqnarray} \Big ( {{d s_i} \over {d \tau}} \Big )_{coll} \fx & = & \fx
-\nu_i \int \frac{d^3 \P}{(2\pi)^3} \Big ( {{\mbox{$\partial$} f_i} \over {\mbox{$\partial$} \tau}} \Big )_{coll}
\ln \Big ({{f_i} \over {1 \pm f_i}} \Big ) \\
\fx & = & \fx -\nu_i \int \frac{d^3 \P}{(2\pi)^3} \; C_i (p_\perp,p_z,\tau)
\ln \Big ({{f_i} \over {1 \pm f_i}} \Big )
\label{eq:s_rate1} \\
\fx & = & \nu_i \int \frac{d^3 \P}{(2\pi)^3} {{f_i -f_{eq\; i}} \over \theta_i}
\ln \Big ({{f_i} \over {1 \pm f_i}} \Big ) \; .
\label{eq:s_rate2}
\end{eqnarray}
By using the explicit expression for the collision terms $C_i$'s
constructed from the interactions \etrref{eq:ggi}{eq:gqi}{eq:qqi}
within perturbative QCD, \etrref{eq:e_trans}{eq:s_rate1}{eq:s_rate2}
allow us to solve for $\theta_i$'s and $T_{eq\; i}$'s.
The gluon multiplication contribution to $C_g$ is constructed from
the infrared regularized Bertsch and Gunion formula \cite{bert&gun}
for the amplitude with partial incorporation of
Landau-Pomeranchuk-Migdal suppression (LPM) for gluon emission and
absorption \cite{biro&etal1,gyul&wang,gyul&etal,baier&etal} as
in the previous work \cite{wong}. The explicit form of the
gluon multiplication collision term and a discussion of the
problem regarding how to incorporate the LPM effect correctly
can be found there also. The remaining binary interaction
contributions to $C_i$ for particle $1$ is, as usual, given by
\begin{eqnarray} C_{i\; 1}^{binary} \fx & = & \fx - \sum_{{\cal P}_i}
{{S_{{\cal P}_i} \nu_2} \over {2 p_1^0}} \prod^4_{j=2}
{{d^3 {\mathbf p}_j} \over {(2\pi)^3 2 p_j^0}} (2\pi)^4 \delta^4(p_1+p_2-p_3-p_4)
|{\cal M}^{{\cal P}_i}_{1+2 \rightarrow 3+4}|^2 \nonum
& & \times
[f_1 f_2(1 \pm f_3)(1 \pm f_4) -f_3 f_4 (1 \pm f_1)(1 \pm f_2)]
\end{eqnarray}
where the ${\cal P}_i$ runs over all the binary processes in
\etrref{eq:ggi}{eq:gqi}{eq:qqi} which involve species $i$,
$|{\cal M}^{{\cal P}_i}|^2$ is the sum over final states and averaged over
initial state squared matrix element, $S_{{\cal P}_i}$ is a symmetry
factor for any identical particles in the final states for the
process ${\cal P}_i$ and $\nu_2$ is the multiplicity of particle 2.
We take $|{\cal M}^{{\cal P}_i}|^2 \; $'s from \cite{cut&siv} and infrared
regularized them using either the Debye mass $m_D^2$ for gluons
or the quark medium mass $m_q^2$ for quarks to cut off any infrared
divergence. These masses are now time-dependent quantities
in a non-equilibrium environment. With non-isotropic momentum
distribution, both the Debye mass \cite{biro&etal2,esk&etal}
and the gluon medium mass, $m^2_g$, are directional dependent. This
is, however, not the case for the quark medium mass, $m^2_q$, which
remains directional independent as in equilibrium. The directional
dependence arises out of the cancellations between identical type
of distribution functions similar to those one finds in the
derivation of hard thermal loops \cite{bra&pis,fre&tay}.
To keep things simple, we removed the directional dependence from
$m_D^2$ and use, for SU(N=3), to leading order in $\alpha_s$,
\begin{equation} m^2_D (\tau)=-8 \pi \alpha_s \int \frac{d^3 \P}{(2\pi)^3}
{\mbox{$\partial$} \over {\mbox{$\partial$} |{\mathbf p}|}} \Big (N \, f_g +n_f \, f_q \Big )
\; .
\end{equation}
For the quark medium mass, to the same order, we use
\begin{equation} m_q^2 (\tau)= 4\pi \alpha_s \; \Big ({{N^2-1} \over {2\, N}}\Big )
\int \frac{d^3 \P}{(2\pi)^3} {1 \over {|{\mathbf p}|}} \; \big (f_g +f_q \big ) \; ,
\end{equation}
which is just the equilibrium expression but with non-equilibrium
distribution functions.
With these masses, we regularize the squared matrix elements by hand
and inserting the masses as follows.
\begin{eqnarray} |{\cal M}_{gg \rightarrow gg}|^2 \fx &=& \fx {{9\; g^2} \over 2} \; \bigg (
3-{{u t} \over {(s+m_D^2)^2}} -{{u s} \over {(t-m_D^2)^2}}
-{{s t} \over {(u-m_D^2)^2}} \bigg )
\\
|{\cal M}_{gg \rightarrow q \bar q}|^2 \fx &=& \fx {g^2 \over 6} \; \bigg (
{t \over {(u-m_q^2)}} +{u \over {(t-m_q^2)}} \bigg )
- {3 \over 8} \; {{u^2+t^2} \over {(s+4 m_q^2)^2}}
\\
|{\cal M}_{gq \rightarrow gq}|^2 = |{\cal M}_{g \bar q \rightarrow g \bar q}|^2
\fx &=& \fx g^2 \bigg ( 1- {{2 u s} \over {(t-m_D^2)^2}}
-{4 \over 9} \; \bigg ( {u \over {(s+m_q^2)}}
+{s \over {(u-m_q^2)}} \bigg ) \bigg )
\\
|{\cal M}_{qq \rightarrow qq}|^2 = |{\cal M}_{\bar q \bar q \rightarrow \bar q \bar q}|^2
\fx &=& \fx {{2\; g^2} \over 9} \;
\bigg ( {{2(s^2+t^2)} \over {(u-m_D^2)^2}}
+ \delta_{12} \; {{2(u^2+s^2)} \over {(t-m_D^2)^2}} \nonum
\fx & & \fx \; \; \; \; \; - \delta_{12} \;
{4 \over 3} \; {s^2 \over {(t-m_D^2)(u-m_D^2)}} \bigg )
\\
|{\cal M}_{q \bar q \rightarrow q \bar q}|^2 \fx &=& \fx {{2\; g^2} \over 9} \;
\bigg ( \delta_{13} \delta_{24} {{2(s^2+t^2)} \over {(u-m_D^2)^2}}
+ \delta_{12} \delta_{34} \; {{2(t^2+u^2)} \over {(s+4 m_q^2)^2}} \nonum
\fx & & \fx \; \; \; \; \; - \delta_{12} \delta_{13} \delta_{34} \;
{4 \over 3} \; {t^2 \over {(u-m_D^2)(s+4 m_q^2)}} \bigg )
\end{eqnarray}
where the $\delta_{ij}$ signifies that the $i$ and $j$ quark or
antiquark must be of the same flavour. This regularization amounts
to screening spacelike and timelike infrared gluons by $m^2_D$ and
$4 m^2_q$, respectively and infrared quarks by $m^2_q$. We stress
that this regularization is done in a very simple manner and with the
right order of magnitude for the cutoffs. Its aim is to get some
estimates to the collision rates without involving too much with
the exact and necessarily complicated momentum dependent form
of the true infrared screening self-energies in an out-of-equilibrium
plasma when their infrared screening effects should be in action.
They should be the extension of the 2-point gluon and quark hard
thermal loops \cite{bra&pis,fre&tay,weld,klim,tay&wong} to a
non-thermalized environment.
We should mention here that the choice of the pair of
equations for solving the two time-dependent unknowns $\theta_i$
and $T_{eq\; i}$ for each particle species is not unique.
One can equally use, for example, the rate equations for the
particle number density instead of the collision entropy
density. With these other choices, the values of
the different quantities are shifted somewhat due to the way
that the initial conditions are extracted but there
is no qualitative different in the result. Our present choice
has the distinct advantage that we can explicitly compare the
different processes using the collision entropy density rates.
This will become clear when we show the results in
Sect. \ref{sec:result}.
\section{Initial Conditions}
\label{sec:ic}
To start the evolution, we use the same initial conditions for
the gluon plasma as before \cite{wong} based on HIJING result for
Au+Au collision. The initial conditions for the quarks (antiquarks)
are obtained by taking a ratio of $0.14$ for the number of initial
quark (antiquark) to the initial total number of partons as done in
\cite{biro&etal1,lev&etal,wang}. The initial conditions are shown
in Table 1. One sees that the initial quark collision times
are long compared to those of the gluons both at RHIC and LHC.
Especially at RHIC, the quark collision time is exceedingly long
and so these particles are essentially free streaming initially.
Taking these numbers as guides to how fast each particle
species is going to equilibrate, we can be sure already of a
two-stage equilibration scenario \cite{shury2}.
\begin{center}
\begin{tabular}{|c|c|c|} \hline\hline
\multicolumn{3}{|c|}{Initial Conditions} \\ \hline
\emph{\ } & \ \ \emph{RHIC}\ \ & \ \ \emph{LHC}\ \ \\ \hline
$\tau_0$ (fm/c) & 0.70 & 0.50 \\
$T_0$ (GeV) & 0.50 & 0.74 \\
$\epsilon_{0\; g}$ (GeV/$\mbox{\rm fm}^3$) & 3.20 & 40.00 \\
$\epsilon_{0\; q}$ (GeV/$\mbox{\rm fm}^3$) & 0.63 & 7.83 \\
$n_{0\; g} (\mbox{\rm fm}^{-3})$ & 2.15 & 18.00 \\
$n_{0\; q} (\mbox{\rm fm}^{-3})$ & 0.42 & 3.53 \\
$l_{0\; g}$ & 0.08 & 0.21 \\
$l_{0\; q}$ & 0.017 & 0.044 \\
$\theta_{0\; g}$ (fm/c) & 2.18 & 0.73 \\
$\theta_{0\; q}$ (fm/c) & 239.72 & 30.92 \\
\hline\hline
\end{tabular}
\end{center}
\begin{center}
TABLE 1. Initial conditions for the evolution of a QCD
plasma created in Au+Au collision at RHIC and at LHC
\end{center}
\vspace{0.5cm}
Using the standard initial picture of heavy ion collisions as
before, our evolution is started when the momentum distribution
in the central region of the collision becomes, for
the first time, isotropic due to longitudinal cooling.
The subsequent development is determined by the interactions
\etrref{eq:ggi}{eq:gqi}{eq:qqi}. In the case of a pure gluon plasma
\cite{wong}, it is clear that interactions bring the system towards
equilibrium and not towards some free streaming final state
which is a possible alternative as can be inferred from the
analysis in Sect. \ref{sec:therm}. That is the interactions
dominate over the expansion. In the present situation, we will see
that the same can certainly be said for the gluons and for the
quarks at LHC but at RHIC, it is less clear for the latters.
The equilibration time for quarks is at least several times
longer than that of the gluons.
Details for the procedure of the computation can be found in
\cite{wong}. The values for the numerical parameters are the same
and in addition, we use $n_f=2.5$ to take into account of the reduced
phase space of strange quark. All time integrations are discretized
and the rates are obtained at each time step necessary for forming
the two pairs of equations \etrref{eq:e_trans}{eq:s_rate1}{eq:s_rate2}.
One then solves the two equilibrium temperatures $\Tg$ and $\Tq$
from two 4th degree polynomials, one for each of the temperatures.
From these solutions, $\qg$ and $\qq$ are obtained and everything
is then fed back into the equations for the next time step.
\section{Equilibration of the QCD Plasma}
\label{sec:result}
We show the results of our computation in this section. They
show clearly the collision times $\qg$ and $\qq$ hold the keys
to equilibration as have been analysed in Sect. \ref{sec:therm}.
We will see shortly that as a result of the disparity between their
magnitudes at finite values of $\tau$, the equilibration of quarks
and antiquarks lags behind that of the gluons both chemically
and kinetically. We will also identify the dominant processes
responsible for equilibration. They are {\em not} the commonly
assumed elastic scattering processes as already mentioned in
the introduction.
When dealing with two particle species, one has several choices
as to when should the evolution be stopped. We choose to do this
when both the quark and the gluon temperature estimates drop
to 200 MeV. For gluons, this estimate is obtained by the near
equilibrium energy and number density expression
\begin{equation} \epsilon_g = a_2 \, l_g \, T^4_g \mbox{\hskip 1cm and \hskip 1cm}
n_g = a_1 \, l_g \, T^3_g \; ,
\end{equation}
which are valid when the fugacity $l_g$ is near $1.0$ i.e. when
the distribution functions can be approximated by
$f_g({\mathbf p},\l_g,\tau) = l_g f_g({\mathbf p},\l_g=1,\tau)$. For quarks and antiquarks,
we cannot do the same as $l_q$ has not time to rise above $0.5$ so
instead, the temperature is estimated from the same quantities in
kinetic equilibrium but at small values of $l_q$
\begin{equation} \epsilon_q = 3 \, \nu_q \, l_q \, T^4_q / \pi^2
\mbox{\hskip 1cm and \hskip 1cm}
n_q = \nu_q \, l_q \, T^3_q / \pi^2 \; .
\end{equation}
\begin{figure}
\centerline{
\hbox{
{\psfig{figure=fig1a.ps,width=3.4in}} \
{\psfig{figure=fig1b.ps,width=3.4in}}
}}
\caption{The time-dependence of the estimated temperatures for
quarks and for gluons and their fugacities at (a) LHC and (b) RHIC.
The solid lines are the estimated temperatures $T_g$ (thick line)
and $T_q$. The dashed lines are the fugacities $l_g$ (thick line)
and $l_q$. Gluon chemical equilibration is much faster
than that of the quarks. The curves are stopped when all the
temperature estimates drop to 200 MeV. The vertical line indicates
when the gluon temperature reaches this value.}
\label{gr:T&l}
\end{figure}
\noindent
These estimates are plotted in \fref{gr:T&l}. The vertical line
marks the point when the gluon temperature estimate (thick solid
line) drops to $200$ MeV. At this point, $\tau \sim 6.25$ fm/c,
the fugacity (thick dashed line) is $l_g \sim 0.935$ at LHC and is
$l_g \sim 0.487$ at $\tau \sim 2.85$ fm/c at RHIC. On the same
plots, the quark temperature (solid line) drops at a slower rate
and the fermionic fugacity (dashed line) is also increasing much
slower given the less favourable initial conditions and initially
much slower quark-antiquark pair creation than gluon
multiplication rate. In the end, the fermions are not too well
chemically equilibrated and in fact, are still quite far away
from $1.0$. This is especially bad at RHIC. We note that comparing
to \cite{biro&etal1,lev&etal,wang}, in our case, gluons chemically
equilibrate faster but quarks are slower.
Unlike chemical equilibration, kinetic equilibration has no
simple indicators like the fugacities that can allow itself to be
simply quantified. One has to, instead, use the anisotropy of
momentum distribution as well as various reaction rates to
get an idea of the degree of kinetic equilibration. The
former can be deduced from the ratios of the longitudinal pressure
and a third of the energy density to the transverse pressure,
$p_L/p_T$ and $\epsilon/3 p_T$ respectively. Whereas from the elastic
scattering rates, one can deduce roughly how close the distribution
functions are to their equilibrium forms by virtue of the fact that
in local kinetic equilibrium, these rates are zero. The pressure
ratios $p_L/p_T$ (solid line) and $\epsilon/3 p_T$ (dashed line) are
plotted in \fref{gr:press} (a) and (a') for gluons, (b) and (b') for
quarks and (c) and (c') for the total sum. These ratios are indeed
approaching $1.0$, the expected value after thermalization, but at
different rates. Gluons are clearly equilibrating much faster than
quarks which proceed rather slowly.
\begin{figure}
\centerline{
\hbox{
{\psfig{figure=fig2.ps,width=3in}} \ \
{\psfig{figure=fig2da.ps,width=3in}}
}}
\caption{The ratios of the longitudinal pressure (solid line)
and a third of the energy density (dashed line) to the transverse
pressure, $p_L/p_T$ and $\epsilon/3 p_T$ respectively for (a) gluons,
(b) quarks and (c) the total sum at LHC. Graphs (a'), (b')
and (c') are the same at RHIC.}
\label{gr:press}
\end{figure}
To show that these behaviours, although slow, are indeed the signs
of equilibration and that the plasma is not approaching
some free streaming final states, we can work out what their
behaviours should be in the latter case by taking the extreme
and let $\theta_i \rightarrow \infty$. From \eref{eq:pres}, as $\tau \rightarrow \infty$,
\vspace{2ex}
\hbox{\hspace{4.9cm}
\raisebox{-2.5ex}{\vbox{
\hbox{
\(
p_L \rightarrow \pi \, \tau_0^3 \, \epsilon_0 /4 \, \tau^3
\)
}
\hbox{
\(
p_T \rightarrow \pi \, \tau_0 \, \epsilon_0 /8 \, \tau
\)
}
\hbox{
\(
\; \; \epsilon \: \rightarrow \pi \, \tau_0 \, \epsilon_0 /4 \, \tau
\)
}
}}
\(
\bigg \} \; \; \Longrightarrow \; \; \bigg \{
\)
\raisebox{-1ex}{\vbox{
\hbox{
\(
p_L/p_T \rightarrow 2 \, \tau^2_0/ \tau^2 \rightarrow 0
\)
}
\hbox{
\(
\epsilon/3 \: p_T \rightarrow 2/3
\)
}
}}
\vbox{\parbox{4cm}{
\begin{equation} \;
\end{equation}
}}
}
\vspace{1ex}
\noindent
where $\epsilon_0$ is the initial energy density and the above ratios
are valid for both quarks and gluons in this extreme. Therefore
in the free streaming case, the first ratio should approach
zero and the second should approach 2/3. These are clearly not
what we see in our plots.
\begin{figure}
\centerline{
\hbox{
{\psfig{figure=fig3.ps,width=3in}} \ \
{\psfig{figure=fig3da.ps,width=3in}}
}}
\caption{The scaled products of the collective variables
(a) energy density, (b) number density and (c) entropy density
and their expected inverse time-dependence in equilibrium
$\tau^{4/3}$, $\tau$ and $\tau$ respectively at LHC. Graphs (a'), (b')
and (c') are the same at RHIC. The solid and dashed lines are
for gluons and quarks respectively. The thick solid line
in (c) and (c') is the scaled product of the total entropy
density and $\tau$.}
\label{gr:scaled}
\end{figure}
To best get an idea of how close the distribution functions are
to the equilibrium forms, the $gg$ and $qq$ or $\bar q\bar q$
elastic scattering processes are ideal for this. These are shown
in \fref{gr:s_g-lhc} and \fref{gr:s_g-rhic} (b) for gluon and
\fref{gr:s_q-lhc} and \fref{gr:s_q-rhic} (c) for
quark. Note that the peaks of these collision entropy rates
coincide with the corresponding mininum points of the pressure
ratios. As expected, the rates maximize at maximum anisotropy
in momentum distribution. They all rise rapidly from zero at $\tau_0$
when the interactions are turned on. The subsequent return to zero
or the approach of the distribution functions to their equilibrium
forms are, however, much less rapid. They only do so
progressively as can be deduced already from the pressure ratio
plots.
Having shown chemical and kinetic equilibrations separately, we
present now the actual approach of the collective variables towards
the equilibrium values. Since we are more interested in the
behaviour of their time-dependence than their absolute magnitudes,
we multiplied them by their expected time-dependence and
scaled these by taking a guess at the corresponding asymptotic
values from the tendency of the curves. The results are plotted
in \fref{gr:scaled}. They are
$\epsilon_i \tau^{4/3}/\epsilon_{s\, i} \tau_{s\, i}^{4/3}$,
$n_i \tau/n_{s\, i} \tau_{s\, i}$ and $s_i \tau/s_{s\, i} \tau_{s\, i}$
in the figures (a) and (a'), (b) and (b') and (c) and (c')
respectively. All these should be nearly constant
with respect to time at large $\tau$. The solid lines
are for gluons and the dashed ones are for quarks.
They showed that the curves do behave in such a way
for the eventual constant behaviour. This feature
is much clearer at LHC than at RHIC which only
reconfirms the previously deduced result of faster equilibration
at LHC than at RHIC. Note that for gluons, the quantities are
approaching the corresponding asymptotic values from above,
whereas for quarks, this approach is from below. This is
because of the simple reason that there is a net conversion
of gluons into quark-antiquark pairs via $gg \longleftrightarrow q\bar q$.
The corresponding collision entropy density rate is negative as
shown in \fref{gr:s_g-lhc} and \fref{gr:s_g-rhic} (c).
We will see that this same interaction becomes dominant
in the later part of the evolution
later on when we compare the importance of the different
processes. So gluons are losing energy, number and
entropy to the fermions. This has to be so before the
system as a whole can settle into complete equilibrium.
The thick solid lines in \fref{gr:scaled} (c) and (c')
show the scaled total entropy per unit area in the central
region which give an idea of the state of the system as
a whole. They show that although the entropy of the individual
subsystem can decrease, the total value must increase in
accordance with the second law of thermodynamics.
\begin{figure}
\centerline{
\hbox{
{\psfig{figure=fig4.ps,width=3in}} \ \
{\psfig{figure=fig4da.ps,width=3in}}
}}
\caption{The time-dependence of the collision time
(a) for gluons $\qg$ and (b) for quarks $\qq$ at LHC.
Their values are compared in (c). $\tau$ overtakes first
$\qg$ and later $\qq$ also. Graphs (a'), (b') and (c')
are the same at RHIC. In this case, $\tau$ only has time
to overtake $\qg$ but not $\qq$.}
\label{gr:relax_t}
\end{figure}
The figures discussed above show that the plasma
is indeed approaching equilibrium and that interactions are
fast enough to dominate over the Bjorken type one-dimensional
scaling expansion.
As we analysed in Sect. \ref{sec:therm}, thermalization is governed
by the $\theta_i$'s. How fast this will proceed depends on their magnitudes
and what is the actual final state depends on their time-dependent
behaviours. For thermalization, the $\theta_i$'s must behave in such a way
such that $x_i \rightarrow \infty$ as $\tau \rightarrow \infty$. That means they
must grow less fast than $\tau$. In \fref{gr:relax_t}, we show these
$\theta_i$'s as a function of $\tau$. Initially, $\theta_i >\tau$ for both quarks
and gluons, and $\qq$ starts off very large (see Table 1) but drops
extremely rapidly back down to within hadronic timescales. The
subsequent expected increase in time \cite{baymetal1,baymetal2,heis}
is sufficiently slow for $\tau$ to get past $\qg$ and $\qq$
at LHC, \fref{gr:relax_t} (a) and (b) but at RHIC,
\fref{gr:relax_t} (b'), $\qq$ is still too large for $\tau$ to overtake
it before the temperature reaches $200$ MeV. Nevertheless, the
$\tau$-dependence is slow enough that $x_i$ should go to infinity
as $\tau \rightarrow \infty$.
We have mentioned in Sect. \ref{sec:relax}, for the system to
equilibrate as one, the target equilibrium temperatures
$\Tg$ and $\Tq$ and also $\qg$ and $\qq$ must approach each other
at large $\tau$. We strongly suspect that the convergence of the
temperatures will proceed in an oscillating fashion where the two
curves intersect each other several times before the final convergence
at very large $\tau$. We can see this in \fref{gr:T_eq} (a) and (b).
At LHC, the initial condition is more favourable for equilibration
and so $\Tg$ intersects $\Tq$ twice already. This is not so at RHIC.
In fact, all indications point to the fact that a plasma created
at LHC will equilibrate better than one created at RHIC. By letting
the plasma to continue its evolution and ignoring the deconfinement
phase transition, we have seen that the collective variables
like the gluon and quark energy densities, gluon entropy
density etc. do show tendency to pass from below to above or vice
versa, the corresponding equilibrium target values i.e.
tendency to overshoot the equilibrium values and hence oscillation.
As to the convergence of $\theta_i$'s, it is not so clear
in \fref{gr:relax_t} (c) and (c'), especially at RHIC in
\fref{gr:relax_t} (c'). $\qq$ is much too large in comparison
with $\qg$ for any clear sign of convergence within the
time available. On the other hand, at LHC, although
there is still a large gap between the magnitudes, there is
a clear tendency that the rate of increase of $\qq$ with $\tau$ is
slowing down in \fref{gr:relax_t} (b) while $\qg$ still increases at
approximately the same rate. It is simply too early for the system
to equilibrate as one. Even near the end, the quarks and gluons
can only be considered as two linked subsystems approaching
equilibrium at very different rates. Hence we have a two-stage
equilibration.
\begin{figure}
\centerline{
\hbox{
{\psfig{figure=fig5a.ps,width=3.2in}} \ \
{\psfig{figure=fig5b.ps,width=3.2in}}
}}
\caption{The time development of the equilibrium target
gluon (solid line) and quark (dashed line) temperatures
$\Tg$ and $\Tq$ respectively at (a) LHC and (b) RHIC.
They should converge in an oscillating fashion at large
$\tau$ in order for the system to equilibrate as one towards
a single temperature. The convergence is less good at RHIC
than at LHC.}
\label{gr:T_eq}
\end{figure}
Having shown that interactions can indeed dominate over the
one-dimensional expansion of the parton gas in the central region
of relativistic heavy ion collisions and hence bring the
plasma into equilibrium. We can now look at the individual processes
and compare their relative importance. These are the processes
\etrref{eq:ggi}{eq:gqi}{eq:qqi}. We have labelled their contributions
to the gluon and quark collision entropy rate $ds_g/d\tau$ and
$ds_q/d\tau$ by $d s_{gi}/d\tau$, $i=1,\dots, 4$ and $d s_{qi}/d\tau$,
$i=1,\dots,3$ in the order that they appear in
\etrref{eq:ggi}{eq:gqi}{eq:qqi}. Processes that give the same rate
due to quark-antiquark symmetry are considered as the same process.
Hence $gq \longleftrightarrow gq$ and $g\bar q \longleftrightarrow g\bar q$ give identical
contribution to gluon and quark collision entropy density rate as
$ds_{g4}/d\tau$ and $ds_{q2}/d\tau$ respectively. Also we have
combined fermion elastic scattering processes as one rate
$ds_{q3}/d\tau$ for convenience. These are shown in
\fref{gr:s_g-lhc}, \fref{gr:s_g-rhic}, \fref{gr:s_q-lhc} and
\fref{gr:s_q-rhic}. The elastic processes have a characteristic
shape, i.e. an initial rapid rise to a peak at maximum anisotropy
before returning to zero progressively. The sharper the peak,
the quicker the kinetic equilibration (compare \fref{gr:s_g-lhc}
and \fref{gr:s_g-rhic} (b), (d) and \fref{gr:s_q-lhc} and
\fref{gr:s_q-rhic} (b), (c) and \fref{gr:press}). Note the negative
rate of \fref{gr:s_g-lhc} and \fref{gr:s_g-rhic} (c) which
is because there are net quark-antiquark pair creations from
gluon-gluon annihilations and entropy decreases with the
number of gluons as already mentioned in the previous paragraphs.
We compare the different processes by plotting the ratio of the
magnitude of each contribution to that of gluon multiplication
for gluons in \fref{gr:s_g-lhc} and \fref{gr:s_g-rhic} (e) and the
ratio of each rate to that of quark-antiquark creation for quarks
(antiquarks) in \fref{gr:s_q-lhc} and \fref{gr:s_q-rhic} (d).
In the (e) figures, gluon multiplication clearly dominates initially
at $\tau \lsim 2$ fm/c at LHC and $\tau \lsim 4$ fm/c at RHIC since all
three ratios in each plot are less than 1. After these times,
$q\bar q$ creation becomes dominant (thick solid line) and rises
to several times larger than gluon multiplication. The $gg$ elastic
scattering, on the other hand, tends to maintain a small, nearly
constant ratio with gluon multiplication (solid line), which
supports the claim made in \cite{wong}. That is, in a pure gluon
plasma, gluon multiplication dominates over $gg$ elastic
scattering in driving the plasma towards equilibrium. This
remains the case even when $l_g \sim 0.93$
which shows that this dominance is not sensitive to the
value of $l_g$. The remaining ratio of quark-gluon scattering
to gluon multiplication continues to rise but not as rapidly
as the first ratio. For quark entropy, \fref{gr:s_q-lhc} and
\fref{gr:s_q-rhic} (d), both ratios of quark-gluon
scattering (solid line) and fermion-fermion scatterings
(dashed line) to $gg \longleftrightarrow q\bar q$ rate remain small
during the time available although they are
both on the rise. So for gluons, gluon multiplication
dominates initially but is later overtaken by $gg\longleftrightarrow q\bar q$
which continues to dominate over other elastic processes.
For quarks (antiquarks), this same process dominates during
the lifetime of the plasma.
These behaviours can be understood in the following way.
Gluon branching dominates initially over any other processes
so long as gluons are not near equilibrium. Once they
approach saturation (the $l_g$ estimates slow down their
approach towards $1.0$ in \fref{gr:T&l} (a) and (b) at about
the times mentioned above), gluon-gluon annihilation to
quark-antiquark takes over as the dominant one because
the fermions are still far from full equilibration.
Because of the latter reason, the other ratios involving
quark or antiquark to gluon branching continue to rise.
\begin{figure}
\centerline{
{\psfig{figure=fig6.ps,width=6.7in}}
}
\caption{ The time development of the different
contributions to the total gluon collision entropy
density rate at LHC. They are (a) $gg\longleftrightarrow ggg$,
(b) $gg\longleftrightarrow gg$, (c) $gg\longleftrightarrow q\bar q$ and (d) $gq\longleftrightarrow gq$
or $g\bar q\longleftrightarrow g\bar q$. The curves of the elastic
scattering processes in (b) and (d) have typical peaks at
maximum anisotropy in momentum distributions. The ratios
of the contribution (b) (thick line), (c) (solid line) and
(d) (dashed line) to that of (a) are plotted in (e). This
shows that first gluon multiplication dominates initially
but is later overtaken by gluon annihilations into
quark-antiquark pairs.}
\label{gr:s_g-lhc}
\end{figure}
So contrary to common assumption, inelastic processes are
dominant in equilibration. This should have consequences in
the perturbative calculations of transport coefficients or
relaxation times \cite{baymetal1,baymetal2,heis} of system
that are not subjected to external forces. These calculations
are based essentially, up to the present, on elastic binary
interactions. As we have seen, they are not the dominant
processes in equilibration.
To the surprising result of gluon multiplication dominates
over elastic gluon-gluon scattering, we provide the following
explanation. If one only looks at the scattering cross-sections,
it is indeed true that gluon-gluon scattering has a larger
value and gluon multiplication processes are down by $\alpha_s$
for each extra gluon produced. The $(n-2)$ extra gluon production
cross-section can be expressed in terms of the elastic scattering
cross-section as \cite{gold&rosen,shury&xion}, in the double
logarithmic approximation,
\begin{equation} \sigma_{gg \rightarrow (n-2) g} \propto \sigma_{gg \rightarrow gg} \;
[\alpha_s \ln^2 (s/s_{cut})]^{n-4}
\end{equation}
where $s_{cut}$ is the cutoff for the mininum binary invariant
$(p_i+p_j)^2 > s_{cut}$ of the 4-momenta of each gluon pair.
In the present problem, $s_{cut}=m^2_D$, the double logarithm
is not large and certainly does not compensate for the smallness
of $\alpha_s$. However, as we have mentioned at the beginning,
the collision term on the r.h.s. of \eref{eq:baymeq}
consists of the sum of the differences of the reactions in
a QCD medium going forward and backward, so a large
cross-section does not automatically imply dominance of
the corresponding process in the approach to equilibrium.
\begin{figure}
\centerline{
{\psfig{figure=fig7.ps,width=6.7in}}
}
\caption{The time development of the same contributions
to the total gluon collision entropy density rate
as in \fref{gr:s_g-lhc} but at RHIC. The same ratios
between the different contributions as at LHC are plotted
in (e).}
\label{gr:s_g-rhic}
\end{figure}
\begin{figure}
\centerline{
{\psfig{figure=fig8.ps,width=5in}}
}
\caption{ The time development of the different
contributions to the total quark collision entropy density
rate at LHC. They are (a) $gg\longleftrightarrow q\bar q$
(b) $gq\longleftrightarrow gq$ or $g\bar q\longleftrightarrow g\bar q$ and
(c) the sum of the contributions of all fermion elastic
scattering processes $qq\longleftrightarrow qq$, $q\bar q\longleftrightarrow q\bar q$ and
$\bar q\bar q\longleftrightarrow \bar q\bar q$. The ratios of the contribution
(b) (solid line), (c) (dashed line) to that of (a) is plotted
in (d). This shows that throughout the lifetime of the
QCD plasma, gluon annihilations into quark-antiquark
pairs dominates in the equilibration of the fermions.}
\label{gr:s_q-lhc}
\end{figure}
\noindent
Similarly, $gg \longleftrightarrow q\bar q$ is not that different from
$gq \longleftrightarrow gq$ or $g\bar q \longleftrightarrow g\bar q$ because the
two matrix elements are related simply by a swapping
of the Mandelstam variables. So why should the first
dominates over the second? Except the different ways that
the infrared divergences are cut off in the processes,
the main reason is $gg \longrightarrow q\bar q$ dominates over
the backward reaction $q\bar q \longrightarrow gg$ due to the
simple fact that there are less fermions than gluons
present in the plasma. An extreme example of this
phenomenon would be the forward and backward reaction
balance out each other for all the elastic interactions as
in a kinetically equilibrated plasma when only inelastic
processes remain in the collision terms. In this extreme,
all the ratios of elastic to inelastic collision entropy
rate vanish.
We can now return to the question of whether other inelastic
processes such as $gg \longleftrightarrow q\bar qg$, $gq \longleftrightarrow gqg$,
$g\bar q \longleftrightarrow g\bar qg$, $gq \longleftrightarrow qq\bar q$,
$g\bar q \longleftrightarrow q\bar q\bar q$, $qq \longleftrightarrow qqg$ etc. should be
included. Although they are non-leading compared to
$gg \longleftrightarrow ggg$ and $gg \longleftrightarrow q\bar q$ due to colour, they
should be significant when one sizes them with the elastic
processes in view of the cancellation between the forward and
backward reactions. In \cite{wong}, the question of the
dominance of inelastic over elastic processes was raised.
Here it is sufficient to include the two leading inelastic
processes to show this explicitly. Had one included these other
processes, then equilibration should be faster and one could
end up with a more reasonable quark-antiquark content in
the plasma. However, we are doubtful that the equilibration
time can be reduced dramatically from what we have shown here.
\begin{figure}
\centerline{
{\psfig{figure=fig9.ps,width=5in}}
}
\caption{The time development of the same contributions
to the total quark collision entropy density
rate as in \fref{gr:s_q-lhc} but at RHIC. The same ratios
as at LHC are plotted in (d). They show again inelastic
process dominates.}
\label{gr:s_q-rhic}
\end{figure}
As we argued in \cite{wong}, it is hard to perturb a parton
system from thermal equilibrium without doing so chemically.
Therefore inelastic processes are always active in the
approach to equilibrium whereas the same is not true for
elastic processes. From our figures, it can be seen
that inelastic processes are not there only for chemical
equilibration or for minor contributions to thermalization
as is commonly assumed due to their possible higher powers
in $\alpha_s$, they contribute even more significantly to
equilibration than elastic processes. Changing the initial
conditions will only vary the dominancy but not remove
the dominance.
Before closing, we would like to point out some differences
of our results with that of PCM. In PCM, there appears to be
no early momentaneous isotropic particle momentum distribution
in either S+S or Au+Au collisions. The first time that
there is approximate isotropy, it is already thermalization
according to \cite{geig}. It was claimed that there was
no further significant change in the total momentum
distribution after $\tau=2.4$ fm/c for Au+Au collision at
RHIC. We assume that they mean the shape of the distribution
with the exception of the slope which should continue to change
due to cooling. However, when the total distribution is broken
down into that of the parton components, the approximate
isotropy or thermalization becomes less obvious. We have
shown that thermalization in the strict sense is slow and
isotropy of gluon momentum distribution can be argued to be
approximate but that of the fermions is not so good.
As to chemical equilibration, PCM shows little chance of
that for the fermions. The corresponding fugacity estimates
are approaching the ``wrong direction'' with increasing time.
This is due to a net outflow of particles from the defined
central region. The net flux of outgoing particles is
arguably more important for fermions than for gluons because
the formers have a larger mean free path. The result is the
gluon (fermion) fraction of the particle composition rises
(drops) with increasing time. Therefore even if there is no
phase transition and the parton plasma is allowed to continue
its one-dimensional expansion indefinitely, chemical
equilibration will never be achieved.
Then according to PCM, the expansion is slow enough for
kinetic equilibration for all particle species but too fast
for chemical equilibration of the quarks and antiquarks.
The boundary effect is too important and is affecting
equilibration. In our case, this effect is not incorporated.
Although equilibration is slow, full equilibration will be
reached given sufficient time.
We find it surprising that although the gluon fugacity
estimate in PCM \cite{geig&kap} overshoots and stays
above or at $1.0$ nearly all the time except at
the beginning, $R_g$ is still positive or an order
of magnitude larger than $R_q+R_{\bar q}$ when the fugacities
of the latter are well below $1.0$ and decreasing. One would
expect rather gluon absorption or conversion into quark-antiquark
should take a significant toll on the gluon production so that
there should be a diminution of gluons. At least, this
should be the case when local kinetic equilibrium has been
or nearly been reached which PCM claimed to be so at the
end of the program run but this is not the case in the
plot of the production rate of the different particle species!
This is counter-intuitive and opposite to what we have shown.
To conclude, we have shown that inelastic processes dominate
in the approach towards equilibrium. In particular, gluon
branching is most important. Gluon-gluon annihilation into
quark-antiquark becomes more important only when the gluons
are near saturation and equilibrium. The lower power in
$\alpha_s$ of the gluon-gluon elastic scattering as compared to the
inelastic gluon emission process is more than compensated for by
the cancellation of the reaction going forward and backward.
The recovery of isotropy in momentum distribution is slow
and so is chemical equilibration. The latter is partly due
to the small initial fugacities that we used. As an intrinsic
feature of perturbative QCD, the quarks and antiquarks
are lagging behind the gluons in equilibration and hence a
two-stage equilibration scenario.
\section*{Acknowledgements}
The author would like to thank M. Fontannaz, D. Schiff and
everyone at Orsay for kind hospitality during his stay
there, R.D. Pisarski and A.K. Rebhan for raising interesting
questions. Thanks also go to R. Baier and everyone at
Bielefeld for hospitality during the author's short stay
there where this work is completed. The author acknowledges
financial support from the Leverhulme Trust.
| proofpile-arXiv_065-440 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
The method of effective field theory has repeatedly been used in the
analysis of the symmetry breaking sector of the Standard Model. It
provides a model independent parametrization of various scenarios for
the spontaneous breakdown of the electroweak symmetry. The unknown
physics is then hidden in the low energy constants of an effective
Lagrangian.
The physics in the low energy region of a full theory is adequately
described by an effective field theory if corresponding Green
functions in both theories have the same low energy structure. One can
take this matching requirement as the definition of the effective
field theory. It determines functional relationships between the low
energy constants of the effective Lagrangian and the parameters of the
underlying theory.
The special role of gauge theories is readily understood. The
effective field theory analysis should not make any particular
assumptions about the underlying theory -- apart from symmetry
properties and the existence of a mass gap. This also requires
parametrizing low energy phenomenology by a gauge-invariant effective
Lagrangian. The definition of Green's functions, on the other hand,
usually does not reflect the symmetry properties of a gauge theory.
Gauge invariance is broken, and the off-shell behaviour of Green's
functions is gauge-dependent. Therefore, if the Green functions
which enter the matching relations do not reflect the symmetry
properties of the full theory, the effective field theory will also
include the corresponding gauge artifacts.
Without resolving these issues, different approaches to determine the
low energy constants at order $p^4$ for the Standard Model with a
heavy Higgs boson where presented in several recent
articles~\cite{SM_Heavy_Higgs}. For Higgs masses below about~$1 TeV$
the effective Lagrangian can be evaluted explicitly with perturbative
methods. These works show clearly that the matching of gauge-dependent
quantities causes all kinds of trouble. This calls for a new
manifestly gauge-invariant technique which avoids these problems, yet
maintains the simplicity and elegance of matching Green's functions as
in the ungauged case.
\section{A gauge-invariant approach}
Any approach to determine the effective Lagrangian for a given
underlying gauge theory should match only gauge-invariant quantities.
Then one does not have to worry about any gauge artifacts which
otherwise might enter the effective field theory. Perhaps the most
straightforward idea that comes to mind is to match only $S$-matrix
elements. However, this approach is quite cumbersome. In particular,
it involves a detailed treatment of infrared physics.
Functional techniques like those described in Ref.~\cite{LSM} provide
a much easier approach. In this case one matches the generating
functionals of Green's functions in the full and the effective theory.
Infrared physics drops out at a very early step of the calculation.
The remaining contributions all involve the propagation of heavy
particles over short distances. Hence, they can be evaluated with a
short distance expansion. The computation of loop-integrals is not
necessary.
Gauge invariance is broken as soon as Green's functions of
gauge-depend\-ent operators are considered. Hence, any manifestly
gauge-invariant approach must confine itself to analyze Green's
functions of gauge-invariant operators, such as the field strength of
an Abelian gauge field or the density of the Higgs field. In the
following we summarize a manifestly gauge-invariant technique to
evaluate the effective Lagrangian describing the low-energy region of
the gauged linear sigma model in the spontaneously broken phase. It
involves only Green's functions of gauge-invariant operators. For any
details the reader is referred to Ref.~\cite{Abelian} where it has
been applied to the Abe\-lian case. The Higgs sector of the Standard
model with the non-abelian group $SU_L(2)\times U_Y(1)$ can be treated
in the same way~\cite{NonAbelian}. We would like to point out, that
all $S$-matrix elements of the theory can be evaluated from these
Green functions as well.
At tree-level the generating functional is given by the classical
action. Since the external sources are gauge invariant, i.e., do not
couple to the gauge degrees of freedom, the equations of motion can be
solved without gauge-fixing. As a result they have a whole class of
solutions. Every two representatives are related to each other by a
gauge transformation. In order to determine the leading contributions
to the low energy constants one merely has to solve the equation of
motion for the Higgs boson field.
To incorporate higher order corrections one may evaluate the
path-integral representation of the generating functional with the
method of steepest descent. In this case they are described by
Gaussian integrals. Since gauge invariance is manifest, these
integrals can also be evaluated without gauge-fixing. As a consequence
the gauge degrees of freedom manifest themselves through zero-modes of
the quadratic form in the exponent of the gaussian factor. The
integration over these modes yields the volume of the gauge group,
which can be absorbed by the normalization of the path-integral. The
remaining integral over the non-zero modes contains all the physics.
We would like to point out one difference between this approach to
evaluate a path-integral and the method of Faddeev and Popov: if the
gauge is not fixed the evaluation of the path-integral does not involve
ghost fields. Hence, the number of diagrams to compute is reduced.
The effective Lagrangian of the linear sigma model is a sum of
gauge-invariant terms with an increasing number of covariant
derivatives and gauge-boson mass factors, corresponding to an
expansion in powers of the momentum and the masses. Note that the
covariant derivative, the gauge-boson masses and the gauge couplings
all count as quantities of order $p$. Thus, the low energy expansion
is carried out such that all light-particle singularities are
correctly reproduced. Furthermore, if the coupling $\lambda$ of the
scalar field is small enough, the low energy constants in the
effective Lagrangian admit an expansion in powers of this quantity,
corresponding to the loop expansion in the full theory. $n$-loop
Feynman diagrams in the Abelian Higgs model yield corrections of order
$\lambda^{n-1}$ to the low energy constants.
\newpage
\section*{Acknowledgments}
Work supported in part by Deutsche Forschungsgemeinschaft Grant Nr.\
Ma 1187/7-2, DOE grant \#DE-FG02-92-ER40704, NSF grant PHY-92-18167,
and by Schweizerischer Nationalfonds.
\section*{References}
| proofpile-arXiv_065-441 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{INTRODUCTION}
In the last four years the Ape group has been extensively studying lattice QCD
in the quenched approximation. Several simulations have been done to study weak
matrix elements such as $f_D$, $f_B$, $B_K$ and to study semileptonic decays
\cite{latI}-\cite{latV}.
These simulations have allowed to study the dependence of the results on the
spacing $a$ and to investigate finite size effects. Here we present results
for light meson masses and decay constants
and baryon spectroscopy. The results have
been obtained from eight sets of data at $\beta=6.0$, $6.2$ and $6.4$ using
either the Wilson action or the ``improved'' SW-Clover action, in order to reduce
$O(a)$ effects. The parameters used in each simulation
are listed in Table~\ref{tab:latparams}.
The Ape group has performed extensive comparisons on data
extracted from smeared and non-smeared propagators and found no real
improvement for lattices with a time extent of 64 at $\beta=6.0$ and
$\beta=6.2$ \cite{latII_a2}.
Note that the simulations have been performed at $\beta$ values of
$6.0$ or larger.
\setlength{\tabcolsep}{.65pc}
\begin{table}
\caption{Predicted meson masses in GeV for all lattices.}
\label{tab:mesonsGeV}
\begin{tabular}{llll}
\hline
& $M_\rho$ & $M_{\eta'}$ & $M_\phi$ \\
\hline
Exper. & 0.770 & 0.686 & 1.019 \\
\hline
Lat~I & 0.809(7) & 0.6849(3) & 0.977(7) \\
Lat~II & 0.808(3) & 0.6849(1) & 0.978(3) \\
Lat~III & 0.81(1) & 0.6849(5) & 0.98(1) \\
Lat~IV & 0.803(6) & 0.6851(2) & 0.984(6) \\
Lat~V & 0.79(1) & 0.6856(5) & 1.00(1) \\
Lat~VI & 0.797(7) & 0.6853(3) & 0.989(7)\\
Lat~VII & 0.796(4) & 0.6853(2) & 0.990(4)\\
Lat~VIII& 0.792(4) & 0.6855(2) & 0.994(4)\\
\hline
\end{tabular}
\end{table}
This is to negate
the large systematic error present in lattice data for $\beta \;\raisebox{-.5ex}{\rlap{$\sim$}} \raisebox{.5ex}{$<$}\;
6.0$ due to lattice artifacts \cite{chris}. All the results we
obtained will be described in greater detail in a forthcoming paper \cite{lavoro}.
\section{MESON MASSES AND DECAY CONSTANTS}
Meson masses and decay constants have been extracted from two-point correlation
functions of the following local operators
\[
P_5(x) = i\bar{q}(x)\gamma_5q(x), \;\; V_k(x)=\bar{q}(x)\gamma_k q(x)\; ,
\]
\[
A_\mu(x) = \bar{q}(x)\gamma_\mu\gamma_5q(x)
\]
in the standard APE way \cite{lavoro}.
We fit the correlation functions of these operators to a
single particle propagator with a $sinh$ in the case of
axial-pseudoscalar function and with a $cosh$ in other cases.
\begin{table*}[hbt]
\setlength{\tabcolsep}{.55pc}
\newlength{\digitwidth} \settowidth{\digitwidth}{\rm 0}
\catcode`?=\active \def?{\kern\digitwidth}
\label{tab:latparams}
\caption{Summary of the parameters of the runs analyzed in this work and
time windows used in the fits.}
\begin{tabular}{lllllllll}
\hline
&Lat~I&Lat~II&Lat~III&Lat~IV&Lat~V&Lat~VI&Lat~VII&Lat~VIII\\
\hline
Ref & \cite{latI}& \cite{latI,latII_a} & \cite{latIII} & \cite{latIII}
& \cite{latV} & \cite{latII_a,latII_a2} & \cite{latIII} & \cite{latIII} \\
$\beta$&$6.0$&$6.0$&$6.2$ &$6.2$&$6.2$&$6.2$&$6.4$&$6.4$\\
Action & SW & Wil & SW & Wil& SW & Wil & Wil & SW \\
\# Confs&200&320&250&250&200&110&400&400\\
Volume&$18^3\times 64$&$18^3\times 64$&$24^3\times 64$& $24^3\times 64$
&$18^3\times 64$&$24^3\times 64$&$24^3\times 64$&$24^3\times 64$\\
\hline
$K$& - & - &0.14144&0.1510& - & - &0.1488&0.1400\\
&0.1425&0.1530&0.14184&0.1515& 0.14144&0.1510&0.1492&0.1403\\
&0.1432&0.1540&0.14224&0.1520& 0.14190&0.1520&0.1496&0.1406\\
&0.1440&0.1550&0.14264&0.1526& 0.14244&0.1526&0.1500&0.1409\\
\hline
\multicolumn{9}{c}{Mesons with zero momentum} \\
$t_1 - t_2$ & 15-28 & 15-28 & 18-28 & 18-28 & 18-28 & 18-28 & 24-30 & 24-30 \\
\hline
\multicolumn{9}{c}{Baryons with zero momentum} \\
$t_1 - t_2$ & 12-21 & 12-21 & 18-28 & 18-28 & 18-28 & 18-28 & 22-28 & 22-28 \\
\hline
\end{tabular}
\end{table*}
The pseudoscalar decay constant $f_{PS}$ has been
extracted by combining the fit of $\langle A_0P_5\rangle$ with the ratio
$\langle A_0P_5 \rangle /\langle P_5P_5\rangle $.
The errors have been estimated by a jacknife procedure, blocking the data in
groups of 10 configurations and we have checked that there are no relevant
changes in the error estimate by blocking groups of
configurations of different size. We have fitted the correlation
functions in time windows
reported in Table~\ref{tab:latparams}.
The time fit intervals are chosen with the following criteria: we fix the
lower limit
of the intervals as the one at which there is a stabilization of the
effective mass
and as the upper limit the furthest possible point before the error overwhelms
the signal. We discard the possibility of fitting in a restricted region
where a plateau is present, as the definition of such a region is highly
questionable \cite{fukugita}.
For lattices with highest number of configurations, i.e. LatII, LatVII and
LatVIII, we
confirm that higher statistics do not lead to a longer or better
(relative to the statistical error) plateau
\cite{fukugita}.
\setlength{\tabcolsep}{.3pc}
\begin{table}
\caption{Extrapolated/interpolated meson decay constants}
\label{tab:decaysGeV}
\begin{tabular}{llllll}
\hline
&$\displaystyle\frac{f_\pi}{Z_A m_\rho}$&
$\displaystyle\frac{1}{f_\rho Z_V}$&$\displaystyle\frac{f_K}{Z_A m_{K^*}}$
&$\displaystyle\frac{1}{f_{K^*} Z_V}$& \\
\hline
Lat~I & 0.17(1) & 0.42(3) & 0.172(9) & 0.39(2) & \\
Lat~II & 0.25(1) & 0.51(2) & 0.239(8) & 0.48(2) & \\
Lat~III & 0.16(1) & 0.39(3) & 0.164(9) & 0.36(2) & \\
Lat~IV & 0.21(1) & 0.47(2) & 0.214(8) & 0.45(1) & \\
Lat~V & 0.19(3) & 0.30(4) & 0.18(2) & 0.30(3) & \\
Lat~VI & 0.21(1) & 0.49(3) & 0.21(1) & 0.46(2) & \\
Lat~VII & 0.23(2) & 0.39(2) & 0.22(1) & 0.38(1) & \\
Lat~VIII& 0.19(1) & 0.30(2) & 0.18(1) & 0.29(1) & \\
\hline
\end{tabular}
\end{table}
Once the hadronic correlation functions have been fitted and the lattice
masses and matrix elements extracted,
we extract as much physics as possible from the
``strange'' region so that the chiral extrapolation will be
needed only in few cases. The method we use is outlined below:\\
{\bf -} We define the lattice planes for meson masses and decay constants
($M_V a$, $(M_{PS} a)^2$), ($f_{PS}
a/Z_A$, $(M_{PS} a)^2$) and ($1/(f_{V} Z_V)$, $(M_{PS} a)^2$) where
the subscripts $PS$ and $V$ stand for pseudoscalar and vector meson.
We plot the Monte Carlo data for each kappa used
in the simulation on these planes;\\
{\bf -} On the vector meson plane ($M_V a$, $(M_{PS} a)^2$)
we impose the physical ratios $M_{K^*}/M_{K}$, $M_{\rho}/M_{\pi}$ and
find the values of $M_\pi a$, $M_\rho a$ (only one independent),
$M_K a$, $M_{K^*} a$ (only one independent), $M_{\eta '}a$ and $M_\phi a$;\\
{\bf -} We now use the value of meson masses determined above to read off the
lattice meson decay constants, $(f_\pi a/Z_A)$, $(f_K a/Z_A)$,
$(f_{\rho} Z_V)^{-1}$ and $(f_{K^*} Z_V)^{-1}$
from the corresponding $f_{PS}$ and $f_V$ planes. \\
This procedure to extract physical quantities only requires meson
masses and not unphysical quantities such as quark masses or $k$
values. It allows us to study the $strange$ physics and fix the lattice spacing
directly in the region where data have been simulated
without chiral extrapolation to zero quark mass. This approach
therefore reduces the
errors on physical quantities induced by the chiral extrapolation.
Using the values of $a^{-1}$ from $M_{K^*}$ we have obtained the physical
value in GeV of meson masses reported in Table~\ref{tab:mesonsGeV}.
Comparing the vector meson mass (in lattice units)
from lattices LatIII, LatV and \cite{ukqcd93} we
infer that there is the possibility of some residual finite volume effects
on the $18^3$ lattice at $\beta=6.2$ \cite{lavoro}. This problem may also
be present
in our $\beta=6.4$ data for which the physical volume is
the same as in Lat V. Further investigations at larger
lattice sizes are necessary to make the situation clearer.
Turning to the continuum limit, any dependence of the meson
spectrum on $a$ is small and difficult to interpret unambiguously at this
stage.
In Table~\ref{tab:decaysGeV} we report results for the meson decay
constants without including the renormalization constants $Z_V$ and $Z_A$.
For both the pseudoscalar and vector decay constants we notice a difference
between the Wilson and SW-Clover data. This is presumably due to
the different renormalization constants and the smaller
$O(a)$ effects in the latter case. There may also be a small residual
finite lattice spacing effect in the vector decay constant in the SW-Clover
data which needs further study.
Overall our results agree with experimental
data to $\sim 5\%$ for meson spectrum and to $\sim 10\%-15\%$ for the
pseudoscalar decay constants.
In our opinion the vector decay constant deserves a much more careful study
at larger volume and $\beta$.
\section{BARYON MASSES}
Baryon masses have been extracted from two-point correlation
functions of the following local operators
\begin{eqnarray}
N & = &\epsilon_{abc}(u^aC\gamma_5 d^b)u^c\nonumber\\
\Delta_\mu & = & \epsilon_{abc}(u^aC\gamma_\mu u^b)u^c\nonumber
\end{eqnarray}
in the standard way by fitting the two point correlation functions
to a single particle propagator with an $exp$ function. The errors have been
estimated as in the meson case.
\setlength{\tabcolsep}{.19pc}
\begin{table}
\caption{Predicted baryon masses in GeV for all lattices.}
\label{tab:baryonsGeV}
\begin{tabular}{llllll}
\hline
& $M_N $ & $M_{\Lambda\Sigma}$ & $M_\Xi $& $M_\Delta $ & $M_\Omega $\\
\hline
Exper. & 0.9389 & 1.135 & 1.3181 & 1.232 & 1.6724 \\
\hline
Lat~I & 1.09(5) & 1.21(4) & 1.32(4) & 1.3(1) & 1.60(9) \\
Lat~II & 1.19(5) & 1.29(4) & 1.40(4) & 1.46(7) & 1.71(4) \\
Lat~III & 1.1(1) & 1.22(8) & 1.34(7) & - & - \\
Lat~IV & 1.17(7) & 1.28(6) & 1.39(5) & - & - \\
Lat~V & 1.1(2) & 1.2(2) & 1.4(1) & 1.6(3) & 1.9(2) \\
Lat~VI & 1.2(1) & 1.3(1) & 1.40(9) & 1.50(9) & 1.72(5) \\
Lat~VII & 1.21(9) & 1.32(8) & 1.43(6) & 1.4(2) & 1.72(9) \\
Lat~VIII& 1.2(1) & 1.29(8) & 1.41(7) & 1.3(2) & 1.7(1) \\
\hline
\end{tabular}
\end{table}
We have used the value of meson masses to read off the
lattice baryon masses from the planes ($M_N a$, $(M_{PS} a)^2$) and
($M_\Delta a$, $(M_{PS} a)^2$). Using the same values of $a^{-1}$ used for
meson masses we have obtained the results reported in Table~\ref{tab:baryonsGeV}.
For baryons we find very good agreement with the old APE \cite{parisi}
data, while we find slightly
larger values when comparing with JLQCD \cite{fukugita} and LANL \cite{gupta}.
Also for baryon masses we can conclude that we do not see
a dependence on $a$ and that we have an agreement with the
experimental data of $\sim 10\%-15\%$.
| proofpile-arXiv_065-442 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
Almost all astronomers will agree that most of the mass in the
Universe is nonluminous. The nature of this dark matter remains
one of the great mysteries of science today.
Dynamics of cluster of galaxies suggest
a universal nonrelativistic-matter density of
$\Omega_0\simeq0.1-0.3$. If the luminous matter were all
there was, the duration of the epoch
of structure formation would be very short, thereby requiring
(in almost all theories of structure formation) fluctuations in
the microwave background which would be larger than those observed.
These considerations imply $\Omega_0\gtrsim0.3$ \cite{marccmb}.
Second, if the current value of $\Omega_0$ is of order unity today,
then at the Planck time it must have
been $1\pm10^{-60}$ leading us to believe that $\Omega_0$ is
precisely unity
for aesthetic reasons. A related argument comes from
inflationary cosmology, which provides the most satisfying
explanation for the smoothness of the microwave background
\cite{inflation}. To account for this isotropy, inflation must set
$\Omega$ (the {\it total} density, including a cosmological
constant) to unity.
\begin{figure}[htbp]
\centerline{\psfig{file=rotation.ps,width=3.3in}}
\bigskip
\caption{
Rotation curve for the spiral galaxy NGC6503. The points
are the measured circular rotation velocities as a
function of distance from the center of the galaxy.
The dashed and dotted curves are the contribution to
the rotational velocity due to the observed disk and
gas, respectively, and the dot-dash curve is the
contribution from the dark halo.}
\label{rotationfigure}
\end{figure}
However, the most robust observational evidence for the
existence of dark matter involves galactic dynamics. There is
simply not enough luminous matter ($\Omega_{\rm lum}\la0.01$)
observed in spiral galaxies to account for their observed
rotation curves (for example, that for NGC6503 shown in
Fig.~\ref{rotationfigure} \cite{broeils}). Newton's
laws imply a galactic dark halo
of mass $3-10$ times that of the luminous component. With the
simple and plausible assumption that the halo of our galaxy is
roughly spherical, one can determine that the local dark-matter
density is roughly $\rho_0 \simeq 0.3$ GeV~cm$^{-3}$.
Furthermore, the velocity distribution
of the halo dark matter should be roughly Maxwell-Boltzmann with
a velocity dispersion $\simeq270$ km~s$^{-1}$.
On the other hand, big-bang nucleosynthesis
suggests that the baryon density is $\Omega_b\la0.1$ \cite{bbn}, too
small to account for the dark matter in the Universe.
Although a neutrino species of mass ${\cal O}(30\, {\rm eV})$ could
provide the right dark-matter density, N-body simulations of
structure formation in a neutrino-dominated Universe do a poor
job of reproducing the observed structure \cite{Nbody}.
Furthermore, it is difficult to see (essentially from the Pauli
principle) how such a neutrino could make up the dark matter in
the halos of galaxies \cite{gunn}. It appears likely then, that some
nonbaryonic, nonrelativistic matter is required.
The two leading candidates from particle theory are the axion
\cite{axion},
which arises in the Peccei-Quinn solution to the strong-$CP$
problem, and a weakly-interacting massive particle (WIMP), which
may arise in supersymmetric (or other) extensions of the
standard model \cite{jkg}.
Here, I review the axion solution to the strong-$CP$ problem, the
astrophysical constraints to the axion mass, and prospects for
detection of an axion. I then review the WIMP solution to the
dark-matter problem and avenues toward detection. Finally, I
briefly discuss how measurements of CMB anisotropies may in the
future help determine more precisely the amount of exotic dark
in the Universe.
\section{Axions}
Although supersymmetric particles seem to get more attention in
the literature lately, we should not forget that the axion also
provides a well-motivated and promising alternative dark-matter
candidate \cite{axion}. The QCD Lagrangian may be written
\begin{equation}
{\cal L}_{QCD} = {\cal L}_{\rm pert} + \theta {g^2 \over 32
\pi^2} G \widetilde{G},
\end{equation}
where the first term is the perturbative Lagrangian responsible
for the numerous phenomenological successes of QCD. However,
the second term (where $G$ is the gluon field-strength tensor
and $\widetilde{G}$ is its dual), which is a consequence of
nonperturbative effects, violates $CP$. However, we know
experimentally that $CP$ is not violated in the strong
interactions, or if it is, the level of strong-$CP$ violation is
tiny. From constraints to the neutron electric-dipole moment,
$d_n \la 10^{-25}$ e~cm, it can be inferred that $\theta
\la 10^{-10}$. But why is $\theta$ so small? This is the
strong-$CP$ problem.
The axion arises in the Peccei-Quinn solution to the strong-$CP$
problem \cite{PQ}, which close to twenty years after it was proposed still
seems to be the most promising solution. The idea is to
introduce a global $U(1)_{PQ}$ symmetry broken at a scale
$f_{PQ}$, and $\theta$ becomes a dynamical field which is the
Nambu-Goldstone mode of this symmetry.
At temperatures below the QCD phase transition,
nonperturbative quantum effects break explicitly the symmetry
and drive $\theta\rightarrow 0$. The axion is the
pseudo-Nambu-Goldstone boson of this near-global symmetry. Its
mass is $m_a \simeq\, {\rm eV}\,(10^7\, {\rm GeV}/ f_a)$, and its
coupling to ordinary matter is $\propto f_a^{-1}$.
{\it A priori}, the Peccei-Quinn solution works equally well for
any value of $f_a$ (although one would generically expect it to
be less than or of order the Planck scale). However, a variety
of astrophysical observations and a few laboratory experiments
constrain the axion mass to be $m_a\sim10^{-4}$ eV, to within a
few orders of magnitude. Smaller masses would lead to an
unacceptably large cosmological abundance. Larger masses
are ruled out by a combination of constraints from supernova
1987A, globular clusters, laboratory experiments, and a search
for two-photon decays of relic axions \cite{ted}.
One conceivable theoretical difficulty with this axion mass
comes from generic quantum-gravity arguments \cite{gravity}. For
$m_a\sim10^{-4}$ eV, the magnitude of the explicit symmetry
breaking is incredibly tiny compared with the PQ scale, so the
global symmetry, although broken, must be very close to exact.
There are physical arguments involving, for example, the
nonconservation of global charge in evaporation of a black hole
produced by collapse of an initial state with nonzero global
charge, which suggest that global symmetries should be violated
to some extent in quantum gravity. When one writes down a
reasonable {\it ansatz} for a term in a low-energy effective
Lagrangian which might arise from global-symmetry violation at
the Planck scale, the coupling of such a term is found to be
extraordinarily small (e.g., $\la 10^{-55}$). Of course,
we have at this point no predictive theory of quantum gravity,
and several mechanisms for forbidding these global-symmetry
violating terms have been proposed \cite{solutions}. Therefore,
these arguments by no means ``rule out'' the axion solution.
In fact, discovery of an axion would provide much needed clues
to the nature of Planck-scale physics.
Curiously enough, if the axion mass is in the relatively small viable
range, the relic density is $\Omega_a\sim1$ and may therefore
account for the halo dark matter. Such axions would be produced
with zero momentum by a misalignment mechanism in the early
Universe and therefore act as cold dark matter. During the process of
galaxy formation, these axions would fall into the Galactic
potential well and would therefore be present in our halo with a
velocity dispersion near 270 km~s$^{-1}$.
Although the interaction of axions with ordinary matter is
extraordinarily weak, Sikivie proposed a very clever method of
detection of Galactic axions \cite{sikivie}. Just as the axion couples to
gluons through the anomaly (i.e., the $G\widetilde{G}$ term),
there is a very weak coupling of an axion to photons through the
anomaly. The axion can therefore decay to two
photons, but the lifetime is $\tau_{a\rightarrow \gamma\gamma}
\sim 10^{50}\, {\rm s}\, (m_a / 10^{-5}\, {\rm eV})^{-5}$ which
is huge compared to the lifetime of the Universe and therefore
unobservable. However, the $a\gamma\gamma$ term in the
Lagrangian is ${\cal L}_{a\gamma\gamma} \propto a {\vec E} \cdot
{\vec B}$ where ${\vec E}$ and ${\vec B}$ are the electric and
magnetic field strengths. Therefore, if one immerses a resonant
cavity in a strong magnetic field, Galactic axions which pass
through the detector may be converted to fundamental excitations
of the cavity, and these may be observable \cite{sikivie}. Such
an experiment
is currently underway and expects to probe the entire acceptable
parameter space within the next five years
\cite{axionexperiment}. A related experiment, which looks for
excitations of Rydberg atoms, may also find dark-matter axions
\cite{rydberg}.
Although the sensitivity of this technique is supposed to be
excellent, it can only cover a limited axion-mass range.
It should be kept in mind that there are no accelerator tests
for axions in the acceptable mass range. Therefore, these
dark-matter axion experiment are actually our {\it only}
way to test the Peccei-Quinn solution.
\section{Weakly-Interacting Massive Particles}
Suppose that in addition to the known particles of the
standard model, there exists a new, yet undiscovered, stable (or
long-lived) weakly-interacting massive
particle (WIMP), $\chi$. At temperatures
greater than the mass of the particle, $T\gg m_\chi$, the
equilibrium number density of such particles is $n_\chi \propto
T^3$, but for lower temperatures, $T\ll m_\chi$, the equilibrium
abundance is exponentially suppressed, $n_\chi \propto
e^{-m_\chi/T}$. If the expansion of the Universe were so slow
that thermal equilibrium was always maintained, the number of
WIMPs today would be infinitesimal. However, the Universe is
not static, so equilibrium thermodynamics is not the entire story.
\begin{figure}[htbp]
\centerline{\psfig{file=yyy.ps,width=3.3in}}
\bigskip
\caption{
Comoving number density of a WIMP in the early
Universe. The dashed curves are the actual abundance,
and the solid curve is the equilibrium abundance.}
\label{YYY}
\end{figure}
At high temperatures ($T\gg m_\chi$), $\chi$'s are abundant and
rapidly converting to lighter particles and {\it vice versa}
($\chi\bar\chi\leftrightarrow l\bar l$, where $l\bar l$ are quark-antiquark and
lepton-antilepton pairs, and if $m_\chi$ is greater than the mass of the
gauge and/or Higgs bosons, $l\bar l$ could be gauge- and/or Higgs-boson
pairs as well). Shortly after $T$ drops below $m_\chi$ the number
density of $\chi$'s drops exponentially, and the rate for annihilation of
$\chi$'s, $\Gamma=\VEV{\sigma v} n_\chi$---where $\VEV{\sigma v}$ is the
thermally averaged total cross section for annihilation of $\chi\bar\chi$
into lighter particles times relative velocity $v$---drops below
the expansion rate, $\Gamma\mathrel{\mathpalette\fun <} H$. At this point, the $\chi$'s cease to
annihilate, they fall out of equilibrium, and a relic cosmological
abundance remains.
Fig.~\ref{YYY} shows numerical solutions to the Boltzmann
equation which determines the WIMP abundance. The
equilibrium (solid line) and actual (dashed lines) abundances
per comoving volume are plotted as a
function of $x\equiv m_\chi/T$ (which increases with increasing time).
As the annihilation cross section is increased
the WIMPs stay in equilibrium longer, and we are left with a
smaller relic abundance.
An approximate solution to the Boltzmann equation yields the
following estimate for the current cosmological abundance of the
WIMP:
\begin{equation}
\Omega_\chi h^2={m_\chi n_\chi \over \rho_c}\simeq
\left({3\times 10^{-27}\,{\rm cm}^3 \, {\rm sec}^{-1} \over
\sigma_A v} \right),
\label{eq:abundance}
\end{equation}
where $h$ is the Hubble constant in units of 100
km~s$^{-1}$~Mpc$^{-1}$. The result is to a first approximation
independent of the WIMP mass and is fixed primarily by its
annihilation cross section.
The WIMP velocities at freeze out are typically some appreciable
fraction of the speed of light. Therefore, from
equation~(\ref{eq:abundance}), the WIMP will have a cosmological
abundance of order unity today if the annihilation cross section
is roughly $10^{-9}$ GeV$^{-2}$. Curiously, this is the order
of magnitude one would expect from a typical electroweak cross
section,
\begin{equation}
\sigma_{\rm weak} \simeq {\alpha^2 \over m_{\rm weak}^2},
\end{equation}
where $\alpha \simeq {\cal O}(0.01)$ and $m_{\rm weak} \simeq
{\cal O}(100\, {\rm GeV})$. The value of the cross section in
equation~(\ref{eq:abundance}) needed to provide $\Omega_\chi\sim1$
comes essentially from the age of the Universe. However, there
is no {\it a priori} reason why this cross section should be of
the same order of magnitude as the cross section one would
expect for new particles with masses and interactions
characteristic of the electroweak scale. In other words, why
should the age of the Universe have anything to do with
electroweak physics? This ``coincidence'' suggests that if a
new, yet undiscovered, massive particle with electroweak
interactions exists, then it should have a relic density of
order unity and therefore provides a natural dark-matter
candidate. This argument has been the driving force behind a
vast effort to detect WIMPs in the halo.
The first WIMPs considered were massive Dirac or Majorana
neutrinos with masses in the range of a few GeV to a few TeV.
(Due to the Yukawa coupling which gives a neutrino its mass, the
neutrino interactions become strong above a few TeV, and it no
longer remains a suitable WIMP candidate \cite{unitarity}.) LEP ruled out
neutrino masses below half the $Z^0$ mass. Furthermore, heavier
Dirac neutrinos have been ruled out as the primary component of
the Galactic halo by direct-detection experiments (described
below) \cite{heidelberg}, and heavier Majorana neutrinos have
been ruled out by indirect-detection experiments
\cite{kamiokande} (also described below) over much
of their mass range. Therefore, Dirac neutrinos cannot comprise
the halo dark matter \cite{griestsilk}; Majorana neutrinos can,
but only over a
small range of fairly large masses. This was a major triumph
for experimental particle astrophysicists:\ the first
falsification of a dark-matter candidate. However, theorists
were not too disappointed: The stability of a fourth generation
neutrino had to be postulated {\it ad hoc}---it was not
guaranteed by some new symmetry. So although heavy neutrinos
were plausible, they certainly were not very well-motivated from
the perspective of particle theory.
A much more promising WIMP candidate comes from supersymmetry
(SUSY) \cite{jkg,haberkane}. SUSY was
hypothesized in particle physics to cure the naturalness problem
with fundamental Higgs bosons at the electroweak scale.
Coupling-constant unification at the GUT scale seems to be
improved with SUSY, and it seems to be an essential ingredient
in theories which unify gravity with the other three fundamental
forces.
As another consequence, the existence of a new symmetry,
$R$-parity, in SUSY theories guarantees that the lightest
supersymmetric particle (LSP) is stable.
In the minimal supersymmetric extension of the
standard model (MSSM), the LSP is usually the neutralino, a linear
combination of the supersymmetric partners of the photon, $Z^0$,
and Higgs bosons. (Another possibility is the sneutrino, but
these particles interact like neutrinos and have been ruled out
over most of the available mass range \cite{sneutrino}.) Given
a SUSY model, the cross section for
neutralino annihilation to lighter particles is straightforward,
so one can obtain the cosmological mass density. The
mass scale of supersymmetry must be of order the weak scale to
cure the naturalness problem, and the neutralino will have only
electroweak interactions. Therefore, it is to be expected that
the cosmological neutralino abundance is of order unity. In
fact, with detailed calculations, one finds that the neutralino
abundance in a very broad class of supersymmetric extensions of
the standard model is near unity and can therefore account for
the dark matter in our halo \cite{ellishag}.
If neutralinos reside in the halo, there are several avenues
for detection \cite{jkg}. One of the most promising techniques currently
being pursued involves searches for the ${\cal O}({\rm keV})$ recoils
produced by elastic scattering of neutralinos from nuclei in
low-background detectors \cite{witten,labdetectors}. Another strategy is
observation of energetic neutrinos produced by annihilation of
neutralinos in the Sun and Earth in converted proton-decay and
astrophysical-neutrino detectors (such as MACRO, Kamiokande,
IMB, AMANDA, and NESTOR) \cite{SOS}. There are also searches for
anomalous cosmic rays which would be produced by
annihilation of WIMPs in the halo. Of course, SUSY particles
should also show up in accelerator searches if their mass falls
within the experimentally accessible range.
Although supersymmetry provides perhaps the most promising
dark-matter candidate (and solves numerous problems in particle
physics), a practical difficulty with supersymmetry is that we
have little detailed predictive power. In SUSY models, the
standard-model particle spectrum is more than doubled, and we
really have no idea what the masses of all these superpartners
should be. There are also couplings, mixing angles, etc.
Therefore, what theorists generally do is survey a huge
set of models with masses and couplings within a
plausible range, and present results for relic abundances and
direct- and indirect-detection rates, usually as scatter plots
versus neutralino mass.
Energetic neutrinos from WIMP annihilation in the Sun or Earth
would be inferred by observation of neutrino-induced upward
muons coming from the direction of the Sun or the core of the
Earth. Predictions for the fluxes of such muons in SUSY models
seem to fall for the most part between $10^{-6}$ and 1
event~m$^{-2}$~s$^{-1}$ \cite{jkg}, although the numbers may be a bit
higher or lower in some models. Presently, IMB and Kamiokande
constrain the flux of energetic neutrinos from the Sun to be
less than about 0.02 m$^{-2}$~s$^{-1}$ \cite{kamiokande,imb}.
MACRO expects to be
able to improve on this sensitivity by perhaps an order of
magnitude. Future detectors may be able to improve on this
limit further. For example, AMANDA expects to have an area of
roughly $10^4$ m$^2$, and a $10^6$-m$^2$
detector is being discussed. However, it should be kept in
mind that without muon energy resolution, the sensitivity of
these detectors will not approach the inverse exposure; it will
be limited by the atmospheric-neutrino background. If a
detector has good angular resolution, the signal-to-noise ratio
can be improved, and even moreso with energy resolution, so
sensitivities approaching the inverse exposure could be
achieved \cite{joakim}. Furthermore, ideas for neutrino detectors with
energy resolution are being discussed \cite{wonyong}, although
at this point these appear likely to be in the somewhat-distant future.
The other possibility is direct detection of a WIMP via
observation of the nuclear recoil induced by WIMP-nucleus
elastic scattering in a low-background detector. The predicted
rates depend on the target nucleus adopted. For example, in a
broad range of SUSY models, the predicted scattering rates in a
germanium detector seem to fall for the most part between
$10^{-4}$ to 10 events~kg$^{-1}$~day$^{-1}$ \cite{jkg}, although again,
there may be models with higher or lower rates. Current
experimental sensitivities in germanium detectors are around 10
events~kg$^{-1}$~day$^{-1}$ \cite{heidelberg}. To illustrate
future prospects, consider the CDMS experiment \cite{cdms} which
expects to soon have a kg germanium detector with an
background rate of 1 event~day$^{-1}$. After a one-year
exposure, their sensitivity would therefore be ${\cal O}(0.1\,
{\rm event~kg}^{-1}\,{\rm day}^{-1})$; this could be improved
with better background rejection. Future detectors will
achieve better
sensitivities, and it should be kept in mind that numerous other
target nuclei are being considered by other groups. However, it
also seems clear that it will be quite a while until a good
fraction of the available SUSY parameter space is probed.
Generally, most theorists have just plugged in SUSY parameters
into the machinery which produces detection rates and plotted
results for direct and indirect detection. However, another
approach is to compare, in a somewhat model-independent although
approximate fashion, the rates for direct and indirect
detection \cite{jkg,taorich,bernard}. The underlying
observation is that the rates for the
two types of detection are both controlled primarily by the WIMP-nucleon
coupling. One must then note that WIMPs generally undergo one
of two types of interaction with the nucleon: an axial-vector
interaction in which the WIMP couples to the nuclear spin
(which, for nuclei with nonzero angular momentum is roughly 1/2
and {\it not} the total angular momentum), and a scalar
interaction in which the WIMP couples to the total mass of the
nucleus. The direct-detection rate depends on the WIMP-nucleon
interaction strength and on the WIMP mass. On the other hand,
indirect-detection rates will have an additional dependence on
the energy spectrum of neutrinos from WIMP annihilation. By
surveying the various possible neutrino energy spectra, one
finds that for a given neutralino mass and annihilation rate in
the Sun, the largest upward-muon flux is roughly three times as
large as the smallest \cite{bernard}. So even if we assume the
neutralino-nucleus interaction is purely scalar or purely
axial-vector, there will still be a residual model-dependence of
a factor of three when comparing direct- and indirect-detection
rates.
For example, for scalar-coupled WIMPs, the event rate in a kg
germanium detector will be
equivalent to the event rate in a $(2-6)\times 10^6$ m$^2$
neutrino detector for 10-GeV WIMPs and $(3-5)\times10^4$ m$^2$
for TeV WIMPs \cite{bernard}. Therefore, the relative
sensitivity of indirect detection when compared with the
direct-detection sensitivity increases with mass.
The bottom line of such an analysis seems to be that
direct-detection experiments will be more sensitive to
neutralinos with scalar interactions with nuclei, although
very-large neutrino telescopes may achieve comparable
sensitivities at larger WIMP masses. This should
come as no surprise given the fact that direct-detection
experiments rule out Dirac neutrinos \cite{heidelberg}, which
have scalar-like interactions, far more effectively than
do indirect-detection experiments \cite{bernard}.
Generically, the sensitivity of
indirect searches (relative to direct searches) should be better
for WIMPs with axial-vector interactions, since the Sun is
composed primarily of nuclei with spin (i.e., protons).
However, a comparison of direct-
and indirect-detection rates is a bit more difficult for
axially-coupled WIMPs, since the nuclear-physics uncertainties
in the neutralino-nuclear cross section are much greater, and
the spin distribution of each target nucleus must be modeled.
Still, in a careful analysis, Rich and Tao found that in 1994,
the existing sensitivity of energetic-neutrino searches to
axially-coupled WIMPs greatly exceeded the sensitivities of
direct-detection experiments \cite{taorich}.
To see how the situation may change with future
detectors, let us consider a specific axially-coupled
dark-matter candidate, the light Higgsino recently put forward by
Kane and Wells \cite{kanewells}. In order to explain the
anomalous CDF $ee\gamma
\gamma + \slashchar{E}_T$ \cite{CDF}, the $Z\rightarrow b\bar b$ anomaly,
and the dark matter, this Higgsino must have a mass between
30--40 GeV. Furthermore, the coupling of this Higgsino to
quarks and leptons is due primarily to $Z^0$ exchange with a
coupling proportional to $\cos 2\beta$, where $\tan\beta$ is the
usual ratio of Higgs vacuum expectation values in
supersymmetric models. Therefore, the usually messy cross
sections one deals with in a general MSSM simplify for this
candidate, and the cross sections needed for the cosmology of
this Higgsino depend only on the two parameters $m_\chi$ and
$\cos2\beta$. Furthermore, since the neutralino-quark
interaction is due only to $Z^0$ exchange, this Higgsino will
have only axial-vector interactions with nuclei.
The Earth is composed primarily of spinless nuclei, so WIMPs
with axial-vector interactions will not be captured in the Earth,
and we expect no neutrinos from WIMP annihilation therein.
However, most of the mass in the Sun is composed of nuclei with
spin (i.e., protons). The flux of upward muons induced by
neutrinos from
annihilation of these light Higgsinos would be $\Gamma_{\rm
det}\simeq 2.7\times10^{-2}\, {\rm m}^{-2}\, {\rm yr}^{-1}\,
\cos^2 2\beta$ \cite{katie}. On the other hand, the rate for scattering from
$^{73}$Ge is $R\simeq 300\, \cos^2 2\beta\, {\rm kg}^{-1}\,
{\rm yr}^{-1}$ \cite{kanewells,katie}. For illustration, in
addition to their kg of
natural germanium, the CDMS experiment also plans to
run with 0.5 kg of (almost) purified $^{73}$Ge. With a
background event rate of roughly one event~kg$^{-1}$~day$^{-1}$,
after one year, the $3\sigma$ sensitivity of the experiment will
be roughly 80 kg$^{-1}$~yr$^{-1}$. Comparing the predictions
for direct and indirect detection of this axially-coupled WIMP,
we see that the enriched-$^{73}$Ge sensitivity should improve on
the {\it current}
limit to the upward-muon flux ($0.02$ m$^{-2}$ yr$^{-1}$)
roughly by a factor of 4. When we compare this with the forecasted
factor-of-ten improvement expected in MACRO, it appears that the
sensitivity of indirect-detection experiments looks more
promising. Before
drawing any conclusions, however, it should be noted that the
sensitivity in detectors with other nuclei with spin may be
significantly better. On the other hand, the sensitivity of
neutrino searches increases relative to direct-detection
experiments for larger WIMP masses. It therefore seems at this
point that the two schemes will be competitive for detection of
light axially-coupled WIMPs, but the neutrino telescopes may
have an advantage in probing larger masses.
A common question is whether
theoretical considerations favor a WIMP which has predominantly
scalar or axial-vector couplings. Unfortunately, there is no
simple answer. When detection of supersymmetric dark matter was
initially considered, it seemed that the neutralino in most
models would have predominantly axial-vector interactions. It
was then noted that in some fraction of models where the
neutralino was a mixture of Higgsino and gaugino, there could be
some significant scalar coupling as well \cite{kim}. As it became evident
that the top quark had to be quite heavy, it was realized that
nondegenerate squark masses would give rise to scalar couplings
in most models \cite{drees}. However, there are still large regions of
supersymmetric parameter space where the neutralino has
primarily axial-vector interactions, and in fact, the Kane-Wells
Higgsino candidate has primarily axial-vector interactions. The
bottom line is that theory cannot currently reliable say which
type of interaction the WIMP is likely to have, so
experiments should continue to try to target both.
\section{Dark Matter and the Cosmic Microwave Background}
The key argument for nonbaryonic dark matter relies on the
evidence that the total nonrelativistic-matter density $\Omega_0
\mathrel{\mathpalette\fun >} 0.1$, outweighs the baryon density $\Omega_b\mathrel{\mathpalette\fun <} 0.1$ allowed by
big-bang nucleosynthesis. With the advent of a new
generation of long-duration balloon-borne and ground-based
interferometry experiments and NASA's MAP \cite{MAP} and ESA's
COBRAS/SAMBA \cite{COBRAS} missions, CMB measurements will usher
in a new era in cosmology. In forthcoming years, the cosmic
microwave background (CMB) may provide a precise inventory of
the matter content in the Universe and confirm the discrepancy
between the baryon density and the total nonrelativistic-matter
density, if it indeed exists.
The primary goal of these experiments is recovery of the
temperature autocorrelation function or angular power spectrum
of the CMB. The fractional temperature perturbation
$\Delta T({\bf \hat n})/T$ in a given direction ${\bf \hat n}$ can be expanded
in terms of spherical harmonics,
\begin{equation}
{\Delta T({\bf \hat n}) \over T} = \sum_{lm} \, a_{(lm)}\,
Y_{(lm)}({\bf \hat n}),
\label{Texpansion}
\end{equation}
where the multipole coefficients are given by
\begin{equation}
a_{(lm)} = \int\, d{\bf \hat n}\, Y_{(lm)}^*({\bf \hat n}) \, {\Delta
T({\bf \hat n}) \over T}.
\label{alms}
\end{equation}
Cosmological theories predict that these multipole coefficients
are statistically independent and are distributed with variance
$ \VEV{a_{(lm)}^* a_{(l'm')} } = C_l \, \delta_{ll'} \,
\delta_{mm'}$.
Roughly speaking, each $C_l$ measures the square of the mean temperature
difference between two points separated by an angle $\theta\sim
\pi/l$.
\begin{figure}[htbp]
\centerline{\psfig{file=models.eps,width=3.7in}}
\bigskip
\caption{
Theoretical predictions for CMB spectra as a function
of multipole moment $\ell$ for models with primordial
adiabatic perturbations. In each case, the
heavy curve is that for the standard-CDM values,
a total density $\Omega=1$, cosmological constant
$\Lambda=0$, baryon density $\Omega_b=0.06$, and
Hubble parameter $h=0.5$. Each graph shows the effect
of variation of one of these parameters. In (d),
$\Omega=\Omega_0+\Lambda=1$.}
\label{curves}
\end{figure}
Theoretical predictions for the $C_l$'s can be made given a
theory for structure formation and the values of several
cosmological parameters. For example, Fig.~\ref{curves} shows predictions
for multipole moments in models with primordial adiabatic
perturbations.
The peaks in the spectra come from oscillations in the
photon-baryon fluid at the surface of last scatter, and the
damping at small angles is due to the finite thickness of the
surface of last scatter. Each panel shows the
effect of independent variation of one of the cosmological
parameters. As illustrated, the height, width, and spacing
of the acoustic peaks in the angular spectrum depend on these
(and other) cosmological parameters. The CMB
spectrum also depends on the model (e.g., inflation or
topological defects) for structure formation, the ionization
history of the Universe, and the presence of gravity waves.
However, no two of the classical cosmological parameters affects
the CMB spectrum in precisely the same way. For example, the
angular position of the first peak depends primarily on the
geometry ($\Omega=\Omega_0 +\Lambda$ where $\Lambda$ is the
contribution of the cosmological constant) of the Universe
\cite{kamspergelsug}, but is relatively insensitive to
variations in the other parameters. Assuming that the
primordial perturbations were adiabatic, we could fit for all of
these parameters if the angular spectrum could be measured
precisely.
COBE normalizes the amplitude and slope of the CMB spectrum to
$\sim10\%$. However, the angular resolution was not fine enough to probe
the detailed shape of the acoustic peaks in the
power spectrum, so COBE was unable to capitalize on this wealth
of information. Nor can it discriminate between scalar and
tensor modes. A collection of recent ground-based
and balloon-borne experiments seem to confirm
a first acoustic peak, but they still cannot
determine its precise height, width, or location.
In the next few years, long-duration balloon
flights (e.g., BOOMERANG and TOPHAT) and ground-based
interferometry experiments (e.g., CAT and CBI) will begin
to discern the first and higher few peaks.
Subsequently, future satellite experiments, such as NASA's MAP
mission \cite{MAP} and then ESA's COBRAS/SAMBA \cite{COBRAS} will
accurately map the CMB temperature over most of the sky with
good angular resolution and will therefore be able to recover
the CMB power spectrum with precision.
Of course, the precision attainable is ultimately limited by
cosmic variance and practically by
a finite angular resolution, instrumental noise, and
partial sky coverage in a realistic CMB mapping experiment.
Assuming that primordial perturbations are adiabatic, one
finds that with future
satellite missions, $\Omega$ may potentially be determined to
better than 10\% {\it after} marginalizing over all other undetermined
parameters, and better than 1\% if the other
parameters can be fixed by independent observations or
assumption \cite{jkksone}. This would be far more accurate
than any traditional
determinations of the geometry. (Of course, if primordial
perturbations turn out to be isocurvature or due to topological
defects, this may not be the case.) The cosmological
constant $\Lambda$ will be determined with a similar accuracy, so
the nonrelativistic-matter density $\Omega_0$ will also be
accurately determined \cite{jkkstwo}. Small variations in the
baryon density
have a dramatic effect on the CMB spectrum, so $\Omega_b$ will
be determined with even greater precision. Therefore, if there
is more nonrelativistic matter in the Universe than the baryons
can account for, as current evidence suggests, it should become
clear with these future CMB experiments.
The CMB will also measure the Hubble constant and perhaps be
sensitive to a small neutrino mass \cite{gds}. Temperature maps
will also begin to disentangle the scalar and tensor (i.e.,
long-wavelength gravity-wave) contributions to the CMB and
determine their primordial spectra, and this could be used to
test inflation \cite{jkkstwo}. CMB polarization maps may also help
isolate the tensor contribution \cite{polarization}. Therefore,
the CMB will become an increasingly powerful probe of the early
Universe.
\acknowledgements{This work was supported by the D.O.E. under
contract DEFG02-92-ER 40699, NASA under NAG5-3091, and the
Alfred P. Sloan Foundation.}
| proofpile-arXiv_065-443 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{INTRODUCTION}
A precise theoretical framework is needed for
the study of the quark mass effects in
physical observables because quarks are not free particles.
In fact, the quark masses should be seen more
like coupling constants than like physical parameters.
The perturbative pole mass and the running mass are
the two most commonly used quark mass definitions.
The perturbative pole mass, $M(p^2=M^2)$,
is defined as the pole of the
renormalized quark propagator in a strictly perturbative sense.
It is gauge invariant and scheme independent. However,
it appears to be ambiguous due to non-perturbative renormalons.
The running mass, $\bar{m}(\mu)$, the
renormalized mass in the $\overline{MS}$ scheme, does not suffer
from this ambiguity.
Both quark mass definitions can be related perturbatively through
\begin{equation}
M = \bar{m}(\mu) \left\{ 1 + \frac{\as(\mu)}{\pi} \left[
\frac{4}{3}-\log \frac{\bar{m}^2(\mu)}{\mu^2} \right] \right\}.
\label{relates}
\end{equation}
Heavy quark masses, like the bottom quark mass, can be
extracted using QCD Sum Rules or lattice calculations
from the quarkonia spectrum, see \cite{UIMP}
and references therein. The bottom quark perturbative pole mass
appears to be around $M_b = 4.6-4.7 (GeV)$ whereas
the running mass at the running mass scale reads
$\bar{m}_b(\bar{m}_b) = (4.33 \pm 0.06) GeV$.
Performing the running until the $Z$-boson mass scale
we find $\bar{m}_b(M_Z) = (3.00 \pm 0.12) GeV$.
Since for the bottom quark the difference between the
perturbative pole mass and the
running mass at the $M_Z$ scale
is quite significant it is crucial
to specify in any theoretical perturbative
prediction at $M_Z$ which mass should we use.
The relative uncertainty in the strong coupling
constant decreases in the running from low to
high energies as the ratio of the strong coupling constants
at both scales. On the contrary,
if we perform the quark mass running with the
extreme mass and strong coupling constant values
and take the maximum difference as the propagated error,
induced by the strong coupling constant error,
the quark mass uncertainty increases following
\begin{equation}
\varepsilon_r (\bar{m}(M_Z)) \simeq
\varepsilon_r (\bar{m}(\mu))
\end{equation}
\[ + \frac{2 \gamma_0}{\beta_0}
\left(\frac{\as(\mu)}{\as(M_Z)}-1 \right)
\varepsilon_r (\as(M_Z)),
\]
where $\gamma_0=2$ and $\beta_0=11-2/3 N_F$, see
figure~\ref{running}. We use the world average \cite{bethke} value
$\as = 0.118 \pm 0.006$ for the strong coupling constant.
It is interesting to stress, looking at figure~\ref{running},
that even a big uncertainty in
a possible evaluation of the bottom quark mass
at the $M_Z$ scale can be competitive with
low energy QCD Sum Rules and lattice calculations
with smaller errors~\footnote{A recent lattice
evaluation \cite{lattice}
has enlarged the initial estimated error on the bottom quark
mass, $\bar{m}_b(\bar{m}_b) = (4.15 \pm 0.20) GeV$, due to
unknown higher orders in the perturbative matching of the HQET
to the full theory.}.
Furthermore, non-perturbative contributions are expected to
be negligible at the $Z$-boson mass scale.
The running mass holds another remarkable feature.
Total cross sections can exhibit potentially dangerous terms
of the type $M^2 \log M^2/s$ that however can be absorbed
\cite{chety} using Eq.~(\ref{relates}) and expressing the
total result in terms of the running mass.
\mafigura{5.5cm}{running.ps}
{Running of the bottom quark mass from low energies to the
$M_Z$ scale.
Upper line is the run of $\bar{m}_b(\bar{m}_b)=4.39(GeV)$ with
$\as (M_Z)=0.112$. Bottom line is the run of
$\bar{m}_b(\bar{m}_b)=4.27(GeV)$ with $\as (M_Z)=0.124$.
Second picture is the difference of both, our estimate for
the propagated error.}
{running}
\section{THREE JETS OBSERVABLES AT LO}
Quark masses can be neglected for many observables
at LEP because usually they appear as the ratio
$m_q^2/M_Z^2$. For the heaviest quark produced at LEP,
the bottom quark, this means a correction of 3 per mil
for a quark mass of 5 (GeV).
Even if the coefficient in front is 10 we get at most a
3\% effect, 1\% if we use the bottom quark running mass
at $M_Z$.
This argument is true for total cross section. However,
jet cross sections depend on a new variable, $y_c$,
the jet-resolution parameter that defines the jet
multiplicity. This new variable introduces a new scale
in the analysis, $E_c = M_Z \sqrt{y_c}$, that
for small values of $y_c$ could enhance the effect
of the quark mass as $m_b^2/E_c^2 = (m_b^2/M_Z^2)/y_c$.
The high precision achieved at LEP makes these effects
relevant. In particular, it has been shown \cite{Juano}
that the biggest systematic error in the measurement of
$\as(M_Z)$ from $b\bar{b}$-production at LEP from the
ratio of three to two jets comes from the uncertainties
in the estimate of the quark mass effects.
We are going to study the effect of the bottom quark mass
in the following ratios of three-jet decay rates and
angular distributions
\begin{eqnarray}
R_3^{bd} &\equiv& \frac{\Gamma^b_{3j}(y_c)/\Gamma^b}
{\Gamma^d_{3j}(y_c)/\Gamma^d}, \label{r3bd}\\
R^{bd}_\vartheta &\equiv& \left.
\frac{1}{\Gamma^b}\frac{d\Gamma^b_{3j}}{d\vartheta}\right/
\frac{1}{\Gamma^d}\frac{d\Gamma^d_{3j}}{d\vartheta},
\label{rtheta}
\end{eqnarray}
where we consider massless the $d$-quark and
$\vartheta$ is the minimum of the angles formed between
the gluon jet and the quark and antiquark jets. Both
observables are normalized to the total decay rates
in order to cancel large weak corrections dependent on the
top quark mass \cite{top}.
\def1.5{1.5}
\begin{table}[t]
\caption{The jet-clustering algorithms}
\label{algorithms}
\begin{center}
\begin{tabular}{ll}
\hline
Algorithm & Resolution \\
\hline
EM & $2(p_i \cdot p_j)/s$ \\
JADE & $2(E_i E_j)(1-\cos \vartheta_{ij})/s$ \\
E & $(p_i+p_j)^2/s$ \\
DURHAM & $2 \min(E_i^2,E_j^2)(1-\cos \vartheta_{ij})/s$ \\
\hline
\end{tabular}
\end{center}
\end{table}
\mafigura{6cm}{LO.ps}
{Feynman diagrams contributing to the three-jets decay rate
of $Z\rightarrow b\bar{b}$ at order $\as$.}
{feynmanLO}
At LO in the strong coupling constant
we must compute the amplitudes of the Feynman diagrams
depicted in figure \ref{feynmanLO} plus the interchange of
a virtual gluon between the quark and antiquark that
only contributes to the two-jet decay rate.
In addition to renormalized UV divergences,
IR singularities, either collinear or soft,
appear because of the presence of massless
particles like gluons. Bloch-Nordsiek
and Kinoshita-Lee-Nauemberg theorems\cite{IR} assure
IR divergences cancel for inclusive cross section.
Technically this means, if we use
DR to regularize the IR divergences of the loop diagrams we
should express the phase space for the tree-level diagrams
in arbitrary $D$-dimensions. The IR singularities
cancel when we integrate over the full phase space.
Another delicate question is the problem of
hadronization. Perturbative QCD gives results at the level
of partons, quarks and gluons, but in nature one observes
hadrons, not partons, and hadronization can shift the QCD
predictions.
We apply to the parton amplitudes the same jet clustering
algorithms applied experimentally to the real observed particles,
see table~\ref{algorithms}.
Starting from a bunch of particles of momenta $p_i$ we calculate,
for instance, $y_{ij}=2(p_i \cdot p_j)/s$, the scalar product
of all the possible momenta pairs.
If the minimum is smaller than a fixed $y_c$
we combine the two involved particles in a new pseudoparticle
of momentum $p_i+p_j$. The procedure starts again until all
the $y_{ij}$ are bigger than $y_c$. The number of pseudoparticles
at the end of the procedure defines the number of jets.
The jet clustering algorithms
automatically define IR finite quantities.
For the moment, we do not enter in the question of which
is the best jet clustering algorithm
although the main criteria followed to choose
one of them should be based in two requirements: minimization
of higher order corrections and insensitivity to hadronization.
If we restrict to the three-jet decay rate the
IR problem can be overcome and everything can be calculated
in four dimensions because
the jet clustering algorithms automatically exclude
the IR region from the three-body phase space.
For massless particles and at the lowest order the EM~\cite{LO},
JADE and E algorithms give the same answers.
Analytical results for the massless
three-jet fraction exist for both JADE-like~\cite{KN} and
DURHAM~\cite{Durham} algorithms.
A complete analysis for the ratios of three-jet decay
rates and the angular distributions quoted in Eq.~\ref{r3bd}
and \ref{rtheta} can be found in \cite{LO}.
For practical purposes a parametrization of the result in terms of
a power series in $\log y_c$ gives a good description \cite{KN,LO}.
\mafigura{6cm}{NLO.ps}
{Feynman diagrams contributing to the three-jets decay rate
of $Z\rightarrow b\bar{b}$ at order $\as^2$.
Self-energies in external legs have not been shown.}
{feynmanNLO}
\section{THREE JETS OBSERVABLES AT NLO}
The effect of the bottom quark mass has been studied
experimentally by~\cite{Joan} on the $R_3^{bd}$ ratio.
As we have seen the running of
the bottom quark mass from low energies to the $M_Z$ scale is
quite strong. The LO QCD prediction for
$R_3^{bd}$ does not allow us to distinguish which mass we
should use in the theoretical expressions, either the pole mass
or the running mass at some scale. The computation of the
NLO is mandatory if we want to extract information about
the bottom quark mass from LEP data.
At the NLO we have to calculate
the interference of the loop diagrams depicted in
figure~\ref{feynmanNLO} with the lowest order Feynman diagrams
of figure~\ref{feynmanLO} plus the square of the tree-level
diagrams of figure~\ref{feynmanNLO}.
The amplitudes in the massless case were
calculated by~\cite{ERT,KL}.
The implementation of the jet clustering algorithms
was performed by~\cite{KN}.
The main problem that now we can not avoid is the appearance
of IR singularities. With massive quarks
we loose all the quark-gluon collinear divergences.
The amplitudes behave better in the IR region.
The disadvantage however is the mass itself.
We have to perform quite more complicated
loop and phase space integrals. Furthermore, we still conserve the
gluon-gluon collinear divergences leading to IR double poles.
The three-jet decay rate can be written as
\begin{equation}
\Gamma^{b}_{3j} = C [g_V^2 H_V(y_c,r_b) + g_A^2 H_A(y_c,r_b)],
\end{equation}
where $r_b=m_b^2/M_Z^2$,
$C=M_Z \: g^2/(c_W^2 64 \pi) (\as/\pi)$ is a normalization constant
that disappear in the ratio and
$g_V$ and $g_A$ are the vector and the axial-vector
neutral current quark couplings.
At tree-level and for the bottom quark
$g_V = -1 + 4 s_W^2/3$ and $g_A = 1$.
Now we can expand the functions $H_{V(A)}$
in $\as$ and factorize the leading dependence on the quark mass
as follows
\begin{eqnarray}
H_{V(A)} &=& A^{(0)}(y_c) + r_b B_{V(A)}^{(0)}(y_c,r_b) \\
&+& \frac{\as}{\pi} \left( A^{(1)}(y_c) + r_b B_{V(A)}^{(1)}(y_c,r_b)
\right), \nonumber
\end{eqnarray}
where we have taken into account that for massless quarks vector
and axial contributions are identical\footnote{We do not consider the
small $O(\as^2)$ triangle anomaly \cite{triangle}.
With our choice of the normalization $A^{(0)}(y_c)=A(y_c)/2$ and
$A^{(1)}(y_c)=B(y_c)/4$, where $A(y_c)$ and $B(y_c)$ are
defined in \cite{KN}.}.
First steep in the calculation is to show the cancellation of
the IR divergences in order to build matrix elements free of
singularities. It is possible to do it analytically.
However, we knew from the beginning IR divergences should
disappear \cite{IR}. The challenge is in the calculation of the
finite parts. This calculation is rather long, complex and full
of difficulties. Strong cancellations occur between different
groups of diagrams making difficult even a numerical approach.
We have taken as guide line the massless result of \cite{ERT,KN}
although the IR structure of the massive case is completely
different from the massless one.
\mafigura{7cm}{Jtest.ps}
{NLO vector contribution to the three-jet decay rate of
$Z\rightarrow b\bar{b}$ for bottom quark masses from
$1$ to $5(GeV)$ and fixed $y_c$ in the JADE algorithm.
Big circle is the massless case.}
{Jtest}
\mafigura{7cm}{Etest.ps}
{NLO vector contribution to the three-jet decay rate of
$Z\rightarrow b\bar{b}$ for bottom quark masses from
$1$ to $5(GeV)$ and fixed $y_c$ in the E algorithm.
Big circle is the massless case.}
{Etest}
In figures \ref{Jtest} and \ref{Etest} we present our
preliminary result for the vectorial contribution to the
$O(\as^2)$ three-jet decay rate of the Z-boson into
bottom quarks. We have performed the calculation for
different values of the bottom quark mass from $1$ to $5(GeV)$
for fixed $y_c$.
We want to show we can recover the massless
result \cite{KN}, depicted as a big circle, i.e., in the limit
of massless quarks we reach the $A^{(1)}(y_c)$ function.
This is our main test to have confidence in our calculation.
In the JADE algorithm we can see that for big values of
$y_c$ the NLO corrections due to the quark mass are
very small and below the massless result.
Notice they increase quite a lot for small values of $y_c$
and give a positive correction that will produce a change
in the slope of the LO prediction for $R_3^{bd}$.
In any case we recover
the massless limit and a linear
parametrization in the quark mass squared could
provide a good description.
The E algorithm behaves also linearly in the quark mass squared
although only for big values of $y_c$.
Corrections in the E algorithm are always very
strong. The reason is the following, the resolution parameter
for the E algorithm explicitly incorporates the quark mass,
$y_{ij}=(p_i+p_j)/s$, i.e., for the same value of $y_c$
we are closer to the two-jet IR region and the difference
from the other algorithms is precisely the quark mass.
This phenomenon already manifest
at the LO. The behaviour of the E algorithm is completely
different from the others for massive quarks.
It is difficult to believe in the E algorithm as a
good prescription for physical applications
since mass corrections as so big. However for the same reason,
it seems to be the best one for testing massive calculations.
\section{CONCLUSIONS}
We have presented the first results for the NLO strong
corrections to the three-jet decay rate of the Z-boson into
massive quarks. In particular, extrapolating our result
we have shown we can recover previous calculations with
massless quarks.
Their application to LEP data,
together with the already known LO,
can provide a new
way for determining the bottom quark mass
and to show for the first time its running.
\vskip 5mm
\noindent{\bf Acknowledgements.}
I would like to thank
J.~Fuster for very encouraging comments during the
development of this calculation and for carefully
reading this manuscript, A.~Santamaria
for very useful discussions and
S. Narison for the very kind atmosphere created
at Montpellier.
| proofpile-arXiv_065-444 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
The recently reported Bose-Einstein condensates of trapped neutral atoms
\cite{AE} -- \cite{KM} represent the first unambiguous
observations of a weakly interacting bose condensed gas.
Quantitatively, we can characterize the strength
of the interaction by the expansion parameter in the perturbation
treatment of the homogeneous bose gas , $\sqrt{n a^{3}}$,
where $n$ is the density and $a$ the scattering length of the inter-atomic
potential. In the atomic-trap experiments, typically n $\sim 10^{12}$
$-$ $10^{14} \rm{cm}^{-3}$,
and a $\sim 1-5 $ nm, so that $\sqrt{n a^{3}} \sim 10^{-2}$$\; - \;
$ $3 \times 10^{-5}$.
Thus, in the sense of perturbation theory, the
observed condensates are indeed textbook examples of weakly interacting
systems.
For the uniform bose gas, perturbation theory leads to simple analytical
results. Although the trapped condensates can be described by means of
the Hartree-Fock Bogolubov equations \cite{GG}-\cite{HP},
the latter approach does not lend itself to an analytical perturbation
treatment.
Intuitively, one expects that a many-body system whose density varies
slowly in space can be described locally as a homogeneous system.
Based on this picture, the Thomas-Fermi method
\cite{F} \cite{T} was proposed
for the calculation of the electron density in a heavy atom. Lieb and Simon
\cite{LS} showed that the treatment is exact in the limit when the atomic
number goes to infinity. Application to a confined Bose condensate was
pioneered by Goldman, Silvera and Legget \cite{GSL}, and recently
reconsidered by Chou, Yang, and Yu \cite{CN}-\cite{CL}.
As pointed out by Kagan, Shlyapnikov, and Walraven\cite{KSW},
the local-density description is valid when
\begin{equation}
\mu/\hbar\omega\gg 1 \; ,
\end{equation}
where $\mu$ is the mean-field energy per particle, or chemical potential,
and $\hbar\omega$ is the zeropoint energy in the trap.
In this paper, we derive such a description from first principles within
the framework of the variational technique.
We emphasize that, unlike the practice of neglecting the
kinetic energy term in the Gross-Pitaevski equation which in the recent
literature is sometimes called the Thomas-Fermi approximation,
the resulting variational description is not limited to the condensate, but
describes the depletion, pressure and all other thermodynamic quantities.
Furthermore, like the uniform gas, the Thomas-Fermi theory leads to a
perturbation treatment of the weakly interacting condensates, giving
simple analytical expressions for these quantities.
Another important advantage of the
Thomas Fermi treatment is that it can be generalized to describe
finite temperature systems, as we shall discuss
in future work. In this paper we focus on the bose gas at zero temperature.
The paper is organized as follows. In section 2, we generalize the usual
Bogolubov transformation to describe spatially inhomogeneous condensates.
In section 3, we introduce the Wigner representation and gradient
expansion, which provide the tools to describe the nearly homogeneous
systems and make the Thomas-Fermi approximation. The advantage of this
systematic approach to the Thomas-Fermi approximation is that it
provides an estimate of the error incurred by the inhomogeneity of
the condensate, allowing one to estimate the accuracy of the Thomas
Fermi results.
We consider this point to be very important in
view of the
fact that some traps, depending on the potential and
the number of trapped
atoms, are too far from homogeneity to be described by a Thomas-Fermi
description. In addition, even if the Thomas-Fermi
description is valid in the middle of the trap, it
breaks down at the edge of the condensate.
In sections 4 and 5, we obtain the mean-field description of the bose
system in the Thomas-Fermi approximation. The equations, derived within
the framework of the variational principle, provide a fully self-consistent
description, indicating that the Thomas-Fermi decription is by no means
limited to weakly interacting systems.
This remark can be expected to be of future importance
in the light of recent experimental efforts to obtain condensates of
higher density.
Nonetheless, because of the special interest in the weakly interacting
systems, we proceed in section 6 to derive a perturbation treatment and
obtain analytical
results for quantities
such as the chemical potential, the local depletion, pairing and pressure.
With the experimental atomic-traps in mind,
we apply the results of the general theory to the special case
of a trapping potential that is of the type of
a simple spherically symmetric harmonic oscillator in section 7.
Finally, in section 8, we derive a density of states of the
trapped weakly interacting condensate within the spirit of the
Thomas-Fermi approximation.
\section{Generalized Bogolubov Transformation}
The Bogolubov quasi-particle concept \cite{NP} provides a very elegant
description of the interacting Bose-Einstein condensate.
The quasi-particles are represented by
creation ($\eta^{\dagger}$) and annihilation ($\eta$) operators
that are linear combinations of regular
single-particle creation ($a^{\dagger}$) and annihilation
($a$) operators. In treating a homogeneous system, for which we can work
in a basis of single-particle plane-wave states of momentum ${\bf k}$,
the Bogolubov transformation which relates the quasi-particle
and regular particular operators, takes on a particularly simple form,
\begin{eqnarray}
\eta^{\dagger}_{\bf k} &=& x_{\bf k} a^{\dagger}_{\bf k} +
y_{\bf k} a_{-{\bf k}} \; , \nonumber \\
\nonumber \\
\eta_{\bf k} &=& x_{\bf k} a_{\bf k} + y_{\bf k} a^{\dagger}_{-{\bf k}} \; ,
\label{e:BT}
\end{eqnarray}
where for the purpose of describing the static properties of a
condensate in equilibrium, we can limit the transformation
parameters, $x_{\bf k},y_{\bf k}$
to real numbers.
Furthermore, the isotropy of the many-body system suggests
that the transformation parameters only depend on the magnitude
of the momentum, $x_{\rm{\bf k}}$ = $x_{\rm{k}}$ and $y_{\rm{\bf k}}$ =
$y_{\rm{k}}$.
Requiring the quasi-particle operators to be canonical,
$\left[ \eta_{\bf k},\eta^{\dagger}_{{\bf k}'} \right]=
\delta_{{\bf k},{{\bf k}'}},
\left[ \eta_{\bf k},\eta_{\bf k'} \right] =
\left[ \eta^{\dagger}_{\bf k}, \eta^{\dagger}_{{\bf k}'} \right] = 0$,
gives an additional constraint to $x_{\rm{k}}$ and
$y_{\rm{k}}$,
\begin{equation}
x^{2}_{k} - y^{2}_{k} = 1 ,
\label{e:cans}
\end{equation}
from which we can see that a single parameter $\sigma_{k}$,
with $x_{k}$ = $\cosh\sigma_{k}$ and $y_{k}$ = $\sinh\sigma_{k}$,
suffices to parametrize the Bogolubov transformation (\ref{e:BT}).
In addition, with Eq.(\ref{e:cans}),
we can also write the
Bogolubov transformation as
\begin{eqnarray}
a^{\dagger}_{\bf k} &=& x_{k} \eta^{\dagger}_{\bf k} -
y_{k} \eta_{-{\bf k}} \; , \nonumber \\
\nonumber \\
a_{\bf k} &=& x_{k} \eta_{\bf k} - y_{k} \eta^{\dagger}_{-{\bf k}} \; ,
\label{e:BTI}
\end{eqnarray}
which is the inverse transformation of (\ref{e:BT}).
It is useful to define the following quantities, the ``distribution
function'' $\rho$ and the ``pairing function'' $\Delta$:
\begin{eqnarray}
\rho_{\bf k} &=& \langle a_{\bf k}^{\dagger} a_{\bf k} \rangle
= \frac{1}{2} \left[ \cosh 2 \sigma_{\bf k} -1 \right] \; , \nonumber\\
\Delta_{\bf k} &=& - \langle a_{\bf k} a_{-\bf k} \rangle
= \frac{1}{2} \sinh 2 \sigma_{\bf k} \; ,
\end{eqnarray}
where the brackets $\langle \; \rangle$ represent the ground state
expectation value.
The best values for $x_{\rm{k}}$ and $y_{\rm{k}}$ are obtained
variationally by minimizing the ground state free energy.
As stated, the above description (1)-(3)
only applies to homogeneous
systems, whereas the treatment of a general (inhomogeneous)
condensate, as we shall
show below, involves a Bogolubov transformation that is
quite
different in appearance, from the homogeneous case.
However, we can expect the results of the
homogeneous treatment to describe the
`local' behavior of an inhomogeneous condensate, provided the
spatial variations of the condensate are sufficiently slow.
In describing many-particle Fermion systems, this intuitive
picture forms one of the key ingredients of the well-known
Thomas-Fermi description of slowly varying many-particle systems.
To arrive at a general treatment, we choose to work with
boson-field operators, $\hat{\Psi}({\bf x})$ and
$\hat{\Psi}^\dagger({\bf x})$, an approach that
offers the advantage of not having to specify a basis a priori. Furthermore,
in the presence of a condensate, it is convenient to work
with the fields $\hat{\psi}({\bf x})$ and
$\hat{\psi}^{\dagger}({\bf x})$, which are displaced from
the original fields $\hat{\Psi}({\bf x})$ and $\hat{\Psi}^{\dagger}({\bf x})$
by the expectation value $\phi({\bf x})$ of $\hat{\Psi}({\bf x})$,
\begin{eqnarray}
\hat{\Psi}({\bf x}) &=& \hat{\psi}({\bf x}) + \phi({\bf x}),
\nonumber \\
\hat{\Psi}^{\dagger}({\bf x}) &=& \hat{\psi}^{\dagger}({\bf x}) +
\phi^{\ast}({\bf x}),
\end{eqnarray}
where, for the purpose of describing the static properties of a condensate in
equilibrium,
$\phi$ can be taken to be real, and where
$\hat{\psi}({\bf x})$ and $\hat{\psi}^{\dagger} ({\bf x})$ are the
displaced field operators which satisfy the canonical commutation
relation, $\left[ \hat{\psi}({\bf x}), \hat{\psi}^{\dagger} ({\bf x}')
\right] $ $= \delta ({\bf x}-{\bf x}')$, and furthermore,
\begin{equation}
\langle \hat{\psi}({\bf x}) \rangle = \langle \hat{\psi}^{\dagger}
({\bf x}) \rangle = 0.
\label{e:expo}
\end{equation}
We introduce the Bogolubov transformation as a general
linear transformation relating the displaced fields to the
quasi-particle fields, $\hat{\xi}({\bf x})$
and $\hat{\xi}^{\dagger}({\bf x})$,
\begin{eqnarray}
\hat{\psi}({\bf x}) &=& \int d^{3}z \left[X({\bf x},{\bf z})\hat{\xi}({\bf z})
-Y({\bf x},{\bf z})\hat{\xi}^{\dagger}({\bf z})\right] , \nonumber \\
\nonumber\\
\hat{\psi}^{\dagger}(\bf x) &=& \int d^{3}z \left[X^{\ast}({\bf x}
,{\bf z})\hat{\xi}^{\dagger}({\bf z})
-Y^{\ast}({\bf x},{\bf z})\hat{\xi}({\bf z})\right],
\label{e:GBT}
\end{eqnarray}
which is the generalization of Eq.(\ref{e:cans}).
The non-local nature of the generalized Bogolubov transformation
(\ref{e:GBT}) should not be surprising $-$ the `homogeneous' Bogolubov
transformation (\ref{e:BT}) can be written in the same form
with the special feature, due
to the homogeneity of the system, that $X({\bf x}, {\bf y})$ and $Y({\bf x},
{\bf y})$ only depend on ${\bf x}
- {\bf y}$.
Requiring the quasi-particle fields to be canonical,
leads to
\begin{equation}
\int d^{3} z \left[ X({\bf x},{\bf z}) X({\bf z},{\bf y}) -
Y({\bf x},{\bf z})
Y({\bf z},{\bf y})\right] = \delta({\bf x}-{\bf y}) \; ,
\label{e:cang}
\end{equation}
which is the generalization of (\ref{e:cans}).
It is possible to derive equations for the inhomogeneous bose systems by
variationally determining the best transformations $X$ and $Y$,
minimizing the free energy. This however, is not the path we
choose to follow here. Instead, we manipulate the generalized
Bogolubov transformations in a manner similar to the procedure
to obtain the Wigner distribution from the off-diagonal
single-particle density function. Once this is achieved, the
steps that lead to a Thomas-Fermi description are known from
quantum transport theory. One interesting new aspect of this
treatment is that the central object of the theory
is not a distribution
function, which in some sense can still be regarded as an observable,
but a transformation.
Although this transformation determines the value of all observables,
it is clearly not an observable quantity by itself.
\section{Wigner Representation and Gradient Expansion}
Wigner showed that a quantum mechanical single-particle system,
costumarily characterized by its wave function
$ \Psi ({\bf x}) $,
can alternatively be fully characterized by a different function,
\begin{equation}
\rho_{\rm{W}}({\bf R},{\bf p}) =
\int d^{3} r \;
\Psi^{\ast}({\bf R}+{\bf r}/2) \Psi({\bf R}-
{\bf r}/2)
\exp (i{\bf p}\cdot{\bf r}) ,
\label{e:WR}
\end{equation}
where here $-$ as in the rest of the paper $-$ we work in units in
which $\hbar$=1. This function (\ref{e:WR}),
known as the Wigner Distribution function,
can be interpreted as a phase space distribution function \cite{WG} and
leads to a description that is remarkably close to classical mechanics.
The analogy with a classical phase space distribution function
is not complete (for example $ \rho_{\rm W} $ can take on
negative values), but can be justified by the fact that the quantum mechanical
expectation value of observables are
equal to the `phase space
integrals' of the corresponding classical quantities, weighted by
$(2 \pi)^{-3} \rho_{{\rm W}} $,
\begin{eqnarray}
\langle \Psi | f | \Psi \rangle & = &
\int d^{3} x \; \Psi^{\ast}({\bf x})
f({\bf x}) \Psi({\bf x})\nonumber \\
& = & (2 \pi)^{-3}
\int d^{3} p \; \int d^{3} R \; f({\bf R}) \;
\rho_{\rm{W}}({\bf R},{\bf p}),
\nonumber\\
\langle \Psi |\hat{{\bf p}} | \Psi \rangle & = &
\int d^{3} x \; \Psi^{\ast}({\bf x})
\hat{{\bf p}} \Psi({\bf x}) \nonumber\\
& = & (2 \pi)^{-3}
\int d^{3} p \; \int d^{3} R \; {\bf p} \;
\rho_{\rm{W}}({\bf R},{\bf p}),
\end{eqnarray}
etc. More recently, the many-particle generalization of the Wigner
distribution has found many important applications in diverse areas such
as nuclear \cite{DL} and solid state physics \cite{DD}.
An important motivation to work in the transformed
representation of Eqs.(\ref{e:WR}),
$({\bf x},{\bf x}') \rightarrow ({\bf R},{\bf p})$,
\begin{equation}
A_{W}({\bf R},{\bf p}) = \int d^{3} r \;
A({\bf R} + {\bf r}/2, {\bf R} - {\bf r}/2)
\exp (i{\bf p}\cdot{\bf r}) ,
\label{e:wrep}
\end{equation}
and its inverse
\begin{equation}
A({\bf x} , {\bf x}') =
(2 \pi)^{-3} \int d^{3} p \; A_{W} (\left[ {\bf x}+{\bf x}' \right] /2,{\bf p})
\; \exp(-i{\bf p}\cdot\left[ {\bf x}-{\bf x}' \right]),
\label{e:wrepi}
\end{equation}
which shall henceforth be referred to as the Wigner representation, is
that it is extraordinarily well suited to describe
nearly homogeneous systems. This convenient feature
follows from the gradient expansion \cite{DL} -- \cite{DD}.
The gradient expansion
shows that, to first
order, a `product' operator $C({\bf x},{\bf x}') = \int
d^{3} z \; A({\bf x},{\bf z}) B({\bf z},{\bf x}') $, in
the Wigner representation simply gives the algebraic product of A
and B, $C_{W}({\bf R},{\bf p}) \approx A_{W}({\bf R},{\bf p})
B_{W}({\bf R},{\bf p})$. The higher-order corrections
to this approximation can be written as a series of
terms containing successively higher-order derivatives
in the (${\bf R},{\bf p}$)$-$ coordinates,
\begin{eqnarray}
C_{W}({\bf R},{\bf p}) \approx
A_{W}({\bf R},{\bf p}) B_{W}({\bf R},{\bf p}) + \frac{1}{2 \rm{i}}
\sum_{j=1}^{3} \left[ \frac{\partial A_{W}}{\partial R_{j}}
\frac{\partial B_{W}}{\partial p_{j}} - \frac{\partial A_{W}}{\partial p_{j}}
\frac{\partial B_{W}}{\partial R_{j}} \right]\nonumber\\
- \frac{1}{8} \sum_{j=1}^{3} \left[
\frac{\partial^{2} A_{W}}{\partial R_{j}^{2}}
\frac{\partial^{2} B_{W}}{\partial p_{j}^{2}}
+ \frac{\partial^{2} A_{W}}{\partial p_{j}^{2}}
\frac{\partial^{2} B_{W}}{\partial R_{j}^{2}}
- 2 \frac{\partial^{2} A_{W}}{\partial R_{j} \partial p_{j}}
\frac{\partial^{2} B_{W}}{\partial R_{j} \partial p_{j}} \right] + \cdots
\; .
\label{e:gradexp}
\end{eqnarray}
The first order correction in the gradient expansion (\ref{e:gradexp})
is $\left\{ A_{W},B_{W} \right\}_{\rm{{PB}}}$,
the Poisson bracket of $A_{W}$ and $B_{W}$. If we know that the range
of $A_{W}$ and $B_{W}$ in ${\bf p}$-space is of the order of $p_{c}$, then
the magnitude of the derivatives
$\partial B_{W} / \partial p $ and $\partial^{2} B_{W} / \partial p^{2}$
in (\ref{e:gradexp}) can
be estimated to be of the order of $B_{W}/p_{c}$ and $B_{W}
/p_{c}^{2}$ respectively.
This approximation will allow us to obtain a very simple estimate of
the `inhomogeneity'
error.
At this point, we return to the generalized Bogolubov transformation,
$X({\bf x},{\bf y}),Y({\bf x},{\bf y})$, of the previous section.
Working in the Wigner representation and expanding the `canonicity'
relation (\ref{e:cang})
between between $X$ and $Y$ in the manner of the gradient
expansion, we find up to first order in the spatial derivatives,
a relation that is similar to the constraint equation (\ref{e:cans})
of the homogeneous Bogolubov transformation,
\begin{equation}
X^{2}_{W}({\bf R},{\bf p}) - Y^{2}_{W}({\bf R},{\bf p}) \approx 1 .
\label{e:canw}
\end{equation}
Consequently, the general Bogolubov transform can be parametrized in the
same way as the Bogolubov transform for the homogeneous bose gas,
$X_{W}({\bf R},{\bf p})$ = $\cosh[\sigma ({\bf R},{\bf p})]$,
$Y_{W}({\bf R},{\bf p})$ = $\sinh[\sigma ({\bf R},{\bf p})]$,
where for the slowly varying condensate,
the $\sigma$ $-$ parameters depend on the momentum {\sl and}
position : $\sigma ({\bf R},{\bf p})$.
The distribution and pairing functions,
$\rho({\bf{x}},{\bf{x}}')$ = $\langle \hat{\psi}^{\dagger} (\bf{x}) \hat{\psi}
(\bf{x}') \rangle $ and $\Delta({\bf{x}},{\bf{x}}')$
= $ - \langle \hat{\psi} (\bf{x})
\hat{\psi}
(\bf{x}') \rangle$ take on the following form in the Wigner representation:
\begin{eqnarray}
\rho_{W} ({\bf R},{\bf p}) &=&
\frac{1}{2} \left[ \cosh (2 \sigma ({\bf R},{\bf p})) -1 \right] ,
\nonumber\\
\Delta_{W} ({\bf R},{\bf p}) &=&
\frac{1}{2} \sinh (2 \sigma ({\bf R},{\bf p})) .
\label{e:rds1}
\end{eqnarray}
The local $\sigma$ parametrization of the Bogolubov transformation
is crucial to the Thomas-Fermi description and it is
upon the validity of (\ref{e:canw}) that the Thomas-Fermi theory rests.
The error introduced to (\ref{e:canw}) due to the inhomogeneity of
the system can be estimated by the lowest order
non-vanishing term in the gradient expansion (\ref{e:gradexp}).
Notice that the first order term in the gradient expansion
of (\ref{e:canw}) vanishes since it is the
the sum of Poisson brackets of quantities with themselves.
Consequently, the error has to be estimated from the second order term.
\section{Energy Density}
In the variational method, the quantity to minimize
is F, the ground state
free energy, which we can put in a `local' form,
$F = \int d^{3} R \; f({\bf R})$, where
$f({\bf R})$ is
the energy density. We achieve this
result in two steps. In the first step, we shift to the Wigner
representation in the integrand for the mean field expression
for the ground state
energy. In the second step, we notice that the short-range
nature of the inter-atomic interaction renders the resulting
integrand essentially `local', i.e. the integrand contains
only $\sl single$ (not double) integrals over the position
variables.
The ground state free energy is the
expectation value of $\hat{H} - \mu \hat{N}$, where
$\hat{H}$ is the many-body hamiltonian of the boson
system, $\hat{N}$, the number operator and $\mu$, the
chemical potential:
\begin{equation}
{\hat{H}} - \mu \hat{N}
= \int d^3 x \; \hat{\Psi}({\bf x})^{\dag} \hat{h}({\bf x})
\hat{\Psi}({\bf x}) +
\frac{1}{2}\int d^3 x \; d^3y \;
\hat{\Psi}({\bf y})^{\dag}\hat{\Psi}({\bf x})^{\dag} V(|{\bf x}-{\bf y}|)
\hat{\Psi}({\bf x}) \hat{\Psi}({\bf y}),
\label{e:ham}
\end{equation}
where $V(|{\bf x}-{\bf y}|)$ represents the inter-atomic potential
and $\hat{h}({\bf x})$ is the one-body part of the free energy,
\begin{equation}
\hat{h}({\bf x}) = -\frac{\nabla^{2}}{2 m} + V_{\rm ext}({\bf x}) -\mu,
\end{equation}
where $V_{\rm ext}({\bf x})$ is the external potential.
The presence of a condensate displaces
the field operators ${\hat{\Psi}} ({\bf x})$ by their expectation
value $\phi ({\bf x})$.
To generate the variational free energy, we shall use
the mean field approximation, in which terms of first and
third order in $\hat{\psi}$ and $\hat{\psi}^{\dagger}$ vanish
(\ref{e:expo}) and the fourth order term factorizes as follows:
\begin{eqnarray}
&& \langle \hat{\psi}({\bf y})^{\dag}\hat{\psi}({\bf x})^{\dag}
\hat{\psi}({\bf x}) \hat{\psi}({\bf y}) \rangle
\approx \nonumber\\
\nonumber\\
&& \Delta ^{\ast}({\bf y},{\bf x}) \Delta ({\bf y},{\bf x})
+ \rho ({\bf y},{\bf x}) \rho ({\bf x},{\bf y})
+ \rho ({\bf x},{\bf x}) \rho ({\bf y},{\bf y}) .
\end{eqnarray}
The variational nature of this procedure is insured by
the existence of a variational ground state that gives this type of
factorization. In fact, the variational ground state corresponds to the choice
of the gaussian wave functional \cite{HP}.
The displacement of the fields and the factorization of the
expectation values, although straightforward, gives rise to
a somewhat lengthy expression for the free energy. It is then
convenient to classify the different contributions by their
order in $\phi$ and their functional dependence on $\rho$ and
$\Delta$:
\noindent
1. $h_{1}$ is the one-body contribution of zeroth
order in $\phi$ to the ground state energy,
\begin{equation}
h_{1} = \frac{1}{2} \int d^{3}x \; d^{3}y \;
\hat{h}({\bf x}) \rho ({\bf y},{\bf x})
\delta ({\bf x}-{\bf y}) .
\end{equation}
2. In analogy with the Hartree-Fock theory, we call $ V_{\rm{dir}} $
given below, the direct energy contribution to the energy,
\begin{equation}
V_{\rm{dir}} = \frac{1}{2} \int d^{3}x \; d^{3}y \;
\rho ({\bf y},{\bf y}) \rho ({\bf x},{\bf x}) V(|{\bf x}-{\bf y}|).
\end{equation}
3. Using the same analogy to the Hartree-Fock treatment, the
exchange energy, $V_{\rm{exch}} $, is equal to
\begin{equation}
V_{\rm{exch}} = \frac{1}{2} \int d^{3}x \; d^{3}y \;
\rho ({\bf x},{\bf y}) \rho ({\bf y},{\bf x}) V(|{\bf x}-{\bf y}|).
\end{equation}
4. Standard Hartree-Fock theory does not describe pairing and the pairing
energy, $ V_{\rm pair} $,
\begin{equation}
V_{\rm{pair}} = \frac{1}{2} \int d^{3}x \; d^{3}y \;
\Delta^{\ast} ({\bf y},{\bf x}) \Delta ({\bf y},{\bf x})
V(|{\bf x}-{\bf y}|) \; ,
\end{equation}
is consequently absent from
the Hartree-Fock expressions.
In second order in $\phi$, we find contributions
that can be
obtained from the above terms by replacing either
$\Delta ({\bf x},{\bf y})$ or $\rho ({\bf x},{\bf y})$
by $\phi ({\bf x}) \phi ({\bf y})$.
\noindent
5. For example,
the one-body contribution, due to the kinetic and potential energy
of the condensate is $h_{1}^{\phi}$, where
\begin{equation}
h_{1}^{\phi} = \int d^{3}x \; \phi ({\bf x})
\hat{h}({\bf x}) \phi ({\bf x}) .
\end{equation}
6. $ V_{\rm{dir}}^{\phi} $ is the direct contribution
to the interaction energy, stemming from the interaction of the condensate with
the particles that have been `forced' out of the condensate (depletion),
\begin{equation}
V_{\rm{dir}}^{\phi} = \frac{1}{2} \int d^{3}x \; d^{3}y \;
\phi ({\bf y}) \phi ({\bf y}) \rho ({\bf x},{\bf x}) V(|{\bf x}-{\bf y}|).
\end{equation}
7. Similarly, $ V_{\rm{exch}}^{\phi} $
is the exchange contribution of
second order in $\phi$,
\begin{equation}
V_{\rm{exch}}^{\phi} = \frac{1}{2} \int d^{3}x \; d^{3}y \;
\phi ({\bf y}) \phi ({\bf x}) \rho ({\bf y},{\bf x}) V(|{\bf x}-{\bf y}|).
\end{equation}
8. We represent the pairing energy of the condensate with the particles out
of the condensate by $ V_{\rm{pair}}^{\phi} $,
\begin{equation}
V_{\rm{pair}}^{\phi} = \frac{1}{2} \int d^{3}x \; d^{3}y \;
\phi ({\bf y}) \phi ({\bf x}) \Delta ({\bf y},{\bf x}) V(|{\bf x}-{\bf y}|).
\end{equation}
9. Finally, we denote the contribution of fourth order in $\phi$,
representing the interaction energy of the condensate with
itself, by $ V^{\phi\phi} $ :
\begin{equation}
V^{\phi\phi} = \frac{1}{2} \int d^{3}x \; d^{3}y \;
\phi^{2} ({\bf y}) \phi^{2} ({\bf x}) V(|{\bf x}-{\bf y}|).
\end{equation}
With this notation, the mean-field expression for the ground state energy reads
\begin{eqnarray}
F &=& \langle \hat{H} - \mu \hat{N} \rangle \nonumber\\
&=& h_{1} +
V_{\rm{dir}} +
V_{\rm{exch}} +
V_{\rm{pair}} + \nonumber\\
&& h_{1}^{\phi} + 2
V_{\rm{dir}}^{\phi} + 2
V_{\rm{exch}}^{\phi} - 2
V_{\rm{pair}}^{\phi} +
V^{\phi\phi} ,
\end{eqnarray}
where the minus sign of the $V_{pair}^{\phi}$ term stems from the definition
of $\Delta$ = $- \langle \hat{\psi} \hat{\psi} \rangle$.
At this point, we introduce the Wigner representation into the integrands
of the above contributions to the mean-field expressions for the ground
state free energy. The resulting expressions
resemble the corresponding terms
for the homogeneous gas, with an additional label ${\bf R}$ over which
is integrated.
For the sake of notational convenience we introduce the following
integration symbol $\int_{\bf R}$ or $\int_{\bf p}$, which
represents the usual integral over all of space, $\int d^{3} {\bf R}$,
if ${\bf R}$
is a position variable or
$(2\pi)^{-3} \int d^{3} p $, if ${\bf p}$ is a momentum variable:
\begin{eqnarray}
\int_{\bf p} &\equiv& (2\pi)^{-3} \; \int d^{3} p \; ,
\nonumber\\
\int_{\bf R} &\equiv& \int d^{3} R \; .
\end{eqnarray}
The terms of zero order in $\phi$ then give
\begin{eqnarray}
h_{1}
&=&
\int_{\bf R} \; \int_{\bf p} \left[ \frac{p^{2}}{2m} + V({\bf R}) - \mu
\right] \rho_{W} ({\bf R},{\bf p}), \nonumber\\
\nonumber\\
V_{\rm{exch}} &=&
\int_{\bf R} \; \int_{\bf p} \; \int_{{\bf p}'} \; \rho_{W} ({\bf R},{\bf p})
v({\bf p}-{\bf p}') \rho_{W} ({\bf R},{\bf p}') \; ,\nonumber\\
V_{\rm{pair}} &=&
\int_{\bf R} \; \int_{\bf p} \; \int_{{\bf p}'} \; \Delta_{W} ({\bf R},{\bf p})
v({\bf p}-{\bf p}') \Delta_{W} ({\bf R},{\bf p}') \; ,
\nonumber\\
V_{\rm{dir}} &=&
\int_{\bf R} \; \int_{\bf r} \; \int_{\bf p} \; \int_{{\bf p}'} \; \int_{\bf q}
\rho_{W} ({\bf R} - {\bf r}/2 ,{\bf p} )
\rho_{W} ({\bf R} + {\bf r}/2 ,{\bf p}')
\exp (i {\bf q}\cdot{\bf r})
v({\bf q})\; ,
\end{eqnarray}
where $v$ is the Fourier transform of the interaction potential,
$v({\bf q})$ = $\int d^{3} r V({\bf r}) \exp (-i{\bf q}\cdot{\bf r})$.
The terms that are of second order in $\phi$ can be obtained
by replacing one $\rho$ or $\Delta$ by $\phi \phi$. In the Wigner
representation, this procedure yields expressions that are similar
to the corresponding terms of zero order in $\phi$ with
$\rho_{W} ({\bf R},{\bf p})$ or $\Delta_{W} ({\bf R},{\bf p})$ replaced
by a function $Q_{W}({\bf R},{\bf p})$, where
\begin{equation}
Q_{W}({\bf R},{\bf p}) =
\int_{\bf r} \; \phi ({\bf R} + {\bf r}/2)
\phi ({\bf R} - {\bf r}/2) \exp (i{\bf p}\cdot{\bf r}) .
\end{equation}
Notice that the contributions of second order in $\phi$,
are non-local in the sense that
their expressions contain integrals over more than one position
variable. Nevertheless, if we consider the scale
on which the physical quantities vary in space,
or in momentum space, it becomes apparent that
the non-local integrals can be approximated by local expressions.
We illustrate this point by considering the exchange
($V_{exch}^{\phi}$) and pairing ($V_{pair}^{\phi}$) energies.
The key to obtain local expressions is to notice that $Q_{W}({\bf R},{\bf p})$
varies with respect to ${\bf p}$ on the scale of ${\bf R}_{0}^{-1}$, where
${\bf R}_{0}$ is the size
of the condensate. On the other hand, $v({\bf p}-{\bf p}')$
varies on the scale of $l_{ r} ^{-1}$ where $l_{ r}$ is the
range of the atom-atom interaction. Typically ${\bf R}_{0} \gg l_{ r}$
so that $Q_{W}({\bf R},{\bf p})$ varies much more rapidly with respect
to ${\bf p}$ than $v({\bf p}-{\bf p}')$. In fact, when ${\bf p}$
is large enough to make $v({\bf p}-{\bf p}')$ significantly different
from $v({\bf p}')$, $Q_{W}({\bf R},{\bf p}) \approx 0$. Thus, we can
replace $v({\bf p}-{\bf p}')$ by $v({\bf p}')$ in the integrands :
\begin{eqnarray}
V_{\rm{exch}}^{\phi} &\approx&
\frac{1}{2}
\int_{\bf R} \int_{\bf r} \int_{\bf p} \int_{{\bf p}'} \;
\phi ({\bf R} + {\bf r}/2)
\phi ({\bf R} - {\bf r}/2) \exp (i{\bf p}\cdot{\bf r})
v({\bf p}') \rho_{W} ({\bf R},{\bf p}')
\nonumber\\
&=& \frac{1}{2} \phi^{2} ({\bf R})
\int_{\bf R} \int_{{\bf p}'} v({\bf p}') \rho_{W} ({\bf R},{\bf p}'),
\nonumber\\
V_{\rm{pair}}^{\phi} &\approx&
\frac{1}{2} \phi^{2} ({\bf R}) \int_{\bf R} \int_{{\bf p}'} \;
v({\bf p}') \rho_{W} ({\bf R},{\bf p}') .
\end{eqnarray}
The same considerations regarding the relative magnitude of the
relevant length scales show that we can similarly simplify the expression
of the $\phi^{4}$ interaction energy,
$V^{\phi\phi} $, and the direct interaction
energies, $V_{\rm{dir}}$ and
$V_{\rm{dir}}^{\phi} $.
The local expressions are most easily obtained by considering the difference
in length scales before introducing
the Wigner representation. In coordinate space, we notice
that $\rho ({\bf x},{\bf x}) \approx \rho ({\bf y},{\bf y})$ if
$|{\bf x}-{\bf y}| \le l_{ r}$. Thus, we can replace
$\rho ({\bf x},{\bf x})$ by $\rho ({\bf y},{\bf y})$ in an
integrand if it is accompanied by $V(|{\bf x}-{\bf y}|)$:
\begin{eqnarray}
V_{\rm{dir}} &\approx& \frac{1}{2}\int d^{3} x \; d^{3} y \;
\rho^{2} ({\bf x},{\bf x}) V(|{\bf x}-{\bf y}|) \nonumber\\
&=& \frac{1}{2} v(0) \int_{\bf R} \int_{\bf p} \; \int_{{\bf p}'} \;
\rho_{W} ({\bf R},{\bf p}) \rho_{W} ({\bf R},{\bf p}') \; , \nonumber\\
V_{\rm{dir}}^{\phi} &\approx& \frac{1}{2} v(0)
\int_{\bf R}
\int_{\bf p} \; \phi^{2} ({\bf R}) \rho_{W} ({\bf R},{\bf p}) \nonumber\\
V^{\phi\phi} &\approx& \frac{1}{2} v(0)
\int_{\bf R}
\phi^{4} ({\bf R}) .
\end{eqnarray}
To conclude this section, we summarize the results by remarking
that the Wigner representation and the length scale considerations
bring the free energy in an almost-local form. We need
to qualify that statement because of
the appearance of the Laplacian, a non-local operator, in the
$ h^{\phi} $ -contribution to the energy.
In fact, it is the non-locality of this term that gives
rise to a generalized Gross-Pitaevski or non-linear Schrodinger equation (NLSE).
The resulting (almost-local) ground state free energy is
$F = \int d^{3} R \; f({\bf R})$, where
\begin{eqnarray}
f({\bf R}) &=& \int_{\bf p} \; \left[ \frac{p^{2}}{2m} + V_{\rm{ext}}
({\bf R}) - \mu \right] \rho({\bf R},{\bf p}) + v_{\rm{exch}} ({\bf R})
+ v_{\rm{dir}} ({\bf R}) + v_{\rm{pair}} ({\bf R}) \nonumber\\
& &+ \phi ({\bf R}) \left[ \frac{-\nabla ^{2}}{2m} + V_{\rm{ext}}
({\bf R}) - \mu \right] \phi ({\bf R}) + 2 v_{\rm{exch}}^{\phi} ({\bf R})
+ 2 v_{\rm{dir}}^{\phi} ({\bf R}) - 2 v_{\rm{pair}}^{\phi} ({\bf R}) \nonumber\\
& & + \frac{1}{2} v(0) \phi^{4} ({\bf R}),
\end{eqnarray}
where the exchange, direct and pairing energy densities are the integrands
of the corresponding interaction energy contributions to the free energy :
\begin{eqnarray}
v_{\rm{exch}} ({\bf R}) &=& \frac{1}{2}
\int_{\bf p} \; \int_{{\bf p}'} \; \rho ({\bf R},{\bf p})
v({\bf p}-{\bf p}') \rho_{W} ({\bf R},{\bf p}') \; , \nonumber\\
v_{\rm{pair}}({\bf R}) &=& \frac{1}{2}
\int_{\bf p} \; \int_{{\bf p}'} \; \Delta_{W} ({\bf R},{\bf p})
v({\bf p}-{\bf p}') \Delta_{W} ({\bf R},{\bf p}') \; , \nonumber\\
v_{\rm{dir}} ({\bf R}) &=& \frac{1}{2}
v(0) \int_{\bf p} \; \int_{{\bf p}'} \;
\rho_{W} ({\bf R},{\bf p}) \rho_{W} ({\bf R},{\bf p}') \; , \nonumber\\
v_{\rm{exch}}^{\phi} ({\bf R}) &=& \frac{1}{2}
\phi^{2} ({\bf R})
\int_{\bf p} \; \rho_{W} ({\bf R},{\bf p})
v({\bf p}) \; ,
\nonumber\\
v_{\rm{pair}}^{\phi}({\bf R}) &=& \frac{1}{2}
\phi^{2} ({\bf R})
\int_{\bf p} \; \Delta_{W} ({\bf R},{\bf p})
v({\bf p}) \; ,
\nonumber\\
v_{\rm{dir}}^{\phi} ({\bf R}) &=& \frac{1}{2}
\phi^{2} ({\bf R})
v(0) \int_{\bf p} \;
\rho_{W} ({\bf R},{\bf p}) \; .
\end{eqnarray}
Notice that the free energy and free energy density are functionals
of $\Delta ({\bf R},{\bf p})$, $\rho ({\bf R},{\bf p})$
and $\phi ({\bf R})$. In the next section we determine
the equilibrium values of $\Delta ({\bf R},{\bf p})$,
$\rho ({\bf R},{\bf p})$ and $\phi ({\bf R})$
by minimizing $F\left[ \rho,
\Delta, \phi ; \mu \right]$.
\section{Self-Consistent Mean Field Theory}
In this section we derive the self-consistent mean-field
equations that describe the
nearly-uniform bose condensate at zero temperature.
In the variational method, one minimizes
the mean-field ground state free
energy $F\left[ \rho ,
\Delta , \phi ; \mu \right]$.
Writing the integrands of the different contributions to the
mean-field free energy in the
Wigner representation, followed by the length scale arguments of the previous
section showed that $F\left[ \rho_{W},
\Delta_{W}, \phi ; \mu \right]$ is essentially a local quantity. Finally, in
the Thomas-Fermi limit of a nearly-homogeneous system,
$\rho_{W} ({\bf R},{\bf p})$ and $\Delta_{W} ({\bf R},{\bf p})$
are parametrized
by a single Bogolubov transformation parameter $\sigma ({\bf R},{\bf p})$
in the manner of Eq.(\ref{e:rds1}).
Thus, to describe a nearly-homogeneous system, we minimize the Thomas-Fermi
ground state free energy, which is obtained from
the mean-field free energy,
assuming that $\rho_{W}$ and $\Delta_{W}$ are parametrized
by $\sigma$ (\ref{e:rds1}),
$F \left[ \sigma, \phi ; \mu \right]$ $=$ $F\left[ \rho_{W}(\sigma),
\Delta_{W}(\sigma), \phi ; \mu \right]$.
We obtain the condensate wave function $\phi_{0} ({\bf R})$
and Bogolubov parameter $\sigma_{0} ({\bf R},{\bf p})$ that describe the
condensate
by varying $\sigma$ and $\phi$ independently to get a minimum in F :
\begin{eqnarray}
\left. \frac{\delta F }{\delta \phi ({\bf R})}
\right|_{\sigma = \sigma_{0},\phi=\phi_{0}} &=&
0 \; , \; \; \; \; (\rm{NLSE})\; \nonumber \\
\left. \frac{\delta F }
{\delta \sigma ({\bf R},{\bf p})} \right|_{\sigma = \sigma_{0},\phi=\phi_{0}}
&=&
0.
\label{e:sfe}
\end{eqnarray}
The $\phi$ variation, $\delta F / \delta \phi =0$, gives
the non-linear Schrodinger Equation (NLSE).
The $\sigma$ variation, $\delta F / \delta \sigma = 0, $
gives an equation for $\sigma_{0} ({\bf R},{\bf p})$.
From $ \rho = \frac{1}{2} \left[ \cosh (2 \sigma) -1 \right]$ and
$\Delta = \frac{1}{2} \sinh (2 \sigma)$ (\ref{e:rds1}),
we find that
$\partial \rho / \partial \sigma = \sinh (2 \sigma)$
and $\partial \Delta / \partial \sigma = \cosh (2 \sigma)$,
so that $\delta F / \delta \sigma = 0$ is equivalent to
\begin{equation}
\tanh (2 {\sigma}_{0}) =
\frac{ - \delta F / \delta \Delta_{W}}
{\delta F / \delta \rho_{W}} .
\label{e:tan}
\end{equation}
Now, several terms of the NLSE, as well as the functional derivatives
$\delta F / \delta \Delta$ and $\delta F / \delta \rho$ (\ref{e:tan}), depend on
$\sigma_{0}$ and $\phi_{0}$ so that the resulting equations have to be solved
self-consistently.
To make the self-consistent nature of the equations more explicit, we consider
the $\sigma$-dependent contributions to the functional derivatives,
$\delta V_{\rm{exch}} / \delta \rho$,
$\delta V_{\rm{dir}} / \delta \rho$ and
$\delta V_{\rm{pair}} / \delta \Delta$, which we shall call
the generalized potentials,
\begin{eqnarray}
U_{\rm{exch}} ({\bf R},{\bf p}) &=&
\delta V_{\rm{exch}} / \delta \rho_{W}
({\bf R},{\bf p}) \; \; =
\int_{{\bf p}'} \; v({\bf p}-{\bf p}') \rho_{W} ({\bf R},
{\bf p}') , \nonumber\\
\nonumber\\
U_{\rm{dir}} ({\bf R}) &=&
\delta V_{\rm{dir}} / \delta \rho_{W}
({\bf R},{\bf p}) \; \; =
v(0) \int_{{\bf p}'} \; \rho_{W} ({\bf R},
{\bf p}') , \nonumber\\
\nonumber\\
U_{\rm{pair}} ({\bf R},{\bf p}) &=&
\delta V_{\rm{pair}} / \delta \Delta_{W}
({\bf R},{\bf p}) \; \; =
\int_{{\bf p}'} \; v({\bf p}-{\bf p}') \Delta_{W} ({\bf R},
{\bf p}') ,
\label{e:genp}
\end{eqnarray}
where we name the generalized potentials after the respective
interaction
energies of which they are the functional derivatives,
$U_{\rm{exch}}$ is the exchange potential, $U_{\rm{dir}}$
the direct potential and $U_{\rm{pair}}$ the pairing potential.
Writing the distribution and pairing function in the integrands
of the generalized potentials in terms of $2\sigma$,
we find with (\ref{e:tan}) that the generalized potentials
implicitly depend on the functional derivatives of F :
\begin{eqnarray}
U_{\rm{exch}} ({\bf R},{\bf p}) &=&
\int_{{\bf p}'} \; v({\bf p}-{\bf p}')
\frac{1}{2} \left[
\frac{\delta F / \delta \rho}
{\sqrt{(\delta F / \delta \rho)^{2} - (\delta F / \delta \Delta)^{2}}}
-1 \right] , \nonumber\\
\nonumber\\
U_{\rm{dir}} ({\bf R}) &=&
v(0) \int_{{\bf p}'} \; \frac{1}{2} \left[
\frac{\delta F / \delta \rho }
{\sqrt{(\delta F / \delta \rho)^{2} - (\delta F / \delta \Delta)^{2}}}
-1 \right] , \nonumber\\
\nonumber\\
U_{\rm{pair}} ({\bf R},{\bf p}) &=&
\int_{{\bf p}'} \; v({\bf p}-{\bf p}') \; \frac{1}{2} \left[
\frac{ - \delta F / \delta \Delta}
{\sqrt{(\delta F / \delta \rho)^{2} - (\delta F / \delta \Delta)^{2}}}
\right],
\label{e:genpo}
\end{eqnarray}
where it is understood that the functional derivatives in the integrands
are evaluated at ${\bf R}$ and ${\bf p}'$.
Functional differentiation shows that the functional derivatives of F
in turn depend on the generalized potentials,
\begin{eqnarray}
\frac{\delta F}{\delta \Delta_{W} ({\bf R},{\bf p})} &=&
U_{\rm{pair}} ({\bf R},{\bf p}) - \phi^{2} ({\bf R})
v({\bf p}) \; , \nonumber\\
\nonumber\\
\frac{\delta F}{\delta \rho_{W} ({\bf R},{\bf p})} &=&
\frac{p^{2}}{2m} + V_{\rm{ext}} ({\bf R}) - \mu \nonumber\\
&& + U_{\rm{exch}} ({\bf R},{\bf p}) + U_{\rm{dir}}
({\bf R},{\bf p}) + \phi^{2} ({\bf R}) \left[ v({\bf p})+v(0)
\right] \; .
\label{e:fund}
\end{eqnarray}
Thus, equations (\ref{e:genpo}) and (\ref{e:fund})
self-consistently determine the generalized potentials.
Furthermore, there is a dependence on the condensate wave function $\phi$.
The latter has to be obtained from the NLSE :
\begin{equation}
\left[ - \frac{\nabla^{2}}{2m} + V_{\rm{ext}} ({\bf R})
- \mu + U({\bf R}) + v(0) \phi^{2} ({\bf R}) \right] \phi({\bf R}) = 0 ,
\label{e:nlse}
\end{equation}
where the potential $U({\bf R})$, is equal to :
\begin{equation}
U({\bf R}) = U_{\rm{dir}} ({\bf R}) + U_{\rm{exch}} ({\bf R},0)
- U_{\rm{pair}} ({\bf R},0) \; .
\label{e:unlse}
\end{equation}
This potential term, which stems from the interaction of the condensate
with the particles out of the condensate, is absent in the simplest
(low density limit) form of the NLSE, usually encountered in the
literature.
The equations, (\ref{e:genpo}) (\ref{e:fund}) (\ref{e:nlse}) and
(\ref{e:unlse}), are the full set of self-consistent mean field equations
that describe the condensate in the Thomas-Fermi approximation.
The self-consistent equations for the homogeneous gas
\cite{GR}, are recovered by
putting $V_{\rm{ext}} = 0$ and by assuming that $\phi$ is independent
of position so that the kinetic energy contribution to the NLSE vanishes.
Regarding the connection with the intuitive Thomas-Fermi model, we note
that $\mu$ and $V_{\rm{ext}}$ in the self-consistent mean field equations
always appear as $\mu - V_{\rm{ext}} ({\bf R})$, so that it is natural
to define a local effective chemical potential:
\begin{equation}
\mu_{\rm{eff}} ({\bf R}) = \mu - V_{\rm{ext}} ({\bf R}).
\label{e:mueff}
\end{equation}
In fact, this is the essence of the Thomas-Fermi description : the system
is described locally as a homogeneous system with a position dependent
effective chemical
potential (\ref{e:mueff}).
The solutions to the fully self-consistent equations
determine the expectation
value of all (static) physical observables as a function of the
chemical potential $\mu$. One observable we can obtain
in this manner is N, the number of trapped particles,
\begin{equation}
N(\mu) = \frac{\partial F}{\partial \mu} = \int_{\bf R} \int_{\bf p}
\rho ({\bf R},{\bf p})
+ \int_{\bf R} \; \phi^{2} ({\bf R}) ,
\end{equation}
the inversion of which yields $\mu$(N), from which we can
cast the results for the thermodynamic quantities in terms of the
parameter that is controled
or measured in the experiment $-$ the number of atoms N.
\section{Low Density Limit}
The self-consistent equations,
(\ref{e:genpo}) (\ref{e:fund}) (\ref{e:nlse}) and
(\ref{e:unlse}) can be solved iteratively.
In the low density regime, where $\sqrt{n a^{3}} \ll 1$, we
approximate the result by the expressions obtained after
a single iteration,
starting from $\sigma^{(0)}_{0} =0$ ($U^{(0)}_{\rm{exch}}= $
$U^{(0)}_{\rm{dir}} = $ $U^{(0)}_{\rm{pair}} = 0$, where
the superscript indicates the order of the iteration).
With this first guess we solve the NLSE and obtain the
functional derivatives (\ref{e:genpo}), $\delta F/\delta \rho$,
$\delta F/ \delta \Delta$, yielding
the first-order $\sigma -$parameter (\ref{e:tan}), $\sigma^{(1)}$,
and the generalized potentials (\ref{e:genp}), $U^{(1)}_{\rm{dir}}$,
$U^{(1)}_{\rm{exch}}$, $U^{(1)}_{\rm{pair}}$. With these
single iteration expressions we compute the expectation values of
the observable quantities.
In solving the NLSE, we shall assume that $\phi ({\bf R})$ varies slowly enough
that we can also neglect the kinetic energy operator.
To make the dependence
on the scattering length explicit, we replace the potential
by a pseudopotential,
\begin{equation}
V_{\rm{pseudo}} ({\bf r}) = \lambda \delta({\bf r}) \frac{\partial}
{\partial r} r ,
\label{e:pseudo}
\end{equation}
where $\lambda = 4 \pi \hbar^{2} a /m$ and the derivative operator
is necessary to remove the divergency in the ground state free energy
\cite{HY}.
Furthermore, we shall assume that $\phi ({\bf R})$ varies slowly enough
that we can also neglect the kinetic energy operator in solving the NLSE
(\ref{e:nlse}) :
\begin{equation}
\lambda \left[ \phi^{{(1)}} ({\bf R}) \right] ^{2}
= \mu_{\rm{eff}} ({\bf R}) ,
\label{e:phio}
\end{equation}
where $\mu_{\rm{eff}}$ is the effective chemical potential
(\ref{e:mueff}).
The functional derivatives (\ref{e:fund}) are
\begin{eqnarray}
\frac{\delta F^{\rm{{(1)}}}}{\delta \Delta} &=&
- \lambda \left[ \phi^{{(1)}} ({\bf R}) \right] ^{2} \; , \nonumber\\
\nonumber\\
\frac{\delta F^{\rm{{(1)}}}}{\delta \rho} &=&
\frac{p^{2}}{2m} - \mu_{\rm{eff}} ({\bf R}) +
2 \lambda \left[ \phi^{{(1)}} ({\bf R}) \right] ^{2}.
\end{eqnarray}
Consequently, the single iteration value for the Bogolubov
transformation parameter $\sigma$ is equal to
\begin{eqnarray}
\tanh \left[ 2 \sigma^{\rm{{(1)}}}_{0} ({\bf R},{\bf p}) \right]
&=& \frac{ \lambda \phi^{2} ({\bf R})}
{(p^2/2m) - \mu_{\rm{eff}} ({\bf R}) +
2 \lambda \phi^{2} ({\bf R})} \nonumber\\
\nonumber\\
&=& \frac{\mu_{\rm{eff}} ({\bf R})}
{(p^2/2m) + \mu_{\rm{eff}} ({\bf R})} \; \; ,
\label{e:sigm1}
\end{eqnarray}
which can be recognized as the dilute uniform gas result if we put
$\mu_{\rm{eff}} = \mu$.
The expression for the Bogolubov parameter $\sigma^{{(1)}}_{0}$
from Eq.(\ref{e:sigm1}) is what we would have obtained with
an effective energy density neglecting the interaction energies of the
particles out of the condensate,
$V_{\rm{dir}} $, $ V_{\rm{exch}} $ and
$ V_{\rm{pair}} $. In other words, the effective ground state
energy is
$F_{\rm{eff}}$ = $\int d^{3} R \; f_{\rm{eff}} ({\bf R})$, where
\begin{eqnarray}
f_{\rm{eff}} ({\bf R}) &=&
\int_{\bf p} \; \left[
\left[ \frac{p^{2}}{2m} - \mu_{\rm{eff}} ({\bf R})
+ 2 \lambda \phi^{2} ({\bf R}) \right] \;
\rho_{W} ({\bf R},{\bf p})
- \lambda \phi^{2} ({\bf R}) \Delta_{W} ({\bf R},{\bf p}) \right]
\nonumber\\
&& - \mu_{\rm{eff}} ({\bf R}) \phi^{2} ({\bf R})
+ \frac{\lambda}{2} \phi^{4} ({\bf R}) .
\label{e:feff}
\end{eqnarray}
We obtain the results for the observable quantities
by calculating their expectation values
from the single iteration $\sigma^{{(1)}}_{0}$ of Eq.(\ref{e:sigm1}).
For example, the condensate wave function is determined from the
NLSE :
\begin{equation}
\lambda \phi^{2} ({\bf R}) \approx
\mu_{\rm{eff}} ({\bf R}) - U^{(1)} ({\bf R}),
\label{e:gap1}
\end{equation}
where the potential $U({\bf R})$ is the sum of the generalized potentials
at zero momentum (\ref{e:unlse}),
\begin{equation}
U^{(1)} ({\bf R}) = U^{(1)}_{\rm{exch}}
({\bf R},0) + U^{(1)}_{\rm{dir}} ({\bf R})
- U^{(1)}_{\rm{pair}} ({\bf R},0) ,
\nonumber\\
\end{equation}
evaluated with the single-iteration value for $\sigma$.
The single-iteration values for the generalized potentials are computed to be :
\begin{eqnarray}
U^{(1)}_{\rm{exch}} ({\bf R},0) &=&
U^{(1)}_{\rm{dir}} ({\bf R}) =
\frac{\lambda}{3\pi^2} \left[ \mu_{\rm{eff}} ({\bf R}) \right]^{3/2}
m^{3/2} \; , \nonumber\\
U^{(1)}_{\rm{pair}} ({\bf R},0) &=&
-{\lambda\over\pi^2} \; \left[ \mu_{\rm{eff}} ({\bf R}) \right]
^{3/2} m^{3/2} \; .
\label{e:u1}
\end{eqnarray}
Thus, the condensate density is (\ref{e:gap1})
\begin{equation}
\phi^{2} ({\bf R}) \approx {1\over\lambda}\mu_{\rm eff} ({\bf R})
- \frac{5}{3 \pi^{2}} \left[\mu_{\rm{eff}} ({\bf R}) \right] ^{3/2}
m^{3/2}\; \; .
\label{e:phi1}
\end{equation}
The total density n(${\bf R}$), including the correction
to $\phi^{2} ({\bf R})$ (\ref{e:phi1}) and the local depletion, is
equal to
\begin{eqnarray}
n({\bf R}) &=&
\phi ^{2} ({\bf R}) + \; \int_{\bf p} \;
\rho ({\bf R},{\bf p})\nonumber\\
&\approx& \phi ^{2} ({\bf R}) + \frac{1}{3\pi^2}
\left[\mu_{\rm{eff}} ({\bf R}) \right] ^{3/2}
m^{3/2}\nonumber\\
&\approx& {1\over\lambda}\mu_{\rm{eff}} ({\bf R})
- \frac{4}{3 \pi^{2}} \left[\mu_{\rm{eff}} ({\bf R}) \right] ^{3/2}
m^{3/2} ,
\label{e:npert}
\end{eqnarray}
resulting in an expression for the density, n(${\bf R}$),
in terms of the effective
chemical potential $\mu_{\rm{eff}} ({\bf R})$. Inverting this
relation up to first order in $\sqrt{n a^{3}}$, we obtain
\begin{equation}
\mu_{\rm{eff}} ({\bf R}) \approx \lambda n({\bf R})
\left[ 1 + \frac{32}{3} \sqrt{\frac{n({\bf R}) a^{3}}{\pi}}
\right] ,
\label{e:mu1}
\end{equation}
which, for the homogeneous case, reduces to the well-known perturbation result.
Finally, in a similar manner, we obtain the local pressure
$P({\bf R})$ from
the expression for the effective free energy density (\ref{e:feff}),
$P({\bf R})$ = $ - f^{(1)}_{\rm{eff}} ({\bf R})$,
\begin{equation}
P({\bf R}) = \frac{\lambda \phi^{4} ({\bf R})}{2}
\left[ 1 - \frac{128}{15\pi^2} \sqrt{ n({\bf R}) a^{3}} \right].
\label{e:pres}
\end{equation}
We can then replace $\phi^{2}$ in (\ref{e:pres}) by its
single-iteration value (\ref{e:phi1}). Furthermore, replacing
$\mu_{\rm{eff}} ({\bf R})$ in the resulting expression
by (\ref{e:mu1}) results in a local equation of state.
The above results illustrate an important advantage of the Thomas-Fermi
description $-$ by neglecting the kinetic energy operator in the NLSE we
recover simple analytical expressions
for most quantities. These expressions are the analogues of the
perturbation results for the dilute homogeneous bose gas.
It is then of course very important to determine
the regime and the conditions under which these results can be trusted.
One source of error in the theory stems from neglecting the Laplacian operator
in the NLSE.
This approximation, although convenient, is
not part of the Thomas-Fermi description.
It is always
possible to calculate the condensate wave function numerically from the NLSE
and proceed from there with the iteration of the
self-consistent Thomas-Fermi
equations (\ref{e:sfe})!
Nevertheless, if we omit
the Laplacian term, we can estimate the error by calculating $e_{L}$, the ratio
of the kinetic energy term, $-\nabla^{2} \phi /2m$, and the non-linear
potential energy in the NLSE, $\lambda \phi^{3}$,
\begin{equation}
e_{L} ({\bf R}) = | -\nabla^{2} \phi / 2m \lambda \phi^{3} | =
\left| \frac{ -\nabla^{2} \phi ({\bf R}) / \phi
({\bf R}) }{k_{c}^{2}({\bf R})} \right|,
\label{e:el}
\end{equation}
where $k_{c}({\bf R})$ = $\left[ 8 \pi a n({\bf R}) \right] ^{\frac{1}{2}}$ is
the inverse of the local coherence length, $k_{c}$
= $\lambda_{c}^{-1}({\bf R})$.
Another source of error, which cannot be remedied but is truly inherent to the
Thomas-Fermi approximation, stems from the inhomogeneity of the system.
This error is also more difficult to estimate, and one benefit of our
approach is that the gradient expansion offers a `handle' on this quantity.
Indeed, we use the lowest order non-vanishing term in the gradient expansion
to estimate the error. This term is of second order because the first order
term vanishes.
We estimate its magnitude (\ref{e:gradexp}) by
replacing the partial derivatives with respect to the momentum
variables by $k_{c}^{-1}$, since $k_{c}$ is a measure of the range
in ${\bf p}$ of the observable at zero temperature. The
relative error for the general product of two arbitrary operators
A and B, $e_{\rm{i}} \left[ AB \right] $, is then given by
\begin{equation}
e_{\rm{i}} \left[ AB \right] \approx
\frac{1}{8 k^{2}_{c} ({\bf R})}
\left[ \frac{\nabla^{2} A_{W} }{ A_{W} } + \frac{ \nabla^{2} B_{W} }{ B_{W}}
- 2
\frac{ \nabla A_{W}\; \cdot \nabla B_{W} }{ A_{W} B_{W} } \right] ,
\end{equation}
The validity of the Thomas-Fermi description depends on
$X_{W}^{2}-Y_{W}^{2}=1$
(\ref{e:canw}), so that we use the accuracy of this equality to test the
validity of the local homogeneity description.
The expression (\ref{e:canw}) can also be written as
$\exp \left[ \sigma({\bf R},{\bf p}) \right]
\exp \left[ -\sigma({\bf R},{\bf p}) \right] =1 $, so that we choose
$A_{W}$ as $\exp \left[ \sigma({\bf R},{\bf p}) \right]$ and
$B_{W}$ as $\exp \left[ -\sigma({\bf R},{\bf p}) \right]$ to estimate
the relative error $e_{\rm i}$. In fact, it is more
convenient to work with $\exp(4\sigma)$ then $\exp(\sigma)$, so that we
compute the inhomogeneity error $e_{i} \left[
\exp(4\sigma) \exp(-4\sigma)\right] $ of $\exp(4\sigma)$ and divide by 4
(since the relative error of $f^{n}$ is simply n $\times$ the relative error
of f).
In this manner, we find that
\begin{eqnarray}
e_{i} \left[ \exp(\sigma) \exp(-\sigma) \right] &=&
\frac{1}{4} e_{i} \left[ \exp(4\sigma) \exp(-4\sigma) \right]
\nonumber\\
&\approx& \frac{1}{8 k_{c}^{2} ({\bf R})} \left| \frac{\nabla \exp(4\sigma)}
{\exp(4\sigma)} \right| ^{2} \; .
\label{e:inh}
\end{eqnarray}
With the single-iteration value for the low-density condensate,
\begin{equation}
\exp \left[ 4\sigma({\bf R},{\bf p}) \right] =
1 + \frac{2 \mu_{\rm{eff}} ({\bf R})}{p^{2}/2m} ,
\end{equation}
we find that the inhomogeneity error, $e_{i} ({\bf R})$ (\ref{e:inh}),
is equal to
\begin{equation}
e_{i} ({\bf R}) = \frac{1}{2} \left|
\frac{{\bf F}_{\rm{ext}}({\bf R}) \lambda_{c}
({\bf R})} {(p^{2}/{2m}) + 2 \mu_{\rm{eff}} ({\bf R})} \right| ^{2} ,
\end{equation}
where ${\bf F}_{\rm{ext}}$ is the force of the external potential,
${\bf F}_{\rm{ext}}$ = $- \nabla V_{ext}$.
As expected, the error is largest for ${\bf p} = 0$, and using the
${\bf p} = 0$ $-$ value, we obtain a simple position dependent estimate
for the inhomogeneity error, $e_{i} ({\bf R})$,
\begin{equation}
e_{i} ({\bf R}) = \frac{1}{8} |{\bf F}_{\rm{ext}}({\bf R})
\lambda_{c}({\bf R}) / \lambda \phi^{2} ({\bf R}) |^{2} ,
\label{e:inhf}
\end{equation}
where we replaced $\mu_{\rm{eff}}$ by $ \lambda \phi^{2}$.
By equating this error (\ref{e:inhf}) to a chosen value, $e_{\rm{cut}} \ll 1$,
reflecting the accuracy we demand from the theory, we can
determine the spatial boundary beyond which the Thomas Fermi Theory
is less accurate than $e_{\rm{cut}}$.
\section{Spherically Symmetric Harmonic Oscillator Trap}
We now specialize $V_{\rm{ext}} ({\bf R})$ to a harmonic oscillator potential,
\begin{equation}
V_{\rm ext} ({\bf R}) = \frac{1}{2} \hbar \omega (R/L)^2 \; ,
\end{equation}
where $L$ is the size of the harmonic oscilator ground state,
\begin{equation}
L = \sqrt{\frac{\hbar}{m \omega}},
\end{equation}
and compute the expectation value of important quantities
in the low density limit of the previous section.
In zeroth order in the iteration, we recover the results of Baym and Pethick
\cite{GB}. From (\ref{e:phio}) we see that
\begin{eqnarray}
\left[ \phi^{(0)} ({\bf R}) \right] ^{2}
&=& \left[ \mu - V_{\rm{ext}} ({\bf R}) \right] / \lambda
\nonumber\\
&=& \frac{R^{2}_{0}}{8 \pi a L^{4}} \left[ 1 - (R/R_{0})^{2} \right]
\label{e:phios}
\end{eqnarray}
where $R_{0}$ is the size of the condensate,
$R_{0} = \sqrt{\frac{2 \mu}{\hbar \omega}}
L$. In zeroth order, all particles are in the condensate, so that
N $= \int_{\bf R} \phi^{2} ({\bf R}) $, and
\begin{equation}
\mu^{(0)} = \frac{\hbar \omega}{2} \left(\frac{ 15 a N}{L} \right) ^{2/5} ,
\end{equation}
and consequently,
\begin{equation}
R_{0} = L \left( \frac{15 a N}{L} \right) ^{1/5} .
\end{equation}
The local coherence length, $\lambda_{c} ({\bf R})$ is given by
\begin{equation}
\lambda_{c} ({\bf R}) = \frac{L^{2}}{\sqrt{R_{0}^{2} - R^{2}}} \; \; .
\label{e:cohl}
\end{equation}
Before we proceed to calculate the perturbation corrections to the
observables, we test the validity of the low density Thomas-Fermi
formalism by calculating the errors. The error due to neglecting the
Laplacian in the NLSE, $e_{L} ({\bf R})$ (\ref{e:el}),
is easily computed with
(\ref{e:cohl}):
\begin{equation}
e_{L} ({\bf R}) = \left( \frac{L}{R_{0}} \right) ^{4}
\frac{ \left[ 3 - 2 (R/R_{0})^{2} \right] }{(1 - (R/R_{0})^{2})^{3}} \; \; ,
\end{equation}
from which we see that the laplacian can be omitted in the NLSE on
condition that the size of the condensate is much larger than the size
of the ground state, $ R_{0} \gg L $, or $ \left( 15 a N / L \right)^{1/5}
\gg 1$.
The error due to the departure of the BEC from homogeneity, $e_{i} ({\bf R})$
(\ref{e:inh}), is
\begin{equation}
e_{i} ({\bf R}) = \frac{1}{2} \left( \frac{L}{R_{0}} \right) ^{4}
\frac{(R/R_{0})^{2}}{(1 - (R/R_{0})^{2})^{3}} \; \; .
\end{equation}
Again, notice that $e_{i}$ is small over most of the condensate region
($R < R_{0}$) if $ R_{0} \gg L $.
In Fig.~(1) we show the density, $[\phi^{(0)} (R)]^{2}
/ [ \phi^{(0)} (R=0) ]^{2}$ (\ref{e:phios}), and both errors,
$e_{L}$ and $e_{i}$, as a function
of the distance to the middle of the trap.
\begin{figure}[htbp]
\centerline{\BoxedEPSF{fig1.eps scaled 600}}
\caption{{\sf (a) Condensate density for N = $10^{3}$ and $10^{6}$; (b) Error
incurred in neglecting kinetic term in NLSE; (c) Error incurred in
Thomas-Fermi approximation. Length scale on horizontal axis is in units of L,
the extend of the ground state wave function. Calculations are done for
L = $10^{-4}$ cm, scattering length a = $5 \times 10^{-7}$ cm.}}
\end{figure}
The curves are
calculated for a harmonic oscillator trap
of $L = 1 \mu m$ and an inter-atomic interaction with
scattering length a = 5 nm. The dotted lines
correspond to $N = 10^{3}$ atoms in the trap, and the full line gives the
results for $N = 10^{6}$ atoms. Notice that for $10^{3}$ particles,
the Laplacian error is already substantial ($\sim 10 \%$) in the middle of
the trap. In contrast, only at
$ R = 1.8 L $ (to be compared to $ R_{0} = 2.4 L $) does the
inhomogeneity error become of comparable magnitude. This indicates
that even for as few as 1000 particles in the trap, the Thomas-Fermi
description could be reasonably accurate for these parameters,
provided one keeps
the kinetic energy term
in solving the NLSE. For $10^{6}$ atoms, $e_{i}$ and $e_{L}$ only become of
the order of $ 10 \%$ at $ R = 9.0 L$, whereas $ R_{0} = 9.4 L$, which
shows
that the Thomas-Fermi description and neglecting the Laplacian operator
are valid approximations in almost all of the condensate region.
Under this condition, it is meaningful to calculate the perturbation
corrections to the expectation values of the observable quantities.
Including the perturbation correction, the local density (\ref{e:npert})
is equal to
\begin{equation}
n(R) = \frac{R_{0}^{2}}{8 \pi a L^{4}}
\left[ 1 - (R/R_{0})^{2} \right]
\left[ 1 - \frac{2 \sqrt{2}}{3 \pi} \frac{a R_{0}}{L^{2}}
\sqrt{1 -
\left(R/R_{0}\right)^{2} } \right].
\end{equation}
The number of trapped particles, N, is obtained by integrating over the
density $n({\bf R})$,
\begin{equation}
N = \int_{\bf R} \; n(R) = 4 \pi \int_{0}^{R_{0}} d R \; R^{2} \; n(R) \; ,
\label{e:nint}
\end{equation}
which leads to
\begin{equation}
N = \frac{1}{15} \frac{L}{a} \left(\frac{2 \mu}{\hbar \omega}\right)^{5/2}
- \frac{\sqrt{2}}{24} \left(\frac{2 \mu}{\hbar \omega}\right)^{3} .
\end{equation}
The inverse relation, $\mu$ as a function of N, can be obtained
by solving for $\mu$ iteratively in the previous equation,
wich gives up to second iteration, the following result :
\begin{equation}
\mu = \frac{\hbar\omega}{2} \left(\frac{15 a}{L} \right)^{2/5} N^{2/5} \left[
1 + \frac{\sqrt{2}}{60} \left(\frac{15 a}{L} \right)^{6/5} N^{1/5} \right].
\end{equation}
Similarly, we obtain the condensate density from (\ref{e:phi1}) or the local
depletion, $d({\bf R}) = \left[ n^{(1)} ({\bf R}) \right.$
$-\left.
\left[ \phi^{1} ({\bf R}) \right] ^{2} \right]
/ \left[ \phi^{(1)} ({\bf R}) \right] ^{2} $ :
\begin{equation}
d(R) = \frac{2 \sqrt{2}}{3 \pi} \frac{a R_{0}}{L^{2}}
\sqrt{ 1 - ( R/R_{0} )^{2} } .
\end{equation}
In Fig.~(2), we show the local depletion as a function of position
for the same parameters as those of Fig.~(1).
\medskip
\begin{figure}[htbp]
\centerline{\BoxedEPSF{fig2.eps scaled 600}}
\caption{{\sf Depletion, defined as $d(R) = \left[ n({\bf R}) - \phi^{2} ({\bf R})
\right] / \phi^{2} ({\bf R})$, for the same systems as Fig.~(1).}}
\end{figure}
\medskip
The local pressure is
shown in Fig.~(3).
\medskip
\begin{figure}[htbp]
\centerline{\BoxedEPSF{fig3.eps scaled 600}}
\caption{{\sf Pressure for the same systems as Fig.~(1).}}
\end{figure}
\medskip
To conclude this section, we repeat that the condition for
the validity of the Thomas-Fermi description is that the size
of the condensate exceeds the size of the ground state of the trap,
$R_{0} \gg L$. An equivalent condition is that the coherence length
in the middle of the condensate is smaller than the size of the ground
state $\lambda_{c} (R=0) \ll L$, or that the chemical potential exceeds the
ground state energy, $\mu \gg (\hbar \omega /2)$. These statements do not
depend on the details of the trapping potential. Of course, the shape of the
condensate, the boundary where the Thomas-Fermi description breaks down,
and the expectation values of the
local observables do depend on the shape of the potential.
In this section, we gave the results for a spherically symmetric harmonic
oscillator potential.
For the convenience of the reader we tabulate several
of the results up to first non-vanishing order in table I.
\section{Density of States}
In the Thomas-Fermi picture, the system is locally equivalent to a
uniform system. Therefore, there are `local' excitations
which in the low-density regime are described by the following
energy spectrum :
\begin{equation}
{\epsilon}_{p} ({\bf R}) =
\sqrt{(p^{2}/2m+\mu_{\rm{eff}}({\bf R}))^{2}-\mu_{\rm{eff}}^{2} ({\bf R}) }
+ \mu \; ,
\label{e:bspec}
\end{equation}
which is well known from the Bogolubov treatment of the uniform case.
The local dispersion relation (\ref{e:bspec}) describes a phonon with
position dependent sound velocity.
To obtain the excitation of the whole system we compute the density of states
using the formula
\begin{equation}
g(\epsilon) = \sum_{i} \; \delta(\epsilon - \epsilon_{i}),
\label{e:dsc}
\end{equation}
where $\sum_{i}$ represents the sum over all excited states.
In the spirit of the Thomas-Fermi approximation we take
\begin{equation}
g(\epsilon) = \int_{\bf R} \int_{\bf p} \;
\delta \left(\epsilon - \epsilon_{\bf p} ({\bf R}) \right) \; .
\label{e:dstf}
\end{equation}
After integration over the momentum variable, we obtain
\begin{equation}
g(\epsilon) = \frac{1}{2 \pi^{2}} \int_{\bf R} \; p_{\epsilon}^{2}
({\bf R}) \; \left| \frac{\partial \epsilon}{\partial p} \right| ^{-1} ,
\label{e:dss}
\end{equation}
where $p_{\epsilon} ({\bf R})$ is the momentum of a particle at position
${\bf R}$ with energy $\epsilon$.
When calculating the remaining integral over space, we need to distinguish
between spatial region (I) with condensate and a second region (II) without
condensate, shown schematically in Fig.~(4).
\medskip
\begin{figure}[htbp]
\centerline{\BoxedEPSF{fig4.eps scaled 600}}
\caption{{\sf Schematic representation of the region with (region I),
and without condensate (region II)
for a BEC in a harmonic trap. The condensate density is
proportional to $\mu_{\rm{eff}} ({\bf R})$, which is a `mirror image' of
the trapping potential. Particles in the condensate have energy $\mu$ and
a particle excited up to energy $\epsilon$ can move into region (II) as far
as the classical turning point $R_{\epsilon}$.}}
\end{figure}
\medskip
It is necessary to break
up the integral (\ref{e:dss}) over the different integration regions,
because the dispersion relations for the excitations are different.
In region (I), we use the Bogoliubov spectrum
(\ref{e:bspec}), whereas in region (II), the atoms are essentially free
particles moving in the trap:
\begin{equation}
\epsilon_{p} ({\bf R}) = \frac{p^{2}}{2m} + V_{\rm{ext}} ({\bf R}) \; .
\label{e:fspec}
\end{equation}
The density of states is then the sum of the integrals over region (I)
and (II):
\begin{eqnarray}
g(\epsilon) = \frac{\sqrt{2}}{2} \frac{m^{3/2}}{\pi^{2}}
& & \left[
(\epsilon - \mu) \int_{(I)} d^{3}R \;
\frac{\sqrt{ \sqrt{\left[ \epsilon - \mu \right] ^{2} +
\mu^{2}_{\rm{eff}}({\bf R})} - \mu_{\rm{eff}} ({\bf R})}}
{\sqrt{\left[ \epsilon - \mu \right] ^{2} +
\mu^{2}_{\rm{eff}}({\bf R})}}
\right.
\nonumber\\
&&
\left.
+ \int_{(II)} d^{3} R \; \sqrt{\epsilon
- V_{\rm ext}({\bf R})} \right] .
\end{eqnarray}
For the special case of a spherically symmetric harmonic oscillator
trap, we find the following expression for the density of states :
\begin{eqnarray}
g(\epsilon) = \frac{4}{\pi} \; \frac{\mu^{2}}{(\hbar \omega)^{3}}
& & \left[ (\epsilon / \mu -1) \int_{0}^{1} dr \sqrt{1 - r}
\frac{ \sqrt{ \sqrt{ (\epsilon / \mu -1)^{2} + r^{2}} - r }}
{ \sqrt{ (\epsilon / \mu -1)^{2} + r^{2}}}
\right.
\nonumber\\
& & \left. \; + \; 2 \int_{1}^{\epsilon / \mu} dr \; r^{2} \;
\sqrt{ \epsilon / \mu - r^{2}} \;
\right] \; .
\end{eqnarray}
In Figs. (5) and (6) we show the density of states for the system discussed
in the previous section, L = $ 1 \mu m$, a = 5 nm, N = $10^{3}$ (Fig.~(5)) and
N = $10^{6}$ (Fig.~ (6)).
\medskip
\begin{figure}[htbp]
\centerline{\BoxedEPSF{fig5.eps scaled 600}}
\caption{{\sf Density of states calculated in the Thomas-Fermi approach described
in the paper. The system is a BEC of N = $10^{3}$ particles interacting
with a scattering length a = $5 \times 10^{-7}$ cm,
in a harmonic trap with ground state of extend L = $10^{-4}$ cm.}}
\end{figure}
\medskip
\medskip
\begin{figure}[htbp]
\centerline{\BoxedEPSF{fig6.eps scaled 600}}
\caption{{\sf Density of states for the same system as in Fig.~(6), but with N =
$10^{6}$ particles.}}
\end{figure}
\medskip
The dotted lines show the result for
the interacting bose gas, the full line shows the density of states of
the ideal gas in the same trap.
The density of states starts from the chemical potential
$\mu$, consistent with (\ref{e:bspec}), which implies that the energies
are measured from the bottom of the potential well so that a particle
of zero momentum in the condensate has energy $\mu$. If we were to set out the
density of states as a function of excitation energy $\epsilon - \mu$,
the density of states curves for the interacting BEC-systems
would be shifted to the left by an amount $\mu$.
In contrast to the homogeneous BEC, the density of states
for the interacting case, as a function of the excitation energy, grows
faster than the density of states of the ideal gas. The reason
is purely geometrical:
the phonon
has a much larger volume in coordinate space available
(at least the volume of the condensate)
than the non-interacting boson that received the same amount of energy
and can only move near the bottom of the potential well.
This effect outweighs the fact that
the momentum space volume available to the phonon
is less than the momentum space volume available to the non-interacting
particle with the same energy.
Of course, the sharpness of the boundary between region (I)
and (II),
is an artifact of neglecting the Laplacian operator in the NLSE.
Nevertheless, except for a region near the boundary, we argue
that the rest of space is well-described and that
the contribution of the near-boundary region is comparatively
small so that the error that is introduced in the integral (\ref{e:dss})
is small provided the Thomas-Fermi
description is valid in most of the condensate region.
\section*{Acknowledgments}
This work was supported in part by funds provided by
the U.S. Department of Energy under cooperative agreement
\# DE-FC02-94ER40818. P.T. was supported by Conselho Nacional de
Desenvolvimento Cientifico e Tecnologico (CNPq), Brazil.
The work of E.T. is supported by the NSF through a grant for the
Institute for Atomic and Molecular Physics at Harvard University
and Smithsonian Astrophysical Observatory.
\newpage
| proofpile-arXiv_065-445 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
Recently it was shown$^{1,2}$ that
the Abrikosov-Gor'kov (AG)$^{3}$ theory of impure superconductors
predicts a large decrease of $T_{c}$, linear in the
nonmagnetic impurity concentration, which is
not consistent with Anderson's theorem.$^{4}$
In their response$^{5}$ on ref. 1, AG argued that the frequency cutoff makes the AG theory
compatible with Anderson's theorem.
This argument is based on Tsuneto's application$^{6}$ of the AG theory
to the Eliashberg equation.
To settle this controversy, we first need to understand
the limitation of Anderson's theorem.
It was pointed out that Anderson's theorem is
valid only up to the first order of the impurity concentration,$^{1,7}$
and the phonon-mediated interaction is strongly decreased by Anderson
localization.$^{1,8,9}$
However, Tsuneto's theory fails to show the existence of localization
correction to the phonon-mediated interaction.
The failure comes from the intrinsic pairing problem$^{10,11}$ in Gor'kov's
formalism.$^{12}$
The kernel of the self-consistency equation should be set by the
physical constraint of the Anomalous Green's function.
The resulting equation is nothing but another form
of the BCS gap equation.
Using the equation, the localization correction
to the phonon-mediated interaction may be calculated.$^{1,11}$
A correct strong coupling theory has been reported by Kim.$^{8}$
For magnetic impurity effects, Kim and Overhauser (KO)$^{7}$ proposed a BCS type
theory with different predictions: (i) The initial slope of $T_{c}$
decrease depends on the superconductor and is not the universal
constant proposed by Abrikosov and Gor'kov(AG).$^{3}$ (ii) The $T_{c}$ reduction
by exchange scattering is partially suppressed by potential scattering
when the overall mean free path is smaller than the coherence length.
This compensation has been confirmed in several experiments.$^{13-15}$
The difference comes again from the pairing problem.
If we impose a correct pairing condition
on the self-consistency equation, or the AG's calculation, we can find KO's result.
\section{Phonon-Mediated Interaction in BCS Theory and Gor'kov's Formalism }
\subsection{BCS Theory}
For a homogeneous system, BCS introduced
a reduced version of the Fr\"ohlich phonon-mediated interaction,
\begin{eqnarray}
H_{red} = \sum_{{\vec k} {\vec k}'} V_{{\vec k} {\vec k}'} c_{{\vec k}'}^{\dagger}c_{-{\vec k}'}^{\dagger}
c_{-{\vec k}}c_{{\vec k}},
\end{eqnarray}
where
\begin{eqnarray}
V_{{\vec k}{\vec k}'}= \cases{-V, &if $|\epsilon_{{\vec k}}|,|\epsilon_{{\vec k}'}|\leq \omega_{D}$\cr
0, &otherwise.\cr}
\end{eqnarray}
This reduction procedure is recognizing in advance which eigenstates
will be paired and so contribute to the BCS condensate.
In the presence of impurities, we can derive the phonon-mediated
interaction by transforming the Fr\"ohlich interaction
using the relation,
\begin{eqnarray}
\psi_{n\sigma} = \sum_{{\vec k}}\phi_{{\vec k}\sigma}<{\vec k}|n>.
\end{eqnarray}
$\psi_{n}$ and $\phi_{{\vec k}}$ denote the scattered state and
the plane wave state.
The reduced version of this interaction anticipates that $\psi_{n}$ (having
spin up) will be paired with its time-reversed counterpart
$\psi_{\overline {n}}$ (having spin down). The new reduced Hamiltonian is
\begin{eqnarray}
H_{red}' = \sum_{n n'} V_{n n'}c_{n'}^{\dagger}c_{\overline {n}'}^{\dagger}c_{\overline
{n}}c_{n},
\end{eqnarray}
where
\begin{eqnarray}
V_{n n'} = -V\sum_{{\vec k} {\vec k}' {\vec q}} <{\vec k} - {\vec q}|n'>
<{\vec k}' + {\vec q}|\overline {n}'><{\vec k}'|\overline {n}>^{*}<{\vec k}|n>^{*}.
\end{eqnarray}
Anderson's theorem is valid only when ${\vec k}'$ can be set equal to
$-{\vec k}$.
\subsection{Gor'kov's formalism}
In Gor'kov's formalism, a point
interaction $-V\delta({\bf r}_{1}-{\bf r}_{2})$ is used for the
pairing interaction between electrons.
For a homogeneous system, the pairing interaction is
\begin{eqnarray}
H_{G} &=& - {1\over 2}V\int d{\bf r}\sum_{\alpha\beta}\Psi^{\dagger}({\bf r}\alpha)
\Psi^{\dagger}({\bf r}\beta)\Psi({\bf r}\beta)\Psi({\bf r}\alpha) \nonumber \\
&=&-{1\over 2}V\sum_{{\vec k}{\vec k}'{\vec q}\sigma\sigma '}
c_{{\vec k} - {\vec q}, \sigma}^{\dagger}c_{{\vec k}' + {\vec q}, \sigma '}^{\dagger}c_{{\vec k} '\sigma '}
c_{{\vec k}, \sigma},
\end{eqnarray}
and
\begin{eqnarray}
V_{{\vec k}{\vec k}'}&=&-V\int \phi_{{\vec k}'}^{*}({\bf r}) \phi_{-{\vec k}'}^{*}({\bf r})
\phi_{-{\vec k}}({\bf r}) \phi_{{\vec k}}({\bf r})d{\bf r}\nonumber \\
&=&-V.
\end{eqnarray}
Eq. (6) is the same as the Fr\"ohlich interaction within the BCS
approximation.
Note that the two points are not clear in Gor'kov's formalism, i.e.,
the BCS reduction procedure and the retardation cutoff.
To obtain the same result as that of the BCS theory,
these two ingredients should be taken care of in some way.
As will be shown later, the negligence of the BCS reduction procedure
causes a serious pairing problem especially in impure superconductors.
In the presence of impurities,
the matrix element of the pairing interaction is
\begin{eqnarray}
V_{nn'}=-V\int \psi_{n'}^{*}({\bf r}) \psi_{\bar n'}^{*}({\bf r})
\psi_{\bar n}({\bf r}) \psi_{n}({\bf r})d{\bf r}.
\end{eqnarray}
Substituting Eq. (3) into Eq. (8) we find that
\begin{eqnarray}
V_{nn'}=-V \sum_{{\vec k} {\vec k}' {\vec q}} <{\vec k} - {\vec q}|n'>
<{\vec k}' + {\vec q}|\overline {n}'><{\vec k}'|\overline {n}>^{*}<{\vec k}|n>^{*}.
\end{eqnarray}
Notice that Eq. (9) is the same as Eq. (5).
\section{Pairing Constraint on Gor'kov's Formalism}
\subsection{Inhomogeneous System: Nonmagnetic Impurity Case}
Near the transition temperature,
the usual self-consistency equation is
\begin{eqnarray}
\Delta({\bf r}) &=& VT\sum_{\omega}\int \Delta({\bf l})G^{\uparrow}_{\omega}({\bf r,l})
G^{\downarrow}_{-\omega}({\bf r,l})d{\bf l}\nonumber \\
&=&\int K({\bf r},{{\bf l}})\Delta({{\bf l}})d{{\bf l}}.
\end{eqnarray}
Note that Kernel $K({\bf r},{{\bf l}})$ is not for Anderson's
pairing.
It includes the extra pairings between $n\uparrow$ and $n'(\not={\bar n})\downarrow$.
The kernel for Anderson's pairing is
\begin{eqnarray}
K^{A}({\bf r},{{\bf l}})&=&VT\sum_{\omega}
\{G^{\uparrow}_{\omega}({\bf r}, {{\bf l}})
G^{\downarrow}_{-\omega}({\bf r'}, {{\bf l}})\}_{p.p.}\nonumber \\
&=&V\sum_{n}{1\over 2\epsilon_{n}}tanh{\epsilon_{n}\over 2T}\psi_{n}({\bf r})
\psi_{\bar n}({\bf r}) \psi^{*}_{\bar n}({{\bf l}})\psi^{*}_{n}({{\bf l}}),
\end{eqnarray}
where p.p. means proper pairing constraint, which dictates
pairing between $n\uparrow$ and ${\bar n}\downarrow$.
It can be shown$^{10,11}$ that the extra pairings violate the physical
constraint of the Anomalous Green's function, i.e.,
\begin{eqnarray}
\overline{F({\bf r},{\bf r'},\omega)}^{imp}&\sim&
\overline{\psi_{n\uparrow}({\bf r})\psi_{n'\downarrow}({\bf r'})}^{imp}\nonumber \\
&\not=& \overline{F({\bf r}-{\bf r'},\omega)}^{imp}.
\end{eqnarray}
These extra pairings should have been eliminated by the BCS reduction
procedure in the Hamiltonian.
Consequently, the revised self-consistency equation is
\begin{eqnarray}
\Delta({\bf r}) = VT\sum_{\omega}\int \Delta({{\bf l}})\{G^{\uparrow}_{\omega}({\bf r,{\bf l}})
G^{\downarrow}_{-\omega}({\bf r,{\bf l}})\}_{p.p.}d{\bf l}.
\end{eqnarray}
Notice that Eq. (13) is nothing but another form of the
BCS gap equation,
\begin{eqnarray}
\Delta_{n}=\sum_{n'}V_{nn'}{\Delta_{n'}\over 2E_{n'}}tanh
{E_{n'}\over 2T}.
\end{eqnarray}
\subsection{Inhomogeneous System: Magnetic Impurity Case}
KO's $^{7}$ theory employed degenerate
scattered state pairs.
It has been claimed that the inclusion of the extra pairing
is the origin of the
so-called pair-breaking of the magnetic impurities.$^{16,17}$
However, the extra pairing terms cause the violation of the physical
constraint of the pair potential and the Anomalous Green's function.$^{11}$
It can be shown$^{11}$ that the homogeneity condition of the
Anomalous Green's function, after the impurity average, requires pairing
between the
degenerate scattered state partners. Then the revised self-consistency equation
gives rise to KO's result.$^{7}$
\subsection{Homogeneous System}
Near the transition temperature, the Anomalous Green's function is given by
\begin{eqnarray}
F({\bf r}, {\bf r'}, \omega) = \int \Delta({{\bf l}})G^{\uparrow}_{\omega}({\bf r}, {{\bf l}})
G^{\downarrow}_{-\omega}({\bf r'}, {{\bf l}})d{{\bf l}},
\end{eqnarray}
Gor'kov$^{12}$ pointed out that $F({\bf r},{\bf r'})$ should depend only on ${\bf r}-{\bf r'}$, i.e.,
\begin{eqnarray}
F({\bf r},{\bf r'}) = F({\bf r}-{\bf r'}).
\end{eqnarray}
Note that Eq. (15) includes the extra pairing terms
between ${\vec k}\uparrow$ and ${\vec k}'\downarrow(\not= -{\vec k}\downarrow)$, which do not
satisfy the homogeneity condition of Eq. (16). In this case, the self-consistency
condition of the pair potential happens to eliminate the extra pairing
in Eq. (15) because of the orthogonality of the wavefunctions.
However, it is important to eliminate the extra pairing
in the Anomalous Green's function from the beginning.
Note that the kernel $K({\bf r},{{\bf l}})$ is not for the pairing
between ${\vec k}\uparrow$ and $ -{\vec k}\downarrow$, but for the pairing between
the states which are the linear combination of the plane
wave states $\phi_{{\vec k}}({\bf r})$.$^{18}$
The inclusion of the extra pairings
hindered our correct understanding of the
impure superconductors and the relation between the
pair potential and the gap parameter.
\section{Pairing Constraint on the Bogoliubov-de Gennes Equations}
\subsection{Inhomogeneous system: Nonmagnetic Impurity Case}
By performing a unitary transformation,
\begin{eqnarray}
\Psi({\bf r}\uparrow) & =& \sum_{n}(\gamma_{n\uparrow}u_{n}({\bf r}) -
\gamma^{\dagger}_{n\downarrow}v^{*}_{n}({\bf r})) \nonumber \\
\Psi({\bf r}\downarrow) & =& \sum_{n}(\gamma_{n\downarrow}u_{n}({\bf r}) +
\gamma^{\dagger}_{n\uparrow}v^{*}_{n}({\bf r})),
\end{eqnarray}
we obtain the well-known Bogoliubov-de Gennes equations.
To understand the physical meaning of the transformation (17), we
express $\gamma_{n\uparrow}$ and $\gamma_{n\downarrow}$ by
the creation and destruction operators for an electron in the scattered
state,$^{18}$
\begin{eqnarray}
\gamma_{n\uparrow} &= &\sum_{n'}\bigl( u_{n,n'}^{*}c_{n'\uparrow} + v_{n,n'}c_{n'\downarrow}^{\dagger}\bigr),\nonumber \\
\gamma_{n\downarrow} &= &\sum_{n'}\bigl( u_{n,n'}^{*}c_{n'\downarrow} - v_{n,n'}c_{n'\uparrow}^{\dagger}\bigr),
\end{eqnarray}
where
\begin{eqnarray}
u_{n,n'} & =& \int \psi^{*}_{n'}({\bf r}) u_{n}({\bf r})d{\bf r}\nonumber \\
v_{n,n'} & =& \int \psi^{*}_{n'}({\bf r})v^{*}_{n}({\bf r})d{\bf r}.
\end{eqnarray}
Accordingly, we obtain a vacuum state where $u_{n}({\bf r})\uparrow$ and
$v_{n}^{*}({\bf r})\downarrow$ (instead of $\psi_{n}({\bf r})\uparrow$ and $
\psi_{\bar n}({\bf r})\downarrow$) are paired.
They are the superpositions of the scattered states.
It is clear that the Bogoliubov-de Gennes equations cannot give rise to
the correct Anderson's pairing because of the position dependence
of the pair potential.
We need to supplement a pairing condition.
If we assume a constant pair potential,$^{16,17}$ Anderson's
pairing is obtained. However, then, the impurity effect on the
phonon-mediated interaction is gone.
In the presence of magnetic impurities, even the constant
pair potential gives a pairing between the states which are the
linear superpositions of the scattered states.
\subsection{Homogeneous System }
As in Sec. IV. A, the unitary transformation generates a vacuum state
and the self-consistency equation for the pairing between
$u_{n}({\bf r})\uparrow$ and $v^{*}_{n}({\bf r})\downarrow$, (instead of
$\phi_{{\vec k}}({\bf r})\uparrow$ and $\phi_{-{\vec k}}({\bf r})\downarrow$).
Note that $u_{n}({\bf r})\uparrow$ and $v^{*}_{n}({\bf r})\downarrow$ are
the linear superpositions of the plane wave states until
we constrain them. In this case, setting the pair potential
gives a pairing between the plane wave states.
However, the kernel of the self-consistency
equation has not been set accordingly.
\section{Pair Potential and Gap Parameter}
For a homogeneous system, it was shown$^{19}$
\begin{eqnarray}
\Delta({\bf r}-{\bf r'})=\int d{\vec k} e^{i{\vec k}\cdot({\bf r}-{\bf r'})}\Delta_{{\vec k}}.
\end{eqnarray}
But this relation is not exact because of the retardation
cutoff.
Correct relation may be obtained only after incorporating the pairing constraint
into the self-consistency equation.
It is given
\begin{eqnarray}
\Delta({\bf r}-{\bf r'})=V\sum_{{\vec k}}{\Delta_{{\vec k}}\over 2E_{{\vec k}}}tanh
{E_{{\vec k}}\over 2T}\phi_{{\vec k}}({\bf r})\phi_{-{\vec k}}({\bf r'}).
\end{eqnarray}
Comparing Eq. (21) with the BCS gap equation and
using Eq. (7), we also find
\begin{eqnarray}
\Delta_{{\vec k}}=\int \phi_{{\vec k}}^{*}({\bf r})\phi^{*}_{-{\vec k}}({\bf r})
\Delta({\bf r})d{\bf r}.
\end{eqnarray}
In the presence of impurities, one finds that
\begin{eqnarray}
\Delta({\bf r})=V\sum_{n}{\Delta_{n}\over 2E_{n}}tanh
{E_{n}\over 2T}\psi_{n}({\bf r})\psi_{\bar n}({\bf r}),
\end{eqnarray}
and
\begin{eqnarray}
\Delta_{n}=\int \psi_{n}^{*}({\bf r})\psi^{*}_{\bar n}({\bf r})
\Delta({\bf r})d{\bf r}.
\end{eqnarray}
Eq. (24) was obtained first by Ma and Lee.$^{11}$
\section{Reinvestigation of Inhomogeneous Superconductors}
Now we need to reinvestigate the inhomogeneous superconductors
studied by Gor'kov's formalism or the Bogoliubov-de Gennes
equations.
In particular, Gor'kov's microscopic derivation$^{12}$ of the
Gizburg-Landau equation is not valid. The gradient term
may not be derived microscopically.
To tackle the inhomogeneous problems, we must choose a correct
pairing and calculate the correct kernel for each
problem. Then the BCS theory may be more easy to apply.
It is not clear whether the pair potential is the more
appropriate quantity than the gap parameter in inhomogeneous
superconductors.
The following problems are required to restudy:
1. Dirty superconductors and localization
2. Magnetic impurities
3. Proximity effect, Andreev reflection and Josephson effect
4. Mesoscopic superconductivity
5. Type II superconductors, vortex problem
6. Non-equilibrium superconductivity
7. High $T_{c}$ superconductors, heavy fermion superconductors
\subsection{Theory of Dirty Superconductors}
Table I lists the theories of impure superconductors.
Notice that Suhl and Matthias$^{20}$ and Abrikosov-Gor'kov$^{3}$
theories for $\Delta T_{c}$ are essentially equivalent.
Both theories
over-estimate the change of the density of states caused by impurity scattering,
because they apply a retardation cutoff to
the energy of the plane wave states and not of the scattered states.
\vskip 4pt
\centerline{{\bf TABLE I}. Theories of impure superconductors}
\vskip 4pt
\begin{tabular}{lll}\hline\hline
& Ordinary impurity & Magnetic impurity\\ \hline
Anderson & $T_{c} =T_{co}$ \qquad & \\
AG & $T_{c} = T_{co}-{T_{co}\over \pi\omega_{D}\tau}({1\over \lambda}+{1\over 2})$ \qquad & $T_{c}=T_{co}-{\pi\over 4}{1\over \tau_{s}}$\\
Suhl and Matthias & $T_{c}\cong T_{co}-{T_{co}\over \lambda\omega_{D}\tau}$ \qquad &
$T_{c}=T_{co}-{\pi\over 3.5}{1\over \tau_{s}}$\\
Baltensperger & \qquad & $T_{c}=T_{co}-{\pi\over 4}{1\over \tau_{s}}$\\
Tsuneto & $T_{c}=T_{co}$ \qquad & \\
KO & $T_{c}=T_{co}-{T_{co}\over \pi\lambda E_{F}\tau}$ \qquad & $T_{c}
=T_{co}-{0.18\pi\over \lambda\tau_{s}}$\cr
& \qquad $-$ localization correction (Kim) \qquad & \\ \hline\hline
\end{tabular}
\vskip 8pt
Now we consider strong coupling theories of dirty
superconductors.
Tsuneto$^{6}$ obtained the gap equation
\begin{eqnarray}
\Sigma_{2}(\omega) = {i\over (2\pi)^{3}p_{o}}\int dq\int d\epsilon \int d\omega '
{qD(q,\omega-\omega')\eta(\omega')\Sigma_{2}(\omega')\over \epsilon^{2} -
\eta^{2}(\omega')\omega'^{2}},
\end{eqnarray}
where $\eta=1 +{1\over 2\tau|\omega|}$, and $\tau$ is the collision time.
On the other hand, Kim$^{8}$ obtained a gap equation
\begin{eqnarray}
\Delta^{*}(\omega_{n}, m) =
\sum_{n'}\lambda(\omega_{n}-\omega_{n'})
\sum_{m'}V_{mm'}{\Delta^{*}(\omega_{n'},m')\over [-i\omega_{n'}
-\epsilon_{m'}][i\omega_{n'}-\epsilon_{m'}]},
\end{eqnarray}
where
\begin{eqnarray}
V_{mm'} = g^{2}\int |\psi_{m}({\bf r})|^{2}
|\psi_{m'}({\bf r})|^{2}d{\bf r}.
\end{eqnarray}
Comparing Eqs. (25) and (26), we find that Tsuneto's result misses
the most important factor $V_{mm'}$, which gives the change of the
phonon-mediated interaction due to impurities.
This factor is exponentially small for the localized
states.
\subsection{Suppression of Magnetic Impurity Effect by Ordinary
Impurities}
KO's results$^{7}$ can be explained physically.
People used to notice that the impurity effect is stronger
in the low $T_{c}$ material than in the high $T_{c}$ cuprate
materials. Because the size of Cooper pair is very much smaller in high $T_{c}$
material, it sees a very small number of the impurities and
so is not much influenced.
Consequently, $T_{c}$ change and its initial slope depend on
the material, which is predicted by KO.
However, the AG theory predicts the universal slope.
Because the AG theory pairs the states, which are the superpositions of the
normal states of the material, the very nature of the
material does not play an important role.
When the conduction electrons have a mean free path that is smaller than
the size of the Cooper pair (for a pure superconductor),
the effective size is reduced. Accordingly, if we add
ordinary impurities to the superconductors with
the magnetic impurities enough to reduce the size of the Cooper pair,
the magnetic impurity effect is partially suppressed.
This compensation phenomena has been confirmed in several
experiments.$^{13-15}$
The notion of gapless superconductor is based on the misunderstanding
of the relation between the pair potential and the gap parameter.
>From Eq. (23), it is clear that the pair potential cannot be
finite when the gap parameter is zero.
It seems that the long range order between the magnetic
impurities or conduction electrons (especially in lead) caused
the gaplesslike behavior in the experiments.
Then it would rather be called zero gap superconductors or
superconductors with nodes.
\subsection{Weak Localization Correction to the Phonon-mediated
Interaction}
For the strongly localized states, the phonon-mediated interaction
is exponentially small like the conductance.
It is then expected that the same weak localization correction terms may occur
in both quantities.
Using the wavefunction obtained by Mott and Kaveh,$^{21}$
it can be shown that$^{11}$
\begin{eqnarray}
V_{nn'}^{3d} &\cong&
-V[1-{1\over (k_{F}\ell)^{2}}(1-{\ell\over L})],\nonumber \\
V_{nn'}^{2d} &\cong&
-V[1-{2\over \pi k_{F}\ell}ln(L/\ell)],\nonumber \\
V_{nn'}^{1d} &\cong&
-V[1-{1\over (\pi k_{F}a)^{2}}(L/\ell-1)],
\end{eqnarray}
where $a$ is the radius of the wire.
There are many experimental results which show the reduction of $T_{c}$
caused by weak localization.$^{22,23}$
Previously, it was interpreted by the enhanced Coulomb repulsion.
However, Dynes et al.$^{23}$ found a decrease of the Coulomb
pseudo-potential $\mu^{*}$ with decreasing $T_{c}$.
We believe that this signals the importance of weak localization
correction to the phonon-mediated interaction.
\subsection{Other Inhomogeneous Superconductors}
We call attention to a few remarks against the conventional
real space formalism.
``By the use of the Gor'kov technique, Abrikosov and Gor'kov have succeeded
in $\cdots$,
\quad it is {\sl entirely incorrect} as far as any physical results are concerned."$^{19}$
``Certain {\sl inconsistencies} are seen to develop with the use of
this approach (the de Gennes-Werthamer model), calling into
question previous results obtained."$^{24}$
``Is the discrepancy $\cdots$ indicative of the
certain {\sl fundamental inconsistencies} in the Green's function
formulation of the theory of nonstationary superconductivity?"$^{25}$
\section{Conclusions}
It is shown that Gor'kov's formalism and
the Bogoliubov-de Gennes equations need a pairing
constraint.
The resulting self-consistency equation is nothing but
another form of the BCS gap equation.
Most inhomogeneous superconductors should be reinvestigated.
I am grateful to Professor A. W. Overhauser for discussions.
This work was supported by the National Science Foundation, Materials Theory
Program.
| proofpile-arXiv_065-446 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section*{I. Introduction}
\end{flushleft}
The inclusive rare decay $B\rightarrow X_s\gamma$ has been studied several
years before [1]. Recently the physics of $B_c$ meson has caught intensive
attentions [2]. The $B_c$ meson is believed to be the next and the final
family of $B$ mesons. It provides unique opportunity to examine various
heavy quark fragmentation models, heavy quark spin-flavor symmetry, different
quarkonium bound state models and properties of inclusive decay channels.
Being made of two heavy quarks of different flavors, $B_c$ radiative weak
decay also offer a rich source to measure elements of Cabibbo-Kobayashi-
Maskawa ( CKM ) matrix of the standard
model ( SM ).
Different from the general rare $B$ decays $B\rightarrow X_s\gamma$ which is
mainly induced by the flavor-changing $b\rightarrow s\gamma$ neutral currents
[3], in $B_c\rightarrow D_s^*\gamma$, the bound state effects could
seriously modify the results from the assumption. Bound state effects include
modifications from weak annihilation which involve no neutral flavor-changing
currents at all. The effects of weak annihilation mechanism are expected
large due to the large CKM amplitude.
Unfortunately, the well known chiral-symmetry [4] and the heavy quark
symmetry [5] can not be applied to $B_c\rightarrow D_s^*\gamma$ process.
Recently, a perturbative
QCD ( PQCD ) analysis of $B$ meson decays seems to give a good prediction
[6]. As it is argued in Ref.[7], $B_c$ two body nonleptonic decay can be
conveniently studied within the framework of PQCD suggested by Brodsky-Lepage
[8] and then developed in Ref.[6]. Here, we summarize their idea:
In the subprocess $b\rightarrow s\gamma$,
s quark obtains a large momentum by recoiling, in order to form a bound state
with the spectator $\overline c$ quark, the most momentum of s must be
transferred to $\overline c$ by a hard scattering process. In the final
bound state ( i.e. $D_s^*$ ), since the heavy charm should share the most
momentum of $D_s^*$, the hard scattering is suitable for PQCD calculation
[6, 8].
In $B_c\rightarrow D_s^* \gamma$, the
subprocess $b\rightarrow s\gamma$, taken as a free decay, is usually
controlled by the one-loop electromagnetic penguin diagrams which are in
particular sensitive to contributions from those new physics beyond the SM.
This situation is similar with $B\rightarrow K^*\gamma$.
Recently, the modification of the electromagnetic penguin interaction
to the decay $b\rightarrow s\gamma$ from PGBs in the one generation
technicolor model
( OGTM ) has been estimated in Ref.[9]. In the recent literatures [10, 11],
the technicolor ( TC ) models with scalars have been
studied extensively. The phenomenological studies have shown that the TC
models with scalars do not generate unacceptably large FCNCs and are
consistent with the
experimental constraints on oblique electroweak radiative corrections.
Among these models, the TC with a massless scalar doublet model ( TCMLSM )
[10], presented by C. D. Carone and H. Georgi
is the simplest nontrivial extension of the SM with only two new free
parameters
( $h_+$, $h_-$ or $h$, $\lambda$ ). The phenomenology of the TCMLSM has
been discussed in the literatures [11]. In this paper we express these two
new free parameters as $m_{\pi_p}$ and $\frac{f}{f'}$ differently from Ref.
[10]. The relation of them can be found in Ref.[10], where $m_{\pi_p}$ is the
mass of $\pi_p$ ( the physical pions in this model ), $f$ and $f'$ are the
technipion decay constant and scalar vacuum expectation value ( VEV ),
respectively. The couplings of charged $\pi_p$ with ordinary fermions are
given as
$$
[\pi_p^+--u_i--d_j]=-\frac{\sqrt2}{2}V_{u_id_j}\frac{f}{vf'}[m_{d_j}(1
+\gamma_5)-m_{u_i}(1-\gamma_5)],
\eqno{(1)}
$$
$$
[\pi_p^---u_i--d_j]=-\frac{\sqrt2}{2}V_{u_id_j}\frac{f}{vf'}[m_{u_i}(1
+\gamma_5)-m_{d_j}(1-\gamma_5)],
\eqno{(2)}
$$
where u=( u, c, t ), d=( d, s, b ),
$V_{u_id_j}$ is the
element of CKM matrix, and $v\sim$ 250 GeV is the electroweak scale.
In this paper, applying the above PQCD method, we address $B_c\rightarrow
D_s^*\gamma$ in the TCMLSM to examine the virtual effects of
$\pi_p$ in the TCMLSM and compare the results with which estimated in the
SM.
This paper is organized as follows: In Sec.II, we display our calculations
in the SM and TCMLSM. We present the final numerical results in Sec.III.
Sec.IV contains the discussion.
\begin{flushleft}
\section*{II. Calculation}
\end{flushleft}
Using the factorization scheme [8] within PQCD, the momentum of quarks
are taken as some fractions $x$ of the total momentum of the meson weighted
by a soft physics distribution functions $\Phi_{H}(x)$. The meson wave
functions of $B_c$ and $D_s^*$ take the simple form of $\delta$ function (
the so-called peaking approximation ) [12, 13]:
$$
\Phi_{B_c}(x)=\frac{f_{B_c}\delta(x-\epsilon_{B_c})}{2\sqrt{3}},
\eqno{(3)}
$$
$$
\Phi_{D_s^{*}}(y)=\frac{f_{D_s^{*}}\delta(y-\epsilon_{D_s^{*}})}{2\sqrt{3}},
\eqno{(4)}
$$
The normalization [13] is
$$
\int_{0}^{1} dx\Phi_{B_c}(x)=\frac{f_{B_c}}{2\sqrt{3}},
\eqno{(5)}
$$
$$
\int_{0}^{1} dy\Phi_{D_s^*}(y)=\frac{f_{D_s^*}}{2\sqrt{3}},
\eqno{(6)}
$$
where $x$, $y$ denote the momentum fractions of $c$, $s$ quarks in the $B_c$
and $D_s^*$ mesons, $f_{B_c}$ and $f_{D_s^*}$ are decay constants of $B_c$
and $D_s^*$ respectively,
$$
\epsilon_{B_c}=\frac{m_c}{m_{B_c}},
\eqno{(7)}
$$
$$
\epsilon_{D_s^*}=\frac{m_{D_s^*}-m_c}{m_{D_s^*}}.
\eqno{(8)}
$$
The spinor part of $B_c$ and $D_s^{*}$ [14] are
$$
\frac{(\not p+m_{B_c})\gamma_5}{\sqrt{2}},
\eqno{(9)}
$$
$$
\frac{(\not q-m_{D_s^*})\not \epsilon}{\sqrt{2}},
\eqno{(10)}
$$
which come from the matrix structures of $B_c$ and $D_s^*$ meson wave
functions, while $p$ and $q$ are the momenta of the $B_c$ and $D_s^*$
respectively, and $\epsilon$ is the polarization vector of $D_s^{*}$.
\begin{flushleft}
\subsection*{II ( i ). Electromagnetic penguin contribution}
\end{flushleft}
The relevant Feynman diagrams which contribute to the short distance
electromagnetic penguin process $b\rightarrow s\gamma$ are illustrated as
the blob of Fig.1.
In the evaluation, we at first integrate out the top quark and the weak $W$
bosons at $\mu=m_W$ scale, generating an effective five-quark theory. By
using the renormalization group equation, we run the effective field theory
down to b-quark scale to give the leading log QCD corrections.
After applying the full QCD equations of motion [15], a complete set of
dimension-6 operators relevant for $b\rightarrow s\gamma$ decay can be
chosen as $O_1$ - $O_8$, which have been given in the Refs.[1, 9]. The
effective Hamiltonian appears at the $W$ scale is given as
$$
H_{eff}=\frac{4G_F}{\sqrt{2}}V_{tb}V_{ts}^* \sum\limits_{i=1}\limits^{8}C_i
(m_W)O_i(m_W).
\eqno{(11)}
$$
The coefficients of 8 operators are:
$$
C_i(m_W) = 0, i= 1, 3, 4, 5, 6, C_2(m_W) = -1,
\eqno{(12)}
$$
$$
C_7(m_W) =\frac{1}{2}A(x)-(\frac{f}{f'})^2[B(y)-\frac{1}{6}A(y)],
\eqno{(13)}
$$
$$
C_8(m_W)=\frac{1}{2}D(x)+(\frac{f}{f'})^2[\frac{1}{6}D(y)-E(y)],
\eqno{(14)}
$$
where functions $A$, $B$, $D$ and $E$ are defined in the Ref.[1],
$x=(\frac{m_t}{m_W})^2$, $y=(\frac{m_t}{m_{\pi_p}})^2$.
The running of the coefficients of operators from $\mu=m_W$ to $\mu= m_b$
was well described in Ref.[16]. After renormalization group running we
have the QCD corrected coefficients of operators at $\mu= m_b$ scale.
$$
C_7^{eff}(m_b) =\varrho^{-\frac{16}{23}}C_7(m_W)+\frac{8}{3}
(\varrho^{-\frac{14}{23}}-\varrho^{-\frac{16}{23}})C_8(m_W)+C_2(m_W)\sum
\limits_{i=1}\limits^{8}h_i
\varrho^{-a_i},
\eqno{(15)}
$$
with
$$
\varrho = \frac{\alpha_s(m_b)}{\alpha_s(m_W)},
\eqno{(16)}
$$
$$
h_i=(\frac{626126}{272277}, -\frac{56281}{51730}, -\frac{3}{7}, -\frac{1}{14}
, -0.6494, -0.0380, -0.0186, -0.0057),
\eqno{(17)}
$$
$$
a_i=(\frac{14}{23}, \frac{16}{23}, \frac{6}{23}, -\frac{12}{23}, 0.4086,
-0.4230, -0.8994, 0.1456).
\eqno{(18)}
$$
Now we write down the amplitude of Fig.1 as
$$
\begin{array}{ll}
M_a=&\int^1_0 dx_1 dy_1 \Phi_{D_s^*}(y_1)\Phi_{B_c}(x_1)\frac{-iG_F}
{\sqrt{2}}V_{tb}
V_{ ts}^*C_7^{eff}(m_b)m_be\frac{\alpha_s(m_b)}{2\pi}C_F \\
&\{T_r[(\not q-m_{D_s}^{*})\not\epsilon\sigma_{\mu\nu}(1+\gamma_5)
k^{\nu}\eta^{\mu}(\not p-y_1\not q+m_b)\gamma_{\alpha}(\not p+m_{B_c})
\gamma_5\gamma^{\alpha}]\frac{1}{D_1D_3} \\
&+Tr[(\not q-m_{D_s^{*}})\not\epsilon\gamma_{\alpha}(\not q-x_1\not p)
\sigma_{\mu\nu}(1+\gamma_5)k^{\nu}\eta^{\mu}(\not p+m_{B_c})
\gamma_5\gamma^{\alpha}]\frac{1}{D_2D_3}\},
\end{array}
\eqno{(19)}
$$
where $\eta$ is the polarization vector of photon, $x_1$, $y_1$ are the
momentum fractions shared by charms in $B_c$ and $D_s^*$ respectively, and
the color factor $C_F$ = $\frac{4}{3}$. The factors $D_1$, $D_2$ and $D_3$
in eq.(19) are the forms of
$$
\begin{array}{l}
D_1=(1-y_1)(m_{ B_c}^{ 2}-m_{D_s^*}^2y_1)-m_{ b}^{ 2},{\hskip 8.5cm} (20)\\
D_2=(1-x_1)(m_{ D_s^*}^2-m_{B_c}^2 x_1),{\hskip 9.5cm}(21) \\
D_3=(x_1-y_1)(x_1m_{B_c}^{ 2}-y_1m_{D_s^*}^{ 2}).{\hskip 9cm}(22)
\end{array}
$$
Now the amplitude $M_a$ can be written as
$$
M_a=i\varepsilon_{\mu\nu\alpha\beta}\eta^{\mu}k^{\nu}\epsilon^{\alpha}
p^{\beta}f_{ 1}^{ peng} +\eta^{\mu}[\epsilon_{\mu}(m_{B_c}^2-m_{D_s}^2)
-(p+q)_{\mu}(\epsilon\cdot k)]f_2^{ peng},
\eqno{(23)}
$$
with form factors
$$
\begin{array}{ll}
f_1^{peng}&=2f_2^{peng}=C\int_{0}^{1}dx_1dy_1\delta(x_1-\epsilon_{B_c})
\delta(y_1-\epsilon_{D_s^*}) \\
&\{[m_{B_c}(1-y_1)(m_{B_c}-2m_{D_s^*})
-m_b(2m_{B_c}-m_{D_s^*})] {\frac{1}{D_1D_3}} - m_{B_c}m_{D_s^*}(1-x_1)
{\frac{1}{D_2D_3}}\},
\end{array}
\eqno{(24)}
$$
where
$$
C=\frac{em_bf_{B_c}f_{D_s^*}C_7^{eff}(m_b)C_F\alpha_s(m_b)G_FV_{tb}
V_{ts}^*}{12\pi\sqrt{2}}.
\eqno{(25)}
$$
\begin{flushleft}
\subsection*{II ( ii ). The weak annihilation contribution}
\end{flushleft}
As mentioned in Sec.I, $B_c$ meson is also the unique probe of
the weak annihilation mechanism. The leading log QCD-corrected effective weak
Hamiltonian of $W$ annihilation is
$$
H^{(W)}_{eff}= \frac{G_F}{2\sqrt{2}}V_{cb}V_{cs}^*( c_+O_+ + c_-O_- ) + H.c,
\eqno{(26)}
$$
with $O_{\pm}=(\overline sb)(\overline cc)\pm(\overline sc)(\overline cb)$,
where $(\overline q_1q_2)
\equiv \overline q_1\gamma_{\mu}(1-\gamma_5)q_2$, $c_{\pm}$ are Wilson
coefficient functions.
Using the method developed by H. Y. Cheng $et \ \ al.$ [17], we can get the
amplitude of $W$ annihilation diagrams ( see Fig.2 ):
$$
M_{ b}^{(W)}=i\varepsilon_{\mu\nu\alpha\beta}\eta^{\mu}k^{\nu}\epsilon^
{\alpha}
p^{\beta}f_{1(W)}^{anni}+\eta^{\mu}[\epsilon_{\mu}(m_{B_c}^2-m_{D_s}^2)-
(p+q)_{\mu}(\epsilon\cdot k)]f_{2(W)}^{anni},
\eqno{(27)}
$$
with
$$
f_{1(W)}^{anni}=2\zeta[(\frac{e_s}{m_s}+\frac{e_c}{m_c})
\frac{m_{D_s^*}}{m_{B_c}}+(\frac{e_c}{m_c}+\frac{e_b}{m_b})]
\frac{m_{D_s^*}m_{B_c}}{m_{B_c}^2-m_{D_s^*}^2},
\eqno{(28)}
$$
$$
f_{2(W)}^{anni}=-\zeta[(\frac{e_s}{m_s}-\frac{e_c}{m_c})\frac{m_{D_s^*}}
{m_{B_c}}
+(\frac{e_c}{m_c}-\frac{e_b}{m_b})]\frac{m_{D_s^*}m_{B_c}}
{m_{B_c}^2-m_{D_s^*}^2},
\eqno{(29)}
$$
where
$$
\zeta=ea_2\frac{G_F}{\sqrt{2}}V_{cb}V_{cs}^*f_{B_c}f_{D_s^*},
\eqno{(30)}
$$
and $a_2=\frac{1}{2}(c_--c_+)$ is a calculable coefficient in the nonleptonic
$B$ decays.
Using the Feynman rules given in eq. ( 1 ) and eq. ( 2 ), the leading log
QCD-corrected effective weak Hamiltonian
of $\pi_p^{\pm}$ annihilation is given as
$$
H^{(\pi_p)}_{eff}=-V_{cb}V_{cs}^*(\frac{f}{vf'})^2 \frac{1}{2m^2_{\pi_P}}(
c_+ O_+ + c_-O_-) + H.c.
\eqno{(31)}
$$
Using the same method as the above, we can write down
the amplitude of $\pi_p^{\pm}$ annihilation diagrams ( see Fig.2 ) as
$$
M_b^{(\pi_p)}=i\varepsilon_{\mu\nu\alpha\beta}\eta^{\mu}k^{\nu}\epsilon^
{\alpha}p^{\beta}f_{1(\pi_p)}^{anni} + \eta^{\mu}[\epsilon_{\mu}(m_{B_c}^2
-m_{D_s^*}^2)-(p+q)_{\mu}(\epsilon \cdot k)]f_{ 2(\pi_p)}^{ anni},
\eqno{(32)}
$$
with
$$
f_{1(\pi_p)}^{anni}=2\zeta^{'}[(\frac{e_s}{m_s}+\frac{e_c}{m_c})
\frac{m_s-m_c}{m_{B_c}}+(\frac{e_b}{m_b}+\frac{e_c}{m_c})
\frac{m_b-m_c}{m_{B_c}}]\frac{m_{B_c}m_{D_s^*}}{m_{B_c}^2- m_{D_s^*}^2},
\eqno{(33)}
$$
$$
f_{2(\pi_p)}^{ anni}=\zeta^{'}[(\frac{e_s}{m_s}+\frac{e_c}{m_c})
\frac{m_{D_s^*}}{m_{B_c}}+(\frac{e_b}{m_b}+\frac{e_c}{m_c})
]\frac{m_{D_s^{*}}m_{B_c}}{m_{ B_c}^{ 2}
-m_{ D_s^{*}}^{ 2}},
\eqno{(34)}
$$
where
$$
\zeta^{'}=ea_2V_{cb}V_{cs}^*(\frac{f}{vf'})^2\frac{1}{4m_{\pi_p}^2}f_{B_c}
f_{D_s^*}(m_{B_c}^2+m_{D_s^*}^2).
\eqno{(35)}
$$
The total annihilation amplitude ( Fig.2 ) is the form of
$$
\begin{array}{lll}
M_b&=&M_b^{(W)}+M_b^{(\pi_p)} \\
&=&i\varepsilon_{\mu\nu\alpha\beta}\eta^{\mu}k^{\nu}\epsilon^{\alpha}
p^{\beta}f_1^{anni}+\eta^{\mu}[\epsilon_{\mu}(m_{B_c}^2-m_{D_s}^2)
-(p+q)_{\mu}(\epsilon \cdot k)]f_2^{anni},
\end{array}
\eqno{(36)}
$$
where
$$
f_1^{anni} = f_{1(W)}^{anni} +f_{1(\pi_p)}^{anni},
\eqno{(37)}
$$
$$
f_2^{anni} = f_{2(W)}^{anni} +f_{2(\pi_p)}^{anni}.
\eqno{(38)}
$$
Finally, we estimate another possible long-distance effect, namely the vector
-meson-dominance ( VMD ) contribution which was advocated by Golowich and
Pakvasa [18]. VMD implies that a possible contribution to $B_c\rightarrow
D_s^*\gamma$ comes from the $B_c\rightarrow D_s^*J/\psi(\psi')$ followed by
$J/\psi(\psi')\rightarrow\gamma$ conversion.
As discussed in Refs.[19, 20], using factorization approach,
the VMD amplitude ( Fig.3 ) is
$$
M^{VMD}=i\varepsilon_{\mu\nu\alpha\beta}\eta^{\mu}k^{\nu}\epsilon^
{\alpha}
p^{\beta}f_1^{VMD}+\eta^{\mu}[\epsilon_{\mu}(m_{B_c}^2-m_{D_s}^2)-
(p+q)_{\mu}(\epsilon\cdot k)]f_2^{VMD},
\eqno{(39)}
$$
with
$$
\begin{array}{ll}
f_1^{VMD}&=eG_FV_{cb}V_{cs}^* \{\sqrt{2}a_2\frac{1}{m_{B_c}+m_{D_s^*}}
(\frac{f_{J/\psi}m_{J/\psi}}{g_{\gamma J/\psi}}+\frac{f_{\psi'}m_{\psi'}}
{g_{\gamma\psi'}})\\
& + a_1f_{D_s^*}m_{D_s^*}[\frac{1}{(m_{B_c}+m_{J/\psi})
g_{\gamma J/\psi}}+\frac{1}{(m_{B_c}+m_{\psi'})g_{\gamma\psi'}}]\} V^{BD}(0),
\end{array}
\eqno{(40)}
$$
$$
f_2^{VMD}=-\frac{1}{2}\frac{A_2^{BD}(0)}{V^{BD}(0)}f_1^{VMD},
\eqno{(41)}
$$
where $a_1=\frac{1}{2}(c_++c_-)$, and $V^{BD}(0)$ and $A_2^{BD}(0)$ are
form factors.
\begin{flushleft}
\section*{III. Numerical results}
\end{flushleft}
We will use the following values for various quantities to carry on our
calculations.
( i ). Decay constants for mesons. Here we use
$$
f_{D_s^*}=f_{D_s}=344 MeV [21], \ \ f_{B_c}=500 MeV [22], \ \ f_{J/\psi}=395
MeV [20], \ \ f_{\psi'}=293 MeV [20].
$$
( ii ). Meson mass and the constituent quark mass [23, 24]
$$
m_{B_c}=6.27 GeV, \ \ m_{D_s^*}=2.11 GeV, \ \ m_b=4.7 GeV, \ \ m_c=1.6 GeV,
$$
$$
m_s=0.51 GeV, \ \ m_{J/\psi}=3.079 GeV, \ \ m_{\psi'}=3.685 GeV.
$$
We also use $m_{B_c}\approx m_b+m_c$, $m_{D_s^*}\approx m_s+m_c$ in our
calculations.
( iii ). $a_1$ and $a_2$ have been estimated very recently in Ref.[17]
according to the CLEO data [25] on $B\rightarrow D^* \pi(\rho)$ and
$B\rightarrow J/\psi K^*$. Here we take
$$
a_1=1.01, \ \ a_2=0.21.
$$
( iv ). CKM matrix elements [24]. Here we use
$$
\vert V_{cb}\vert=0.04, \ \ \vert V_{ts}\vert=\vert V_{cb}\vert, \ \ \vert
V_{cs}\vert=0.9745, \ \ \vert V_{tb}\vert=0.9991.
$$
( v ). The QCD coupling constant $\alpha_s(\mu)$ at any renormalization
scale, can be calculated from $\alpha_s(m_Z)=0.117$ via
$$
\alpha_s(\mu)=\frac{\alpha_s(m_Z)}{1-(11-\frac{2}{3}n_f)\frac{\alpha_s(m_Z)}
{2\pi}\ln(\frac{m_Z}{\mu})}.
$$
We obtain
$$
\alpha_s (m_b)=0.203, \ \ \ \alpha_s(m_W)=0.119.
$$
( vi ). The Ref.[10] gives a constraint on $m_{\pi_p}$ in the
allowed parameter space of the model: $\frac{1}{2}m_Z<m_{\pi_p}<1 TeV$.
Here we take
$$
m_{\pi_p}=( 50\sim 1000 ) GeV.
$$
( vii ). With the constraint of $f^2+f^{'2}=v^2$ and the
chiral perturbation theory in Ref.[10], we can get $0.115\leq \frac{f}{f'}
\leq 1.74$. Here we take
$$
\frac{f}{f'}=0.115
$$
in our calculations.
( viii ). The form factors $V(0)$ and $A_2(0)$ appearing in the
two-body decays of $B$. From Ref.[26], we take
$$
V^{BD}(0)=0.30, \ \ A_2^{BD}(0)=0.20.
$$
We present the form factors $f_i$ ( $f_1^{peng}$, $f_2^{peng}$, $f_1^{anni}$,
$f_2^{anni}$ ) in the SM and TCMLSM in
Table 1, so do the decay widths in Table 2 using the amplitude formula
$$
\Gamma(B_c\rightarrow D_s^*\gamma) = \frac{(m_{B_c}^2-m_{D_s^*}^2)^3}
{32\pi m_{B_c}^3}(f_1^2 +4f_2^2).
$$
The calculated results indicate that the VMD effects are large near the pole
and which can not be neglected. The calculated results are
$$
f_1^{VMD}=6.73\times 10^{-10}, \ \ f_2^{VMD}=-2.24\times 10^{-10},
$$
$$
\Gamma^{VMD}=1.12\times 10^{-18} GeV.
$$
The lifetime of $B_c$ is given in Ref.[27]. In this paper we use
$$
\tau_{B_c}=( 0.4 ps\sim 1.35 ps )
$$
to estimate the branching ratio BR ( $B_c\rightarrow D_s^*\gamma$ ) which
is a function of $\tau_{B_c}$. The results are given in Table 3.
\begin{flushleft}
\section*{IV. Discussion }
\end{flushleft}
Applying PQCD, we have studied two mechanisms which contribute to the
process $B_c\rightarrow D_s^*\gamma$. For the short-distance one ( Fig.1 )
induced by electromagnetic penguin diagrams,
the momentum square of the hard scattering exchanged by gluon is about
$3.6 GeV^2$ which is large enough for PQCD analyzing. The hard scattering
process can not
be included conveniently in the soft hadronic process described by the
wave-function of the final bound state, which is one important reason that we
can not apply the commonly used models with spectator [28] to the two body
$B_c$ decays. There is no phase-space for the propagators appearing in
Fig.1 to go on-shell, consequently, unlike the situation in the Ref.[6], the
imaginary part of $M_a$ is absent. Another competitive mechanism is the weak
annihilation. In the SM, we
find this mechanism is as important as the former one ( they can
contribute with the same order of magnitude ). This situation
is distinct from that of the
radiative weak $B^{\pm}$ decays which is overwhelmingly dominated by
electromagnetic penguin.
This is due to two reasons: one is that the compact
size of $B_c$ meson enhances the importance of annihilation decays; the
other comes from the Cabibbo allowance: in
$ B_c \rightarrow D_s^* \gamma$, the CKM amplitude of weak
annihilation is $\vert V_{cb}V_{cs}^*\vert $, but in $ B^{\pm}
\rightarrow K^{\pm}\gamma$, the CKM part is $ \vert V_{ub}V_{us}^* \vert $
which is much smaller than $\vert V_{cb}V_{cs}^*\vert$.
Particularly, the VMD contribution is found not small. This situation is
quite different from the cases $B\rightarrow J/\psi K(K^*)$ and $B\rightarrow
J/\psi\rho$ [19, 20]. The reason comes from that although the coupling of
$J/\psi(\psi') - \gamma$ is small ( $e/g_{\gamma J/\psi(\psi')}\approx
0.025(0.016)$ ), the $J/\psi(\psi')$ resonance effect can be very large.
In addition, we find that the modification of $B_c\rightarrow D_s^* \gamma $
from $\pi _p$ in the TCMLSM is small for the allowed range of
mass of $\pi_p$ ( with $\frac{f}{f'}$
fixed ). This situation is quite different from that of Ref.[9], in which
the size of contribution to the rare decay of $b\rightarrow s\gamma$ from
the PGBs strongly depends on the values of the masses of the charged PGBs.
The difference is mainly due to the small value of $\frac{f}{f'}$
which leads to the small modification from $\pi_p$ in the TCMLSM. However,
in the OGTM, such suppression factor $\frac{f}{f'}$ does not exist. In our
calculations, we take $\frac{f}{f'}$=0.115 as the input parameter. When
$\frac{f}{f'}$ is taken properly larger ( without exceeding the constraint:
$0.115\leq \frac{f}{f'} \leq 1.74$ ), the calculated results remain unchanged
basically. In view of the above
situation, it seems to indicate that the window of process $B_c\rightarrow
D_s^*\gamma$ is close for the TCMLSM. But in our
calculations, besides the peaking approximation of the meson wave functions,
the theoretical uncertainties are neglected, such as that of $\alpha ( m_Z )
$, next-to-leading log QCD contribution [29], QCD correction from $m_t$ to
$m_W$ [30], etc. When the more reliable estimation
is available within the next few years, one can, in principle, make
the final decision whether the window for TCMLSM is open or close.
\vspace{1cm}
\noindent {\bf ACKNOWLEDGMENT}
This work was supported in part by the National Natural Science
Foundation of China, and by the funds from
Henan Science and Technology Committee.
\vspace{1cm}
\begin {center}
{\bf Reference}
\end {center}
\begin{enumerate}
\item
B. Grinstein $et \ \ al.$, Nucl. Phys. B 339 ( 1990 ) 269.
\item
D. S. Du and Z. Wang, Phys. Rev. D 39 ( 1989 ) 1342; K. Cheng, T. C. Yuan,
Phys. Lett. B 325 ( 1994 ) 481, Phys. Rev. D 48 ( 1994 ) 5049; G.R. Lu,
Y.D. Yang and H.B. Li, Phys. Lett. B341 (1995)391, Phys. Rev. D51 (1995)2201.
\item
J. Tang, J. H. Liu and K. T. Chao, Phys. Rev. D51 (1995)3501; K. C.
Bowler $ et \ \ al.$, Phys. Rev. Lett. 72 ( 1994 ) 1398.
\item
H. Leutwyler and M. Roos, Z. Phys. C 25 ( 1984 ) 91.
\item
M. Neubert, Phys. Rep. 245 ( 1994 ) 1398.
\item
A. Szczepaiak $et \ \ al.$, Phys. Lett. B 243 ( 1990 ) 287; C. E. Carlson
and
J. Milana, Phys. Lett. B 301 ( 1993 ) 237, Phys. Rev. D 49 ( 1994 ) 5908,
$ibid$ 51 ( 95 ) 450; H-n. Li and
H. L. Yu, Phys. Rev. Lett. 74 ( 1995 ) 4388.
\item
D. S. Du, $et \ \ al.$, BIHEP-TH-95-38 ( hep-ph/9603291 )
\item
S. J. Brodsky and G. P. Lepage, Phys. Rev. D 22 ( 1980 ) 2157.
\item
Cai-Dian L\"u and Zhenjun Xiao, Phys. Rev. D 53 ( 1996 ) 2529.
\item
C. D. Carone and H. Georgi, Phys. Rev. D 49 ( 1994 ) 1427.
\item
C. D. Carone and E. H. Simmons, Nucl. Phys. B 397 ( 1993 ) 591;
E. H. Simmons, Nucl. Phys. B 312 ( 1989 ) 253; S. Samuel, $ibid$,
B 347 ( 1990 ) 625; A. Kagan and S. Samuel, Phys. Lett.
B 252 ( 1990 ) 605, B 270 ( 1991 ) 37.
\item
S. J. Brodsky and C. R. Ji, Phys. Rev. Lett. 55 ( 1985 ) 2257.
\item
Hsiang-nan Li, Phys. Rev. D 52, ( 1995 ) 3958.
\item
C. E. Carlson, J. Milana, Phys. Rev. D 51, ( 1995 ) 4950.
\item
H. D. Politzer, Nucl. Phys. B 172 ( 1980 ) 349; H. Simma, Preprint, DESY
93-083.
\item
M. Misiak, Phys. Lett. B 269 ( 1991 ) 161; K. Adel, Y. P. Yao, Mod. Phys.
Lett. A 8 ( 1993 ) 1679; Phys. Rev. D 49 ( 1994 ) 4945; M. Ciuchini $ et
\ \ al.$, Phys. Lett. B 316 ( 1993 ) 127.
\item
H. Y. Cheng $et \ \ al.$, Phys. Rev. D 51 ( 1995 ) 1199.
\item
E. Golowich and S. Pakvasa, Phys. Lett. B 205 ( 1988 ) 393.
\item
M. Bauer, B. Stech and M. Wirbel, Z. Phys.C 34 ( 1987 ) 103;
E. Golowich and S. Pakvasa, Phys. Rev. D 51 ( 1995 ) 1215.
\item
H. Y. Cheng, Phys. Rev. D 51 ( 1995 ) 6228.
\item
A. Aoki $et \ \ al.$, Prog. Theor. Phys. 89 ( 1993 ) 137; D. Acosta
$et \ \ al.$, CLNS 93/1238; J. Z. Bai $ et \ \ al.$, BES Collaboration
Phys. Rev. Lett. 74 ( 1995 ) 4599.
\item
W. Buchm\"uiller and S-H. HTye, Phys. Rev. D 24 ( 1994 ) 132; A. Martin,
Phys. Lett. B 93 ( 1980 ) 338; C. Quigg and J. L. Rosner, Phys. Lett. B 71
( 1977 ) 153; E. Eicheten $et \ \ al.$, Phys. Rev. D 17 ( 1978 ) 3090.
\item
W. Kwong and J. L. Rosner, Phys. Rev. D 44 ( 1991 ) 212.
\item
Particle Data Group, L. Montanet $et \ \ al.$, Phys. Rev. D 50
( 1994 ) 1173.
\item
CLEO Collaboration, M. S. Alam $et \ \ al.$, Phys. Rev. D 50 ( 1994 ) 43.
\item
M. Neubert, CERN-TH/96-55 (hep-ph/9604412).
\item
C. Quigg, FERMILAB-Conf-93/265-T; C. H. Chang and Y. Q. Chen,
Phys. Rev. D49 ( 1994 ) 3399; P. Colangelo $et \ \ al.$, Z. Phys. C 57
( 1993 ) 43.
\item
M. Bauer $et \ \ al.$, Z. Phys. C29 ( 1985 ) 637; B. Grinstein $et \ \ al.$,
Phys. Rev. D39 ( 1987 ) 799.
\item
A. J. Buras $et \ \ al.$, Nucl. Phys. B 370 ( 1992 ) 69; Addendum, $ibid$, B
375 ( 1992 ) 501, B 400 ( 1993 ) 37 and B 400 ( 1993 ) 75; M. Ciuchini $et \
\ al.$, Phys. Lett. B 301 ( 1993 ) 263, Nucl. Phys. B 415 ( 1994 ) 403.
\item
C. S. Gao, J. L. Hu, C. D. L\"u and Z. M. Qiu,
Phys. Rev. D52 (1995)3978.
\end{enumerate}
\newpage
\begin{table}[h]
\caption{Form factors in the SM and TCMLSM. $f^{peng}$ and $f^{anni}$
represent form factors through electromagnetic penguin process and through
weak annihilation process respectively.}
\begin{center}
\begin{tabular}{|c|c|c|} \hline
$f_i$ & SM & TCMLSM \\ \hline
$f_1^{peng}$ & $-3.05\times 10^{-10}$ & $(-3.02\sim -3.05)\times 10^{-10}$
\\ \hline
$f_2^{peng}$ & $-1.52\times 10^{-10}$ &$ -1.52\times 10^{-10}$
\\ \hline
$f_1^{anni}$ & $7.10\times 10^{-10}$ & $ 7.10\times10^{-10}$
\\ \hline
$f_2^{anni}$ & $-1.70\times 10^{-10}$ & $ -1.70\times 10^{-10}$
\\ \hline
\end{tabular}
\end{center}
\end {table}
\begin{table}[h]
\caption{The decay rates in the SM and TCMLSM. The $\Gamma^{peng}$, $\Gamma^
{anni}$ and $\Gamma^{total}$ represent $\Gamma$ ( $B_c\rightarrow D_s^*
\gamma$ ) through electromagnetic penguin process, through weak annihilation
process and penguin + annihilation respectively.}
\begin{center}
\begin{tabular}{|c|c|c|} \hline
$\Gamma(B_c \rightarrow D_s^* \gamma)$ & SM & TCMLSM \\ \hline
$\Gamma^{peng}$(GeV) &$ 3.18 \times 10^{-19}$ & $ ( 3.12\sim 3.17 )
\times 10^{-19}$ \\ \hline
$\Gamma^{anni}$(GeV) & $1.06\times 10^{-18}$ & $ 1.06
\times 10^{-18}$ \\ \hline
$\Gamma^{total}$(GeV) & $4.03\times 10^{-18}$ & $ 4.03\times
10^{-18}$ \\ \hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[h]
\caption{The branching ratio ( $B_c\rightarrow D_s^*\gamma$ ). The $BR^{SM}_
{total}$ and $BR^{TCMLSM}_{total}$ represent the branching ratio ( $B_c
\rightarrow D_s^*\gamma$ ) in the SM and TCMLSM respectively.}
\begin{center}
\begin{tabular}{|c|c|c|c|} \hline
$\tau_{B_c} $ & 0.4ps & 1.0ps & 1.35ps \\ \hline
$ BR^{SM}_{total} $ & $2.44\times 10^{-6}$ & $
6.12\times 10^{-6}$ & $8.27\times 10^{-6}$ \\ \hline
$BR^{TCMLSM}_{total}$ &$2.44\times 10^{-6}$
&$6.12\times 10^{-6}$ &$8.27\times 10^{-6}$ \\ \hline
\end{tabular}
\end{center}
\end{table}
\vspace{1cm}
\begin{center}
{\bf Figure captions}
\end{center}
Fig.1: The Feynman diagrams which contribute to the rare radiative
decay $B_c\rightarrow D_s^*\gamma$ through electromagnetic penguin process.
The blob represents the electromagnetic penguin operators contributing to
$b\rightarrow s\gamma$, $x_2 p$ and $x_1 p$ are momenta of b and c quarks in
the $B_c$ meson respectively, $y_2 q$ and $y_1q$ are momenta of s and c
quarks in the $D_s^*$ meson, respectively.
Fig.2: The Feynman diagrams which contribute to the rare radiative
decay $B_c\rightarrow D_s^*\gamma$ through weak annihilation process. In the
SM, there is only $W^{\pm}$ annihilation, in the TCMLSM, there are both
$W^{\pm}$ and $\pi_p^{\pm}$ annihilations.
Fig.3: VMD processes which contribute to $B_c\rightarrow D_s^*\gamma$
with the vector-meson intermediate states $J/\psi(\psi')$.
\newpage
\begin{picture}(30,0)
\setlength{\unitlength}{0.1in}
\put(5,-15){\line(1,0){20}}
\put(5,-20){\line(1,0){20}}
\put(3,-18){$B_c$}
\put(26,-18){$D_s^*$}
\multiput(10,-15.8)(0,-0.8){6}{$\varsigma$}
\multiput(14,-15)(1,1){5}{\line(0,1){1}}
\multiput(13,-15)(1,1){6}{\line(1,0){1}}
\put(14,-15){\circle*{3}}
\put(11,-18){$q_G$}
\put(16.2,-14){( $W^{\pm}$, $\pi_p^{\pm}$ , c, t )}
\put(5,-16.5){$x_2 p$}
\put(5,-21.5){$x_1 p$}
\put(23,-16.5){$y_2 q$}
\put(23,-21.5){$y_1 q$}
\put(11,-14.5){$l_1 $}
\put(29,-25.2){Fig.1}
\put(21,-8){k}
\put(19,-10){$\gamma$}
\put(35,-15){\line(1,0){20}}
\put(35,-20){\line(1,0){20}}
\multiput(47,-15.8)(0,-0.8){6}{$\varsigma$}
\multiput(44,-15)(1,1){5}{\line(0,1){1}}
\multiput(43,-15)(1,1){6}{\line(1,0){1}}
\put(48,-18){$q_G$}
\put(35,-16.5){$x_2 p$}
\put(35,-21.5){$x_1 p$}
\put(53,-16.5){$y_2 q$}
\put(53,-21.5){$y_1 q$}
\put(46,-14.5){$l_2 $}
\put(44,-15){\circle*{3}}
\put(51,-8){k}
\put(49,-10){$\gamma$}
\put(5,-35){\line(5,-4){5}}
\put(5,-43){\line(5,4){5.5}}
\multiput(10.4,-39.5)(1,0){7}{V}
\put(17.5,-38.5){\line(5,-4){5.5}}
\put(18,-39.1){\line(5,4){5}}
\multiput(51.7,-41)(1,1){3}{\line(0,1){1}}
\multiput(50.7,-41)(1,1){4}{\line(1,0){1}}
\put(3,-39){$B_c$}
\put(24,-39){$D_s^*$}
\put(11,-42){ $W^{\pm}$, $\pi_p^{\pm}$ }
\put(11,-33){$\gamma$}
\put(35,-35){\line(5,-4){5}}
\put(35,-63){\line(5,4){5.5}}
\multiput(40.4,-59.5)(1,0){7}{V}
\put(47.5,-58.5){\line(5,-4){5.5}}
\put(48,-59.1){\line(5,4){5}}
\multiput(8.5,-61)(1,-1){3}{\line(0,-1){1}}
\multiput(7.5,-61)(1,-1){4}{\line(1,0){1}}
\put(5,-55){\line(5,-4){5}}
\put(5,-63){\line(5,4){5.5}}
\multiput(10.4,-59.5)(1,0){7}{V}
\put(17.5,-58.5){\line(5,-4){5.5}}
\put(18,-59.1){\line(5,4){5}}
\multiput(51.7,-55.5)(-1,1){4}{\line(0,-1){1}}
\multiput(50.7,-55.5)(-1,1){4}{\line(1,0){1}}
\put(35,-55){\line(5,-4){5}}
\put(35,-43){\line(5,4){5.5}}
\multiput(40.4,-39.5)(1,0){7}{V}
\put(47.5,-38.5){\line(5,-4){5.5}}
\put(48,-39.1){\line(5,4){5}}
\multiput(8.5,-37)(1,1){3}{\line(0,1){1}}
\multiput(7.5,-37)(1,1){4}{\line(1,0){1}}
\put(29,-70){Fig.2}
\end{picture}
\newpage
\begin{picture}(30,0)
\setlength{\unitlength}{0.1in}
\put(5,-15){\line(1,0){9.5}}
\put(14,-16.4){\oval(8,3)[r]}
\multiput(9.6,-19)(-0.2,0.5){8}{$v$}
\put(6.5,-18){$W^-$}
\put(14,-20){\oval(8,4.1)[l]}
\put(5,-25){\line(1,0){18}}
\put(14,-22.1){\line (1,0){8.5}}
\put(18,-18.5){$J/{\psi}(\psi^{'})$}
\multiput(19,-16.4)(1,1){3}{\line(0,1){1}}
\multiput(18,-16.4)(1,1){4}{\line(1,0){1}}
\put(22,-13){$\gamma$}
\put(5.5,-14.5){b}
\put(13,-14.5){c}
\put(13,-21.5){s}
\put(13,-17.5){$\overline {c}$}
\put(5.5,-24.5){$\overline {c}$}
\put(3,-20){$B_c$}
\put(23,-24){$D_s^*$}
\put(29,-30){$Fig.3$}
\multiput(40,-22.1)(0.5,0.3){12}{$v$}
\put(41,-20){$W^-$}
\put(50,-18){\oval(8,3)[l]}
\put(35,-25.1){\line(1,0){14}}
\put(35,-22.1){\line (1,0){14}}
\put(49,-23.5){\oval(8,3)[r]}
\put(54,-24.5){$J/{\psi}(\psi^{'})$}
\multiput(54,-24)(1,1){3}{\line(0,1){1}}
\multiput(53,-24)(1,1){4}{\line(1,0){1}}
\put(57,-20){$\gamma$}
\put(35.5,-21.5){b}
\put(48,-21.5){c}
\put(48,-16){s}
\put(48,-19.5){$\overline {c}$}
\put(35.5,-24.5){$\overline {c}$}
\put(33,-24){$B_c$}
\put(51,-18.5){$D_s^*$}
\end{picture}
\end{document}
| proofpile-arXiv_065-447 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Figure captions}
{\bf Fig. 1} Diagrams with four quarks in the final state containing a cut triangular
quark loop.\\[2ex]
{\bf Fig. 2} Contributions to the asymmetry fragmentation function $F_A(x,Q^2)$ \eref{eq:24}
at $Q=M_Z$ using the fragmentation density set of \cite{Bin95}. Solid line: LO.
Dashed line: NLO. Dotted line: NNLO. The exprimental data are taken from
OPAL \cite{Ake95}.\\[2ex]
{\bf Fig. 3} The ratio $R_A(x,Q^2)$ \eref{eq:25} at $Q=M_Z$ using
the fragmentation density set of \cite{Bin95}. Short dashed line: LO.
Solid line: NLO. Dotted line: NNLO. The exprimental data are taken from
OPAL \cite{Ake95}.
| proofpile-arXiv_065-448 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
Galaxies are open systems and the properties of the interstellar gas
are regulated by the internal sources of energy and interactions with
other galaxies. There are a series of different mechanisms, for both
isolated and interacting galaxies, that are able to accumulate large
gas masses and create star-forming clouds in relatively short time
scales. These include cloud collisions, gravitational and thermal
instabilities, Parker instabilities, gas flows in a bar potential,
tidal interactions, direct galaxy collisions, and mergers (some of
these mechanisms are discussed in this volume by Boeker {\it et al.}, Borne
{\it et al.}, Dultzin-Hacyan, Elmegreen, Friedli \& Martinet, Lamb {\it et al.}, and
Moss \& Whittle). Any one, or a combination, of these processes could
be operative in different locations and at different moments in the
host galaxy, and star formation is the end product of a series of
successive condensations of the interstellar medium. Once a cloud is
formed, however, a distinction should be made between molecular and
self-gravitating clouds (see Franco \& Cox 1986). The criterion to
form molecular clouds is high opacity in the UV, to prevent molecule
photo-destruction, and this is achieved at column densities above
$N_{\tau} \sim 5\times 10^{20} (Z/Z_{\odot})^{-1}$ cm$^{-2}$, where
$Z$ is the metallicity and $Z_{\odot}$ is the solar value. In
contrast, self-gravity becomes dominant when the column density
becomes larger than $N_{sg} \sim 5\times 10^{20} (P/P_{\odot})^{1/2}$
cm$^{-2}$, where $P$ is the interstellar pressure and $P_{\odot}$ is
the value at the solar circle. These two values are similar at the
location of the Sun, but $N_{\tau} < N_{sg}$ in the inner Galaxy and
$N_{\tau} > N_{sg}$ in the outer parts of the Milky Way. This
difference has important consequences and may explain the observed
radial trends of molecular gas in spirals: it is easier to form
molecular clouds in the internal, chemically evolved, parts of spiral
galaxies. In any case, the transformation of gas into stars is due to
a gravitational collapse and self-gravity defines the structure of the
star forming clouds.
\section{Self-gravity}
The formation of stellar groups (or isolated stars, if any) occurs in
the densest regions, the cores, of massive and self-gravitating
clouds. In our Galaxy, the $average$ densities for giant molecular
cloud complexes is in the range $10^2$ to $10^3$ cm$^{-3}$, but the
actual densities in the dense cores is several orders of magnitude
above these values: close to about $\sim 10^6$ cm$^{-3}$ ({\it e.g.},\ Bergin
{\it et al.}\ 1996; see recent review by Walmsley 1995). Moreover, recent
studies of young stellar objects suggest the existence of even denser
gas, with values in excess of $10^8$ cm$^{-3}$ (Akeson {\it et al.}\ 1996).
Thus, the parental clouds have complex clumpy (and filamentary)
structures, with clump-interclump density ratios of about $\sim 10^2$,
or more, and temperatures ranging between 10 and $10^2$ K. In
addition, the existence of large non-thermal velocities, of several km
s$^{-1}$, and strong magnetic fields, ranging from tens of $\mu$G to
tens of mG (see Myers \& Goodman 1988 and references therein),
indicate large $total$ internal pressures, up to more than five orders
of magnitude above the ISM pressure at the solar neighborhood (which
is about $10^{-12}$ dyn cm$^{-2}$). A simple estimate for isothermal,
spherically symmetric, clouds (with a central core of constant density
$\rho_c$ and radius $r_c$, and an external diffuse envelope with a
density stratification $\rho = \rho_c (r/r_c)^{-2}$), indicates that
self-gravity provides these large total pressure values (see
Garc\'{\i}a-Segura \& Franco 1996). In hydrostatic equilibrium, the
pressure difference between two positions located at radii $r_1$ and
$r_2$ from the center of the core is given by $\Delta P =
-\int^{r_2}_{r_1} \rho g_r dr$, where $g_r$ is the gravitational
acceleration in the radial direction. The total pressure at the core
center is
\begin{equation}
P(0)=P_0= \frac{2 \pi G}{3} \rho_c^2 r_c^2 + P(r_c)= \frac{8}{5} P(r_c)
\simeq 2\times 10^{-7} \ n_6^2 r_{0.1}^2 \ \ \ \ {\rm dyn \ cm^{-2}},
\end{equation}
where $G$ is the gravitational constant, $P(r_c)$ is the pressure at
the core boundary $r=r_c$, $n_6=n_c/10^6$ cm$^{-3}$, and
$r_{0.1}=r_c/0.1$ pc. The corresponding core mass is
\begin{equation} M_c\simeq \left( \frac{\pi P_0}{G}\right)^{1/2} r_c^2 \sim 10^2 \
P_7^{1/2} r_{0.1}^2 \ \ \ \ {\rm M_{\odot}},
\end{equation}
where $P_7= P_0/10^{-7}$ dyn cm$^{-2}$. For $P_7\sim 1$ and a typical
core size for galactic clouds, $r_{0.1}\sim 1$ (see Walmley 1995),
gives a value similar to the observationally derived core masses; in
the range of 10 to 300 M$_{\odot}$ ({\it e.g.},\ Snell {\it et al.}\ 1993). The
pressure inside the core varies less than a factor of two between the
center and $r=r_c$. Taking $r_{0.1}=1$ and the maximum core density
value, $n_c \sim 5\times 10^6$ cm$^{-3}$ ({\it e.g.},\ Bergin {\it et al.}\ 1996), the
upper bound for the expected core pressures is about $P_0 \simeq
5\times 10^{-6}$ dyn cm$^{-2}$. The large range in observed cloud
properties obviously results in pressure fluctuations of a few orders
of magnitude (both, from cloud to cloud and inside any given cloud),
and it is meaningless to define an ``average'' cloud pressure
value. Actually, given that star forming clouds have nested
structures, in which dense fragments are embedded in more diffuse
envelopes, different cloud locations have different total
pressures. Also, the expected range of cloud pressures in our Galaxy
should probably span from the ISM values, $P_7\sim 10^{-5}$, at the
very external cloud layers, up to $P_7\sim 10$ inside the most massive
star forming cores.
\section{Stellar radiation: HII regions and cloud destruction}
The initial structure and pressure of the gas in a star forming cloud
is defined by self-gravity. Once young stars appear, the new energy
input modifies the structure and evolution of the cloud. Low-mass
stars provide a small energy rate and affect only small volumes,
but their collective action may provide partial support against the
collapse of their parental clouds, and could regulate some aspects of
the cloud evolution (Norman \& Silk 1980; Franco \& Cox 1983; Franco
1984; McKee 1989; see also the paper by Bertoldi \& McKee in this
volume). In contrast, the strong radiation fields and fast stellar
winds from massive stars are able to excite large gas masses and can
even disrupt their parental clouds ({\it e.g.},\ Whitworth 1979; Franco {\it et al.}\
1994). Also, they are probably responsible for both stimulating and
shutting off the star formation process at different scales. The
combined effects of supernovae, stellar winds, and H II region
expansion destroy star-forming clouds and can produce, at some
distance and later in time, the conditions for further star formation
({\it e.g.},\ Franco \& Shore 1984; Palous {\it et al.}\ 1995). Thus, the
transformation of gas into stars may be a self-limited and
self-stimulated process (see reviews by Franco 1991, Ferrini 1992, and
Shore \& Ferrini 1994).
In the case of the dense star-forming cores, the sizes of either HII
regions or wind-driven bubbles are severely reduced by the large
ambient pressure (Garc\'{\i}a-Segura \& Franco 1996). In fact, the
pressure equilibrium radii of ultra-compact HII regions are actually
indistinguishable from those of ultra-compact wind-driven bubbles. When
pressure equilibrium is reached, the UCHII radius is
\begin{equation}
R_{{\rm UCHII,eq}} \approx 2.9 \times 10^{-2} \,\, F_{48}^{1/3}
\,\, T_{{\rm HII},4}^{2/3} \,\, P_7^{-2/3} \ \ \,\,{\rm pc} ,
\label{Rsequnits}
\end{equation}
where $F_{48}$ is the total number of ionizing photons per unit time in units
of $10^{48}$ s$^{-1}$, and $T_{{\rm HII},4} = T / 10^4$ K. For the case of a
strong wind evolving in a high-density molecular cloud core, the equilibrium
radius of a radiative bubble is
\begin{equation}
R_{,{\rm WDB,eq}}=\left[ \frac{\dot{M} \,v_{\infty}}{4 \,\pi \,P_0} \right]^{1/2}
\simeq 2.3\times 10^{-2} \left[ \frac{\dot{M_6} \,\,v_{\infty,8}}{P_7}
\right]^{1/2} \ \ \,\,{\rm pc} , \label{Req}
\end{equation}
where the mass loss rate is $\dot{M_6}=\dot{M}/10^{-6}$ M$_\odot$ yr$^{-1}$,
and the wind velocity is $v_{\infty,8}= v_{\infty}/10^8$ cm s$^{-1}$. Thus,
for dense cores with $r_c\sim 0.1$ pc, the resulting UCHIIs and wind-driven
bubbles can reach pressure equilibrium without breaking out of the core ({\it i.e.},\
they could be stable and long lived). Recently, Xie {\it et al.}\ (1996) have found
evidence indicating that this is probably the case: the smaller UCHII seem
to be embedded in the higher pressure cores.
If the limit to continued star forming activity inside the core is due
to photoionization by these internal H II regions, the maximum number
of OB stars is given by the number of H II regions required to
completely ionize the core (Franco et al 1994),
$N_{OB}=(1-\epsilon)M_{c}/M_{i}$, where $M_{i}$ is the ionized
mass. This means that the maximum number of massive stars that can be
formed within a core is
\begin{equation}
N_{OB} \approx 3 {M_{c,2} n_{6}^{3/7} \over F_{48}^{5/7} (c_{s,15}
t_{MS,7})^{6/7}}
\end{equation}
where $M_{c,2}$ is the core mass in $10^{2}$ M$_{\odot}$, $c_{s,15}$
is the HII region sound speed in units of 15 km s$^{-1}$, and
$t_{MS,7}$ is the mean OB star main sequence lifetime in
$10^{7}$yr. Clearly, for increasing core densities, the value of
$R_{0}$ decreases and the resulting number of OB stars increases. In
the case of the gas in nuclear regions, due to the intrinsic larger
ISM pressures in the inner regions of galaxies, the population of
clouds are denser and more compact. The corresponding star forming
clouds should also be denser than in the rest of the disk, and a
larger number of stars can be formed per unit mass of gas. Thus, {\it
nuclear starbursts can be a natural consequence of the higher pressure
values} (a bursting star formation mode can also be associated to a
delayed energy input, see Parravano 1996).
When stars are located near the edge of the core, and depending on the
slope of the external density distribution, both HII regions and
wind-driven bubbles can accelerate and flare out with a variety of
hydro-dynamical phenomena. These include supersonic outflows, internal
shocks, receding ionization fronts, fragmentation of the thin shell,
etc ({\it e.g.},\ Tenorio-Tagle 1982; Franco {\it et al.}\ 1989, 1990;
Garc\'{\i}a-Segura \& Mac Low 1995a,b). Thus, no static solution
exists in this case and the pressure difference between the HII
regions and the ambient medium begins to evaporate gas from the
cloud. This represents a clear and simple physical mechanism for cloud
destruction and, as the number of OB stars increases, more expanding H
II regions form and limit the rate of new star formation by ionizing
the surrounding molecular gas (Franco et al 1994). Eventually, when
the whole cloud is completely ionized, star formation ceases. The
total cloud mass ionized by an average OB star, integrated over its
main sequence lifetime, is
\begin{equation}
M_{i}(t) \approx \frac{2\pi}{3} R_{0}^{3}\mu_p n_{0}
\left\lbrack\left(1+\frac{5c_{s}t_{MS}}{2R_{0}}\right)^{6/5}-1\right\rbrack.
\end{equation}
where $R_{0}$ is the initial radius at the average cloud density,
$n_{0}$, $\mu_p$ is the mass per gas particle, $c_{s}$ is the sound
speed in the HII region, and $t_{MS}$ is the main sequence lifetime of
the average OB star. For a cloud of mass $M_{GMC}$, with only 10\% of
this mass in star-forming dense cores, the number of newly formed OB
stars required to completely destroy it is
\begin{equation}
N_{OB} \sim 30
\frac{M_{GMC,5}n_{3}^{1/5}}{F_{48}^{3/5}(c_{s,15} t_{MS,7})^{6/5}}.
\end{equation}
where $M_{GMC,5}=M_c/10^5$ \mbox{M$_{\odot}$}, $n_{3}=n_{0}/10^3$ cm$^{-3}$,
$c_{s,15}=c_{s}/15$ km s$^{-1}$, and $t_{MS,7}=t_{MS}/10^7$ yr.
Assuming a standard IMF, this corresponds to a total star forming
efficiency of about $\sim 5$ \%. For the average values of stellar
ionization rates and giant molecular cloud parameters in our Galaxy,
the overall star forming efficiency should be about 5\%. Obviously,
larger average densities and cloud masses can result in higher star
formation efficiencies.
Summarizing, photoionization from OB stars can destroy the parental
cloud in relatively short time scales, and defines the limiting number
of newly formed stars. The fastest and most effective destruction
mechanism is due to peripheral, blister, HII regions, and they can
limit the star forming efficiency at galactic scales. Internal HII
regions at high cloud pressures, on the other hand, result in large
star forming efficiencies and they may be the main limiting mechanism
in star forming bursts and at early galactic evolutionary stages (see
Cox 1983).
\section{Mechanical energy}
As the cloud is dispersed, the average gas density decreases and the
newly formed cluster becomes visible. The individual HII regions merge
into a single photo-ionized structure and the whole cluster now powers
an extended, low density, HII region. The stellar wind bubbles now can
grow to larger sizes and some of them begin to interact. As more winds
collide, the region gets pressurized by interacting winds and the
general structure of the gas in the cluster is now defined by this
mass and energy input (Franco {\it et al.}\ 1996).
Given a total number of massive stars in the cluster, $N_{OB}$, and their
average mass input rate, $<\dot{M}>$, the pressure due to interacting adiabatic
winds is
\begin{equation}
P_i\sim \frac{N_{OB} <\dot{M}> c_i}{4 \pi r^2_{clus}}\sim 10^{-8}
\frac{N_{2} <\dot{M}_6> c_{2000}}{r^2_{pc}} \ \ \ \ {\rm dyn \ cm^{-2}},
\end{equation}
where $N_{2}=N_{OB}/10^2$, $<\dot{M}_6>=<\dot{M}>/10^{-6}$ M$_\odot$
yr$^{-1}$, $r_{pc}=r_{clus}/1$ pc is the stellar group radius, and
$c_{2000}=c_i/2000$ km s$^{-1}$ is the sound speed in the interacting
wind region. This is the central pressure driving the expansion of the
resulting superbubble before the supernova explosion stage. For modest
stellar groups with relatively extended sizes, like most OB
associations in our Galaxy, the resulting pressure is only slightly
above the ISM pressure ({\it i.e.},\ for $N_{2} \sim 0.5$ and $r_{pc}\sim 20$,
the value is $P_i \sim 10^{-11}$ dyn cm$^{-2}$). For the case of rich
and compact groups, as those generated in a starburst, the pressures
can reach very large values. For instance, for the approximate cluster
properties in starbursts described by Ho in this volume, $r_{pc}\sim
3$ and $N_{2}>10$, the resulting pressures can reach values of the
order of $P_1\sim 10^{-7}$ dyn cm$^{-2}$, similar to those due to
self-gravity in star forming cores. At these high pressures, the winds at the
evolved red giant (or supergiant) phases cannot expand much and they reach
pressure equilibrium at relatively small distances from the evolving star.
Thus, the large mass ejected during the slow red giant
wind phase is concentrated in a dense circumstellar shell.
\begin{figure}
\vspace*{43mm}
\begin{minipage}{43mm}
\special{psfile=fig.ps hscale=95. vscale=95. hoffset=-17 voffset=-340 angle=0.
}
\end{minipage}
\caption{
Evolution of a wind-driven bubble in a high pressure medium.
{\bf (a)} Evolution of a stellar wind from a 35\,M$_{\odot}$
star. Left scale: terminal wind velocity (km s$^{-1}$); right scale:
mass loss rate (M$_{\odot}$ yr$^{-1}$); horizontal scale: time
(millions of yr). {\bf (b)-(d)}: Evolution of the wind-driven bubble. The gas
density (cm$^{-3}$) is plotted in a logarithmic scale against the radial
distance (pc). Evolutionary times (shown in the upper-left corner of each
panel) are given in million years.}
\label{fig:1}
\end{figure}
An example of this is shown in Figure 1, where the evolution of a
wind-driven bubble around a 35\,M$_{\odot}$ star is presented. Fig. 1a
shows the wind velocity and mass-loss rate (dashed and solid lines,
respectively: Garcia-Segura, Langer \& Mac Low 1996). We ran the
simulation only over the time spanning the red supergiant and
Wolf-Rayet phases, and assume that the region is already pressurized
by the main sequence winds from massive stars. We used the AMRA code,
as described by Plewa \& R\'o\.zyczka in this volume. During the RSG
phase the wind-driven shell is located very close ($R\sim$ 0.04 pc) to
the star due to a very low wind ram-pressure (Fig. 1b). Later on
(Fig. 1c), the powerful WR wind pushes the shell away from the star to
the maximum distance of $R\approx 0.54$ pc. Still later, when the wind
has variations, the shell adjusts its position accordingly, and
reaches the distance $R\sim 0.3$ pc at the end of simulation
(Fig. 1d). It must be stressed that the series of successive
accelerations and decelerations of the shell motion during the WR
phase will certainly drive flow instabilities and cause deviations
from the sphericity assumed in our model. The role of these
multidimensional instabilities in the evolution of the shell is
currently under study (with 2-D and 3-D models), and the results will
be presented in a future communication.
Regardless of the possible shell fragmentation, however, when the star
explodes as a supernova, the ejecta will collide with a dense
circumstellar shell. This interaction generates a bright and compact
supernova remnant, with a powerful photoionizing emission ({\it i.e.},\
Terlevich {\it et al.}\ 1992; Franco {\it et al.}\ 1993; Plewa \& R\'o\.zyczka this
volume), that may also be a very strong radio source, like SN 1993J
(see Marcaide {\it et al.} 1995). If the shell is fragmented, the
ejecta-fragment interactions will occur during a series of different
time intervals, leading to a natural variability in the emission at
almost any wavelength (see Cid {\it et al.} 1996). This type of interaction
is also currently under investigation, and further modeling will shed
more ligth on the evolution of SN remnants in high-pressure environs.
This work was done during the First ``{\bf Guillermo Haro}'' Workshop,
in April-May 1996, and we thank the hospitality of the staff at INAOE
in Tonantzintla, Puebla. We warmly thank many useful discussions with
Roberto Cid, Jos\'e Rodriguez-Gaspar, Elisabete de Gouveia, Gustavo
Medina-Tanco, Michal R\'o\.zyczka, Sergei Silich, Laerte Sodre,
Guillermo Tenorio-Tagle, Roberto Terlevich, and To\~ni Varela, and the
enthusiasm and support given to the whole project by Alfonso Serrano,
General Director of INAOE. JF and GGS acknowledge partial support to
this project by DGAPA-UNAM grant IN105894, CONACyT grants
400354-5-4843E and 400354-5-0639PE, and a R\&D Cray research
grant. The work of TP was partially supported by the grant KBN
2P-304-017-07 from the Polish Committee for Scientific Research. The
simulations were carried out on a workstation cluster at the
Max-Planck-Institut f\"ur Astrophysik.
{\small
| proofpile-arXiv_065-449 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
Among the most important applications of lattice QCD are the determinations
of the fundamental parameters of the standard model in the quark sector.
Of these, one of the most important is the overall scale of the light
quark masses.
It is one of the most poorly known of the parameters of the standard model
from prelattice methods.
The Particle Data Group estimates a range of a factor of three in allowed
values \cite{RPP96}:
\begin{eqnarray}
100\ {\rm MeV}&<&\overline{m}_s(1\ {\rm GeV}) <300\ {\rm MeV,} \\
5\ {\rm MeV}&<&\overline{m}_d(1\ {\rm GeV}) <15\ {\rm MeV,\ and} \\
2\ {\rm MeV}&<&\overline{m}_u(1\ {\rm GeV}) <8\ {\rm MeV.}
\end{eqnarray}
(I will use $\overline{m}_q$ to represent the running quark mass in
the $\overline{MS}$ scheme.)
Global fits to standard model parameters are very sensitive to
such large variations.
It is also one for which lattice methods provide the only systematically
improvable determination.
This can be contrasted with the strong coupling constant $\alpha_s$,
for example,
for which it is possible to imagine going to higher and higher energy
scattering experiments and extracting $\alpha_s$ with perturbation theory.
Lattice quark mass extractions are harder to do with solid error analysis than
$\alpha_s$ extractions, but they may be more important in the long run.
\section{Prelattice Quark Mass Results}
\subsection{Quark mass ratios}
The ratios of light quark masses can be investigated with
some degree of reliability
using chiral perturbation theory ($\chi$PT),
which becomes asymptotically exact in the zero quark mass, zero energy
limit of QCD~\cite{Leu90}.
One combination of the light quark mass ratios is
especially likely to be reliable, since it has been constructed
to have vanishing leading order corrections in $\chi$PT:
\begin{eqnarray}
&&\frac{m_s^2-m_l^2}{m_d^2-m_u^2}=
\frac{M_{K}^2}{M_{\pi}^2}
\frac{M_K^2-M_\pi^2}{M_{K^0}^2-M_{K^+}^2+M_{\pi^+}^2-M_{\pi^0}^2}
\nonumber \\
&&\times \left[1+{\cal O}(m_s^2)+{\cal O}\left(e^2\frac{m_s}{m_d-m_u}\right)\right]
\end{eqnarray}
The other combination of quark mass ratios may be obtained at leading
order in chiral perturbation theory from the canonical prediction
of $\chi$PT, that the meson mass squared is proportional to the quark mass:
\begin{equation}\label{msml}
\frac{m_s+m_l}{2m_l}=\frac{M_{K^0}^2}{M_{\pi^0}^2}\left[1+\ldots\right].
\end{equation}
The usefulness of this relation has been called into question by the discovery
of a symmetry of the chiral Lagrangian which leaves physical predictions
invariant under simultaneous transformations of the interactions, and of the
quark masses~\cite{kap86}:
\begin{eqnarray}
{\cal L}_\chi &\rightarrow& {\cal L}_\chi' \\
m_u &\rightarrow& m_u + \lambda m_s m_d,
\end{eqnarray}
and cyclic in u,d,s.
The effects on the pseudoscalar spectrum of a nonzero $u$ quark mass can be
mimicked exactly by higher order interactions in the $\chi$PT
Lagrangian and altered $d$ and $s$ quark masses.
In light of this, Leutwyler has assembled a variety of arguments
to test the size of the corrections to Eq. (\ref{msml})
and in particular the possibility of $m_u=0$ \cite{Leu96}.
He concludes that the corrections are small and
obtains
\begin{eqnarray}
m_s/m_d &=& 18.9(8),\\ \label{sd}
m_u/m_d &=& 0.553(43). \label{ud}
\end{eqnarray}
\subsection{Absolute value of the quark masses}
Chiral perturbation theory makes no statement about the
third combination of light quark masses,
the overall scale.
A large range of results has been obtained with a variety of methods.
See, for example, a compilation of results
for $m_s$ from Ref. \cite{RPP96} shown in Table \ref{tab:1}.
The most reliable of these prelattice results
are perhaps those using QCD sum rules,
but even here, the systematic errors of the method are hard to pin down.
It is here that there is the largest spread in existing results and that
lattice methods probably have the most important role to play.
\begin{table}[hbt]
\caption[duh]{Compilation of results for $m_s$ from Ref.~\cite{RPP96}.
If defined,
the renormalization scheme is $\overline{m}_s(1\ {\rm GeV})$.
}
\begin{tabular}{l|l}
\hline
$m_s$ (MeV) & method \\
\hline
194 (4) & quark model \\
118 & sum rules \\
175 (55) & sum rules \\
$>300$ & sum rules \\
112 (66) & $\chi$PT + estimate of $\langle \overline{q}q\rangle$ \\
378 (220) &
$\chi$PT + estimate of $\langle \overline{q}q\overline{q}q\rangle$ \\
150 & strange baryon splittings \\
135 & SU(6) \\
\hline
\end{tabular}\label{tab:1}
\end{table}
\section{Lattice Quark Mass Determinations}
Lattice determinations of standard model parameters require:
1) fixing the bare lattice parameters from physics, and
2) obtaining the $\overline{MS}$ parameters from these with
short-distance matching calculations.
Spin averaged splittings in the $\psi$ and $\Upsilon$ systems
are convenient quantities to set the lattice spacing \cite{El-Khadra+al}.
The mesons are small and easy to understand because the
quarks are nonrelativistic. Light
pseudoscalar meson masses are the most convenient quantities to fix the
light quark masses.
Chiral symmetry makes them very sensitive to the quark masses,
the mesons are small,
the correlators have good statistics and are easy to fit over long time
separations.
The correlators have simple behavior as the quark mass is varied toward zero,
in contrast with unstable vector mesons.
Other quantities must give the same results as the
approximations are removed.
The second piece is the determination of the
parameters of the $\overline{MS}$ Lagrangian
by matching perturbative, dimensionally regularized
short distance amplitudes to their lattice counterparts.
It is desirable to do the lattice part of such calculations nonperturbatively
as much as possible, to test for the presence of nonperturbative
short-distance effects and possible poor convergence of perturbation theory.
Such nonperturbative short-distance calculations
are harder to design for quark masses than
for the strong coupling constant.
Such nonperturbative short distance analysis of quark mass extractions
is currently less advanced than the analogous investigations
for the strong coupling constant $\alpha_s$.
The perturbative expression giving the $\overline{MS}$ mass
from the lattice bare mass $m_0$
may be written
\begin{equation}\label{m0msbar}
\overline{m}(\mu)
= \tilde{m}\left[1+g^2 \left( \gamma_0 \left( \ln \tilde{C}_m-\ln\left( a \mu\right) \right) \right) \right]. \label{eq:star}
\end{equation}
The mean-field-improved bare mass $\tilde{m}$ is given by
$\tilde{m} = m_0/(1-\frac{1}{12}g^2)$
in perturbation theory, and
$\tilde{m}=m_0/\sqrt[4]{\langle U_P\rangle_{MC}}$
nonperturbatively, if the expectation value of the
plaquette,
$\langle U_P\rangle$,
is used to define
mean field improvement\cite{LM93}.
$\gamma_0=1/(2\pi^2)$ is the leading quark mass
anomalous dimension.
This coefficients in this expression are well behaved for
the Wilson action and the ${\cal O}(a)$ improved action
of Sheikholeslami and Wohlert (SW) \cite{She85}
(for which $\tilde{C}_m = 1.67$ \cite{Gro84} and 4.72 \cite{Gab91}
respectively).
For staggered fermions, $\tilde{C}_m$ is 132.9 \cite{Gro84},
leading to renormalization factors of 50--100\%
which are not explained by large tadpole graphs.
The status of quark mass extractions as of a few years ago was reviewed in
Refs.~\cite{Uka93,Gup94}.
Some data tabulated in Ref.~\cite{Uka93} are shown in Fig.~\ref{old}.
Unimproved perturbation theory was used, and the lattice spacing was
determined with the $\rho$ mass.
The quenched results for staggered fermions (dark squares)
are relatively independent of the lattice spacing,
but Eq.~\ref{m0msbar} is untrustworthy because of the very large
quantum correction.
The quenched Wilson results (diamonds)
have much better controlled perturbation theory, but
are much more lattice spacing dependent.
If the results are extrapolated in $a$, they approach the staggered results
more closely.
However, the magnitude and origin of remaining lattice spacing
errors is then unknown.
Improved actions must be used to investigate this.
\begin{figure}
\epsfxsize=0.45 \textwidth
\epsfbox{oldcombined.eps}
\caption[old]{
Old lattice quark mass results were reviewed in Ref.~\cite{Uka93}.
Quenched staggered results (dark squares)
show good cutoff independence, but suffer
from huge perturbative corrections.
Quenched Wilson results (circles) had well-converged perturbation
theory, but large cutoff dependence.
The white squares show some unquenched staggered results.
}
\label{old}
\end{figure}
\section{Recent Quenched Lattice Results}
\subsection{Unimproved Wilson fermion results.}
New unimproved results have been presented in the JLQCD
contribution to this volume by T. Yoshi\'e \cite{yos96}.
They include a data point at $\beta=6.3$, corresponding to
a smaller lattice spacing
($a=(3.29\ {\rm GeV})^{-1}$) than previous results.
Fig. \ref{jlqcd} shows their Wilson fermion results superimposed on
a subset of the world data.
$M_\rho$ has been used to set the lattice spacing.
Improved perturbative theory has been used in the renormalizations,
which makes the Wilson results slightly higher than
in the older analyses, and the staggered results much higher,
about 50\%.
The new JLQCD Wilson results are statistically consistent with
previous results, and so can be combined with them for a consistent
extrapolation of the leading error.
Taken by themselves, however,
they appear to have a somewhat smaller $a$ dependence
then other results.
Wilson data were also presented by Fermilab
(upper points in Fig.~\ref{fermilab}) \cite{Ono96,gou96}.
These points also lie slightly below and have a smaller slope than
most of the world data in Fig.~\ref{jlqcd}.
This is due to the fact that
the Fermilab data use the spin averaged 1P--1S splitting in the $\psi$ system
to determine the lattice spacing,
which is expected to have a small ${\cal O} (a)$ error.
Determining $a$ with the $\rho$ mass is equivalent to calculating
$m_l/m_\rho$.
Since the $ \overline{\psi} \sigma_{\mu\nu} F_{\mu\nu} \psi$
${\cal O} (a)$
correction operator is expected to push spin partners
like the $\pi$ and the $\rho$ apart,
${\cal O} (a)$ errors in both masses contribute to the $a$ dependence.
When the 1P--1S splitting is substituted for the $\rho$,
the slope ought to be reduced, as observed.
If $\rho$ mass lattice spacings are substituted into the Fermilab data,
the results line up with the upper points in Fig.~\ref{jlqcd}.
\begin{figure}
\epsfxsize=.45 \textwidth
\epsfbox{fig-3a.ps}
\caption[jlqcd]{
Recent unimproved Wilson fermion results for $\overline{m}_l(2\ {\rm GeV})$
from JLQCD compared with previous world data (upper points) \cite{yos96}.
Lattice spacing from $M_\rho$.
JLQCD staggered fermion results are the lower $*$'s.
}
\label{jlqcd}
\end{figure}
Gupta and Bhattacharya have performed a linear extrapolation
on the world Wilson data and obtain
$\overline{m}_l(2\ {\rm GeV})= 3.1(3)$ MeV \cite{Gup96}.
They use slightly different analysis methods from those of Ref.~\cite{yos96},
and obtain a result below that which would be obtained from Fig.~\ref{jlqcd},
which is about 3.8 MeV.
However, the authors say that their error estimate
is a ``guess chosen to reflect the various uncertainties discussed \dots
and is not based on systematic analysis''.
JLQCD also presented new results for staggered fermions which left the existing
situation unchanged.
The results are almost lattice independent for $a<(1.5\ {\rm GeV})^{-1}$,
but it is difficult to estimate perturbative uncertainties
since the perturbative
renormalization factor is so large. (At $\beta=6.4$, they use
$Z=1.69$, a 69\% correction of which most is unexplained by mean
field improvement.)
\subsection{Improved quenched Wilson fermion results}
To investigate the size of the remaining cut--off
errors in the Wilson action (${\cal O} (a^2)$, ${\cal O} (a \alpha_s)$,
${\cal O} (\alpha_s^2)$, etc.),
it is necessary to remove the leading error with an improved action.
In the SW action, the leading ${\cal O} (a)$ error is removed with
the addition of the operator
$ \overline{\psi} \sigma_{\mu\nu} F_{\mu\nu} \psi$.
There are several gauge links in the operator, so tadpole improvement
predicts a rather large correction, of order 50\%.
New results with the tadpole improved SW action were presented by
Fermilab \cite{Ono96}.
The lattice spacing dependence observed in the Wilson points is
substantially reduced, but not completely eliminated.
If this arises from a residual ${\cal O} (a)$ error,
the results extrapolate down to rather near the staggered results.
(A recent nonperturbative determination of the coefficient of the
SW improvement operator has indicated the possibility of even larger
corrections than the sizable ones
indicated by perturbation theory and mean field
improvement~\cite{Jan96}.)
If it arises from higher order effects in $a$, the answer is close to the
existing small lattice spacing points, and the discrepancy with staggered
results must be attributed to poorly behaved staggered perturbation theory.
\begin{figure}
\epsfxsize=0.45 \textwidth
\epsfbox{mq.eps}
\caption[fermilab]{
Fermilab light quark masses for Wilson action (circles)
and the tadpole improved SW action (triangles)~\cite{Ono96}.
Lattice spacing from the 1P--1S splitting in the $\psi$ system.
}
\label{fermilab}
\end{figure}
Since a downward trend in $a$ is still present in the tadpole improved
data, we take the improved result at the finest lattice spacing
as an upper bound in the quenched approximation and a linear extrapolation
as a lower bound.
Adding perturbative uncertainties linearly and other uncertainties in
quadrature gives the quenched result:
\begin{eqnarray}\label{eq:qu}
\overline{m}_l(2\ {\rm GeV}) & = & 3.6 (6)\ {\rm MeV} ,\\
\overline{m}_s(2\ {\rm GeV}) & = & 95 (16)\ {\rm MeV}.
\end{eqnarray}
Another determination of the strange quark mass with an ${\cal O}(a)$
improved action has been reported,
in Ref. \cite{All94}.
This determination used a tree-level, rather than a mean-field improved
coefficient for the improvement operator.
They obtained $\overline{m}_s(2\ {\rm GeV})=128 \pm 18$ MeV.
They did not attempt to
correct for the effects of the remaining lattice spacing
dependence
or the effects of the quenched approximation.
Much of the discrepancy with the Fermilab
results arises from fact that we have used
much larger SW improvement
coefficients, and make an allowance for the fact that
we continue to find significant cut-off dependence even so.
\section{Corrections to Quark Mass Ratios}
\subsection{Nonlinearities in $m_q$ vs. $M_\pi^2$}
\begin{figure}
\epsfxsize=0.45 \textwidth
\epsfbox{aoki.ps}
\caption[nonlin]{
$(aM_\pi)^2$ vs. $am_q$ for staggered fermions at $\beta=6.4$
on a $40^3\times 96$ lattice \cite{jlq96}.
Deviations from linearity are a few \% or less.
}
\label{nonlin}
\end{figure}
There is very little lattice evidence for large deviations from linearity
in the quark masses in Eq. (\ref{msml})
in either quenched or unquenched calculations.
In the most accurate data, the question often seems to be why
the predictions of chiral symmetry appear linear
at the few~\% level up to such large values of the quark masses.
(See Fig.~\ref{nonlin}. The largest pion mass is around 850 MeV,
and nonlinearity in $(aM_\pi)^2$ vs. $am_q$ is only a few~\%.)
However, searching for such deviations is tricky,
especially in the quenched approximation where quenched chiral logarithms
may add spurious nonlinearities at small $m_q$ \cite{Sha96}.
Existing unquenched calculations have not yet examined carefully the
case of broken flavor SU(3), $m_s>m_l$, so not all of the higher order
operators of the chiral Lagrangian have yet been tested.
\subsection{Electromagnetic effects} It is in principle simple to determine the contributions of electromagnetism
to hadron masses using numerical simulations with link matrices which
are products of SU(3) and U(1) matrices.
The U(1) phases of electromagnetism are cheap to generate because
to leading order in $\alpha_{em}$
(all that is required for practical purposes),
they can be obtained from Fourier transforms of Gaussian fluctuations
in momentum space.
To see the effects of electromagnetism clearly above fluctuations
in the SU(3) field, one wants to use values of $\alpha_{em}$
which are larger than the physical value.
It then remains only to show that electromagnetic effects are still
linear in this region.
That this indeed holds has been shown recently by Duncan, Eichten,
and Thacker \cite{dun96}.
They have performed a prototype calculation of the $\pi^+ - \pi^0$
splitting at $\beta=5.7$ and with it have obtained $m_u/m_d = 0.51$,
in agreement (so far) with Eq.~\ref{ud}.
\subsection{Can $m_u = 0$?}
The most interesting application of these calculations
is the settling of the question of whether $m_u \equiv 0$
in the real world.
This possibility is fervently desired in spite of all evidence to the
contrary because of its neat solution to the strong CP problem.
Ref.~\cite{kap86} showed that allowing corrections
as large as 30\% to $\chi$PT
equations such as \ref{msml} made $m_u = 0$ compatible with meson mass data.
(See Fig.~\ref{nonlin}, however.)
The possibility of large instanton-induced flavor mixing effects
has been proposed as a mechanism to generate such corrections in
QCD~\cite{Geo81}.
Not all the required lattice calculations have been done, but so far,
the lattice evidence is against $m_u=0$.
\section{The Quenched Approximation}\label{quapp}
As the lattice spacing is reduced, while keeping hadronic physics fixed,
the lattice bare couplings evolve according to their anomalous dimensions.
In the quenched approximation, these anomalous dimensions are slightly
wrong, due to the absence of light quark loops.
The strong coupling constant evolves according to the zero-quark $\beta$
function coefficient $\beta_0^{(0)}=11$ rather than the correct three-quark
coefficient $\beta_0^{(3)}=11-2/3 n_f=9$.
Asymptotically, $\alpha_s(\pi/a)$ in the quenched approximation is expected
to be too small by a factor of about 9/11.
Since the short-distance quark mass evolution is
given by $d\ln m(q)/d \ln(q) = - \gamma_0 \alpha_s/(4\pi)$,
where $\gamma_0 = 8$,
this implies that the quark mass evolves too slowly
in the quenched approximation, and
at small lattice spacings is larger than in real life.
At high energies, running mass evolution is given by
\begin{equation}
\frac{m(q_1)}{m(q_2)} \approx
\left( \frac{\alpha_s(q_1)}
{\alpha_s(q_2)}\right)^{\frac{\gamma_0}{2\beta_0}}.
\end{equation}
To leading log accuracy, therefore, the effect of the absence of quark loops
due to {\em perturbative} effects
on the evolution of the running mass from the strong coupling region
(where $\alpha_s\approx 1$) to the high energy region can be
approximated by
\cite{Mac94}
\begin{eqnarray}
\frac{m(\pi/a)|_{\rm qu.\ \ \ }}{m(\pi/a)|_{\rm unqu.}}
&\approx& \alpha(\pi/a)^{\frac{\gamma_0}{2}(1/\beta_0^{(0)}-1/\beta_0^{(3)})}\\
&\approx& 1.1 {\ \rm to \ } 1.2, \label{eq:1.15}
\end{eqnarray}
for $\alpha(\pi/a)\approx 1/4$ to 1/8.
There are also quenching effects arising from the nonperturbative region.
Unlike the case of quarkonium systems, for pions there is no argument that
these should be smaller than the perturbative effects.
The above expression can't be taken as a correction factor,
only as an indication of the direction and order of magnitude of effects to
be expected.
Nonperturbative calculations are required to investigate quenching
effects quantitatively.
Unquenched results for Wilson fermions appear complicated and hard
to interpret.
They differ by as much as a factor of two from quenched results,
and do not seem to reproduce the lattice spacing dependence of the quenched
results as would have been expected (see Ref. \cite{Uka93}).
Since most other unquenched Wilson results, such as those in thermodynamics,
are also hard to understand compared to staggered results,
I will not attempt to fit unquenched Wilson fermion results into
my general picture.
Some unquenched staggered results summarized in Ref. \cite{Uka93}
are
shown in Fig. \ref{old} (white squares) along with
the quenched results.
The unquenched results indeed lie below the quenched results by roughly
the expected amount, and I will use them
to estimate the effects of quenching.
The large corrections in staggered fermion mass renormalization
cancel out in the ratio of the quenched
and unquenched determinations, making this an useful quantity to examine.
To minimize effects due to differences in analysis methods, we estimate
the ratio from the results of a single group, at similar volumes and
lattice spacings (about 0.4 GeV$^{-1}$) \cite{Ish92,Fuk92}, and obtain
\begin{eqnarray}
\frac{\overline{m}_l(1.0\ {\rm GeV})_{\rm n_f=0}}{\overline{m}_l(1.0\ {\rm GeV})_{\rm n_f=2}}
&\approx& \frac{2.61(9)}{2.16(10)}\\
&=& 1.21(7) \label{eq:unq}
\end{eqnarray}
Since there are, in fact, three flavors of light quarks in the world
and not two, I will this ratio as a lower bound on the actual ratio
and use the square (corresponding to four light quarks) as an upper bound.
\section{Synthesis}
Lattice determinations of light quark masses are more difficult than
the analogous determinations of $\alpha_s$.
Pion masses have worse $a$ dependence than the quarkonium splittings.
Finding nonperturbative ways of eliminating the perturbative
relations with the bare lattice quark mass requires more work.
Nevertheless, with a few exceptions,
most lattice quark mass extractions are consistent with a reasonably
simple picture.
I ignore results with very large $a$ or very small volume.
Unquenched Wilson results, which are also hard to interpret in most
other quantities, do not seem to make sense compared with
quenched Wilson results.
Of the remaining results,
the larger magnitude and lattice spacing
dependence of Wilson results compared with
staggered results is greatly reduced with
${\cal O} (a)$ improved actions.
The magnitudes are reduced more with tadpole improved SW correction terms
than with tree--level correction terms.
If the remaining discrepancy arises mainly from
residual ${\cal O} (a)$ effects in the improved action,
the true quenched answer lies close to the staggered result,
$\overline{m}_l$ a little over 3 MeV.
If it arises mainly from higher order corrections in the staggered fermion
lattice--$\overline{MS}$ mass conversion (where the leading correction
is 50--100\%), the true answer lies closer to the improved result,
$\overline{m}_l \sim 4$ MeV.
Unquenched staggered results lie somewhat below quenched staggered results,
but by an amount which is reasonable.
Taking the ratio from Sec.~\ref{quapp} and using it to make a correction on
our quenched result we obtain
\begin{itemize}
\item $\overline{m}_s(2\ {\rm GeV})$ in the range 54 -- 92 MeV,
\item $\overline{m}_l(2\ {\rm GeV})$ in the range 2.1 -- 3.5 MeV,
\end{itemize}
for the $\overline{MS}$ masses renormalized at 2 GeV.
The uncertainties most in need of further study are
those associated with lattice spacing dependence and the quenched
approximation.
\section*{Acknowledgments}
I thank Brian Gough, Aida El-Khadra, George Hockney,
Andreas Kronfeld, Bart Mertens, Tetsuya Onogi, and Jim Simone
for collaboration on the work of Ref. \cite{Ono96,gou96}.
I thank the Center for Computational Physics in Tsukuba for hospitality
while this paper was written,
and I thank the members of the JLQCD collaboration for useful discussions.
Fermilab is operated by Universities Research
Association, Inc. under contract with the U.S. Department of
Energy.
| proofpile-arXiv_065-450 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{...} is the same as in LaTeX
\def\section#1{\ifblank\sectionnumstyle\then
\else\newsubsectionnum=\z@\@resetnum\sectionnum=\next \fi
\displayhead{\ifblank\sectionnumstyle\then\else\sectionnumformat\ \fi#1}
}
\def\subsection#1{\advance\subsectionnum by 1
{\boldfont \leftline{\ifblank\sectionnumstyle\then\else\sectionnumformat\fi\number\subsectionnum\
#1}} \vskip-10pt \line{\hrulefill}}
\def
\section{Introduction}
\medskip\noindent
Dynamical systems that are naturally close to or on their critical point
have been proposed as an explanation of the appearance of power laws in
nature \cite{1,2}. This `self-organized criticality' still
consists to a large extent
in the study of toy models and has many open or controversial
questions. The lattice models can be grouped at least into three classes.
A first class contains models with a local conservation law like
the famous sandpile-model \cite{1,2}. Recent experiments
on piles of rice \cite{3} partially confirm the general theoretical
predictions by exhibiting power laws, but also show that the existence
of power laws in real systems depends on microscopic details such as the
aspect ratio of grains of rice. Models of evolution \cite{4,5}
are one example in a second class where the dynamics is specified
in terms of a globally selected extremal site. This extremal dynamics
can be used to obtain some general statements for the complete class
of models \cite{6}. A model for self-organized criticality
that is known as the `forest-fire model' is a member of a third class
of models that have parameters which can be tuned close to the critical
point in a natural manner, but unlike in the two previous classes cannot
be entirely discarded (In dealing with this model one should be aware
that it is highly idealized and not expected to describe real
forest fires). The precise version that
we study in this paper has first been introduced in a short note
\cite{7} and arises as a certain limit of the more general model
proposed later independently in \cite{8}.
\medskip\noindent
The two-dimensional forest-fire model has been already discussed
very controversially, mainly on the basis of Monte-Carlo simulations
where usually the accuracy of the predictions was the issue.
An originally proposed version \cite{9} did not show the desired
critical behaviour \cite{10,11}, and it was necessary to introduce
lightnings \cite{8}. Subsequently, Monte-Carlo simulations have
several times lead to values for the critical exponents which
had to be corrected later on \cite{8,12,13,14}.
Here we add to this discussion by re-examining some of the quantities
investigated only in one of the earlier works \cite{12}. We were
motivated by a study of the one-dimensional case \cite{15} where
we had discovered the existence of two different length scales.
The analogous question in two dimensions has been addressed in
\cite{12}, but there it seemed that the two scales are
proportional to each other. Here, we present more accurate simulations
that demonstrate these two length scales to be different also in two
dimensions. As by-products we also check or improve estimates for
other exponents (see in particular \cite{14}). A final subject
is the approach to equilibrium which seems not
to have been addressed in a similar way before. In a second part,
we try to discard the spatial structure and introduce a global
model similar to the one of \cite{15} for one dimension. On the
one hand, some of the qualitative features of the stationary state
at the critical point (e.g.\ the power law of the cluster-size
distribution) can nicely be described by such a global model. On the
other hand, there are discrepancies in quantitative details and the
range of such a simplified model is very limited in comparison to
the full model. We believe that this is an important point, e.g.\ because
it has been suggested in \cite{16} that power laws in nature
might arise from a global (`coherent') driving that does not see
any spatial structure. Our findings here and in the one-dimensional
case \cite{15} demonstrate that the full model is not only different
from but also richer than the simplified model. This is analogous
to the result of \cite{17} that versions of certain models of evolution
with and without spatial structure lead to different results (at least
if examined closely).
\medskip\noindent
We now define the model before we proceed with a presentation of our simulation
results in the next Section. The forest-fire model is defined on a
cubic lattice in $d$ dimensions.
Any site can have two states: It can either be empty or it can be
occupied by a tree. The dynamics of the model is specified by the
following update rules (following \cite{7,12,13,14}
\footnote{${}^{1})$}{
To be precise, the simulations in \cite{13,14} have
been performed according to slightly different rules, because they
aimed at investigating only quantities associated with clusters.
}):
In each Monte-Carlo step first choose an arbitrary site of the lattice.
\item{a)} If it is empty, grow a tree there with probability $p$.
\item{b)} If it is occupied by a tree, delete the entire geometric
cluster of trees connected to it with probability $f$.
This corresponds to a lightning stroke with subsequent
spreading of the fire.
\par\noindent
A rescaling of the probabilities $p$ and $f$ just amounts
to a rescaling of the time scale, and in particular leaves the
stationary state invariant. We exploit this to set $p=1$.
There is a critical point at $f/p = 0$, but the parameter $f/p$ is
relevant and it is not legitimate to consider the forest-fire model
precisely at this critical point (compare \cite{18}).
\bigskip\noindent
\section{Simulation results in two dimensions}
\medskip\noindent
In the following we consider the two-dimensional version
of the forest-fire model on a quadratic lattice with
periodic boundary conditions. The linear size of the lattice
will be denoted by $L$. So the volume $V$ is given by $V = L^2$.
We use a `global' time scale, a unit of which is defined by
the number of Monte-Carlo steps needed in order to visit each
site on average once, i.e.\ a unit of global time consists
of $L^2$ Monte-Carlo steps.
\medskip\noindent
In order to do the simulation efficiently also for large systems
and small $f/p$ one has to be careful and use e.g.\
bitmapping technologies. For details on the implementation
compare the WWW page \cite{19}. Using this program,
the simulations of this paper took about four months of CPU time on
150MHz DEC alpha workstations.
\medskip\noindent
We have investigated mainly systems of linear size $L=16384$
and parameter values $10^{-2} \ge f/p \ge 10^{-4}$.
For a simulation, one random initial condition with density
$\rho = 1 / 2$ was chosen. In order to equilibrate
the system, it was left to evolving freely for at least 15 global
time units. This equilibration time was increased to 25 global
time units for $f/p \le 3 \cdot 10^{-4}$ and to 35 global time units
for $f/p = 1 \cdot 10^{-4}$ (these times were adjusted according
to the observed time evolution of $\rho$, see also Section 2.3
below). After this, the system was iterated
further for another 60 to 90 global time units (120 units for
$f/p = 3 \cdot 10^{-4}$). During this period, measurements
were made as global averages at intervals of usually one
global time unit (for the density 200 times more frequently).
This amounts to at least $60 L^2$ measurements for each quantity
of interest. Note that we use only a single run.
\bigskip\noindent
\subsection{Correlation functions}
\medskip\noindent
Let us first discuss the simulation results for the usual
tree correlation function $\langle T({\vec{x}}) T({\vec{x}}+{\vec{y}})\rangle$
($T({\vec{x}}) = 1$ in a configuration with a tree at site ${\vec{x}}$,
$T({\vec{x}}) = 0$ if the site ${\vec{x}}$ is empty).
We have only investigated displacements ${\vec{y}}$ along the vertical
axis (which can be treated particularly efficiently using bitmaps).
The treatment of the data resulting from the simulation is
straightforward because it can nicely be fitted by
$$C(y) :=
\langle T({\vec{x}}) T({\vec{x}}+y \vec{e}_2)\rangle - \langle T({\vec{x}}) \rangle^2 = a e^{-{y / \xi}}
\label{defTp}$$
in a suitable interval $y_{\rm min} \le y \le y_{\rm max}$.
The parameters $a$ and $\xi$ were estimated by taking the logarithm of
the r.h.s.\ of \ref{defTp} and then performing a linear regression for
$y_{\rm min} \le y \le y_{\rm max}$. These bounds are chosen
such that the approximation \ref{defTp} by a single exponential function
is good. The lower cutoff can be chosen small ($y_{\rm min} \approx 20$)
independent of $f/p$. An upper cutoff $y_{\rm max}$ has to be imposed
at distances of 4 to 7 times the correlation length $\xi$ because then
statistical errors become large.
\medskip\noindent
\centerline{\psfig{figure=length_n.ps}}
\smallskip\noindent
{\par\noindent\leftskip=4.5 true pc \rightskip=4 true pc
{\bf Fig.\ 1:}
The two correlation lengths $\xi$ and $\xi_c$.
For $\xi_c$ all estimates are on lattices with $L=16384$ and
the different symbols correspond to different ways of treating the
data.
\par\noindent}
\medskip\noindent
Fig.\ 1 shows the results for $\xi$ obtained in this manner
on lattices with $L=8192$ (diamonds), $L=16384$ (crosses)
and $L=17408$ (triangles).
One observes that they are in good agreement with the form
$$\xi \sim \left({f \over p}\right)^{-\nu_T} \, .
\label{DEFnuT}$$
Performing linear regression fits on a doubly logarithmic scale
one finds \footnote{${}^{2})$}{
Error estimates are always the $1 \sigma$ confidence interval of
a fit unless discussed explicitly.}
$$\nu_T = 0.541 \pm 0.004\, .
\label{RESnuT}$$
This is close to the result $\nu_T = 0.56$ found in \cite{12}.
Comparing the values for $\xi$ obtained from simulations
on different lattice sizes one can see that they may have a residual
statistical error of around 2\% which is about the same as the
scattering of the individual data points around the line
\ref{DEFnuT}. Since also further
data for larger $f/p$ is consistent with the form \ref{DEFnuT}
and the value \ref{RESnuT}, this result for $\nu_T$ and its error
bound can be considered reliable.
\medskip\noindent
The estimates for the normalization constant $a$ in \ref{defTp} are
compatible with a value $a = 0.030 \pm 0.001$ independent of
$f/p$. So, in the limit $f/p \to 0$ the two-point function seems
to tend to $\langle T({\vec{x}}) \rangle^2 + a$ for $y \age 100$. In terms
of the alternative ansatz $C(y) = a y^{-\eta_{{\rm occ}}} e^{-{y / \xi}}$
used in \cite{12} this corresponds to $\eta_{{\rm occ}} = 0$,
while the result found there was $\eta_{{\rm occ}} = 0.120 \pm 0.015$.
We have also looked into the possibility of a power-law correction factor
using our data for $L=16384$ and $f/p \le 3 \cdot 10^{-4}$. With the
estimates for $\xi$ shown in Fig.\ 1 one finds that
$e^{{y / \xi}} C(y) \sim y^{-0.11}$ for $y \le 20$, i.e.\ for small $y$
there is indeed a power-law correction factor
with an exponent that is consistent with \cite{12}. On the other
hand, for $50 \le y \le 4 \xi$, the function $e^{{y / \xi}} C(y)$
is flat -- its smallest values are around $0.027$ and its maximal
values around $0.031$. This clearly contradicts a power-law correction
factor with $\eta_{{\rm occ}} \approx 0.11$ in the large-distance
asymptotics. Thus, for the large-distance behaviour
$\eta_{{\rm occ}} = 0$ seems to be correct, while the power-law
correction factor observed in \cite{12} applies to small distances.
\medskip\noindent
We now look at a second quantity, namely
the `connected correlation function' $\langle T({\vec{x}}) T({\vec{x}}+{\vec{y}})\rangle_c$
describing the probability to find two trees at positions
${\vec{x}}$ and ${\vec{x}}+{\vec{y}}$ {\it inside the same cluster}. This
quantity is usually referred to as {\it the} two-point function
in the context of self-organized criticality and often is also the
only correlation function that is investigated. We demonstrated in one
spatial dimension \cite{15} that the length scales associated
to this correlation function and $C(y)$ have different critical exponents
while \cite{12} suggested that in two dimensions length scales
associated to different quantities are equivalent. This latter
suggestion was based on simulations with lattice sizes up to
$512 \times 512$ and $f/p \age 5 \cdot 10^{-4}$.
One of the main aims of the simulations presented here is to
check if different length scales can be exhibited also in
two dimensions after sufficiently improving the accuracy.
As before, we restrict to displacements ${\vec{y}}$ along the vertical
axis, i.e.\ we consider
$$K(y) := \langle T({\vec{x}}) T({\vec{x}}+y \vec{e}_2)\rangle_c \, .
\label{DEFconTPF}$$
This quantity can be determined in the same run as
the two-point function $\langle T({\vec{x}}) T({\vec{x}}+{\vec{y}})\rangle$,
but it involves the additional effort of determining all
clusters present in the system.
\medskip\noindent
Following \cite{14} we associate a correlation length $\xi_c$
to the second moment of $K(y)$ via
$$\xi_c^2 := {\sum_{y=1}^{\infty} y^2 K(y) \over
\sum_{y=1}^{\infty} K(y)} \, .
\label{DEFxiC}$$
The squares in Fig.\ 1 show values of $\xi_c$ extracted from
simulations with $L=16384$ using this definition. One sees that these
values are consistent with
$$\xi_c \sim \left({f \over p}\right)^{-\nu} \, .
\label{DEFnuC}$$
and a (preliminary) value of
$$\nu = 0.5738 \pm 0.0013 \, .
\label{RESnuCpre}$$
This agrees roughly with the value $\nu = 0.60$ found in \cite{12},
and also with the more accurate result in \cite{14}
which, including error bounds, is given by $\nu = 0.580 \pm 0.003$
\cite{20}. It should be noted that the lower bound $y=1$
in \ref{DEFxiC} is crucial. Starting the summation instead e.g.\
at $y=10$, one finds $\xi_c \approx 24$ at $f/p = 10^{-2}$
and $\xi_c \approx 219$ for $f/p = 10^{-4}$ instead of
$\xi_c \approx 14$ and $\xi_c \approx 203$, respectively. This in turn
would make the value for $\nu$ smaller. We will argue soon that
the start of summation $y=1$ is indeed the correct choice, but
the impact of a modification here could give rise to a slightly
larger error than the $1\sigma$ interval for the fit given in
\ref{RESnuCpre}.
\medskip\noindent
We now proceed with a more detailed discussion of the
form of $K(y)$ which will also justify the definition
\ref{DEFxiC}. Eq.\ \ref{DEFxiC} is based on the following expected
form of $K(y)$:
$$K(y) = a_c \; y^{-\eta} \; e^{-y/\xi_c} \, .
\label{DEFeta}$$
In particular, if $e^{y/\xi_c} \; K(y)$ agrees well with
a power law, the determination of $\xi_c$ via \ref{DEFxiC} is justified.
Using the values of $\xi_c$ given by the boxes in Fig.\ 1 one finds that
$e^{y/\xi_c} \; K(y)$ does indeed agree well with
$a_c \; y^{-\eta}$ for $y$ between 1 and several times $\xi_c$
\footnote{${}^{3})$}{
The values of $a_c$ are all compatible with the $f/p$-independent
value $a_c = 0.20 \pm 0.01$. Therefore, at the critical point the
power law $K(y) = a_c y^{-\eta}$ is expected to
be valid for all $y$.}.
The crosses in Fig.\ 2 show these estimates for $\eta$.
They obviously depend on $f/p$ contrary to what one would
naively expect. This points to systematic errors in the
determination of $\eta$ which one may expect to become less
important for smaller $f/p$ where the power law is better
visible. This suggests to make a `scaling ansatz'
$\eta(f/p) = \eta + \bar{a} \; (f/p)^{\bar{b}}$
to determine the value of $\eta$ in the limit $f/p \to 0$.
A least-squares fit gives the dotted line in Fig.\ 2.
One finds a critical $\eta = 0.374 \pm 0.011$.
\medskip\noindent
\centerline{\psfig{figure=eta.ps}}
\smallskip\noindent
{\par\noindent\leftskip=4.5 true pc \rightskip=4 true pc
{\bf Fig.\ 2:}
Values for the exponent $\eta$ obtained in two different
ways (crosses and diamonds) and scaling fits (lines).
\par\noindent}
\medskip\noindent
Alternatively, one can fit all three parameters in \ref{DEFeta}
directly from the data. The values for $\eta$ obtained
with this approach are shown in Fig.\ 2 by diamonds.
A scaling analysis (the line with long dashes in Fig.\ 2) yields a
critical $\eta = 0.342 \pm 0.008$. Comparing this with the value
found earlier, one finds a small disagreement. So, the error
bounds of these two estimates for the critical $\eta$ may be a
little too optimistic because they do not include
systematic errors. In order to be careful we give a final result
that includes both estimates for $\eta$ and the two error ranges:
\goodbreak
$$\eta = 0.36 \pm 0.03 \, .
\label{RESeta}$$
This is compatible with the result $\eta = 0.411 \pm 0.02$ of
\cite{12} (although the error estimate of \cite{12}
seems a little too optimistic, possibly because it does not
take systematic errors into account).
One also obtains different estimates for $\xi_c$ from the same
fits that gave the alternative values for $\eta$. These values
are shown in Fig.\ 1 by `$\times$'. One sees that they
are again compatible with the form \ref{DEFnuC}, and obtains
an alternative estimate for the associated critical exponent:
$\nu = 0.5777 \pm 0.0013$. There is a minor difference between
this value and the earlier estimate \ref{RESnuCpre} indicating
that we have indeed neglected systematic errors. Therefore we
give a final result
$$\nu = 0.576 \pm 0.003
\label{RESnuC}$$
which includes the two direct estimates and their error bound. This
final result is in excellent agreement with \cite{14}.
\medskip\noindent
Comparing \ref{RESnuC} with \ref{RESnuT}, we see that
$\nu \ne \nu_T$, so that $\xi_c$ and $\xi$ are basically
different lengths. A difference $\nu - \nu_T \approx 0.04$ was
already observed in \cite{12}, but attributed to numerical
errors because it was unexpected.
\medskip\noindent
It should be mentioned that our results \ref{RESeta} and \ref{RESnuC}
for $\eta$ and $\nu$ do not satisfy the scaling relation $2-\eta = 1/\nu$
\cite{12}. We can only speculate about the reason for the
disagreement. For example, in the derivation of this scaling
relation one assumes that $\eta$ and $\nu$ do not depend on
the direction of the displacement vector ${\vec{y}}$, and we have
not checked whether this is indeed true. Another possibility
is that we have still overlooked systematic errors. Inserting
$\nu$ (which is the more reliable value) according to \ref{RESnuC}
into the scaling relation $2-\eta = 1/\nu$ yields $\eta \approx 0.28$
which is possible if our error estimate in \ref{RESeta}
is by a factor of about 3 too small.
\bigskip\noindent
\subsection{Cluster-size distribution}
\medskip\noindent
The distribution $n(s)$ of clusters with size $s$ arises as
a by-product of the determination of $K(y)$ during the simulations.
We extract an exponent $\tau$ from it following the lines
described in detail in \cite{12,13}. One introduces
the quantity $P(s) = \sum_{s' > s} s' n(s')$. Assuming that
it behaves as $P(s) = \alpha s^{2-\tau} e^{-s/s_{{\rm max}}}$,
one can use three-parameter fits to obtain estimates for $\tau$
and $s_{{\rm max}}$. Extrapolation of the values obtained in
this manner from the simulations with $L=16384$ yields
$\tau = 2.1595 \pm 0.0045$ for $f/p \to 0$.
The values of $\tau$ for $f/p > 0$ approach
this limiting value from above. An alternative way to extract
a value of $\tau$ is to look directly at $n(s)$ and assume
that $n(s) \sim s^{- \tau}$ for intermediate $s$. Applying this
second method to the same data and to some results
for $L=8192$ we obtain the estimate $\tau = 2.159 \pm 0.006$
at the critical point. This limit is now approached from below and
is in excellent agreement with the one obtained
before. To be on the safe side we retain the value with the
larger error bound as the final result:
$$\tau = 2.159 \pm 0.006 \, .
\label{REStau}$$
This value agrees within error bounds with most previous results
\cite{12,21,13,14}, but our error bound is
considerably smaller than in most of them.
\medskip\noindent
When one extracts estimates for $\tau$ from $P(s)$ one also
obtains estimates for $s_{{\rm max}}$. These values are in
good agreement with the form $s_{{\rm max}} \sim (f/p)^{-\lambda}$
with $\lambda = 1.171 \pm 0.004$. To check
the validity of our error estimate we can use the scaling
relation $\lambda = 1/(3-\tau)$ \cite{14}. After inserting
\ref{REStau} one finds the prediction $\lambda = 1.189 \pm 0.008$.
This prediction agrees roughly with the direct estimate. However,
there is a small discrepancy indicating that the estimate
$$\lambda = 1.17 \pm 0.02
\label{RESlam}$$
is more realistic. Within error bounds we find good agreement
with the values of \cite{14} and \cite{12}. From
the results \ref{RESnuC} and \ref{RESlam} we
obtain the fractal dimension $\mu = 2.03 \pm 0.04$ using
the scaling relation $\mu = \lambda/\nu$ \cite{14}. This
does not quite agree with \cite{14} where $\mu = 1.96 \pm 0.01$
was found, but would favour instead the expectation $\mu = 2$
of \cite{13}. Unfortunately, with our simulations we had not
aimed at determining $\mu$ and we are therefore not able to clarify
this interesting point.
\bigskip\noindent
\subsection{Density and time evolution}
\medskip\noindent
Here we study the critical behaviour of the stationary density
as well as the temporal behaviour of the density. The latter yields
the lowest gap in the spectrum of the time-evolution operator which is
given by a straightforward generalization of eq.\ (3.6) in
\cite{15}. Some high relaxational modes of this time-evolution
operator can be written down explicitly as we demonstrate in Appendix A.
In one dimension, the low-lying spectrum of this operator could be
studied numerically \cite{15}, but this is not feasible in
higher dimensions. Fortunately, simulations of $\rho(t)$ exhibit
much clearer features in two dimensions than was the case in
one dimension. This is illustrated by Fig.\ 3 which shows the
initial time evolution of $\rho(t)$ after the system was started
at $\rho(0) = 1/2$.
\medskip\noindent
\centerline{\psfig{figure=rho_t.ps}}
\smallskip\noindent
{\par\noindent\leftskip=4.5 true pc \rightskip=4 true pc
{\bf Fig.\ 3:}
The density $\rho(t)$ as a function of global time $t$. Shown are
results from simulations with $f/p = 10^{-2}$ (bottom, full line),
$f/p = 10^{-3}$ (line with long dashes, middle) and $f/p = 10^{-4}$
(top line with short dashes).
\par\noindent}
\medskip\noindent
First we examine the critical behaviour of the stationary density
of trees $\rho(\infty)$. The values in Table 1 are obtained
by averaging $\rho(t)$ over times $t$ after the equilibration time,
i.e.\ at or beyond the right border of Fig.\ 3.
\medskip\noindent
\centerline{\vbox{
\hbox{
\vrule \hskip 1pt
\vbox{ \offinterlineskip
\defheight2pt&\omit&&\omit&&\omit&\cr{height2pt&\omit&&\omit&&\omit&&\omit&\cr}
\def\tablerule{ height2pt&\omit&&\omit&&\omit&\cr
\noalign{\hrule}
height2pt&\omit&&\omit&&\omit&\cr }
\hrule
\halign{&\vrule#&
\strut \quad\hfil#\hfil \quad\cr
height2pt&\omit&&\omit&&\omit&\cr
height2pt&\omit&&\omit&&\omit&\cr
& $f/p$ && $\rho(\infty)$ && oscillation period $T_{{\rm osc}}$ && decay time $T$
& \cr height2pt&\omit&&\omit&&\omit&\cr \tablerule
& $1 \cdot 10^{-2}$ && 0.376833 && $0.878 \pm 0.028$ && $0.988$ & \cr height2pt&\omit&&\omit&&\omit&\cr
& $7 \cdot 10^{-3}$ && 0.381539 && $0.889 \pm 0.043$ && $1.155$ & \cr height2pt&\omit&&\omit&&\omit&\cr
& $5 \cdot 10^{-3}$ && 0.385406 && $0.890 \pm 0.014$ && $1.312$ & \cr height2pt&\omit&&\omit&&\omit&\cr
& $3 \cdot 10^{-3}$ && 0.390296 && $0.878 \pm 0.012$ && $1.588$ & \cr height2pt&\omit&&\omit&&\omit&\cr
& $2 \cdot 10^{-3}$ && 0.393463 && $0.883 \pm 0.017$ && $1.719$ & \cr height2pt&\omit&&\omit&&\omit&\cr
& $1.4\cdot10^{-3}$ && 0.395793 && $0.883 \pm 0.025$ && $1.966$ & \cr height2pt&\omit&&\omit&&\omit&\cr
& $1 \cdot 10^{-3}$ && 0.397672 && $0.879 \pm 0.026$ && $2.296$ & \cr height2pt&\omit&&\omit&&\omit&\cr
& $7 \cdot 10^{-4}$ && 0.399321 && $0.878 \pm 0.011$ && $2.455$ & \cr height2pt&\omit&&\omit&&\omit&\cr
& $5 \cdot 10^{-4}$ && 0.400673 && $0.881 \pm 0.048$ && $2.603$ & \cr height2pt&\omit&&\omit&&\omit&\cr
& $3 \cdot 10^{-4}$ && 0.402291 && $0.875 \pm 0.015$ && $3.645$ & \cr height2pt&\omit&&\omit&&\omit&\cr
& $2 \cdot 10^{-4}$ && 0.403231 && $0.876 \pm 0.026$ && $3.559$ & \cr height2pt&\omit&&\omit&&\omit&\cr
& $1 \cdot 10^{-4}$ && 0.404696 && $0.868 \pm 0.031$ && $3.951$ & \cr height2pt&\omit&&\omit&&\omit&\cr
}
\hrule}\hskip 1pt \vrule}
\hbox{\quad \hbox{Table 1:} Estimates with $L=16384$ for the stationary density of trees $\rho(\infty)$ and}
\hbox{\quad \phantom{Table 1:} the lowest decay mode in $\rho(t)$ of the two-dimensional forest-fire model.}}
}
\medskip\noindent
These values are in good agreement with
$$\rho_c - \rho(\infty) \sim \left({f \over p}\right)^{1/\delta} \, ,
\label{FORMrho}$$
where the critical density $\rho_c$ and the critical exponent
$\delta$ are given by
$$\eqalignno{
\rho_c &= 0.40844 \pm 0.00011 \, , &\eqnlabel{RESrhoC} \cr
1/\delta &= 0.466 \pm 0.004 \, . &\eqnlabel{RESdelta}
}$$
These results agree within error bounds with the
values obtained in \cite{14}, but our bounds are
smaller. In particular we can now rule out that
$1/\delta = 1/2$ as proposed by \cite{13}.
\medskip\noindent
Having determined $\rho(\infty)$, it is straightforward to extract
the oscillation period and decay time of the slowest decay mode from
$\rho(t)$. First, one determines those $t$ where $\rho(t)$ crosses
the value $\rho(\infty)$. From the distances between these crossings
the oscillation period $T_{{\rm osc}}$ can be determined easily.
Averaging 10 to 15 half oscillation periods estimated in this
manner for a suitable interval of time in Fig.\ 3 leads to the values
given in Table 1. One observes that the oscillation period
equals $T_{{\rm osc}} = 0.88$ within error bounds for all $f/p$,
i.e.\ the slowest relaxational mode oscillates with a constant
frequency. This is to be contrasted with the one-dimensional case where
simulations of $\rho(t)$ clearly demonstrated that the oscillation
period depends on $f/p$ \cite{15}.
\medskip\noindent
Finally, we extract the leading decay time $T$ from $\rho(t)$
according to the following procedure. At times $t$ precisely
in the middle between two subsequent crossings used for the
determination of the oscillation period, the value of
$\abs{\rho(t) - \rho(\infty)}$ is determined. One finds
for these (approximately ten) values of $t$ that
$\abs{\rho(t) - \rho(\infty)} \sim \exp(-t/T)$ from which
it is straightforward to obtain the estimates for $T$
presented in Table 1 \footnote{${}^{4})$}{
These estimates also verify that our
equilibration times are long enough, since we have
equilibrated the system for at least $6 T$ before starting to
collect data. So, the non-stationary modes are damped by
factors of at least $\exp(-6) \approx 2 \cdot 10^{-3}$
during data collection.
}. These values for $T$ are compatible
with a critical behaviour
$$T \sim \left({f \over p}\right)^{-\zeta} \, ,
\label{DEFzeta}$$
where the critical exponent $\zeta$ is determined to be
$$\zeta = 0.314 \pm 0.013 \, .
\label{RESzeta}$$
This exponent probably is another new exponent that is not
related to the ones determined so far \cite{12,14}.
\bigskip\noindent
\subsection{Summary of simulations}
\medskip\noindent
Table 2 summarizes our results for the critical exponents
of the two-dimensional forest-fire model. It also
includes the critical density $\rho_c$ and the global
oscillation period which seems to be
independent of $f/p$. For comparison we have
also included the results for one dimension \cite{15,22}.
In that case, the oscillation period diverges and
we have listed the exponent rather than a period in Table 2.
Similarly, the amplitude $a$ vanishes in one dimension for
$f/p \to 0$ but is roughly constant in two dimensions.
\medskip\noindent
\centerline{\vbox{
\hbox{
\vrule \hskip 1pt
\vbox{ \offinterlineskip
\defheight2pt&\omit&&\omit&&\omit&\cr{height2pt&\omit&&\omit&&\omit&\cr}
\def\tablerule{ height2pt&\omit&&\omit&&\omit&\cr
\noalign{\hrule}
height2pt&\omit&&\omit&&\omit&\cr }
\hrule
\halign{&\vrule#&
\strut \quad\hfil#\hfil \quad\cr
height2pt&\omit&&\omit&&\omit&\cr
height2pt&\omit&&\omit&&\omit&\cr
& quantity && value in $d=2$ && value in $d=1$ & \cr height2pt&\omit&&\omit&&\omit&\cr \tablerule
& $\nu_T$ && $0.541 \pm 0.004$ && $0.8336 \pm 0.0036$ & \cr height2pt&\omit&&\omit&&\omit&\cr
& $\eta_{{\rm occ}}$ && $0$ && $0$ & \cr height2pt&\omit&&\omit&&\omit&\cr
& $\nu$ && $0.576 \pm 0.003$ && $1$ & \cr height2pt&\omit&&\omit&&\omit&\cr
& $\eta$ && $0.36 \pm 0.03$ && $0$ & \cr height2pt&\omit&&\omit&&\omit&\cr
& $\tau$ && $2.159 \pm 0.006$ && $2$ & \cr height2pt&\omit&&\omit&&\omit&\cr
& $\lambda$ && $1.17 \pm 0.02$ && $1$ & \cr height2pt&\omit&&\omit&&\omit&\cr
& $1/\delta$ && $0.466 \pm 0.004$ && $0$ & \cr height2pt&\omit&&\omit&&\omit&\cr
& $\zeta$ && $0.314 \pm 0.013$ && $\approx 0.405$ & \cr \tablerule
& $T_{{\rm osc}}$ && period: $0.88 \pm 0.02$ &&
exponent $\approx 0.194$ & \cr height2pt&\omit&&\omit&&\omit&\cr
& $a$ && $0.030 \pm 0.001$ && $\sim (f/p)^{0.1031 \pm 0.0022}$ & \cr height2pt&\omit&&\omit&&\omit&\cr
& $\rho_c$ && $0.40844 \pm 0.00011$ && $1$ & \cr height2pt&\omit&&\omit&&\omit&\cr
}
\hrule}\hskip 1pt \vrule}
\hbox{\quad \hbox{Table 2:} Summary of our results for the critical behaviour of the}
\hbox{\quad \phantom{Table 2:} forest-fire model.}}
}
\vfill
\eject
\section{Global models}
\medskip\noindent
We now wish to investigate through simplified variants of
the model to what extent one can describe the
stationary properties of the two-dimensional forest-fire model
by global variables in a way similar to \cite{15}. There
it was shown that in order to describe the one-dimensional
critical stationary state it suffices to know the relative weight
of the sum of all configurations with a fixed number of occupied
(or empty) sites. Thus one is dealing with a kind of grand canonical
ideal lattice gas. Such a model has no intrinsic spatial structure
and leads to a two-point function independent of the distance.
It can therefore be used to describe the asymptotic behaviour of
the critical $C(y)$ of Section 2. Working with the (global) density
of trees $\rho$, one has to specify the probability distribution
$p(\rho)$ and obtains $C(y) = \langle \rho^2 \rangle - \langle \rho \rangle^2$
for $y \ne 0$ in the thermodynamic limit. This is positive for all
continuous distributions.
\medskip\noindent
In two dimensions the limits $f/p \to 0$ and $L \to \infty$ do
not commute with each other (in contrast to one dimension).
Therefore, perturbation theory
cannot be used to compute $p(\rho)$, and in fact we do not
know of a good analytic method to determine $p(\rho)$
from the rules described in the Introduction. Therefore
we use heuristic arguments and simulations to discuss it.
\medskip\noindent
The forest-fire model reminds one of site percolation. Thus,
it is natural to try to relate the forest-fire model to
percolation and gain some insight from that. Attempts in
this direction have been made e.g.\ in \cite{8,21}.
We will also try to use some results of percolation theory,
but we will follow a different route. Namely, we try
to interpret the stationary state of the forest-fire model at
the critical point as a suitable ensemble of percolation
problems with a distribution $p(\rho)$.
\medskip\noindent
It is known from percolation theory \cite{23} that in a homogeneous
configuration with a density $\rho$ above the percolation
threshold $\rho_{{\rm perc}}$ two arbitrary sites are connected
with a finite probability. In such configurations, the dynamics
of the forest-fire model with arbitrarily small $f/p > 0$
would very quickly destroy all percolating
clusters and thus drive the density below the percolation
threshold \footnote{${}^{5})$}{
This implies that the density $\rho$ cannot be continuous
at $f/p=0$ for $d>1$ and is the reason why perturbation
theory cannot be used in higher dimensions.
}. Thus, the probability $p(\rho)$ to have a global
density above the percolation threshold $\rho_{{\rm perc}}$
must vanish in the forest-fire model, i.e.\ $p(\rho) = 0$
for $\rho > \rho_{{\rm perc}}$. In one dimension one has
$\rho_{{\rm perc}} = 1$ and in two dimensions
$\rho_{{\rm perc}} = 0.592746$ \cite{23}. In all simulations
$\rho(t)$ was well below this percolation threshold
at all times (compare Fig.\ 3).
\medskip\noindent
It is useful to visualize the stationary state of the
forest-fire model in order to gain some intuition.
Fig.\ 4 shows an area of $760 \times 472$ sites
at $t=66$ during a simulation with $L=16384$ and $f/p = 10^{-4}$
(compare also Fig.\ 2 of \cite{24}, Fig.\ 1 of \cite{14}
and Fig.\ 6 of \cite{13}). One observes that at a certain fixed time
the system consists of rather well-defined patches with different
mean density of trees. These patches are not to be confused with
single forest clusters, they usually contain many such clusters.
Their typical size increases as $f/p$ becomes
smaller, which reflects the divergence of correlation lengths.
Looking at the time evolution of such a state
\footnote{${}^{6})$}{
On X11 platforms, such a visualization is possible with the code
used for the simulations presented in this paper. This code is
available on the WWW \cite{19}.},
one observes an increase of density due to growth of trees
that is constant throughout the patches and that lightning
strikes essentially only the patches with the highest density.
After a lightning has struck such a patch, a new patch with a
low density is created. This new density is not really zero because
there are always some trees in the patch that are not connected
to the cluster which is destroyed. The idea now is
that the important information about the critical stationary state
is given by the distribution
$p(\rho)$ of densities in these patches and that nothing essential
changes if we replace such patchy systems with an ensemble of systems of
{\it global} density $\rho$ occurring with the same probability $p(\rho)$.
Of course, it remains to be tested to what extent this picture
works.
\medskip\noindent
\centerline{\psfig{figure=patch.ps}}
\smallskip\noindent
{\par\noindent\leftskip=4.5 true pc \rightskip=4 true pc
{\bf Fig.\ 4:}
Snapshot of an area with $760 \times 472$ sites in the stationary
state during a simulation with $L=16384$ and $f/p = 10^{-4}$.
Trees are black and empty places white.
\par\noindent}
\bigskip\noindent
\subsection{A simple model}
\medskip\noindent
Let us make a very simple model based on these ideas.
Assume that below the percolation threshold $\rho_{{\rm perc}}$
trees just grow (with probability $p=1$) and no lightning strikes.
As soon as the percolation threshold is exceeded, lightning
strikes immediately and destroys all trees in the system
(not just those in the percolating cluster). In
order to determine $p(\rho)$ we first compute the mean lifetime
of a configuration with density $\rho$, assuming that no lightning
strikes (this consideration will also be useful later on). Such a
configuration lives precisely $n$ local updates if $n-1$
times an occupied place and then an empty place are selected.
Because of the above assumptions there is no correlation and
therefore the probability of this to happen is given by
$\rho^{n-1} (1-\rho)$. This yields the expectation value $t(\rho)$
of the lifetime as
$$t(\rho) = \sum_{n=1}^{\infty} n \rho^{n-1} (1-\rho)
= {1 \over 1-\rho} \, .
\label{timeDens}$$
If no lightning strikes, the probability $p(\rho)$ to find
a configuration with density $\rho$ is proportional to the
lifetime of a state with this density. Taking into account that lightning
strikes all trees at $\rho_{{\rm perc}}$, we find for this simple model
$$p(\rho) =
\cases{{\cal N} (1-\rho)^{-1}\, , & $\rho < \rho_{{\rm perc}}$, \cr
0 \, , & $\rho > \rho_{{\rm perc}}$, \cr}
\label{simpleGmodel}$$
with the normalization constant given by
${\cal N}^{-1} = \int_0^{\rho_{{\rm perc}}} {\rm d}\rho \; (1-\rho)^{-1}$.
In one dimension, \ref{simpleGmodel} agrees with eq.\ (5.4) of \cite{15}
which was derived there using a different argument.
\medskip\noindent
This simple model has the following periodic time evolution: Trees grow
until the density reaches the percolation threshold. Then lightning
strikes and the process is restarted with a completely
empty system. The average time needed for such a cycle is precisely
the global oscillation time and is given by
$\int_0^{\rho_{{\rm perc}}} {\rm d}\rho \; t(\rho)$. Inserting
$t(\rho)$ according to \ref{timeDens} and the value of $\rho_{{\rm perc}}$
in two dimensions yields an oscillation period of
$T_{{\rm osc}} \approx 0.898$ which is in very good agreement with what
we found in simulations (compare Table 1).
The mean density is given by $\int_0^1 {\rm d}\rho \; \rho p(\rho)$
from which one finds a critical density $\rho_c = 0.340\ldots$
(this deviates notably from the result \ref{RESrhoC} found by simulations
of the full model). Finally, the probability to find two trees
at arbitrary places is given by the second moment of $p(\rho)$,
i.e.\ by $\int_0^1 {\rm d}\rho \; \rho^2 p(\rho)$. So,
the two-point function exceeds the value
$\rho_c^2$ by an amount $a=0.0289\ldots$ which agrees within
error bounds with what we found by simulations for the large-distance
asymptotics of the two-point function. Cluster-type quantities
are not accessible as easily and would e.g.\ require again
Monte-Carlo simulations.
\medskip\noindent
Although this simple model yields very good values for two
quantities, its failure to give the correct $\rho_c$ is not
surprising. Firstly, we have neglected the fact that lightning
can already strike configurations with $\rho < \rho_{{\rm perc}}$
even for arbitrarily small $f/p > 0$ (compare Appendix A).
Secondly, lightning does not really lead to the completely
empty system, but usually leaves some isolated trees or
small clusters in the patch behind. Unfortunately we do not
know how to treat either effect analytically.
This lack of knowledge also has the effect that we can
in general not compute an oscillation time from $p(\rho)$
although we do of course still think of the system as
evolving in cycles (compare also section 6.3.2 of \cite{25}).
\bigskip\noindent
\subsection{Realistic distributions}
\medskip\noindent
Next we try to obtain a realistic $p(\rho)$ from
Monte-Carlo simulations.
Measurements of the global density cannot be used to extract
$p(\rho)$ from a simulation, because $\rho$ fluctuates only very little
around its mean value for sufficiently large systems (see Fig.\ 3).
Therefore, one has to look at the distribution of local densities.
In a large system, areas with different local densities coexist
and we are interested precisely in these local fluctuations and
not just the global average.
We have decided to divide the system into $16 \times 16$
plaquettes and use the distribution of the average density
per plaquette. This size of the plaquettes was chosen because
then their linear extent is much smaller than the correlation
lengths and they contain sufficiently many sites to obtain a
fairly smooth distribution. Fig.\ 5 shows a result $p_{{\rm re}}(\rho)$
obtained in this manner using the parameter values
closest to the critical point, namely $f/p = 10^{-4}$
and $L=16384$. Samples were taken at the same
90 times where also the correlation functions were determined,
amounting to a total of almost $10^{8}$ samples for local
densities. The normalization in Fig.\ 5 is such
that $\sum_{r=0}^{256} p_{{\rm re}}(r/256) = 1$.
\medskip\noindent
As explained above,
the first and second moments of $p(\rho)$ are related to the
mean density and the asymptotic value of the two-point function.
From this one finds
$$\rho(\infty) = 0.4044\ldots \, , \qquad a = 0.031\ldots \, .
\label{ParamRealModel}$$
The way we have determined $p_{{\rm re}}(\rho)$ ensures that the first moment
indeed equals the value obtained by directly taking a global
average. The slight difference between \ref{ParamRealModel} and the value
for $\rho(\infty)$ in Table 1 is due to the fact that the latter is based
on a much larger amount of configurations.
As for the simple model presented before,
the prediction for $C(y) = a \approx 0.031$ agrees
within error bounds with what we expect for the asymptotics
of the two-point function at the critical point.
\medskip\noindent
\centerline{\psfig{figure=dens_fluc.ps}}
\smallskip\noindent
{\par\noindent\leftskip=4.5 true pc \rightskip=4 true pc
{\bf Fig.\ 5:}
Result for the distribution $p_{{\rm re}}(\rho)$ on $16 \times 16$
plaquettes in a simulation with $L=16384$ and $f/p = 10^{-4}$.
\par\noindent}
\medskip\noindent
Even though this basic test yields good values, one should
be aware that examining the system only in windows blurs the
distribution in Fig.\ 5 in several ways. Firstly, looking through
a window containing only 256 sites, one finds
a smearing of the density by $\Delta \rho \approx 0.02$
just because of statistical effects. Secondly, such windows may
accidentally intersect the boundary between two patches with low
and high densities. This has the effect that $p(\rho)$ is
estimated too large for intermediate $\rho$ and too small for
the extremal ones. Our choice of $16 \times 16$ plaquettes is
designed to make both effects reasonably small and is about
the best we can do.
\medskip\noindent
Let us now discuss Fig.\ 5 keeping these effects in mind.
For not too small values of $\rho$, $p_{{\rm re}}(\rho)$ increases slowly and
reaches a maximum around $\rho \approx 0.54$. Around
$\rho \approx 0.62$ there is a sharp decrease. The broad
distribution shows that fluctuations of $\rho$ are important.
A peak just below the percolation threshold $\rho_{{\rm perc}}$
and a steep decrease above it correspond to our expectation.
However, there is still a substantial contribution to
$p_{{\rm re}}(\rho)$ above $\rho_{{\rm perc}}$ which is not explained
by windowing effects. This is due to the patchy structure of
the system: Finite patches with $\rho > \rho_{{\rm perc}}$
are not destroyed instantly, but rather live for a time
which is the longer the smaller these patches actually are.
One could also say that the patchy structure demonstrates that
the system is actually correlated. In particular at small
distances ($y \ale 20$), the two-point function $C(y)$ retains
a $y$-depence (compare Section 2.1) for $f/p \to 0$ and
thereby exceeds the asymptotic constant $a$.
This correlation is necessary
to observe a non-trivial distribution of local densities,
but also leads to contributions to $p(\rho)$ above the
percolation threshold.
\medskip\noindent
So, if we want to work with a `realistic' $p(\rho)$, the best we
can do is to proceed with the one shown in Fig.\ 5.
Alternatively, one can work with an approximation to this
distribution where one suppresses the undesired $p(\rho)$
above the percolation threshold by hand. One such approximation
we have examined is a linear one, i.e\ $p_{{\rm lin}}(\rho) \sim \rho$
for $\rho < 0.59$ and $p_{{\rm lin}}(\rho) = 0$ for $\rho > 0.59$.
This linear distribution yields $\rho_c \approx 0.395$ and
$a \approx 0.019$ (both somewhat too small).
\medskip\noindent
\centerline{\psfig{figure=ns_fluc.ps}}
\smallskip\noindent
{\par\noindent\leftskip=4.5 true pc \rightskip=4 true pc
{\bf Fig.\ 6:}
The cluster-size distribution obtained from a simulation of
the full model with $L=16384$ and $f/p = 10^{-4}$ (full line),
the one based on $p_{{\rm re}}(\rho)$ (long dashes) and
the result obtained from $p_{{\rm lin}}(\rho)$ (short dashes).
\par\noindent}
\medskip\noindent
The determination of the cluster distribution $n(s)$ and $K(y)$
is a complicated combinatorial problem which we solve again
using simulations. One creates configurations
with densities distributed according to the
given $p(\rho)$ and then measures $n(s)$ and $K(y)$.
We have created 100000 configurations on a $512 \times 512$
lattice distributed according to $p_{{\rm re}}(\rho)$
and 70000 configurations on a $1024 \times 1024$
lattice for $p_{{\rm lin}}(\rho)$. Fig.\ 6 shows the cluster-size
distribution $n(s)$ obtained in this manner together with
the one obtained from a simulation of the full model.
For small cluster sizes ($s < 100$), all three distributions
are close to each other. However, at larger $s$ the distributions
based on a globally given $p(\rho)$ decay faster than the
true $n(s)$. The corresponding exponent is $\tau \approx 2.48$
for the distribution $p_{{\rm re}}(\rho)$ and $\tau \approx 2.44$ for
$p_{{\rm lin}}(\rho)$ -- both much closer to the mean-field
value $\tau = 5/2$ \cite{21} than to the true value \ref{REStau}.
In Fig.\ 6 one also observes a peak in the cluster-size
distribution corresponding to $p_{{\rm re}}(\rho)$ for $1 \cdot 10^5 \le s
\le 2.5 \cdot 10^5 \approx 512^2$, i.e.\ just below the volume
of the system. This is due to the non-vanishing of $p_{{\rm re}}(\rho)$
for $\rho > \rho_{{\rm perc}}$ which leads to clusters spanning
a finite (and large) fraction of the system.
\medskip\noindent
\centerline{\psfig{figure=cor_fluc.ps}}
\smallskip\noindent
{\par\noindent\leftskip=4.5 true pc \rightskip=4 true pc
{\bf Fig.\ 7:}
The full line shows the correlation function $K(y)$ obtained
from a simulation of the full model with $L=16384$ and $f/p = 10^{-4}$.
The result for $p_{{\rm re}}(\rho)$ is shown by the line with long dashes,
the one obtained from $p_{{\rm lin}}(\rho)$ is indicated by the shorter dashes.
\par\noindent}
\medskip\noindent
Fig.\ 7 shows the probability
$K(y)$ to find two trees at distance $y$ inside the same cluster.
Here, the results obtained from the two $p(\rho)$'s deviate more
notably from the result obtained by simulation of the full
model (full line). As noted above, the distribution $p_{{\rm re}}(\rho)$
gives rise to percolating clusters
and produces a constant background in $K(y)$ of approximately
$0.0552$. After subtracting this background, one can fit $K(y)$
with \ref{DEFeta} for $\xi_c \approx 180$ and $\eta \approx 0.92$.
This value for $\eta$ is more than twice as large the true one \ref{RESeta}.
For $p_{{\rm lin}}(\rho)$ one finds good agreement with the form \ref{DEFeta}
for $\xi_c = L/2$ and $\eta \approx 0.95$.
\bigskip\noindent
\subsection{Summary and outlook on global models}
\medskip\noindent
We have shown that the distributions $p_{{\rm re}}$ and $p_{{\rm lin}}$ lead to
a power law for $n(s)$. One can check that the same is true
for other $p(\rho)$'s that are cut off at $\rho_{{\rm perc}}$
in a way similar to $p_{{\rm lin}}$. Thus, a power law in $n(s)$ arises
automatically from a description in terms of
global quantities and need not be a signal for criticality
in a conventional sense. However, in all examples for $p(\rho)$
discussed so far, we obtained values for $\tau$ and $\eta$ that
are unsatisfactorily larger than those of the full model.
Nevertheless, a description in terms of $p(\rho)$
can still be forced to work because for two-dimensional critical
percolation one has $\tau \approx 2.055$ and $\eta \approx 0.208$
\cite{23} -- values which are smaller than the ones in Table 2.
So, one can obtain the desired value e.g.\ of $\tau$
by peaking the distribution $p(\rho)$ more prominently just
below $\rho_{{\rm perc}}$. Adjusting just $\tau$ to its correct
value can be expected to also give a reasonably good approximation
for $\eta$. Afterwards, both $\rho_c$ and $a$ may be
tuned to the desired values by adding another peak in $p(\rho)$
for smaller densities (which does not contribute to large clusters
and thus affect the asymptotics of cluster quantities). However,
there does not seem to be a natural way to make these adjustments.
In particular, local densities above $\rho_{{\rm perc}}$ do exist
in the full model which would have to be discarded by hand
in a global model in order to obtain the correct $K(y)$.
\medskip\noindent
In one dimension the spatial structure becomes irrelevant
at the critical point \cite{15}. We have seen in this
Section that this can be generalized to the qualitative features
of the critical correlations in two dimensions, but not to the
quantitative details. In contrast to the one-dimensional case we had
no analytical tools at our disposal and have therefore not been
able to derive the distribution $p(\rho)$ of local densities
explicitly. The introduction
of block-spin variables is reminiscent of real-space renormalization
group ideas and it would be interesting to see if they can be used
to find $p(\rho)$. However, one would have to go beyond the
block-spin renormalization-group study of \cite{18}. Firstly,
one would have to admit densities different from 0 or 1 for the block-spin
variables, and moreover the dynamics should not be treated
just in mean-field approximation.
\medskip\noindent
Two-dimensional percolation is believed to be conformally invariant
(see e.g.\ \cite{26}). The globalized models are just ensembles
of percolation problems and should therefore be conformally
invariant as well. It would be interesting to know if also the
stationary state of the full model in two space dimensions
is conformally invariant, even if the standard techniques
of conformal field theory would probably not say much about
quantities like cluster sizes.
\bigskip\noindent
\section{Conclusions}
\medskip\noindent
In this paper we have again looked at several aspects
of the two-dimensional forest-fire model. Firstly,
we have shown that the two length scales $\xi$ and
$\xi_c$ have different critical exponents. That this
might be possible had been suggested by a study of
the one-dimensional model \cite{15} which illustrates
that one-dimensional systems can provide useful insights
because of their relative simplicity even if one is
actually interested in higher-dimensional versions.
This result shows that in generic non-equilibrium systems
geometric objects and the usual (occupancy) correlation
functions can behave completely differently.
For an equilibrium system as the Ising model such an observation
was already made some time ago in \cite{27}. In this case,
percolation occurs away from the critical point, i.e.\
the two length scales are so different that they diverge
at different temperatures (see e.g.\ \cite{28,23}).
\medskip\noindent
In order to show that $\nu_T \ne \nu$ we had to improve
the error bounds of earlier investigations. As a by-product we
have also improved the accuracy of other critical exponents.
It may be possible that one could still improve the error
bounds by another digit using optimized code on today's most
powerful computers. Historically, Monte-Carlo simulations have
already several times lead to values for the critical exponents that
had to be corrected later on \cite{8,12,13,14}
and as we have shown here, some of them were still not
treated adequately. Therefore, it may be desirable to perform
yet another independent verification of the results presented
here, but a further increase of accuracy may not be necessary
for this end.
\medskip\noindent
The second part of the paper focussed on a globalized model.
We found that one can easily obtain power laws in the cluster-size
distribution by discarding the spatial structure, i.e.\
by making the usual two-point function independent of the spatial
coordinates. This generalizes a result obtained for one dimension
in a previous paper \cite{15} and is line with the observation
in \cite{16}, based on a different one-dimensional model,
that one can obtain power laws in clusters or avalanches by
global (`coherent') driving. However, some quantitative
predictions of the globalized model did not work out satisfactorily.
One reason is that the full two-dimensional forest-fire
model has non-trivial two-point functions at least at small distances
which is also reflected by the existence of patches with local densities
above the percolation threshold. In addition, the full model exhibits
many critical exponents that cannot be described by a global model.
\medskip\noindent
A study of the usual correlation functions would also be desirable
in other models of self-organized criticality where they have not
yet been investigated. This could help to clarify to what extent
the process of self-organization can be regarded as a global
phenomenon. Two-point correlation functions would also be important
quantities to examine in experiments. It would e.g.\ be interesting
to extract the spatial correlation functions of the local heights
and slopes from the experimental data of \cite{3}.
\medskip\noindent
Even for the two-dimensional forest-fire model
there are still many issues we have not looked at, including
e.g.\ finite-size effects. We have only looked at the regime
$f V \gg p$ which is close to the thermodynamic limit. One
could also look at a different limit, namely
$f V \ll p$ where the first-order approximation of \cite{15}
applies independent of the spatial dimension. In this limit, trees
grow until the lattice is full and after a certain time of rest, lightning
destroys all these trees and the process starts again.
It would be interesting to see how this behaviour crosses over to
the critical behaviour studied here as the volume of the system
is increased, and if finite-size scaling can be observed.
\vskip 0.8 cm
\displayhead{Acknowledgments}
\medskip\noindent
Useful discussions with S.\ Clar, M.\ Hasenbusch and D.\ Stauffer
are gratefully acknowledged.
The work of A.H.\ has been funded by the Deutsche Forschungsgemeinschaft.
\sectionnumstyle{Alphabetic}
\newsubsectionnum=\z@\@resetnum\sectionnum=0
\vskip 1.6 cm
| proofpile-arXiv_065-451 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
In this work we will study the behaviour of the $\rm Q \cdot Q$
interaction in a nucleus as a function of the strength of the
interaction. Although this is a model study it helps to make things
concrete by focusing on a particular nucleus. We choose $^{10}Be$.
This nucleus is of
particular interest because in a $0p$ space calculation with $\rm Q \cdot
Q$ there are some interesting degeneracies. For example, the $2_1^+$
and $2_2^+$ states are degenerate, and both have orbital symmetry
[400]. The $L=1~S=1$ states with orbital symmetry [330] and [411] are
degenerate. Thus we have two sets of degenerate triplets $J=0^+,~1^+$
and $2^+$ emanating from the above two orbital configurations. These
degeneracies can easily be found by applying the $SU(3)$ formula:
\[E(\lambda \mu)=\bar{\chi}\left[-4(\lambda^2+\mu^2+\lambda\mu+ 3
(\lambda+\mu))+3L(L+1) \right] \]
If we write the interaction as $\chi Q \cdot Q$ then, in the $0p$
space and when we change $\chi$, the degeneracies will not be
removed. All that will happen is that the energies of all states will
be proportional to $\chi$, and so the spectrum will be blown up or
shrunk as $\chi$ is made larger or smaller.
What happens though if we include contributions from other major
shells? The $\rm Q \cdot Q$ interaction will not connect $\Delta N=1$
states because of parity arguments, but there will be $\Delta N=2$
admixtures. Indeed, the concept of an $E2$ effective charge is often
illustrated by using a $\rm Q \cdot Q$ interaction to obtain $\Delta N=2$
admixtures.
\section{The Hamiltonian and Choice of Parameters}
It has been shown by Bes and Sorensen~\cite{bes} that if in the
valence space ($0p$) the appropriate $\rm Q \cdot Q$ interaction is
$-\chi_0 Q \cdot Q$, then in a space which includes $\Delta N = 2$
excitations the appropriate strength is $-\frac{\chi}{2} Q \cdot
Q$. We wish to vary the interaction, and we therefore parameterize it
as
\[V=-t \left (\frac{\chi_0}{2} \right ) Q \cdot Q \]
\noindent so that for $t=1$ we have the standard choice of Bes and
Sorensen~\cite{bes}. For $^{10}Be$, we have $\chi_0=
0.36146~MeVfm^{-1}$.
The Hamiltonian we use is
\[ H=\sum_i T_i + \frac{1}{2}m \omega^2 r_i^2 - \sum_{i < j} t
\frac{\chi_0}{2} Q(i) \cdot Q(j) \]
\noindent We perform shell model calculations using the OXBASH shell
model code~\cite{oxbash} in the space $(0p)^6$ plus $2 \hbar \omega$
excitations. As mentioned in the introduction, when harmonic
oscillator wave functions are used in a single major shell, the single
particle terms in the above Hamiltonian are constant, so
that the only part of the Hamiltonian that affects the spectra is the
two-body term. The separation of energies is linear in $t$ -the wave
functions are unaffected by the choice of (positive) $t$.
However, when $\Delta N =2$ excitations are allowed, the linear terms
are no longer constant, and the behaviour as a function of $t$ is more
complicated. We will now study this behaviour as a function of $t$ in
$^{10}Be$.
\subsection{Removal of Degeneracies}
In Table I we present $t=1$ results for $^{10}Be$ in a large space
$(0p)^6$ plus all $2 \hbar \omega$ excitations. We present the results
for the energies of $J=1^+$ and $2^+$ $T=1$ states, as well as $B(M1)$
and $B(E2)$ transitions from the ground state to these states. More
precisely, it is the isovector $B(M1)$ in units of $\mu_N^2$ and
isoscalar $B(E2)$ ($e_p=1,~e_n=1$) in units of $e^2 fm^4$.
We see that there are many degeneracies still present in the large
space $e.g.$ four $J=1^+~T=1$ states at 12.12 $MeV$ and three at 13.90
$MeV$, as well as three $J=2^+~T=1$ states at 12.12 $MeV$. These
degeneracies clearly correspond to various $S,~T$ combinations for
states of given $L$ and orbital symmetry [$f$].
However, other `accidental' degeneracies which were present in the
small space $(0p)^6$ are no longer present. The $J=1^+_1$ and $1^+_2$
states at 3.74 $MeV$ and 7.31 $MeV$ are linear combinations of the
$L=1$ [330] and $L=1$ [411] configurations (actually they are parts of
$J=0^+,~1^+,~2^+$ triplets). In the small space, these two $J=1^+$
states (or triplets) were degenerate -now one of the states is almost
at twice the excitation energy of the other.
The $2_1^+$ and $2_2^+$ states are no longer degenerate. The $2_1^+$
state is at 2.19 $MeV$ with $B(E2)\uparrow$ from the ground state of
63.8 $e^2 fm^4$, whilst the $2_2^+$ state is at 3.40 $MeV$ with
$B(E2)\uparrow=113.4~e^2 fm^4$. Note that, contrary to experiment, the
second $2^+$ state is the one most strongly excited. When a reasonable
spin-orbit interaction is added to the Hamiltonian the situation is
corrected.
It should be pointed out that in perturbation theory, in which only
the {\em direct} part of the particle-hole interaction of $\rm Q \cdot Q$
is used to renormalize the interaction between two particles in the
valence space, the degeneracies above would {\em not} be removed. The
relevant diagram is the familiar Bertsch-Kuo-Brown bubble (or phonon)
exchange between two nucleons~\cite{bert,kbrown}. For a simple $\rm Q
\cdot Q$ interaction, this diagram simply renormalizes the strength of
the $\rm Q \cdot Q$ interaction. Clearly, changing the strength in the
valence space will not remove the degeneracies.
Thus, the shell model diagonalization implicitly contains effects
beyond the direct bubble diagram. Furthermore, these effects are quite
important.
\subsection{Change in the Nature of the Ground State as $t$ Increases}
We now vary $t$ over the range $0 < t < 2$. In Fig. 1 we plot as a
dot-dash curve the value of $E/t$ for the lowest $2^+$ state, the one with
finite but small $B(E2)$ strength from ground. We also plot $E/t$ as a
solid line for the state with the strongest $B(E2)$ from ground. It
starts off at $t=1$ as the second $2^+$ state. In the $0p$ space, the
$2_1^+$ and $2_2^+$ states would be degenerate, and the curve for
$E/t$ vs. $t$ would be a horizontal line. However, in the
$0p+2\hbar\omega$ space there is a dependence on $t$ (and more so for
the solid curve).
But, for $t\approx 1.8$ and beyond, all the $B(E2)$ strength from
ground state goes to the new lowest $2^+$ state. Furthermore, the
value of $E/t$ becomes constant for $t \geq 1.8$ $i.e.$ the curve
becomes horizontal. Clearly, the nature of the ground state changed
beyond $t=1.7$. We will now examine this change in more detail.
If the only thing that happened was that there was a new $J=0^+$
ground state, then of course there would be a sudden change in the
$B(E2)\uparrow$'s from this new ground state to the $2^+$
states. However, the static quadrupole moments of the $2^+$ states
themselves would not change.
For $t=1.1$ the $B(E2)\uparrow$ to the $2_1^+$ state is 40.61 $e^2
fm^4$ and to the $2^+_2$ state 158.0 $e^2 fm^4$. The calculated
static quadrupole moments are respectively 11.56 $e fm^2$ and -11.46
$e fm^2$. As discussed by Fayache, Sharma and Zamick in
Ref.~\cite{qqt}, this is consistent with the two $2^+$ states being
prolate, with about the same intrinsic quadrupole moment $\rm Q_0$, but
the lower state would have $K=2$ and the upper one $K=0$. Indeed, for
$K=0$ $\rm Q(2^+)=-2/7Q_0$, and for $K=2$ $\rm Q(2^+)=+2/7Q_0$.
The behaviour for $t=1.1$ is maintained for $t=1.3$ and 1.5, but for
$t=1.7$ and up to $t=1.75$ there is a big drop in $B(E2)_{0_1^+
\rightarrow 2^+_2}$. The other three quantities do not change, not even
-strangely enough- $\rm Q(2^+_2)$.
Remember that for $t=1.75$ we are still below the critical $t$ for
which the $J=0^+$ ground state changes its nature. What is clearly
happening is that a third $2^+$ state has crossed over and came below
what was formerly the $2^+_2$ state. This is confirmed by noting that
at $t=1.75$ the $B(E2)_{0^+_1 \rightarrow 2^+_3}$ is very large (162.1
$e^2 fm^4$). Clearly, in going from $t=1.7$ to $t=1.75$, the $2^+_2$
and $2^+_3$ states have interchanged positions. The fact that the
static quadrupole moments of the two states are about the same means
that both states can be associated with two different prolate $K=0$
bands which have the same deformation.
Next we consider $t=1.76$. Here we are just beyond the critical $t$,
and the nature of the ground state has changed. Now the $B(E2)$ to the
$2^+_1$ state is much weaker, and the $B(E2)$ to the $2^+_2$ state is
strong. This suggests that the new $0^+$ ground state and the new
$2^+_2$ state are members of a new rotational band, and that both of
these states have come down in energy together.
When we go from $t=1.76$ to $t=1.78$ there is a big change, but the
results stabilize beyond that. Now the $B(E2)$ to the $2_1^+$ state is
the strongest (140 $e^2 fm^4$), and the {\em signs} of the static
quadrupole moments change. Now $\rm Q(2^+_1)$ is negative, and $\rm Q(2^+_2)$
is positive.
What is clearly happening is that there is another cross-over. What
was formerly the $2^+_3$ state at $t=1.7$ first crosses the the
$2^+_2$ state at $t=1.75$ (as mentioned above), and now crosses the
$2^+_1$ state at $t=1.78$. By $t=1.78$ and beyond, the $0^+$ and $2^+$
members of a new band have become the lowest two states, and the
results stabilize.
\section{Interpretation of the New Band: States with Integer Occupancies}
In Fig. 2 we plot the rapid descent of the $J=0^+$ and $2^+$ members
of the new rotational band. We start from $t=1$, but if we project
backward we see that for small $t$ the band emanates from the $2 \hbar
\omega$ region. To better ascertain the nature of the new band, we
give in Table III the occupancies of the single-particle levels that
were used in this calculation {\em i.e.} $0s$, $0p$, $1s-0d$ and
$1p-0f$.
At $t=1.7$, just before the critical value, the $J=0^+$ ground state
is normal. The occupancy of the four major shells (in the order
mentioned above) is 3.84, 5.78, 0.18 and 0.20. The first excited $0^+$
state at 0.585 $MeV$ has occupancy 4, 4, 2 and 0. Clearly two nucleons
have been excited from $0p$ to $1s-0d$.
What is at first surprising is that, for this state, the occupancies
are {\em precisely integers}. This is not an isolated example. It is
also true at $t=1.7$ for the second $2^+$ state at 4.33 $MeV$.
When we go to $t=1.9$, we have passed the critical value, and things
have settled down. the lowest $0^+$ and $2^+$ states are now the
$2p-2h$ states, both with the integer occupancies 4, 4, 2 and 0.
Whereas most states do not have integer occupancies, there are many
which do. These are at higher energies. For example, for $t=1.7$,
there are other states with the occupancy 4, 4, 2, 0 at 21.64, 22.95,
22.97, 25.32 and 35.75 $MeV$. These are states with occupancy (3, 4,
1, 0) at 43.6 and 44.2 $MeV$. The latter correspond to lifting one
nucleon through two major shells.
Why do we get such a simple behaviour for the $2p-2h$ states? The
answer involves a special feature of the $\rm Q \cdot Q$ interaction: all
matrix elements in which two particles in a major shell $N$ scatter
into a major shell $N \pm 1$ vanish. This is due to a parity selection
rule. For example, $\langle 0p~0p | Q\cdot Q | 0d~0d \rangle$ factors
into $\langle 0p |Q | 0d \rangle \langle 0p |Q | 0d \rangle$, and each
of these factors vanishes because of this parity rule.
Carrying the argument further, there can be no matrix element coupling
the $(0s)^4(0p)^4(0d-1s)^2$ configuration with other configurations
such as $(0s)^4(0p)^6$ or $(0s)^4(0p)^5(0f-1p)$ etc...
Also, in our calculation we have limited the space to 2 $\hbar \omega$
excitations. Once we create the state with occupancy (4,4,2,0) our model
space does not permit further excitations. This explains the integer
occupancy. Presumably if we enlarged the space to include 4 $\hbar
\omega$ excitations we would no longer have the integer occupancies. It
would also be of interest to study the $4p-4h$ states.
This also explains why, as we increase $t$ towards 1.76, the descent
of the new band is so simple. Since there is no mixing with the other
configurations, the $2p-2h$ $J=0^+$ and $2^+$ states can just slip
down below the $(0p)^6$ states.
The rapid descent of the $2p-2h$ states can be understood in terms of
the Nilsson model. To form the $2p-2h$ state, we take two nucleons
from the $0p$ shell and put them in the Nilsson orbit ($Nm_3\Lambda$)
with quantum numbers (220). This orbit comes down rapidly in energy as
the nuclear deformation is increased. The Nilsson one-body deformed
Hamiltonian can be obtained from the $\rm Q \cdot Q$ two-body interaction
by replacing $\rm Q \cdot Q$ by $\rm Q \cdot \langle Q \rangle$ where $\langle
Q \rangle$ is the quadrupole moment of the intrinsic state.
\section{Closing Remarks}
In this work, we have studied the properties of the interaction
$-t\frac{\chi_0}{2} Q \cdot Q$ as a function of the coupling strength
$t$ in an extended model space which includes all $2 \hbar \omega$
excitations beyond the valence space. Using $^{10}Be$ as an example,
we found that states that were `accidentally' degenerate in the $0p$
valence space ($e.g.$ $2_1^+$ and $2_2^+$ of orbital symmetry [42], or
the $J=0^+,1^+,2^+$ triplets of orbital symmetries [411] and [33]),
are no longer degenerate in the extended space. This means that the $2
\hbar \omega$ admixtures do more than renormalize the coupling
strength of $\rm Q \cdot Q$.
The extended model space allows for $2p-2h$ admixtures and indeed, for
sufficiently large $t$ ($t \geq 1.8$), the $J=0^+$ and $2^+$ members
of this new band become the new $0_1^+$ and $2_1^+$ states. We find
that this band, unlike the `normal' ground state band for $t < 1.7$,
has integer occupancies 4, 4, 2 and 0 for $0s$, $0p$, $1s-0d$ and
$1f-0p$ respectively. There is no mixing between the new $2p-2h$ band
and the $0p-0h$ band. This is a special feature of the $\rm Q \cdot Q$
interaction.
On a speculative level we may argue that, for the $2p-2h$ band
above, we should use a value of $t$ considerably larger than 1. We
don't have enough model space to renormalize the two-body interaction
in this band. Just as the interaction between the $(0p)^6$ states
gets renormalized by the configurations in which one nucleon is
excited through 2 major shells, so the interaction for the $2p-2h$
state would get renormalized by allowing at least one nucleon to be
excited through 2 major shells. But this would be a 4 $\hbar \omega$
state which, for practical reasons, we don't have in our model space.
We can use the $\rm Q \cdot Q$ interaction to place these $2p-2h$ bands at
the correct energies, but we will need other components of the
realistic nucleon-nucleon interaction in order to mix these bands with
the $0p-0h$ bands. For example, we could use the dipole-dipole or
octupole-octupole parts of the nucleon-nucleon
interaction. Alternatively one can work directly with realistic
interactions.
There has been much progress in large-basis shell model calculations
in light nuclei. For example, there is the work of W.C. Haxton and
C. Johnson~\cite{hax} where they actually get the superdeformed
$4p-4h$ state in $^{16}O$ at a reasonable energy, although perhaps not
with the full quadrupole collectivity. There is also the work of Zheng
$et. al.$~\cite{zheng} where up to $8 \hbar \omega$ excitations have
been included in calculations of nuclei ranging from $^4He$ to
$^7Li$. Also, Zamick, Zheng and Fayache~\cite{zzf} required
multi-shell admixtures to demonstrate the `self-weakening mechanism'
of the tensor interaction in nuclei.
Nevertheless, schematic interactions like $\rm Q \cdot Q$ still play a
primary role in describing nuclear collectivity throughout the
periodic table. They are of special importance for highly deformed
intruder states.
Surprisingly, there have been very few studies of schematic
interactions in multi-shell spaces. In the Elliott $SU(3)$ model,
momentum terms have been introduced to prevent $N=2$ admixtures in the
valence space~\cite{elliott}. This has lead to great simplicities and
beautiful results. There have been $R.P.A.$ studies with $\rm Q \cdot Q$
which involve $N=2$ mixing. These studies give $E2$ effective charge
renormalizations in the valence space and also the energies of the
giant quadrupole resonances, but they will not give us the highly
deformed states such as the $2p-2h$ state that we have found here.
We therefore feel that careful studies of the schematic interactions
in multi-shell spaces are important, and we hope that others will
agree.
\acknowledgements
This work was supported by a Department of Energy grant
DE-FG02-95ER 40940. One of us (L.Z.) thanks E. Vogt, B. Jennings,
H. Fearing, and P.Jackson for their help and hospitality at TRIUMF.
| proofpile-arXiv_065-452 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\subsection{Light yield measurement}
The absolute light yield for minimum--ionising particles was measured
in a cosmic ray telescope with an effective area of 12~x~12 cm$^{2}$.
The PMT pulse was integrated by a LeCroy 2249A ADC,
triggered by a threefold coincidence of the signals from the cosmic
trigger\footnote{The absolute trigger efficiency was determined to
be 98.9\%}. The ADC gate length was 100 ns.
Figure<~\ref{cos.ps} shows the response to cosmic ray particles
for the final tile/fiber/PMT layout.
\begin{figure}[tbp]
\centerline{\psfig{figure=DESY-96-139_1.ps,bbllx=0pt,bblly=140pt,bburx=550pt,bbury=650pt,height=8cm}}
\caption{\it Response of a scintillator tile to cosmic muons. The peak
around the pedestal value of 45 ADC counts is due to false triggers
and zero response due to photostatistics.}
\label{cos.ps}
\end{figure}
The ratio of triggers with no signal above pedestal
recorded in the ADC to the total number of triggers is equal to
(0.37$\pm$0.03\%), taking into account the trigger efficiency.
From this we obtain
an absolute light yield of 5.6$\pm$.1 photoelectrons.
\subsection{Uniformity measurement}
The tile uniformity was measured with a
collimated $^{106}$Ru source, a computer-controlled x-y scanning table
as scintillator support and an XP2020 photomultiplier.
The ADC was triggered by a coincidence
of two photomultiplier signals reading out a large trigger counter
located underneath the tile to be tested (figure~\ref{tab.ps}). To ensure that the
$^{106}$Ru source simulates minimum--ionising particles,
we measured the light yield of the tile also
with the cosmic telescope. The pulse height spectra of
cosmic muons and electrons from the $^{106}$Ru source are very similar
in shape, with peak values equal within 5\%.
A 3--mm--diameter collimator was used to measure the response
over a 20~x~20 cm$^{2}$ tile in steps of 2~mm in x and y.
The response is uniform within 5\%.
The distribution of the mean values
of the measurements for all positions is shown in figure~\ref{mean.ps}.
The slight bump below 120 channels is an effect occurring near the edge of the tile
where some particles leave the tile before crossing its full thickness.
\begin{figure}[tbp]
\centerline{\psfig{figure=DESY-96-139_2.ps,bbllx=130pt,bblly=55pt,bburx=450pt,bbury=600pt,height=8cm,angle=270}}
\caption{\it Scanning table uniformity measurement}
\label{tab.ps}
\end{figure}
\begin{figure}[tbp]
\centerline{\psfig{figure=DESY-96-139_3.ps,bbllx=15pt,bblly=150pt,bburx=550pt,bbury=670pt,height=8cm}}
\caption{\it Spread of response over a tile. The response is uniform within 5\%.}
\label{mean.ps}
\end{figure}
\section {Mechanical layout}
\subsection {Tile assembly}
The scintillator tiles are assembled in cassettes, made of 0.4~mm
thick stainless steel; they are 20~cm wide and vary in length,
containing between 1 and 10 tiles. In order for
the readout fibers to clear the neighboring scintillator tile
the 5~mm thick tiles are raised
at the readout end, supported by a 2.5~mm thick rohacell strip.
The total thickness of the cassette is
$\simeq$ 11~mm and represents about 5~\% of a radiation length (1.2~\% for the
scintillator and 4~\% for the stainless steel).
The length of readout fibers is $\simeq$ 300~cm.
The six readout fibers of one tile are glued together in a connector which
is fixed to the PMT housing. In addition to the six fibers
a seventh clear fiber is added to guide the light from a laser/LED system
to the PMT.
\subsection {Detector assembly}
The forward and rear
calorimeters are split in two halves, such that they can be withdrawn
from the beam pipe region during injection of the electrons and protons in HERA.
A group of 19~cassettes, which covers one half of each calorimeter face,
is glued on a 2~mm thick aluminium plate (0.02 radiation lengths) of 2~x~4~m$^2$.
Figure~\ref{layout_presam} shows the coverage of the calorimeter by the presampler.
Shown is the segmentation of the electromagnetic sections, which is
finer in the region not shadowed from the nominal interaction point by the
barrel calorimeter. The 20~x~20~cm$^{2}$ towers covered by the presampler tiles are shaded.
\begin{figure}[tbp]
\centerline{
\psfig{figure=DESY-96-139_4.ps,bbllx=0pt,bblly=100pt,bburx=580pt,bbury=680pt,height=8cm}
\psfig{figure=DESY-96-139_5.ps,bbllx=0pt,bblly=100pt,bburx=580pt,bbury=680pt,height=8cm}}
\caption{\it
Front view of the forward (FCAL) and rear calorimeter (RCAL). The 20~x~20~cm$^{2}$
white square in the center corresponds to the hole for the beam pipe. The coverage
of the presampler is indicated by the shaded region.}
\label{layout_presam}
\end{figure}
A 2.5~mm diameter tube is glued over the full length on the outside of each
cassette, positioned at the center of the tiles.
The tube guides a radioactive source for calibrating the light output
of the individual tiles and the gain of the PMT channels (see section 7.2).
\section {Photomultiplier tests}
\subsection{Performance specifications}
Due to the limited space available in the ZEUS detector it was decided
to use multi\-chan\-nel PMT's.
The magnetic field amounts to a few hundred Gauss in the area where
the PMT's are located and therefore adequate shielding is needed.
Since we measure pulse heights, the crosstalk between adjacent
channels in the tube is required to be less than 5\%.
Another requirement is
the size of the photocathode for a single channel, which must match the readout fibers
of one scintillator tile.
The Hamamatsu R4760 16-channel photomultiplier has been extensively tested
for our application (see also \cite{R476ref}). This is a 4~x~4 multichannel
PMT with a front face of 70~mm diameter. Each of the 16 channels
has a 10 stage dynode chain, but they all share the same
voltage divider. The diameter of the photocathode for each channel
is 8~mm. Our PMT's fulfill the following requirements:
\begin{itemize}
\item cathode sensitivity $>$ 45 $\mu$A/lm
\item minimum gain at 1000 V: 1~x~10$^6$
\item gain spread between channels, within one PMT assembly, less than a factor of 3
\end{itemize}
The crosstalk has been measured to be less than 3\%.
\subsection{HV supply and linearity measurement}
A high voltage system based on the use of a
Cockcroft--Walton generator has been developed \cite{henk_hv}.
Power dissipation is negligible compared to that
of a resistive voltage divider. The system can
be safely operated because of the low voltage input and provides
a protection against high currents (light leaks);
the maximum anode current is 100$\mu$A.
The HV units consist of a microprocessor board and the voltage
multiplier boards. The microprocessor performs the HV setting and monitoring.
The R4760 PMT operates in the range 800-1200V.
Tests showed that the optimum linearity is obtained (at the
cost of a slightly lower gain) if the dynode voltage differences
are distributed in the proportions 2:1:1:1:1:1:2:2:4:3, starting at the cathode.
Since a pulse height measurement is required,
good knowledge of the relation between input charge and output signal
is necessary. For the linearity measurement we used
an LED and a linear neutral density filter. As an example we show in
figure~\ref{pmlinea} the deviation from linear behaviour versus
anode charge for a HV setting of 1100V. The dotted line shows
the linear fit through the first four points, the dashed line represents
a polynomial fit through the last four points.
\begin{figure}[tbp]
\centerline{\psfig{figure=DESY-96-139_6.ps,bbllx=0pt,bblly=150pt,bburx=550pt,bbury=670pt,height=8cm}}
\caption{\it Results on the PMT nonlinearity.
The line shows the linear fit
through the first four points. The dashed line represents a
polynomial fit through the last four points.}
\label{pmlinea}
\end{figure}
At an anode charge of
25 pC the measured values are about five percent lower
than expected for linear behaviour. These
values vary strongly from channel to channel and from PMT to PMT.
\section {Readout system}
The readout system is a copy of the existing ZEUS calorimeter
readout system with some minor modifications \cite{caldwell}.
The PMT pulses are amplified and shaped by a pulse shaper circuit
mounted at the detector. The shaped pulse is sampled
every 96 ns (the bunch crossing
rate of the HERA storage ring) and stored in a switched capacitor analog pipeline.
After receipt of a trigger from the ZEUS detector, eight samples are
transferred from the pipeline to an analog buffer and multiplexed to ADCs.
The data are sent to a location outside the detector where the
digitisation and signal processing takes place.
The modifications consist of upgraded versions of the
shaping/amplifier~\cite{iris} and the digital signal processor and a different
mechanical layout of the analog front end cards.
\section {Results from cosmic ray measurements}
A cosmic ray test was performed to measure the light yield of
the 576 tiles ( 264 FCAL and 312 RCAL tiles) assembled in 76 cassettes.
The trigger system consists of eight cosmic ray telescopes.
Each of these consists of two scintillator pads, 20~cm apart,
with an area of 12~x~12~cm$^{2}$ which give, together
with a third scintillator counter of 240~x~12~cm$^{2}$, a trigger system with a
three-fold coincidence. The efficiency of each single telescope was
better than 98\%. The readout PMT was a 16-channel R4760 as used in
the final presampler design. The setup allows the measurement of 16 channels
simultaneously ( e.g. two cassettes with eight tiles each). The
trigger rate of the complete setup was about 50 counts/min
with about 6 counts/min for a single telescope. An example of the
cosmic-test measurement is given in figure~\ref{cos.ps}.
To compare the light yield of different tiles, all 16 channels of the
PMT were calibrated with a reference tile to correct for differences in
quantum efficiency (QE) and gain between the 16 PMT channels.
Figure~\ref{fig_tst} shows
the mean value for all 576 tiles normalized to one of them.
From the RMS value of the distribution we
conclude that the responses of all tiles are equal to within 12\%.
\begin{figure}[htbp]
\centerline{\psfig{figure=DESY-96-139_7.ps,bbllx=45pt,bblly=160pt,bburx=530pt,bbury=650pt,height=5.7cm}}
\caption{\it Average pulse height for cosmic muons normalized to one after having
corrected for the gain differences between individual PMT channels.}
\label{fig_tst}
\end{figure}
Figure~\ref{fig_npe} shows the mean number of photoelectrons for each tile
assuming a QE of 8.5\% which is the minimum value accepted for the
presampler PMT's (the mean QE for all channels is 11.6\%).
\begin{figure}[htbp]
\centerline{\psfig{figure=DESY-96-139_8.ps,bbllx=45pt,bblly=150pt,bburx=530pt,bbury=650pt,height=5.7cm}}
\caption{\it Average number of photoelectrons per tile per MIP}
\label{fig_npe}
\end{figure}
\vskip 3cm
\section {Calibration tools}
\subsection{Minimum--ionising particles in situ}
During the operation of ZEUS, halo muons
and charged hadrons are used to determine the response to single particles for each
individual channel. The high voltage setting common to the sixteen pixels of
one R4760 PMT is chosen such that the pixel with the least gain has
an average response to minimum--ionising particles which is a factor of ten
greater than the RMS noise level of the analog signal--processing front--end
electronics (0.05 pC). The pixel--to--pixel gain variation of about a factor of three
within one PMT results in a similar variation in the saturation levels.
The in situ calibration for the 1995 running period achieved a precision
of better than 5\% per tile.
\subsection{The radioactive source system}
We use an LED/laser system to monitor the gains of the PMT's and
a source system to monitor the combined response of tile, fiber and PMT.
The response to a $^{60}$Co source provides a relative calibration and quality
control of the individual channels of the presampler.
The source scans take place during shutdowns of HERA and provide
information on the long term behaviour of the light output of the
combination of scintillator and wavelength-shifting fiber.
\begin{figure}[htbp]
\centerline{\psfig{figure=DESY-96-139_9.eps,bbllx=203pt,bblly=0pt,bburx=0pt,bbury=608pt,angle=270,width=15cm}}
\caption{\it Example of a cassette with 9 scintillating tiles and source tube
glued on top, connected to source scanning system.}
\label{source_layout}
\end{figure}
Brass tubes with 2.5 mm outer diameter and 0.2 mm wall thickness
run over the full length of the cassette, positioned
in the middle (figure~\ref{source_layout}).
They guide the pointlike (0.8~mm diameter, 1~mm length, 74 MBq) source.
The source is driven in 2~mm steps via a 1.2~mm diameter
steel wire by a stepper motor~\cite{source_ref}
controlled by a PC. The PMT currents are integrated
with a time constant of 24 ms and read into a 16-channel 12-bit ADC card.
Figure~\ref{x3} shows as an example the superposition of the
responses of the tiles within one cassette as a function of
the location of the source. The different heights of the
maxima are mainly due to the different gains of
the 16 channels of the R4760 PMT, all supplied with the same
high voltage. The $^{60}$Co source is not collimated, as can be seen
in the shape of the individual peaks.
\begin{figure}[htbp]
\centerline{\psfig{figure=DESY-96-139_10.eps,width=8cm,height=10cm}}
\caption{\it Responses of the scintillating tiles within one cassette
to a $^{60}$Co source. The step width is approximately 2 mm.}
\label{x3}
\end{figure}
\clearpage
\section {Beam test results}
The influence of material in front of the ZEUS
calorimeter on its energy measurement
has been studied previously in several test beam runs with
the ZEUS forward calorimeter (FCAL) prototype \cite{prot}.
The corrections to the calorimetric measurements that can be derived from
presampling measurements have been studied in subsequent test periods
for both hadrons and electrons~\cite{marcel}. In the following we summarize
the most recent results obtained for electrons with the final presampler
design~\cite{adi}.
\begin{figure}[tbp]
\centerline{\psfig{figure=DESY-96-139_11.eps,bbllx=30pt,bblly=440pt,bburx=565pt,bbury=700pt,height=6cm}}
\caption{\it Experimental setup of the FCAL prototype and presampler in CERN
test beam. The presampler is mounted directly on the FCAL frontplate.}
\label{preprot}
\end{figure}
\subsection{Overview}
\normalsize
The presampler prototype
consists of an array of 4 x 4 scintillator tiles covering
an area of 80 x 80 cm$^2$. As shown in figure~\ref{preprot}
it is positioned directly in front of the ZEUS FCAL
prototype which has the same lateral size.
The depth of the calorimeter is 7 interaction lengths \cite{prot}.
Beam tests were performed in the X5 test beam of the CERN SPS West Area.
The prototype presampler detector is read out via an R4760 multichannel
photomultiplier using the Cockcroft--Walton HV system. Furthermore, the
final readout electronics was used for both the presampler and the FCAL
prototype modules.
The uranium radioactivity was used to
set the relative gains of the calorimeter
phototubes and 15 GeV electrons served to set
the energy scale.
Muons were used
to calibrate the presampler. The combined response of the
presampler and calorimeter was determined for electrons
in the energy range from 3-50 GeV. The amount of
material installed in front of the presampler varied between
0 and 4 radiation lengths (X$_0$) of aluminium.
During these studies, the position of both
calorimeter and presampler relative to the beam was fixed.
A delay wire chamber allowed the determination
of the impact point of the beam particles with an accuracy
of 0.5~mm.
Most of the data were recorded with a
defocussed beam about 10 cm in diameter, facilitating studies of uniformity and
position dependence of the energy correction algorithms.
\subsection{ The uniformity of the presampler response to muons }
Figure~\ref{f9-1} shows the mean presampler response to 75 GeV muons.
The position information was provided by the delay wire
chamber. The presampler signals for the incident
muons are normalized
to the response at the center of the tile and averaged over
uniformly populated rectangles of 90~x~5~mm$^2$. The nonuniformity
in the sum of the two bordering tiles is a few percent in the regions
of the fibers. In the horizontal direction (figure~\ref{f9-1}b) the tiles
are mounted within one cassette with no gaps between them.
In the vertical coordinate (figure~\ref{f9-1}c)
a signal drop is observed between the cassettes
due to the 1.4 mm gap between the scintillator tiles.
The nonuniformity averaged over the surface of a tile is
less than 1\%.
\begin{figure}[hb]
\begin{picture}(400,170)(0,0)
\put(0,18) {\psfig{figure=DESY-96-139_12.ps,bbllx=6pt,bblly=40pt,bburx=580pt,bbury=630pt,height=4.7cm}}
\put(130,0) {\psfig{figure=DESY-96-139_13.eps,height=6cm}}
\put(285,0){\psfig{figure=DESY-96-139_14.eps,height=6cm}}
\put(10,134){\large{a)}}
\put(160,80){\large{b)}}
\put(320,80){\large{c)}}
\end{picture}
\caption{\it
The muon response uniformity of the presampler near tile borders.\hspace{2cm}
a) A sketch of scanning region,
b) a horizontal scan (* represents the response of
tile 6, $\circ$ that of tile 7 and $\bullet$ the sum of both),
c) a vertical scan, perpendicular to the embedded fibers.
The dashed lines indicates the fiber positions.
The data in both plots are averaged over a uniformly populated rectangle
90~x~5~mm$^2$ in area with the long side perpendicular to the scanning direction.}
\label{f9-1}
\end{figure}
\subsection{The presampler response to electrons}
The electron beam used for the energy correction studies was 1 cm wide and 10
cm high, centered horizontally within one FCAL module and vertically on
one of the 20~x~5~cm$^{2}$ electromagnetic sections.
The presampler signal ($E_{pres}$) was obtained by summing all 16 tiles in
order to be sure to get the entire signal and because the
electronic noise contribution
was negligible. The signal from each tile was
normalized to its average response to muons, resulting in units we refer to
as ``MIP''. Aluminium plates of 3 cm thickness were used as the absorber
material. In the following three such plates together are referred to as
one radiation length, an approximation which is accurate to 1\%.
As examples of the calorimeter and presampler signal spectra
we show in figure~\ref{calvpres}
the energy distributions measured with the calorimeter ($E_{cal}$)
and the signal in the presampler for 25 GeV electrons for
aluminium absorber thicknesses ranging between 0 and 4 X$_0$.
The mean value for $E_{cal}$ decreases by more than 20\% but
the shapes of the distributions remain approximately gaussian.
The resolution deteriorates substantially in the presence of more than
2 X$_0$ of absorber material.
\begin{figure}[tbp]
\centerline{\psfig{figure=DESY-96-139_15.ps,bbllx=10pt,bblly=130pt,bburx=550pt,bbury=700pt,height=12cm}}
\caption{\it Signal distributions in the calorimeter and presampler for 25 GeV
electrons having passed through 0, 1, 2, 3, and 4 radiation lengths of
aluminium absorber.}
\label{calvpres}
\end{figure}
\clearpage
Table 1 shows the relative calorimeter signal loss and the average and RMS
values of the presampler signal spectra for the full range of electron
energies and absorber thicknesses.
The uncertainties presented are dominated by the statistical
precision.\\*[3mm]
\begin{tabular}{|c|c|rcl|rcl|rcl|}
\hline
Energy & Absorber & \multicolumn{3}{|c|}{Rel. Energy Loss} & \multicolumn{3}{|c|}{Presampler Avg} & \multicolumn{3}{|c|}{Presampler RMS} \\
(GeV) & ($X_0$) & \multicolumn{3}{|c|}{(\%)} & \multicolumn{3}{|c|}{(MIP)} & \multicolumn{3}{|c|}{(MIP)} \\
\hline
3& 1& \hspace*{1mm} 7.8& $\pm$ &1.0& 5.5& $\pm$ &0.1& \hspace*{2mm} 3.7& $\pm$ &0.1\\
\cline{3-11}
& 2& \hspace*{1mm} 18.0& $\pm$ &1.0& 10.3& $\pm$ &0.2& \hspace*{2mm} 5.3& $\pm$ &0.1\\
\cline{3-11}
& 3& \hspace*{1mm} 32.8& $\pm$ &1.0& 12.0& $\pm$ &0.2& \hspace*{2mm} 5.6& $\pm$ &0.1\\
\cline{3-11}
& 4& \hspace*{1mm} 49.4& $\pm$ &1.0& 12.1& $\pm$ &0.2& \hspace*{2mm} 5.5& $\pm$ &0.1\\
\hline
5& 1& \hspace*{1mm} 6.4& $\pm$ &0.6& 6.2& $\pm$ &0.2& \hspace*{2mm} 4.1& $\pm$ &0.1\\
\cline{3-11}
& 2& \hspace*{1mm} 13.0& $\pm$ &0.7& 12.8& $\pm$ &0.2& \hspace*{2mm} 6.5& $\pm$ &0.2\\
\cline{3-11}
& 3& \hspace*{1mm} 27.4& $\pm$ &0.7& 16.9& $\pm$ &0.3& \hspace*{2mm} 7.3& $\pm$ &0.2\\
\cline{3-11}
& 4& \hspace*{1mm} 42.5& $\pm$ &0.7& 19.5& $\pm$ &0.3& \hspace*{2mm} 7.1& $\pm$ &0.2\\
\hline
10& 1& \hspace*{1mm} 4.8& $\pm$ &0.5& 7.7& $\pm$ &0.2& \hspace*{2mm} 5.0& $\pm$ &0.1\\
\cline{3-11}
& 2& \hspace*{1mm} 10.1& $\pm$ &0.5& 18.2& $\pm$ &0.4& \hspace*{2mm} 8.9& $\pm$ &0.3\\
\cline{3-11}
& 3& \hspace*{1mm} 20.3& $\pm$ &0.5& 27.4& $\pm$ &0.4& \hspace*{2mm} 10.9& $\pm$ &0.3\\
\cline{3-11}
& 4& \hspace*{1mm} 34.6& $\pm$ &0.7& 32.7& $\pm$ &0.6& \hspace*{2mm} 11.3& $\pm$ &0.4\\
\hline
15& 1& \hspace*{1mm} 3.5& $\pm$ &0.4& 7.9& $\pm$ &0.2& \hspace*{2mm} 5.2& $\pm$ &0.2\\
\cline{3-11}
& 2& \hspace*{1mm} 7.8& $\pm$ &0.4& 20.4& $\pm$ &0.5& \hspace*{2mm} 9.6& $\pm$ &0.4\\
\cline{3-11}
& 3& \hspace*{1mm} 17.3& $\pm$ &0.4& 34.9& $\pm$ &0.6& \hspace*{2mm} 13.8& $\pm$ &0.4\\
\cline{3-11}
& 4& \hspace*{1mm} 29.6& $\pm$ &0.6& 46.0& $\pm$ &0.8& \hspace*{2mm} 14.6& $\pm$ &0.6\\
\hline
25& 1& \hspace*{1mm} 2.7& $\pm$ &0.3& 9.8& $\pm$ &0.2& \hspace*{2mm} 6.5& $\pm$ &0.2\\
\cline{3-11}
& 2& \hspace*{1mm} 5.9& $\pm$ &0.3& 26.6& $\pm$ &0.4& \hspace*{2mm} 12.8& $\pm$ &0.3\\
\cline{3-11}
& 3& \hspace*{1mm} 15.4& $\pm$ &0.3& 47.3& $\pm$ &0.5& \hspace*{2mm} 17.9& $\pm$ &0.4\\
\cline{3-11}
& 4& \hspace*{1mm} 24.4& $\pm$ &0.3& 66.1& $\pm$ &0.6& \hspace*{2mm} 20.9& $\pm$ &0.4\\
\hline
50& 1& \hspace*{1mm} 1.6& $\pm$ &0.2& 12.0& $\pm$ &0.3& \hspace*{2mm} 7.9& $\pm$ &0.2\\
\cline{3-11}
& 2& \hspace*{1mm} 4.6& $\pm$ &0.2& 35.6& $\pm$ &0.6& \hspace*{2mm} 16.4& $\pm$ &0.4\\
\cline{3-11}
& 3& \hspace*{1mm} 11.5& $\pm$ &0.5& 72.7& $\pm$ &2.1& \hspace*{2mm} 27.7& $\pm$ &1.5\\
\cline{3-11}
& 4& \hspace*{1mm} 19.8& $\pm$ &0.3&108.1& $\pm$ &1.2& \hspace*{2mm} 31.6& $\pm$ &0.8\\
\hline
\end{tabular}
\vskip 5mm
Table 1: {\it The relative decrease in the calorimeter signal and
the average and RMS values of the presampler signal spectra for
each electron energy and each aluminium absorber thickness used
in the test beam studies. The uncertainties shown are dominated by the
statistical precision.}
\clearpage
\subsubsection {Contribution of backscattering to the presampler signal}
The comparison of presampler signal spectra from incident muons with those
of incident electrons allowed us to estimate the contribution of
backscattering from the electromagnetic showers in the uranium calorimeter.
Figure~\ref{backscat} shows this comparison for 5, 15, 25 and 50 GeV electrons.
We determine
the average relative increase to be 1.45, 1.65, 1.80, and 2.16 respectively,
to an accuracy of 2\%. By adding absorber upstream of the presampler and
displacing both absorber and presampler several meters upstream to measure
the decreased contribution from backscattering, we ascertained that the
backscattering contribution is not increased by the presence of the absorber
in front of the presampler. Thus we can be sure that the backscattering
contribution remains at the level of 1 MIP and can be neglected at the level
of 10\% of the size of the signals we use for the electromagnetic energy
correction. We also measured the backscattering contribution from hadronic
showers and found for 15 and 75 GeV incident pions values for the average
relative increase less than 1.5 and 2.0 respectively.
\begin{figure}[tbp]
\centerline{\psfig{figure=DESY-96-139_16.ps,bbllx=50pt,bblly=270pt,bburx=530pt,bbury=540pt,height=8cm}}
\caption{\it
A comparison of the presampler signal spectra for 5, 15, 25, and 50 GeV
electrons to that for muons. The smooth curve shows the response to muons.
The presampler
signal has been normalized to the average value of the muon spectrum. Each
spectrum contains 2000 entries.}
\label{backscat}
\end{figure}
\subsubsection{Electron energy correction}
Figure~\ref{f9-4} shows the correlation between the presampler signal (normalized to
the average signal of a minimum--ionising particle) and
the calorimeter
signal for 25 GeV electrons as a function of the amount of absorber material.
\begin{figure}[tbp]
\centerline{\psfig{figure=DESY-96-139_17.ps,bbllx=25pt,bblly=270pt,bburx=535pt,bbury=545pt,height=8cm}}
\caption{\it Calorimeter versus presampler response for 25 GeV electrons
and absorber material ranging from 1 to 4 X$_0$. The line represents
the fit to the data according formula (1).}
\label{f9-4}
\end{figure}
We have considered a variety of parametrisations for the relationship between
calorimeter and presampler responses. In the well--defined
environment of a test beam
the correction is straightforward and depends on the incident
energy and the amount of absorber material, both of which are precisely known.
\begin{figure}[tbp]
\vskip -2cm
\centerline{\psfig{figure=DESY-96-139_18.ps,bbllx=50pt,bblly=265pt,bburx=530pt,bbury=540pt,height=8cm}}
\caption{\it Average calorimeter response normalized to the
electron energy versus the electron energy before and after correction}
\label{f9-2}
\end{figure}
In a detector environment the amount of absorber material in
front of the calorimeter is not uniformly distributed, arising from cables,
support structures, etc. One can, however, identify regions where the
average amount of absorber material is roughly known. For this reason
we show here the result obtained with
one set of correction constants common to the 1 and 2 X$_0$
data set and one for the 3 and 4 X$_0$ data set.
The relation between the measured mean values of $E_{cal}$ and $E_{pres}$
has been parametrised in a linear approximation:
\begin{equation}
\label{eq:etrue}
E_{cal}= a_0 + a_1 E_{pres}
\end{equation}
The result for the two data sets for 25 GeV electrons is shown in figure~\ref{f9-4}.
The parameters $a_i$ depend on the amount of
material and on the electron beam energy. This correction algorithm allows
for a linear energy dependence of the
parameters $a_i$: $a_i =
\alpha_i +\beta_i E_{beam}$. We neglect the dependence on the amount of
absorber material in order to estimate the success of the algorithm when
the amount of absorber varies within the data sample.
The parameters $\alpha_i$ and $\beta_i$ are determined by
minimising the difference of the beam energy and the
corrected calorimeter signal,
The results
for the corrected calorimeter response for electrons in the energy
range 3-50 GeV, are
shown in figure~\ref{f9-2}. This procedure provides a
correction accurate to 3\%
for the energy range studied here, but for an overcorrection
of about 10\% for the 3~X$_0$ data points at low energy.
For electron energies greater than
5 GeV and for absorber thickness less than 2 X$_0$, the values relevant to
the operation of the ZEUS detector, this simple correction algorithm
yields a systematic precision of 2\%.
The improvement in the energy resolution as well as in the energy scale
is shown in figure~\ref{mixture}. The energy distribution
of 25 GeV electrons for a merged 1-4 X$_0$ data set is shown before
and after correction.
\begin{figure}[htbp]
\centerline{\psfig{figure=DESY-96-139_19.ps,bbllx=25pt,bblly=270pt,bburx=530pt,bbury=540pt,height=6.5cm}}
\caption{\it Reconstructed energy distributions for 25 GeV electrons for a mixture of the 1-4 X$_0$ data before and after application of the correction algorithm}
\label{mixture}
\end{figure}
\section {Summary and conclusions}
We have designed, built, installed and operated a scintillator-tile presampler
for the forward and rear calorimeter of the ZEUS detector. The scintillation
light from the tiles is collected by wavelength-shifting fibers and guided
to multi-anode photomultipliers via clear fibers. The signals are shaped,
sampled and pipelined in the manner employed for the calorimeter itself,
easing the integration of the presampler in the ZEUS data acquisition system.
The performance of the tiles and fiber readout were monitored with a
cosmic--ray telescope and with collimated sources during production.
The single-particle detection efficiency is greater than 99\% and the
response uniformity over the area of each of the 576 20~x~20~cm$^{2}$ tiles is
better than 5\%. An LED flasher system and scans with radioactive sources
have proven useful diagnostic tools since the installation of the presampler.
The in situ calibration with minimum--ionising particles during the 1995
data--taking period achieved a precision better
than 5\% per tile. Test beam studies of a presampler prototype with the
final geometry and readout in combination with a prototype of the ZEUS
forward calorimeter verified the efficiency and uniformity results
and allowed the determination of backscattering contributions. Tests
with electrons were performed in the energy range 3--50 GeV with
0--4 radiation lengths of aluminium absorber. These studies
have proven the feasibility of an electron energy
correction accurate to better than 2\% in the energy range and for the
configuration of inactive material relevant to the ZEUS detector.
\section {Acknowledgements}
We would like to thank the technical support from the institutes
which collaborated
on the construction of the presampler,
in particular W. Hain, J. Hauschildt, K. Loeff\-ler, A. Maniatis,
H.--J. Schirrmacher (DESY),
W. Bienge, P. Pohl, K.--H. Sulanke (DESY--IfH Zeuthen),
M. Gospic, H. Groenstege, H. de Groot, J. Homma, I. Weverling,
P. Rewiersma (NIKHEF), R. Mohrmann, H. Pause, W. Grell (Hamburg),
M. Riera (Madrid), R. Granitzny, H.--J. Liers (Bonn). We are also grateful for the
hospitality and
support of CERN, in particular the
help of L. Gatignon is much appreciated.
Finally we would like to thank R. Klanner for his enthusiastic support during all
phases of the project.
| proofpile-arXiv_065-453 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Brief introduction}
In the present talk we discuss the spin-dependent structure
functions (SF) of nuclei and their relation to those of nucleon.
Our main focus will be the deuteron, which we study in detail
in the covariant Bethe-Salpeter formalism.
Why it so important and interesting to study the nuclear effects in the SFs?
First, nuclei are the only source of the experimental information
about neutron SFs, including the spin-dependent ones.
To obtain this information, it
is important to understand how nucleons are bound
in the nuclei and how this binding affects
their SFs. An accurate method to extract
the neutron SFs from the nuclear data
must be an essential part of the consistent analysis of the
nucleon SFs.
Second, a physics governing
the processes with the participation of the nuclei is extremely interesting
by itself.
For instance, a spin-1 nucleus, such as a deuteron, has extra
spin-dependent
SFs than nucleons, i.e. $b_{1,2}^D$.
Another example, nuclei as a slightly relativistic and weakly bound
systems allows for more progress
than the hadrons, in studying the covariant
bound state problem.
In certain situations the covariant approach gives results
noticeably different from the nonrelativistic ones.
For the spin-dependent SFs such situation is
a calculation of the $b_{1,2}^D$.
And third, our interest in the study of the reactions with the deuteron
is in part motivated by the future and ongoing experiments. In particular,
very recently we started a study of the chiral-odd SF
$h_1^D$.
\section{Spin-dependent SF of nucleon, $g_1^N$. }
For recent reviews about the nucleon spin-dependent SFs
see refs.~\cite{jaffer,rev2}.
\subsection{Basic formulae}
The differential cross section for the polarized electron-nucleon
scattering has the form:
\begin{eqnarray}
\frac{d^2\sigma}{d\Omega dE'} = \frac{\alpha^2E'}{2mq^4E} L^{\mu\nu}W_{\mu\nu},
\label{crosec}
\end{eqnarray}
where $\alpha = e^2/(4\pi)$, $q=(\nu,0,0,-\sqrt{\nu^2+Q^2})$
is the momentum transfer,
$ Q^2=-q^2$, $m$ is the nucleon mass, $E(E')$ is the energy of the incoming (outgoing)
electron, $L^{\mu\nu}$ and $W_{\mu\nu}$ are the leptonic and hadronic tensors.
The most general expression of $W_{\mu\nu}$ is
\begin{eqnarray}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! W_{\mu\nu}^N(q,p) =\label{htenn} \\
&&\!\!\!\!\!\!\!\!\!
\left ( -g_{\mu\nu} +\frac{q_\mu q_\nu}{q^2}\right ) F_1^N(x_N,Q^2)
+
\left ( p_{\mu} - q_\mu \frac{pq}{q^2} \right )
\left ( p_{\nu} - q_\nu \frac{pq}{q^2} \right )
\frac{F_2^N(x_N,Q^2)}{pq} \nonumber\\
+&&\!\!\!\!\!\!\!\!\!
\frac{im}{pq} \epsilon_{\mu\nu\alpha\beta}
q^\alpha \left \{ S^\beta \left (g_1^N(x_N,Q^2) + g_2^N(x_N,Q^2) \right )
-p^\beta \frac{(Sq)}{pq}g_2^N(x_N,Q^2) \right \} ,
\nonumber
\end{eqnarray}
where
$x_N = Q^2/(2pq)$ (in the rest frame of the nucleon
$x_N = Q^2/(2m\nu)$) and
$S$ is the nucleon spin.
In accordance to the ideology of
the quark-parton model, the SFs $F_{1,2}$ and $g_1$ are
proportional to the appropriate quark distributions on a
part of the total longitudinal momentum of the nucleon, $x$.
For instance, if we denote
the net spin carried by quarks as
$\Delta q(x,Q^2)$ ($q = u,d,s$)
and introduce the following combinations:
\begin{eqnarray}
&&\Delta q_3(x,Q^2) \equiv \Delta u(x,Q^2) - \Delta d(x,Q^2) ,
\label{su33}\\
&&\Delta q_8(x,Q^2) \equiv \Delta u(x,Q^2) + \Delta d (x,Q^2)
-2\Delta s(x,Q^2) ,
\label{su38}\\
&&\Delta \Sigma (x,Q^2) \equiv \Delta u(x,Q^2) + \Delta d(x,Q^2)+ \Delta s(x,Q^2),
\label{su30}
\end{eqnarray}
then the SFs of the proton(neutron) can be written as:
\begin{eqnarray}
g_1^{p,n}(x,Q^2) = \pm \frac {1}{12}\Delta q_3(x,Q^2) +
\frac{1}{36} \Delta q_8(x,Q^2) +\frac{1}{36}\Delta \Sigma(x,Q^2) .
\label{g1pm}
\end{eqnarray}
Important objects of the study of quark structure of the hadrons
are the so-called sum rules for the SFs. The sum rules
relate the moments of the SFs to the fundamental
(or sometimes not very fundamental) constants of the theory.
Integrating eq.~(\ref{g1pm}), the first moments
of the proton (neutron) structure functions can be written in self-explaining
notation:
\begin{eqnarray}
S^{p(n)} &\equiv& \int\limits_0^1 dx g_1^{p(n)}(x,Q^2) =\label{ig1p}\\
&&\!\!\!\! \frac {1}{12}
\left( 1 -\frac{\alpha_s}{\pi}+ {\small \ldots} \right) \left (\pm
\Delta q_3 +
\frac{1}{3} \Delta q_8 \right ) + \frac{1}{9}
\left( 1 -\frac{\alpha_s}{3\pi}+ {\small \ldots} \right)\Delta \Sigma ,
\label{ignuc}\end{eqnarray}
where the perturbative QCD
corrections, to order ${\cal O}(\alpha_s)$, are also presented.
From the current algebra for asymptotic integrals we have ($Q^2 \to \infty $):
\begin{eqnarray}
\Delta q_3 = 1.257\pm 0.003 , \quad
\Delta q_8 = 0.59 \pm 0.02\; (?).
\label{isu38}
\end{eqnarray}
The first constant is from the weak decay of the neutron and
the second constant is from
the decay of the hyperon.
The ``?'' mark is due to the residual questions about SU(3).
The Bjorken sum rule is the most fundamental relation:
\begin{eqnarray}
S^p - S^n &=& \frac {1}{6}
\left( 1 -\frac{\alpha_s}{\pi}+ \cdots \right) \Delta q_3,
\label{bsr}\end{eqnarray}
which numerically gives $0.187 \pm 0.003$ at $Q^2 = 10$~GeV$^2$ and
$0.171 \pm 0.008$ at $Q^2 = 3$~GeV$^2$.
The Ellis-Jaffe sum rule is not so fundamental. Assuming that
$\Delta s = 0$ and, therefore, $\Delta \Sigma = \Delta q_8
\simeq 0.6$, we get at $Q^2=10$~GeV$^2$ ($3$~GeV$^2$):
\begin{eqnarray}
S^p_{EJ} &=& \phantom{-}0.171\pm 0.004 \quad
(\phantom{-}0.161 \pm 0.004) ,\label{ejp}\\
S^n_{EJ} &=& -0.014\pm 0.004 \quad (-0.010 \pm 0.004).\label{ejn}
\end{eqnarray}
The spin-dependent SFs, $g_1$, allow also to study
spin content of the hadrons. Indeed, using eqs.~(\ref{isu38})
and experimental values of $S^{p,n}$
(a fraction of) the nucleon spin carried by quarks, $\Delta \Sigma$,
can be determined. The total angular
momentum (spin) of the nucleon consists not only of $\Delta\Sigma$, but
also:
\begin{eqnarray}
\frac{1}{2} = \frac{1}{2}\Delta \Sigma + \Delta G + L_z^g +L_z^G,
\label{spincont}
\end{eqnarray}
where $\Delta G$ is the gluon spin contribution,
$L_z^{q(G)}$ is the quark (gluon) orbital angular momentum contribution.
In naive quark model $\Delta \Sigma =1$ and others are zeros.
In the relativistic quark model $\Delta \Sigma =0.75$ and $L_z^q=0.125$
and
others are zeros.
From the current algebra $\Delta \Sigma \approx 0.6 \pm 0.1$, others are unknown.
\subsection{Experiments for $g_1^N$}
Both the SFs and the sum rules are the subject of
intensive experimental studies in recent years.
Table~I presents measurements by various experimental
groups.
\begin{center}
{\small
{\normalsize \bf Table I.}
\vskip .25cm
\begin{tabular}{|c|c|c|c|c|}
\hline
Experiment & Year & Target & $\sim Q^2$~GeV$^2$ & $S^{Target} $ \\
\hline
\hline
E80/E130 & 1976/1983 & p & 5 & 0.17 $\pm$ 0.05 \\
EMC & 1987 & p & 11 & 0.123 $\pm $ 0.013 $\pm$ 0.019\\
SMC & 1993 & d & 5 & 0.023 $\pm $ 0.020 $\pm$ 0.015\\
SMC & 1994 & p & 10 & 0.136 $\pm $ 0.011 $\pm$ 0.011\\
SMC & 1995 & d & 10 & 0.034 $\pm $ 0.009 $\pm$ 0.006\\
E142 & 1993& n ($^3He$)& 2 & -0.022 $\pm $ 0.011\\
E143 & 1994& p & 3 & 0.127 $\pm $ 0.004 $\pm$ 0.010\\
E143 & 1995& d & 3 & 0.042 $\pm $ 0.003 $\pm$ 0.004\\
HERMES& 1996& n ($^3He$)& 3 & -0.032 $\pm $ 0.013 $\pm$ 0.017\\
\hline
\end{tabular}
}
\end{center}
\vskip .2cm
From the SMC and E143 data the Bjorken sum rule is:
\begin{eqnarray}
\left (S^p - S^n \right )_{SMC} &\approx&
0.199 \pm 0.038 \quad{\rm at} \quad Q^2 = 10\quad{\rm GeV}^2,\label{bsre10}\\
\left (S^p - S^n \right) _{E143} &\approx&
0.163 \pm 0.010 \pm 0.016 \quad{\rm at} \quad Q^2 = 3\quad{\rm GeV}^2,\label{bsre3}
\end{eqnarray}
i.e. the sum rule is confirmed with 10 \% accuracy.
From Table~I it is clear that the Ellis-Jaffe sum rules are broken.
As to the spin content, (\ref{spincont}), only one piece, $\Delta \Sigma$,
can be extracted from the
data for the integrals of SFs.
The world data from Table~I
gives:
\begin{eqnarray}
\Delta \Sigma \approx 0.3\pm 0.1,
\label{spinconte}
\end{eqnarray}
which is larger than the first result of EMC,
$\Delta \Sigma = 0.12\pm 0.094\pm 0.138\approx 0$, but still
lower than quark model estimates.
In addition to the perturbative corrections in eq.~(\ref{ig1p}),
various other corrections, such as
the kinematic
mass corrections, $\sim m^2/Q^2$ and higher twist corrections, $\sim 1/Q^2$,
are discussed.
\section{Nucleons and nuclei}
Note that actual data for the neutron is not presented
in Table~I, only the data for lightest nuclei.
A simple formula is used to obtain $g_1^n$ from the combined
proton and deuteron
data:
\begin{eqnarray}
g_1^D = \left (1-\frac{3}{2}w_D\right )\left (g_1^p+g_1^n\right),
\label{simple}
\end{eqnarray}
where $w_D$ is the probability of the $D$-wave state in the deuteron.
Depending on the model,
$w_D = 0.04 - 0.06$.
Similarly, the neutron SF is obtained
from the $^3He$ data:
\begin{eqnarray}
g_1^{^3He} = \left ( P_S + \frac{1}{3}P_{S'} -P_D
\right )
g_1^n
+\left (\frac{2}{3}P_{S'} - \frac{2}{3}P_D
\right )
g_1^p,
\label{simple1}
\end{eqnarray}
where $P_S$, $P_{S'}$ and $P_D$ are weights of the $S$, $S'$ and
$D$ waves in $^3He$, respectively. Typical (model-dependent)
values of these weights
are $P_S\approx 0.897$, $P_{S'}=0.017$ and $P_D=0.086$.
It is important to realize that the real connection between the nucleon and
nuclear SFs is more complex than given by formulae like
eq.~(\ref{simple}) and (\ref{simple1}). Studies of the last decade
show importance of the proper separation
of the binding, Fermi motion and the off-mass-shell
effects in the procedure of extracting the neutron SFs from the
nuclear data
(see refs.~\cite{amb,unfo,mtamb} and references therein).
However, effects of the Fermi motion are sometimes
estimated by the experimental groups, other effects are always
neglected.
Such a way of action can be phenomenologically more or less safe
at the present level of accuracy of the experiments, but
not in general.
The deuteron is the most appropriate target to study the neutron
SFs, since it has a well-known structure and
well-studied wave function or relativistic amplitude. Besides
all other effects such as meson exchanges, binding of the nucleons,
off-mass-shell corrections, shadowing, etc, are minimal.
Even in the case of $^3$He the situation is
known to be different~\cite{kubj,fshe}. Indeed,
eq.~(\ref{simple1}) or even more sophisticated convolution formula
violate the fundamental Bjorken sum rule for the
$^3$He-$^3$H pair at the 3-5\% level, which is a serious
indication of other degrees of freedom involved in the process.
Once again, this fact is completely ignored by the experimental
groups reporting the results for the neutron SFs
from experiments with $^3$He.
In what follows we consider the nuclear effects in the
spin-dependent
SFs of the deuteron.
Results of our studies make us certain that an accurate
extraction of the {\em neutron } spin structure
function, $g_1^n$,
is possible.
Considering nuclei as a complex system of interacting nucleons and mesons,
we calculate the nuclear SFs in terms of the structure
functions of its constituents, nucleons and mesons, and in
the Bethe-Salpeter formalism for the deuteron amplitude.
For the spin-independent SFs, $F^D_2$, the mesonic
contributions to the SF is important
(although quite small) for the consistency
of the approach, since the mesons carry a part of the total momentum
of the nuclei (see~\cite{uk} and references therein).
However, for the spin-dependent
SFs explicit contribution of mesons
is not important. Rather,
their presence manifests via binding of nucleons in nuclei.
This is why we consider only nucleon contributions to the
spin-dependent
SFs.
We start with the general form of the hadron tensor
of the deuteron with the total angular momentum projection, $M$,
keeping only
leading twist SFs:
\begin{eqnarray}
W_{\mu\nu}^D(q,P_D,M) &=& \left ( -g_{\mu\nu} +\frac{q_\mu q_\nu}{q^2}\right ) F_1^D(x_D,Q^2,M)
+ \label{htend} \\
&& \left ( P_{D\mu} - q_\mu \frac{P_Dq}{q^2} \right )
\left ( P_{D\nu} - q_\nu \frac{P_Dq}{q^2} \right )
\frac{F_2^D(x_D,Q^2,M)}{P_Dq} \nonumber\\
&& +\frac{iM_D}{P_Dq} \epsilon_{\mu\nu\alpha\beta}
q^\alpha S_D^\beta(M) g_1^D(x_D,Q^2),
\nonumber
\end{eqnarray}
where
$x_D = Q^2/(2P_Dq)$ (in the rest frame of the deuteron
$x_D = Q^2/(2M_D\nu)$),
$S_D(M)$ is the deuteron spin
and $F_{1,2}^D$ and $g_1^D$
are the deuteron SFs.
Averaged over $M$ this expression leads to the well-known form of the
spin-independent hadron tensor which is valid for hadron with any spin:
{\small\begin{eqnarray}
&& W_{\mu\nu}^D(q,P_D) = \frac{1}{3}\sum_M W_{\mu\nu}^D(q,P_D,M) \label{av}\\
&=&\!\!\!\!
\left ( -g_{\mu\nu} +\frac{q_\mu q_\nu}{q^2}\right ) F_1^D(x_D,Q^2)
+
\left ( P_{D\mu} - q_\mu \frac{P_Dq}{q^2} \right )
\left ( P_{D\nu} - q_\nu \frac{P_Dq}{q^2} \right )
\frac{F_2^D(x_D,Q^2)}{P_Dq},
\nonumber
\end{eqnarray}
}
where $F_{1,2}^D(x_D,Q^2)$ are the result of averaging of the SF
$F_{1,2}^D(x_D,Q^2,M)$.
To separate $g_1^D$ we can use one of the following projectors~\cite{ukk}:
\begin{eqnarray}
R^{(1)}_{\mu\nu} \equiv i\epsilon_{\mu\nu\alpha\beta}q^\alpha S^\beta_D(M),
\quad R^{(2)}_{\mu\nu} \equiv \frac{i (S_D(M)q)}{P_Dq}
\epsilon_{\mu\nu\alpha\beta}q^\alpha P_D^\beta
. \label{aw2}
\end{eqnarray}
In the limit $Q^2/\nu^2 \to 0$:
\begin{eqnarray}
g_1^D = \frac{R^{(1)\mu\nu}W_{\mu\nu}^D }{2\nu} =
\frac{R^{(2)\mu\nu}W_{\mu\nu}^D }{2\nu}
. \label{exg1}
\end{eqnarray}
The nucleon contribution to the deuteron SFs
is presented by the triangle graph, written in terms of the Bethe-Salpeter
amplitude of the deuteron~\cite{uk,ukk}:
\let\picnaturalsize=N
\def3.0in{2.10in}
\defh-1d.eps{tre-dia.eps}
\ifx\nopictures Y\else{\ifx\epsfloaded Y\else\input epsf \fi
\let\epsfloaded=Y
\centerline{\ifx\picnaturalsize N\epsfxsize 3.0in\fi \epsfbox{h-1d.eps}}}\fi
\noindent
where $\hat W$ is the appropriate operator, describing the scattering on the
constituent nucleon. Neglecting small correction due to the ``nucleon
deformation''~\cite{mst} it can be written down as:
\begin{eqnarray}
&&\hat W^N_{\mu\nu}(q,p)=\hat W_{\{\mu\nu\}}(q,p) +
\hat W_{[\mu\nu]}(q,p) \label{deftens}\\[3mm]
&&\hat W_{\{\mu\nu\}}(q,p) = \frac{\hat q}{2pq}W^N_{\mu\nu}(q,p),
\label{ops}\\
&&\hat W_{[\mu\nu]}(q,p) =
\frac{ i}{2pq}
\epsilon_{\mu\nu\alpha\beta}
q^{\alpha}
\gamma^\beta \gamma_5 g_1^N (q,p)
,
\label{opa}
\end{eqnarray}
where $\{\ldots \}$ and $[\ldots ]$ denote symmetrization and
antisymmetrization of indices, respectively, $W_{\{\mu\nu\}}^N$
is the spin-independent part of the hadron tensor of the nucleon
and $g_1^N(q,p)=g_1^N(x,Q^2)$ is the spin-dependent
nucleon SF.
The explicit expressions of the deuteron SFs in
terms of the Bethe-Salpeter amplitude, $\Psi_M(p_0,{\bf p})$, are given by:
\begin{eqnarray}
F_2^D(x_N,Q^2,M) &=& i\int \frac{d^4p}{(2\pi)^4}
{F_2^N} \left( \frac{x_N m}{p_{10}+p_{13}}, Q^2\right)\label{f2m} \\
&&\frac{ {\sf Tr}\left\{
\bar\Psi_M(p_0,{\bf p})(\gamma_0+\gamma_3) \Psi_M(p_0,{\bf p}) (\hat p_2-m)
\right \}}{2M_D},
\nonumber \\[2mm]
g_1^D(x_N,Q^2) &=& i
\int \frac{d^4p}{(2\pi)^4}
{g_1^N}\left( \frac{x_N m}{p_{10}+p_{13}}, Q^2\right) \label{g1m}\\
&& \frac{\left. {\sf Tr}\left\{
\bar\Psi_M(p_0,{\bf p})(\gamma_0+\gamma_3)\gamma_5 \Psi_M(p_0,{\bf p}) (\hat p_2-m)
\right \}\right |_{M=1}}{2(p_{10}+p_{13})},
\nonumber
\end{eqnarray}
where
$p_{10}$ and $p_{13}$ are the time and 3-rd components of the
struck nucleon momentum.
Averaging over the projection $M$ has not been done
in eq.~(\ref{f2m}),
since we use the present form later to calculate
the SF $b_{1,2}^D$. Then two
independent ``SFs'', with $M = \pm 1$ and $M=0$ are obtained:
\begin{eqnarray}
&&F_2^D(x_N,Q^2) = \frac{1}{3} \sum_{M=0,\pm 1} F_2^D(x_N,Q^2,M),
\label{f2}\\[1mm]
&& F_2^D(x_N,Q^2,M=+1) = F_2^D(x_N,Q^2,M=-1).\label{pm}
\end{eqnarray}
A method to calculate numerically expressions like
(\ref{f2m}) and (\ref{g1m})
is presented in ref.~\cite{ukk}. The important details of
the calculations are:
\begin{enumerate}
\item A realistic model for the Bethe-Salpeter amplitudes
is essential for a realistic estimate of the
nuclear effects. We use a recent numerical solution~\cite{uk}
of the ladder Bethe-Salpeter equation with a realistic
exchange kernel.
\item The Bethe-Salpeter amplitudes and, therefore,
eqs.~(\ref{f2m})-(\ref{g1m})
have a nontrivial singular structure. These singularities
must be carefully taken into account.
\item The BS amplitudes are numerically calculated with the
help of the Wick rotation. Therefore, the numerical
procedure for inverse Wick rotation must be applied.
\end{enumerate}
\begin{center}
\vspace*{-5cm}
\let\picnaturalsize=N
\def3.0in{4.8in}
\defh-1d.eps{g1.ps}
\ifx\nopictures Y\else{\ifx\epsfloaded Y\else\input epsf \fi
\let\epsfloaded=Y
\centerline{\ifx\picnaturalsize N\epsfxsize 3.0in\fi \epsfbox{h-1d.eps}}}\fi
\vspace*{-5.5cm}
Figure~1.
\end{center}
Calculations with the realistic BS amplitudes result in the
behavior of the $g_1^D$ very similar to
other calculations~\cite{kaemp,mg1}. The result is presented in Fig.~1
in the form of the ratio, $g^D_1/g_1^N$. Dotted curve presents
non-relativistic calculations with the Bonn wave function, solid curve the
BS result. The dashed curve is the illustrative result for the
non-relativistic calculations utilizing the BS axial-vector
density. Last curve, dot-dased, in Fig.~1 corresponds to the
naive formula (\ref{simple}).
Despite seemingly drastic difference in the ratio $g^D_1/g_1^N$ given by
(\ref{simple}) and realistic calculations, the typical experimental
errors today are larger. (Large fluctuations of the ratio
at $x<0.7$ are not too important. They correspond to zeros
of the nucleon SF which are slightly shifted by the convolution formula.)
Indeed, in Fig.~2 we present representative example of data
(SMC-1994), together with two fits of these data (dashed lines).
The solid lines present results of {\em exact}
extraction of the nucleon SF from the present deuteron data.
We see that curves for deuteron and nucleon both do not contradict
the experiment.
However, in certain kinematical conditions effects can be bigger.
For instance, lately much interest is devoted to the discussion
of the Gerasimov-Drell-Hearn sum rule for the proton and the neutron
SFs at small $Q^2$ in general and
at $Q^2 = 0$ in particular (see reviews~\cite{jaffer,rev2}
and references therein). Very important contribution
to the study of the neutron SFs
is expected from the Jefferson Lab groups~\cite{expgdh}, where
experiments with the deuterons and $^3He$ are planned in the intervals
$Q^2\sim 0.15 - 2$~GeV$^2$.
Analysis of the deuteron SFs in this interval of $Q^2$,
the nucleon ``resonances'' region, shows that effect of the binding and Fermi
motion is much larger here than in the deep inelastic regime.
An example of the calculation of the deuteron structure
function, $g_1^D(x,Q^2)$, is presented in Fig.~3a (dashed line)
at $Q^2=1.0$~GeV$^2$. It is compared
with the nucleon SF, $g_1^N(x,Q^2)$, input into the calculation.
In the areas of resonance structures in $g_1^N(x,Q^2)$, the deuteron
SF differs up to 50\%!
In Fig.~3b we present a comparison of the neutron SF,
$g_1^n$ (solid line, input into calculations in Fig.~3a),
with the ``neutron''
SF ``extracted'' by means of the naive formula (\ref{simple})
(dashed line).
We see that these two functions have nothing in common.
\vspace*{-2mm}
\begin{center}
\let\picnaturalsize=N
\def3.0in{4.0in}
\defh-1d.eps{g1-extract.eps}
\ifx\nopictures Y\else{\ifx\epsfloaded Y\else\input epsf \fi
\let\epsfloaded=Y
\centerline{\ifx\picnaturalsize N\epsfxsize 3.0in\fi \epsfbox{h-1d.eps}}}\fi
Figure~2.
\end{center}
The same effects appear in $^3He$~\cite{ciofi}.
The presented example, shows that in every particular situation
one has to consider the nuclear effects and take into account
corresponding corrections to the SFs.
Mathematically the problem of extraction of the neutron
SF from the deuteron data is formulated as a problem to solve
the inhomogeneous integral equation (\ref{g1m})
for the neutron SF with a model kernel
and experimentally measured left hand side\footnote{Depending
on the model, some
additive corrections could be taken into account.}, $g_1^D$.
Recently we proposed a method
to extract the
neutron SF from the deuteron data
within any
model, giving deuteron SF in the form
of a "convolution integral plus/minus additive corrections"~\cite{unfo}.
The principal advantages of the method, compared with
the smearing factor method, are the following.
(i) Only analyticity of the SF need
be assumed, (ii) the
method allows us to elaborate on the spin-dependent SF,
where the traditional smearing factor method does not work.
\begin{center}
\begin{minipage}{15cm}
\vspace*{-7.05cm}
\hspace*{-2.5cm}
\let\picnaturalsize=N
\def3.0in{3.60in}
\defh-1d.eps{gdh-sf1.ps}
\ifx\nopictures Y\else{\ifx\epsfloaded Y\else\input epsf \fi
\let\epsfloaded=Y
\centerline{\ifx\picnaturalsize N\epsfxsize 3.0in\fi \epsfbox{h-1d.eps}}}\fi
\vspace*{-12.7cm}
\hspace*{4.5cm}
\let\picnaturalsize=N
\def3.0in{3.60in}
\defh-1d.eps{gdh-ntr1.ps}
\ifx\nopictures Y\else{\ifx\epsfloaded Y\else\input epsf \fi
\let\epsfloaded=Y
\centerline{\ifx\picnaturalsize N\epsfxsize 3.0in\fi \epsfbox{h-1d.eps}}}\fi
\end{minipage}
\vskip .9cm
a) \hspace*{6cm} b)
Figure~3.
\end{center}
\section{Other spin-dependent structure functions}
\subsection{SFs for spin-1 hadron, ${b_{1,2}^D}$}.
The SF $b_1^D$ is defined by
(see ref.~\cite{ukk,ub} and references therein):
\begin{eqnarray}
&& b_2(x_N,Q^2) = F_2^D(x,Q^2,M=+1)-F_2^D(x,Q^2,M=0),\label{b2}
\end{eqnarray}
Note, the SF $F_2^D(x,Q^2,M)$ is independent of the
lepton polarization, therefore, both SFs, $F_2^D$
and $b_2^D$, can be measured in experiments with an unpolarized
lepton beam and polarized deuteron target. In view of eq.~(\ref{pm}),
only one of the SFs $F_2^D(x,Q^2,M)$ is needed, in addition
to the spin-independent $F_2^D(x,Q^2)$, in order to obtain
$b_2(x,Q^2)$. The other SF, $b_1^D$, is related
to the deuteron SF $F_1^D$, the same way as $b_2^D$ is
related to $F_2^D$, via eqs.~(\ref{f2}), and $b_2^D = 2xb_1^D$.
Sum rules for the deuteron SFs $b_1^D$ and $b_2^D$
are a
result of the fact that the vector charge and energy of the system are
independent of the spin orientation:
\begin{eqnarray}
\int\limits_0^1 dx_D b_1^D(x_D) =0, \quad
\int\limits_0^1 dx_D b_2^D(x_D) =0.
\label{sr2}
\end{eqnarray}
These sum rules
were suggested by Efremov and Teryaev~\cite{et}.
\vspace*{-2.75cm}
\begin{center}
\let\picnaturalsize=N
\def3.0in{3.90in}
\defh-1d.eps{b2.ps}
\ifx\nopictures Y\else{\ifx\epsfloaded Y\else\input epsf \fi
\let\epsfloaded=Y
\centerline{\ifx\picnaturalsize N\epsfxsize 3.0in\fi \epsfbox{h-1d.eps}}}\fi
\vspace*{-3.2cm}
Figure~4.
\end{center}
The SFs $b_1^D$ and $b_2^D$ are calculated within
two approaches as well. The results are shown in Fig.~4 a) and b).
The behavior of the functions in Fig.~4 a) suggests the validity of the
first of sum rules (\ref{sr2}). At the same time, the nonrelativistic
calculation for
$b_2^D$ in Fig.~4 b) (dotted line) obviously
does not satisfy the second sum rule. The main difference of
the relativistic and nonrelativistic calculations is at small $x$, where
these approaches give different signs for the SFs. To check a model
dependence of the nonrelativistic calculations, we also performed
calculations with the
``softer'' deuteron wave function (with cut-off of the realistic
wave function at $|{\bf p}| = 0.7$~GeV).
Corresponding SFs are shown in Fig.~4 a) and b) (dashed line).
It also does not affect the principle conclusion that the nonrelativistic
approach violates the sum rules.
\subsection{Chiral-odd SF ${h_{1}^D}$}.
The spin-dependent SFs $h_1$ of the nucleons and deuteron
can not be measured in the inclusive deep inelastic
scattering, but in the semi-inclusive process~\cite{jj}. In this sense
these SFs are different from
the SFs studied in the present paper.
However, we present the results for these functions,
since (i) they carry important information about the spin structure
of the nucleons~\cite{jj,hj} and the deuteron~\cite{uhk},
(ii) the experiments are planned to measure them~\cite{exp}
and (iii) from the theoretical point of view structure
function of the deuteron, $h_1^D$, is defined in a way very
similar to the usual deep inelastic SFs~\cite{uhk}:
\begin{eqnarray}
h_1^D(x_N) &=& i
\int \frac{d^4p}{(2\pi)^4}
{h_1^N}\left( \frac{x_N m}{p_{10}+p_{13}}\right) \label{h1m}\\
&&\frac{\left. {\sf Tr}\left\{
\bar\Psi_M(p_0,{\bf p})\gamma_5\gamma_3\gamma_0 \Psi_M(p_0,{\bf p})
(\hat p_2-m)
\right \}\right |_{M=1}}{2(p_{10}+p_{13})}.
\nonumber
\end{eqnarray}
To calculate the realistic SF $h_1^D(x)$ we need
the nucleon SFs $h_1^N(x)$. However, so far there is
no existing experimental data for this function, and very little is known
about the form of $h_1^N$ in theory.
In the present paper
we follow the ideas of ref.~\cite{jj} to estimate $h_1^N$.
Since the sea quarks do not contribute to $h_1^N$, its
flavor content is
simple:
\begin{eqnarray}
h_1^N (x) = \delta u(x) +\delta d(x),
\label{hn1}
\end{eqnarray}
where $\delta u(x) $ and $\delta d(x)$ are the contributions
of the u-
and d-quarks, respectively~\cite{jj,hj}.
Since the matrix elements of the operators $\propto \gamma_5 \gamma_3$
and
$\propto \gamma_5 \gamma_3\gamma_0$ coincide in the static limit,
as a crude estimate we can expect that
\begin{eqnarray}
\delta u(x) \sim \Delta u(x), \quad \quad
\delta d(x) \sim \Delta d(x), \label{dud1}
\end{eqnarray}
where $\Delta u(x)$ and $\Delta d(x)$ are contributions
of the u- and d-quarks to the spin of the nucleon, which is
measured through the
SF $g_1^N$. Correspondly, the simplest estimation
for $h_1^N$
\begin{eqnarray}
h_1^N (x) = \alpha \Delta u(x) +\beta \Delta d(x),
\label{hn2}\\
\alpha = \beta=1 \label{ab1}
\end{eqnarray}
should not be too unrealistic. In fact,
the bag model calculation shows that difference between $\delta q$ and
$\Delta q$ is typically only few percent~\cite{jj}. This analysis is
mostly a qualitative one, since it is
limited by the case with one quark flavor and does not pretend to describe
a phenomenology.
To evaluate possible deviations from the simple choice
of $h_1^N$, (\ref{hn2}) with (\ref{ab1}), we
suggest:
\begin{eqnarray}
\alpha = \delta u/\Delta u, \quad \beta= \delta d/\Delta d, \label{ab2}
\end{eqnarray}
where $\delta q$ and $ \Delta q$ are the first moments of
$\delta q(x)$ and $ \Delta q(x)$, respectively ($q = u, d$).
For
$\delta u$ and $ \delta d$ we can adopt the results from the QCD sum rules
and the bag model calculations~\cite{hj}. As to
$\Delta u$ and $ \Delta d$, we
can use the experimental data analysis~\cite{jaffer,rev2}
or theoretical results, e.g.
the QCD sum rule results~\cite{hj}.
Thus, we estimate~\cite{uhk}:
\begin{eqnarray}
\alpha = 1.5 \pm 0.5, \quad \beta= 0.5 \pm 0.5, \label{ab3}
\end{eqnarray}
at the scale of $Q^2=1$~GeV$^2$.
\vskip -.7cm
\begin{center}
\let\picnaturalsize=N
\def3.0in{3.0in}
\defh-1d.eps{h-1d.eps}
\ifx\nopictures Y\else{\ifx\epsfloaded Y\else\input epsf \fi
\let\epsfloaded=Y
\centerline{\ifx\picnaturalsize N\epsfxsize 3.0in\fi \epsfbox{h-1d.eps}}}\fi
Figure~5.
\end{center}
The realistic form of the distributions $\Delta u(x)$ and $\Delta d(x)$
can be taken from a fit to the experimental data for $g_1^N$. In our
calculations we used parametrization from ref.~\cite{shaf}.
At this point we have to realize that, in spite of expected
relations~(\ref{dud1}), distributions $\delta q$ and
$\Delta q$ are very different in their nature. Especially at
$x {\ \lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }0.1$, where $\Delta q$ probably contains a singular contribution
of the polarized sea quarks, but $\delta q$ does not.
Therefore we expect eq.~(\ref{hn2}) to be a reasonable
estimate in the region of the valence quarks dominance, say
$x {\ \lower-1.2pt\vbox{\hbox{\rlap{$>$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }0.1$.
For completely consistent analysis, the parameters $\alpha$ and $\beta$, and
the distributions $\Delta u(x)$ and $\Delta d(x)$
should be scaled to the same value of $Q^2$. However,
for the sake of the unsophisticated estimates we do not
go into such details.
The results of calculation of the nucleon and deuteron
SFs, $h_1^N$ (solid lines) and $h_1^D$ (dashed lines),
are shown in Fig.~5.
The group of curves 1 represents case (\ref{ab1}), which
is a possible lower limit for $h_1^{N,D}$
in accordance with
our estimates (\ref{ab3}). Curves 2 represent the case
$\alpha = 1.5,\quad\beta = 0.5$, which is close
to the mid point results of the bag model and the QCD sum rules.
The upper limit corresponding to the estimates (\ref{ab3})
is presented by curves 3. For all cases
the deuteron
SF is suppressed comparing to the nucleon one, mainly
because of the depolarization effect of the D-wave in the deuteron.
This is quite similar to the case of the SFs $g_1^N$
and $g_1^D$.
Note that our estimate of the nucleon SF $h_1^N$,
(\ref{ab3}),
gives systematically larger function than naive suggestion (\ref{ab1}),
the curves 1 in Fig.~5
which essentially corresponds to the estimate $h_1^N \simeq (18/5) g_1^N$,
neglecting possible negative contribution of the s-quark sea~\cite{jaffer,rev2}.
The large size of the effect suggests that it can be detected in
future experiments
with the deuterons~\cite{exp}.
\section{Brief conclusion}
We have presented the results of our study of the spin-dependent structure
functions of the
deuteron. In particular, the leading twist $g_1^D$, $b_{1,2}^D$ and
$h_1^D$ are considered.
The issue of the extraction of the neutron structure functions
from the deuteron data is addressed. The role of relativistic effects
is studied and can be summarized as: (i) relativistic calculations
give a slightly larger magnitude of the binding effects, (ii) the
relativistic Fermi motion results in ``harder'' SF
at high $x$, and (iii) covariant approach
is internally consistent, while the nonrelativistic approach
is internally inconsistent and violates
important sum rules.
\section*{Acknowledgements}
We wish to thank every one who essentially contributed to studies
included in this presentation, L.P. Kaptari, C. Ciofi degli Atti, Han-xin He and
S. Scopetta.
This work is supported in part by NSERC, Canada, and INFN, Italy.
{\small
| proofpile-arXiv_065-454 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{INTRODUCTION}
\noindent
The occurrence of variable stars radially pulsating in the
fundamental and/or the first overtone modes is a well
known and well established observational evidence based on
both Population I and Population II variables. The possible
occurrence of stars pulsating in the second overtone (SO) is a still
much debated argument.
For a long time the observational scenario concerning such an occurrence
has been limited to the suggestions of several authors
(van Albada \& Baker 1973; Demers \& Wehlau 1977; Clement, Dickens
\& Bingham 1979; Nemec, Wehlau \& Mendes de Oliveira 1988 and references
therein) that some globular clusters RRc variables (first overtone)
characterized by
very short periods (P$\sim$ 0.3 d) and small pulsational amplitudes could
be good SO~ candidates. Only in recent times the large and homogeneous
photometric databases collected by the MACHO project for radial pulsators
in the Large Magellanic Cloud (LMC) have brought out for the first time a
sound evidence of
the occurrence of SO pulsators in classical Cepheids (Alcock et al. 1995).
On the basis of the same extensive photometric survey, from
the period distribution of about 8000 RR Lyrae variables, it has
been also suggested (Alcock et al. 1996) that a peculiar peak located
at P=0.28 d could be due to low mass SO~ variables.
\noindent
From a theoretical standpoint, the limit cycle stability of second overtone~
pulsators in population I and population II variable stars has been an
odd problem for a long time. As a matter of fact, still at present we lack
either firm theoretical predictions concerning the existence of SO~
pulsators, or convincing insights on the
physical mechanisms which could govern their approach to mode stability.
Classical nonlinear analyses of pulsation properties
and modal stability of population II low mass variables (Christy 1966;
Stellingwerf 1975) never produced any firm evidence of unstable SO~
pulsators. By using a simple nonlinear and nonadiabatic
one-zone model Stellingwerf, Gautschy \& Dickens (1987, hereinafter
referred to as SGD) succeeded in producing an unstable SO~
pulsator. However, even though
one-zone pulsating models can be of some help to disclose the
main processes which constrain the physical structure of stellar
envelopes, unfortunately this approach cannot supply definitive
conclusions on the existence of these variables since these models
do not take into account the dynamical effects of the regions which
operate the damping of the pulsation.
An unstable SO model at limiting amplitude was constructed for RR
variables by Stothers (1987) on the basis of a nonlinear radiative approach.
However, the light curve of his model shows a peculiar feature,
i.e. a deep splitting just after the maximum in luminosity,
which is not observed in any known group of radial pulsators and
in the light curve predicted by SGD. Moreover, no evidence of unstable
SOs was found in the homogeneous and systematic survey of
nonlinear, nonlocal and time-dependent convective models provided by
Bono \& Stellingwerf (1994 hereinafter referred to as BS) and by
Bono et al. (1996a hereinafter referred to as BCCM).
\noindent
According to the previous discussion, no SO~ pulsators appear to be
foreseen in old, low mass variable stars. In this context, it is worth
underlining that periods of RRc in globular clusters are not in conflict
with first overtone expectations, and their small amplitudes could
be due to the sudden decrease of the first overtone amplitudes
near the first overtone blue boundary (BCCM). Moreover, the peak in the
period distribution at P=0.28 d of RR Lyrae in LMC can be taken as a
compelling evidence of a moderately metal rich population, with periods
which overlap the characteristic short periods already found both in the
Galactic bulge and in the metal rich field RR Lyrae. The reader interested
to a detailed discussion on the evolutionary and pulsational properties
of metal rich RR Lyrae variables is referred to Bono et al. (1996b).
\noindent
As far as more massive stars are concerned, it has been often
suggested that the Hertzsprung progression of the bump in the light
curve of classical Cepheids could be closely correlated to a {\em spatial
resonance} between the SO~ and the fundamental mode (Simon \& Schmidt 1976;
Buchler et al. 1996). However,
present nonlinear calculations devoted to the modal stability of
these variables did not supply any firm conclusion on the existence
of SO~ pulsators. Although further nonlinear investigations have
not been so far
undertaken to properly tackle the problem, it is worth mentioning
that linear nonadiabatic models provided by Baker (1965) for the
mass value $M/M_{\odot}\,$=1.0 disclosed a SO~ instability over a wide range
of luminosities and temperatures.
Moreover, Deupree \& Hodson (1977), in their study on the instability
strip of Anomalous Cepheids, gave an evaluation of the SO~ linear
blue edge for even larger masses ($1.0 \leq $ $M/M_{\odot}\,$ $\leq 2.0$).
However, the quoted authors did not come to a sound conclusion since
the numerical difficulties caused by the increased efficiency of the
convective overshooting prevented them from evaluating the nonlinear
modal stability of these pulsators.
\noindent
An extensive grid of $\delta$ Scuti~ pulsation properties has been recently provided
by Milligan \& Carson (1992) taking into account a combination of stellar
evolution models and both linear and nonlinear pulsation models.
Although in this investigation they carried out a detailed analysis
of the linear and nonlinear properties of $\delta$ Scuti~ stars, only few models
have been followed at limiting amplitude due to the small value of the
growth rates involved. Therefore the ultimate modal stability of the wide
range of nonlinear models could not be firmly established.
Let us note that $\delta$ Scuti~ variables are most attractive
objects among the several groups of variables presently known since
they also show simultaneous excitation of both radial and nonradial modes.
This peculiar feature is of the utmost importance for properly
addressing several astrophysical questions concerning the physical
mechanisms which rule the driving and the quenching of stellar
oscillations.
\noindent
The observational panorama of $\delta$ Scuti~ stars is rather complex
since both the rapidly oscillating roAp magnetic stars (Shibahashi 1987;
Martinez \& Kurtz 1995) and the large amplitude metal poor SX Phoenicis
stars (Nemec, Linnell Nemec \& Lutz 1994; Nemec et al. 1995) belong to
this group of variable stars as well. However, new high quality photometric
data of $\delta$ Scuti~ variables in the Galactic field (Rodriguez et al. 1995),
in the Baade's window (Udalski et al. 1995), in open clusters
(Frandsen et al. 1995) and in globular clusters (Nemec et al. 1994)
have been recently provided and therefore a proper classification of
these objects based on their evolutionary and pulsational properties
will be soon available.
Another interesting aspect which makes $\delta$ Scuti~ variables worth being investigated
is that during their evolution intermediate mass stars cross the instability
strip in different evolutionary phases. In fact, in contrast with
canonical Cepheid-like variables which are connected with Helium burning
evolutionary phases, in this region of the HR diagram the current evolutionary
theory foresees stars in pre-main-sequence, main sequence and post main
sequence phases. As a consequence, their pulsational properties can supply
useful constraints on their evolutionary properties and at the
same time an independent check of stellar models (Breger 1993,1995).
\noindent
Linear adiabatic and nonadiabatic models of these objects have been
computed by Stellingwerf (1978), Andreasen, Hejlesen \& Petersen (1983) and
Andreasen (1983). Even though these investigations have significantly
improved our knowledge of the location of the linear blue boundaries of the
instability strip, the theoretical scenario of $\delta$ Scuti~ stars presents
several unsettled problems.
According to this evidence, we decided to extend the investigation on
modal stability and pulsational properties already provided for low mass
He burning stars to larger mass values. Canonical evolutionary prescriptions
indicate that stars with
masses of the order of (1.5 - 2.0)$M_{\odot}$, unlike low mass stars,
spend a non negligible portion of their lifetime inside the instability
strip during their evolution off the Main Sequence. Therefore relatively
massive stars are expected to generate, during this phase, short period
radial pulsators.
\noindent
In this paper we investigate the nonlinear pulsational properties of this
kind of variable stars (Eggen 1994; McNamara 1995) by exploring the
pulsational behavior of a sequence of models as a function of the surface
effective temperature. For the first time we show that the adopted
theoretical framework accounts for the modal stability of SO~ pulsators
and allows some predictions concerning the full amplitude behavior of
SO~ light and velocity curves. In \S 2 we describe the numerical and physical
assumptions adopted for constructing both linear and nonlinear pulsation
models, whereas in \S 3 we discuss the full amplitude pulsation behavior
for the first three modes. The shape and the secondary features of both light
and velocity curves are presented in \S 4 together with a plain comparison
with available photometric data. The physical parameter which rules the
SO~ limit cycle stability and their dynamical properties are discussed
in \S 5. Our conclusions are presented in \S 6. In this section the
consequences arising from these new findings and the observative features
worth being investigated are outlined as well.
\section{STANDARD PULSATION MODELS}
\noindent
The theoretical framework adopted for investigating linear and nonlinear
pulsation characteristics of radial pulsators have been previously
described in a series of papers (Stellingwerf 1982; BS and references
therein). The sequence of models presented in this paper was
computed at fixed mass
value ($M/M_{\odot}\,$ =2.0), chemical composition (Y=0.28, Z=0.02) and luminosity
level (log $L/L_{\odot}\,$ =1.7) and by exploring a wide range of effective
temperatures ($7500 \geq T_e \geq 6000$ K).
The static envelope models were constructed by adopting an optical
depth of the outermost zone of the order of 0.001, and the inner boundary
was assumed fixed so that possible destabilization effects due to variations
in the efficiency of the H shell burning are ignored.
Moreover, we adopted the OP radiative opacities provided by Seaton et al.
(1994) for temperatures higher than 10,000 K whereas for lower temperatures
we adopted molecular opacities provided by Alexander \& Ferguson (1994).
The method adopted for handling the opacity tables was already described
in Bono, Incerpi \& Marconi (1996).
\noindent
Each envelope model extends from the surface to 20-10\% of the stellar
radius and the zone closest to the Hydrogen Ionization Region (HIR) was
constrained to $T_{HIR}=1.3 \times 10^4$ K.
Between the HIR and the surface we inserted 20 zones to ensure a good
spatial resolution of the outermost regions throughout the pulsation cycle.
Due to the key role played by spatial resolution for firmly estimating
both linear and nonlinear modal stability of higher modes, the envelopes
were discretized by adopting a detailed zoning in mass. The mass
ratio between consecutive zones ($\theta$) has been assumed equal to
$\theta$=1.1 for temperatures lower than $6 \times 10^5$ K, whereas
for higher temperatures it has been set equal to $\theta$=1.2.
By adopting this type of zoning the ionization
regions and the opacity bump due to iron are covered with a number of
zones lying between 100 and 150.
The fine spatial resolution of the ionization fronts provides, in turn,
an accurate treatment of both the formation and the propagation of shock
fronts during the phase interval between the phase of minimum radius and
the phase of maximum luminosity (for complete details see BS).
On the basis of these assumptions a typical envelope model is characterized
by roughly 20-30\% of the total stellar mass and by 150-250 zones.
\noindent
To clarify matters concerning the dependence of the
pulsation behavior on the spatial resolution we have constructed a
much finer envelope model. This model is located at $T_e=7000$ K and, in
contrast with the standard sequence of linear nonadiabatic models,
it was constructed by adopting a smaller mass ratio ($\theta$=1.08)
for the regions located at temperatures lower than $6 \times 10^5$ K.
Moreover, the inner boundary condition for this model was
chosen in such a way that the base of the envelope was located below
0.1 of the photospheric radius and its temperature was of the order of
$5-6 \times 10^6$ K.
\noindent
As a consequence of these assumptions the detailed model
presents an increase in both the envelope mass ($M_{env}=0.46 M_T$
against $M_{env}=0.21 M_T$) and the number of zones which cover the
ionization regions (180 against 150). We eventually found that
for the first three modes the differences between the pulsation
characteristics of both detailed and standard model are quite negligible.
In fact, the discrepancies range from $10^{-4}$ to $10^{-3}$
for the periods and from $10^{-6}$ to $10^{-4}$ for the growth rates.
As a result we can assume that the spatial resolution of the standard
sequence allows a proper treatment of the radial pulsation of $\delta$ Scuti~ stars
during their off main sequence evolution.
\section{APPROACH TO LIMIT CYCLE STABILITY}
\noindent
According to the usual approach, a sequence of linear nonadiabatic models
was first constructed for supplying the static structure of the envelope
to the nonlinear stability analysis. Then the equations governing both the
dynamical and the convective structures were integrated in time until
the initial perturbations and the nonlinear fluctuations, which result from
superposition of higher order modes, settled down (for more detail see
Bono, Castellani \& Stellingwerf 1995).
The dynamical behavior of the envelope models was computed for
the first three modes and the initial velocity profile was obtained
by perturbing the linear radial eigenfunctions with a constant
velocity amplitude of 20 km$s^{-1}$ which causes a global expansion
of the envelope.
\noindent
As is well known, the linear nonadiabatic $\delta$ Scuti~ models present very small
growth rates and therefore, before radial motions approach the nonlinear
limit cycle stability, it is necessary to carry out extensive calculations.
In fact, the long-term stability of a particular mode, due to the mixture
of both periodic and nonperiodic motions characterized by very small growing
and/or decaying rates, cannot be easily assessed at small amplitude.
As a consequence, in order to find out the possible appearance of a mode
switching or of a mixture of modes we evaluated the asymptotic behavior
of each mode by performing very long runs.
This approach leads to an integration of the governing equations for a
number of periods which ranges from 5,000 to 50,000 for some peculiar
cases. The integration is generally stopped when the nonlinear total
work is vanishing and the pulsation amplitudes present over two consecutive
periods a periodic similarity of the order of or lower than
$10^{-(2 \div 3)}$.
\noindent
Since this is the first time that hydrodynamic calculations are performed
over such a long time interval, Figures 1, 2 and 3 show the time behavior of
period, velocity and magnitude for three cases characterized by a different
approach to nonlinear limit cycle stability. In particular, Fig. 1 shows the
variation
of the quoted quantities for a single pure SO~ model, whereas Fig. 2 shows
that radial motions at $t\approx 7$ yrs experience a mode switching from
the first overtone to the SO~. Figure 3 finally presents the limiting
amplitude behavior of a case which presents a permanent mixture of different
radial modes.
\noindent
As a result of the modal stability analysis, we found stable nonlinear limit
cycles in the fundamental, in the first overtone and, for the first time,
in the second overtone when the effective temperature is increased.
Even though so far the location inside the instability strip of $\delta$ Scuti~ stars
characterized by different pulsation modes has not been firmly
established, the previous finding confirms the distribution originally
suggested by Breger \& Bregman (1975).
In fact, by assuming that $\delta$ Scuti~ variables are radial pulsators, these authors
found that observed second and first overtone variables were located at
effective temperatures higher than the fundamental ones.
\noindent
In Table 1 are listed selected observational parameters for the sequence
of $\delta$ Scuti~ models. As a first result,
data in Table 1 show that theoretical periods appear in general
agreement with the observed range of $\delta$ Scuti~ values.
It is worth underlining that the effective temperature of the fundamental
red edge should be considered an upper limit. As a matter of fact, even
though the model located at 6300 K after 23,000 periods presents
both a constant negative value in the total work term and a very low
pulsational amplitude ($\Delta M_{bol} \approx 4-6 \times 10^{-3}$ mag),
this region of the instability strip, due to the slow approach to limit
cycle stability, should be investigated in more detail before firmly
constraining the location of the red edge.
\noindent
In order to disclose the main features of the modal behavior in $\delta$ Scuti~
stars, Figure 4 shows the bolometric light curves and the surface
radial velocities of SO~ pulsators; Figure 5 shows the same quantities
but they are referred to selected first overtone (solid lines) and
fundamental (dashed lines) pulsators.
Figure 6 shows the light and radial velocity curves
of mixed mode pulsators, i.e. of models which present a permanent mixture
of different radial modes at limiting amplitude.
\noindent
Inspection of light curves discloses the surprising evidence that
the shape of SO~ light curves -sudden increase in the rising branch and slow
decrease in the decreasing branch-
closely resembles canonical fundamental mode rather than first overtone
RR Lyrae pulsators. This finding confirms the original prediction
concerning the shape of SO~ light curves made by SGD.
Moreover, we find that moving from SO~ to lower
pulsational modes the amplitudes progressively decrease and the shape
of the light curves becomes more sinusoidal. It is worth noting that the
pulsation amplitudes of RR Lyrae variables, which are located in the same
region of the instability strip, present an opposite trend.
In fact, for this group of pulsators the RRab variables (fundamental)
show the largest amplitudes. These theoretical prescriptions can be
usefully compared with the observational scenario recently discussed
by McNamara (1995 and references therein).
\noindent
According to this author, $\delta$ Scuti~ stars on the basis of their luminosity
amplitude can be empirically divided into two groups.
The light curves of stars with larger amplitudes appear to be
asymmetrical whereas the light curves for lower amplitudes
tend to be much more symmetrical. However, in the above paper McNamara also
suggests that for low amplitude variables, which are poorly sampled,
it is often difficult to determine whether the light curves are symmetric
or asymmetric. Therefore, for light curves which are only partially
covered by photometric data, several cases of probable asymmetry are brought
forward. By analogy with the behavior of RR Lyrae variables, McNamara (1995)
assumes that stars with asymmetric, large amplitude
light curves are fundamental pulsators whereas symmetric, low amplitude
light curves belong to first overtone pulsators.
The comparison of similar empirical prescriptions with the current
theoretical scenario shows a convincing degree of agreement.
However, theory now tells us that asymmetric, large amplitude pulsators
are good SO~ candidates, whereas low amplitude pulsators could be a
mixture of fundamental and first overtones.
\noindent
To go further on with this comparison, let us refer to the sample of variable
stars recently collected by the OGLE collaboration (Udalski et al. 1995 and
references therein) as the result of their search for evidence of
microlensing in the bulge of the Galaxy.
Inspection of photometric data connected with short period variables
discloses that the observed light curves can be arranged in three typical
classes, as shown in Fig. 7, with class "A" representing McNamara large
amplitude pulsators and classes "B" and "C" the small amplitudes ones.
For a meaningful comparison between theory and observation, the
bolometric light curves have been transformed into the I band according
to Kurucz's (1992) atmosphere models. Figures 8 and 9 show the light
curves for single mode pulsators.
Due to the magnitude scale adopted for plotting data in Fig. 9, the light
curves of fundamental pulsators (dashed lines) seem almost perfectly
sinusoidal. However, even though the luminosity variations throughout
the cycle are quite smooth, a bump appears before the phase of minimum
radius. Taking into account that we explored only one mass value and
only one luminosity level, the comparison should be considered more
than satisfactory.
\section{ LIGHT CURVE MORPHOLOGY}
\noindent
Available observational data hardly allow to detect minor details in the
light curves. However, the quality of both spectroscopic and
photometric data is rapidly improving (see for example Milone, Wilson
\& Fry 1994; Breger et al. 1995 and references therein) and therefore
in this section we discuss even minor features of theoretical light curves
in order to underline the theoretical predictions worth being
investigated with the required accuracy. As a first point, let
us notice that the light and velocity curves of SO~ pulsators
present two further relevant distinctive features:
\noindent
1) like canonical first overtone RR Lyrae variables, the bump does not
appear along the decreasing branch of the light curves and, moving from
higher to lower effective temperatures the dip becomes more and more
evident along the rising branch;
\noindent
2) moving from the blue to the red boundary of the SO instability region
the velocity curves show smooth variations but at phases 0.2-0.3
a bump appears due to the propagation of an outgoing shock.
\noindent
As a second point, we find that the shape of first overtone light curves
presents some features which allow a careful distinction
between different radial modes. In fact for these models the dip is the
main maximum, whereas the "true" maximum takes place along the decreasing
branch. Moreover, the first overtone light curves show that the bump
appears along the increasing branch and that just before the phase of
minimum radius they also display a short stillstand phase
($\phi \approx 0.45$).
\noindent
The scenario concerning the pulsation characteristics of $\delta$ Scuti~ stars can
be now nicely completed by the models which are simultaneously
excited in two or more radial modes. Figure 6 shows a collection of
light and velocity curves of mixed mode pulsators, a glance to these curves
brings out both the expected fluctuation of the pulsation amplitudes between
consecutive cycles and the appearance of the secondary features which
characterize the first three lower single mode pulsators.
\noindent
A thorough comparison with observed period ratios of $\delta$ Scuti~ variables is beyond
the scope of the present study since the "true" period ratios should be
evaluated through a Fourier decomposition of the theoretical light curves.
Moreover, for properly constraining the stellar masses and the luminosity
levels of these objects by means of the Petersen diagram
($P_1$/$P_0$ vs. $P_0$) an extensive set of nonlinear models computed for
different assumptions on astrophysical parameters is necessary
(Bono et al. 1996c).
\noindent
Nevertheless, since period ratios of double mode pulsators can
provide valuable clues on several astrophysical problems involving both the
evolutionary and the pulsational properties of these objects, we constructed
a new sequence of detailed, linear, radiative, nonadiabatic models by adopting
the assumptions already discussed at the end of section 2. This analysis
has been undertaken only for supplying a preliminary but meaningful theoretical
guess concerning the location of this group of variables inside the instability
strip.
\noindent
At first it is important to note that the linear periods
and the related period ratios listed in Table 2 are, within the estimated
uncertainties, in agreement with
observed values (see for example data on double mode $\delta$ Scuti~ stars collected by
Andreasen 1983 and by Petersen 1990). Indeed the observed period ratio between
first overtone and fundamental pulsators ranges, for large amplitude $\delta$ Scuti~
stars, from 0.760 to 0.780, whereas the period ratio between second overtone
and first overtone is approximatively of the order of 0.800.
This agreement is a remarkable result since the period ratios predicted by
pulsation models are the most important
observable adopted for finding out whether the pulsation is "driven" by
radial or by nonradial modes. As a consequence, this finding provides sound
evidence that these variables are mixed mode radial pulsators (Breger 1979).
\noindent
Moreover, previous linear and nonlinear results suggest that, due to the
appearance of three different modal stabilities inside the instability
strip, double mode pulsators belonging to this group of variables are
located close
to the fundamental blue edge when they are a mixture of fundamental and
first overtone and close to the first overtone blue edge when they are
a mixture of first and second overtones.
A detailed investigation on the dependence of this peculiar occurrence among
radial pulsators together with a straightforward analysis of the envelope
structure will be discussed in a forthcoming paper (Bono et al. 1996d).
\section {SECOND OVERTONE INSTABILITY}
\noindent
In order to properly identify the regions of the stellar envelope which
{\em drive} or {\em damp} the pulsation instability of SO~ pulsators,
Fig. 10 shows the nonlinear differential work integrals versus
the logarithm of the external mass for a model located close to
the SO~ blue edge. In this plane the positive areas denote driving
regions (growing oscillations) whereas the negative areas damping
regions (quenching oscillations). The total work curve shows quite
clearly the two driving sources due to the hydrogen and helium
ionization zones as well as the radiative damping due to the inner
regions. Unlike in canonical cluster variables, the second helium
ionization zone provides a stronger destabilization if compared
with the HIR. This effect is mainly due
to the increase in effective temperature which causes a shift of
the HIR toward the surface and therefore a decrease of the mass
which lies above these layers. However, the total work plotted in Fig. 10
clearly shows that this element, in contrast with previous
qualitative arguments, provides a substantial amount of driving to
the pulsation instability of $\delta$ Scuti~ stars.
For the reasons previously discussed all other nonlinear work
terms supply a negligible damping effect on the pulsation.
\noindent
Nevertheless the physical parameter which rules the SO~ instability
is the location of the nodes
\footnote{In the case of a vibrating membrane a {\em node} is the
point where an eigenfunction vanishes or attains a sharp minimum and
its phase changes by almost $\pi$ radians.}
inside the envelope. In fact,
the nodes of temperature, luminosity and radius fall within the region of
radiative damping. As a consequence, the amount of damping is strongly
reduced and the destabilization of the ionization regions pumps up
the pulsation amplitude. In particular, it is worth emphasizing
that among cluster variables the outermost node of radius is located
quite close to the helium driving region (Stellingwerf 1990).
In this context the node of radius has an opposite effect since it
reduces the amount of driving and consequently quenches the oscillations.
\noindent
On the basis of a well-known theorem concerning the Sturm-Liouville
eigenvalue problem, a SO~ eigenfunction should split its domain into
two parts by means of its {\em nodal lines} (Courant \& Hilbert 1989).
Even though during a full pulsation cycle the radial motions never
exactly approach the initial equilibrium configuration, the velocity
curves plotted in Fig. 11 undoubtedly show two different
subdomains characterized by opposite radial motions.
In order to disclose the dynamical structure of the model previously
discussed, Fig. 11 shows the radial displacements of the whole
envelope over two consecutive periods. Dots and pluses mark the phases
during which each zone is contracting or expanding respectively.
It is easy to ascertain from the above figure that at a fixed pulsation
phase the envelope is divided into two different regions and the velocity
curves are exactly 180$^{\circ}$ out of phase at their boundaries.
In particular, the layers located close to the two boundaries define two
zones which virtually remain at rest throughout the pulsation
cycle and in which the radial velocity changes its sign abruptly.
\section{CONCLUSIONS}
\noindent
In this paper we report the first theoretical evidence for the
occurrence of pulsators that when moving inside the instability strip
from lower to higher effective
temperatures show three different stable pulsation modes, namely
fundamental, first and second overtone.
We found that the predicted features of the light curves
appear in general agreement with observational constraints
concerning $\delta$ Scuti~ variables, suggesting that SO~ pulsators have
been already observed and pointing out, at the same time, a revised
approach to observational evidences for different pulsation modes.
Moreover, we show that the location of radius, luminosity and temperature
nodes in the damping region of the stellar envelope is the main
physical parameter which governs the limit cycle stability of SO~
pulsators. This result strongly supports the discussion given in the
introduction to this paper about the absence of SO~ pulsators in low mass
variable stars.
\noindent
The satisfactory agreement between the computed period ratios and the
observed ones casts new light on the problem of limit cycle stability
of mixed mode variables belonging to population II stars and provides
a valuable piece of information for accounting for the pulsation
properties of $\delta$ Scuti~ stars. At the same time there is a growing strong
evidence that these stars during their evolution off the main sequence
are pure or mixed mode radial pulsators.
\noindent
Before being able to provide a sound comparison with available photometric
data on $\delta$ Scuti~ variables, the present sequence of nonlinear models should be
extended to both lower luminosity levels and to smaller mass values.
However, even though the computed models cover a restricted range of
effective temperatures, the theoretical framework currently adopted
accounts for both pure and mixed mode radial pulsators. This new scenario
presents several interesting features since both the modal stability and
the pulsational behavior have been investigated in a homogeneous
physical context without invoking unpleasant {\em ad hoc} physical
mechanisms and/or peculiar characteristics.
\noindent
Plenty new high quality photometric data on $\delta$ Scuti~ stars will be soon
available as a by-product of the international projects involved in the
search for microlensing events and therefore new sequences of nonlinear $\delta$ Scuti~
models at full amplitude, albeit such analysis places nontrivial
computational efforts, are necessary to firmly accomplish the pulsation
properties of these objects.
\noindent
Finally we suggest that short period RR Lyrae-like pulsators found in
the Galactic bulge as well as in dwarf spheroidals like Carina and
Sagittarius should be regarded as an evidence of relatively massive
stars, a witness of the efficiency of star formation until relatively
recent times. On the other hand this finding stresses once again the key
role played by variable stars as tracers of stellar populations which
experienced different dynamical and/or chemical evolutions.
\noindent
It is a pleasure to thank F. Pasian and R. Smareglia for their kind
and useful sharing of computing facilities which made this investigation
possible. We are grateful to J. Nemec for his clarifying suggestions as
referee on an early draft of this paper. We also benefit from the use of
the SIMBAD data retrieval system (Astronomical Data Center, Strasbourg,
France). This work was partially supported by MURST, CNR-GNA and ASI.
\pagebreak
| proofpile-arXiv_065-455 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\subsection*{Acknowledgments}
This work was supported by the US Department of Energy, Nuclear Physics
Division, under contract number W-31-109-ENG-38.
| proofpile-arXiv_065-456 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
\label{intro-sec}
Japanese is a language well-known for grammaticization of discourse
function. It is rich with ways for speakers to indicate the information
status of the discourse entities they are talking about. Japanese allows a
speaker to clearly indicate topic-hood, along with the grammatical
functions such as subject, object and object2, by using the morphological
case markers {\em wa, ga, o, ni}. In addition, it provides morphological
means to indicate speaker's perspective through the use of verbal
compounding, i.e. the addition of suffixes such as {\em kureta, kita} (See
section 3). Unexpressed arguments of the verb are common; these are
known as zero pronouns.
Because there are zero pronouns and because Japanese is a head-final
language with otherwise relatively free word order, there could, in
principle, be a great deal of ambiguity. However this is not the case.
Speakers are assumed to be cooperative, to be collaborating with the hearer
in conversation, and to be ensuring that each utterance is relevant and
coherent in the context of what was said before \cite{Grice75,SSJ74}. We
believe that speakers do not choose to express their thoughts through
arbitrary syntactic constructions, but that there is some correspondence
between choice of syntactic construction, what the speaker wants to convey,
and aspects of the current discourse situation\cite{Prince85}.
Within a theory of discourse, {\sc centering} is a computational model
of the process by which a speaker and hearer make obvious to one
another their assumptions about the salience of discourse entities.
Using pronominal referring expressions is one way for discourse
participants to do this. We propose that the resolution of zero
pronouns is constrained by centering, and ambiguity is thereby
reduced.
Centering has its computational foundations in the work of Grosz and
Sidner\cite{Grosz77,Sidner79,GS85} and was further developed by Grosz,
Joshi and Weinstein\cite{GJW83,GJW86,JW81}. It is formalized as a
system of constraints and rules, which can, as part of a computational
discourse model, act to control inferencing\cite{JW81}. Brennan,
Friedman and Pollard use these rules and constraints to develop an
algorithm for resolving the co-specifiers of
pronouns\cite{BFP87,Walker89b}. Our analysis uses an adaptation of this
algorithm. By making full use of the centering formalism,
we avoid the postulation of additional mechanisms, e.g. property
sharing\cite{Kameyama86a}.
In addition, we propose a notion of {\sc topic ambiguity}, which
characterizes some ambiguities in Japanese discourse that are allowed by
the centering process. Topic ambiguity has been ignored in previous
accounts of Japanese zero pronoun resolution, but it explains the
availability of interpretations that previous accounts would predict as
ungrammatical. Centering gives us a computational way of determining when a
zero pronoun may be assigned {\sc Topic}.
This analysis informs the design of language independent discourse
processing modules for Natural Language systems. We propose that the
centering component of a discourse processing module can be constructed in
a language independent fashion, up to the declaration of a language-specific
value for one variable in the algorithm, i.e., Cf list ranking (see
section \ref{cent-form}). The centering algorithm has been
implemented in an HPSG Natural Language system with both English and
Japanese grammars.
\section{The Centering Formalism}
\label{cent-form}
The modeling of attentional state in discourse by
centering depends on analyzing each pair of utterances in a discourse
according to a set of transitions. These transitions are a measure of
the coherence of the segment of discourse in which the utterance
occurs. Each utterance in a discourse has associated with it a set of
discourse entities called {\sc forward-looking centers}, $\rm{Cf}$,
and a special member of this set called the {\sc backward-looking
center}, $\rm{Cb}$. The {\sc forward-looking centers} are ranked according
to discourse salience; the
highest ranked member of the set is the {\sc preferred center},
$\rm{Cp}$. With these definitions we can give the constraints:
\nopagebreak
\begin{itemize}
\item
{\bf CONSTRAINTS} \\
For each $\rm{U_{i}}$ in a discourse segment $\rm{U_{1}, \ldots ,U_{m}}$:
\vspace{-3.0ex}
\begin{enumerate}
\item There is precisely one $\rm{Cb}$.
\item Every element of $\rm{Cf(U_{i}})$ must be realized\footnote
{An utterance U (of some phrase, not necessarily a full clause), {\em
realizes\/} {\bf c} if {\bf c} is an element of the situation
described by U, or {\bf c} is the semantic interpretation of some
subpart of U.} in $\rm{U_{i}}$.
\item The center, $\rm{Cb(U_{i})}$, is the highest-ranked element
of $\rm{Cf(U_{i-1})}$ that
is realized in $\rm{U_{i}}$.
\end{enumerate}
\end{itemize}
The typology of transitions from one utterance, $\rm{U_{i}}$, to the next
is based on two factors: whether the backward-looking center, $\rm{Cb}$, is
the same from $\rm{U_{i-1}}$ to $\rm{U_{i}}$, and whether this discourse
entity is the same as the preferred center, $\rm{Cp}$ of $\rm{U_{i}}$.
Backward-looking centers are often pronominalized and discourses that
continue centering the same entity are more coherent than those that shift
from one center to another. This means that some transitions are preferred
over others. These two facts give us the rules:
\begin{itemize}
\item
{\bf RULES} \\
For each $\rm{U_{i}}$ in a discourse segment $\rm{U_{1}, \ldots ,U_{m}}$:
\vspace{-3.0ex}
\nopagebreak
\begin{enumerate}
\item If some element of $\rm{Cf(U_{i-1})}$ is realized as a
pronoun in $\rm{U_{i}}$,
then so is $\rm{Cb(U_{i})}$.
\item Transition states are ordered.
{\sc Continuing\/} is preferred to {\sc retaining\/} is preferred to
{\sc shifting-1} is preferred to {\sc shifting}\footnote{\cite{BFP87}
introduces the distinction between SHIFTING-1 and SHIFTING.}.
\end{enumerate}
\end{itemize}
The transition states that are used in the rules are defined in Figure
\ref{state-fig}, ({\sc backward-looking center} $ = \rm{Cb}$, {\sc
preferred Center} $ = \rm{Cp}$).
\begin{small}
\begin{figure}[htb]
\setlength{\unitlength}{.65in}
\begin{flushright}
\begin{picture}(4,2)
\put(0,0){\framebox(2,1){RETAINING}}
\put(0,1){\framebox(2,1){CONTINUING}}
\put(2,1){\framebox(2,1){SHIFTING-1}}
\put(2,0){\framebox(2,1){SHIFTING}}
\put(1,2.2){\makebox(0,0){$\rm{Cb(U_{i}) = Cb(U_{i-1}}$)}}
\put(3,2.2){\makebox(0,0){$\rm{Cb(U_{i}) \neq Cb(U_{i-1}}$)}}
\put(-.33,1.65){\makebox(0,0){$\rm{Cb(U_{i})}$}}
\put(-.25,1.45){\makebox(0,0){$=$}}
\put(-.33,1.25){\makebox(0,0){$\rm{Cp(U_{i})}$}}
\put(-.33,0.70){\makebox(0,0){$\rm{Cb(U_{i})}$}}
\put(-.25,0.5){\makebox(0,0){$\neq$}}
\put(-.33,0.30 ){\makebox(0,0){$\rm{Cp(U_{i})}$}}
\end{picture}
\end{flushright}
\normalsize
\caption{ Transition States}
\label{state-fig}
\end{figure}
\end{small}
The centering algorithm incorporates these rules and constraints in
addition to linguistic constraints on coreference\cite{BFP87}. The
behavior of the centering algorithm for the resolution of pronouns is
largely determined by the ranking of the items on the forward center
list, $\rm{Cf}$, because, as per Constraint 3, this ranking determines
from among the elements that are realized in the next utterance, which
of them will be the $\rm{Cb}$ for that utterance.
Although all of the factors that contribute to the $\rm{Cf}$ ranking
have not been determined, syntax and lexical semantics have an
effect\cite{Prince81,Prince85,Hudson88,Brennan89,GJW86,JW81,BF83}.
We postulate that this ordering will vary from language to language
depending on the means the language provides for expressing discourse
functions. Our adaptation of the algorithm for Japanese consists of
substituting a different ranking of the forward centers list
$\rm{Cf}$. In every other way, the algorithm functions exactly as it
is for English.
\section{Centering in Japanese}
In order to apply the centering algorithm to the resolution of zero
pronouns in Japanese, we must determine how to order the forward
centers list, $\rm{Cf}$. The function {\sc topic} is indicated by the
morphological marker {\em wa}, along with {\sc subject} ({\em ga}),
{\sc object} ({\em o}), and {\sc object2} ({\em ni}). The optional
use of {\em wa} picks out the most salient entity in the discourse.
In addition, Kuno proposed the notion of {\sc empathy}, which is the
perspective from which a speaker describes an event\cite{Kuno73}. The
realization of speaker's empathy is especially important when
describing an event involving some transfer. For example, there is no
way to describe a {\em giving\/} and {\em receiving\/} situation
objectively\cite{Kuno-Kab77}. In (1), the use of the past tense {\em
kureta} of the verb {\em kureru\/}, indicates the speaker's empathy
with the discourse entity realized in object position\footnote{We
use identifiers of all capital letters to denote the discourse entity
realized by the corresponding string. Centers are semantic entities,
not syntactic ones.}.
\noindent (1)\\
\footnotesize
\begin{tabular}{llllll}
Hanako wa & Taroo ni & hon o & kureta. & & \\
top-subj & obj2 & book obj & give-past & & \\
\multicolumn{6}{l}{{\it ``Hanako gave Taroo a book.''}}\\
\multicolumn{6}{l}{EMPATHY=OBJ2=TAROO} \\
\end{tabular}
\normalsize
In (2), the speaker's empathy with the subject entity's
perspective is indicated using {\em yatta}, the past tense of the verb
{\em yaru\/}.
\noindent (2)\\
\footnotesize
\begin{tabular}{llllll}
Hanako wa & Taroo ni & hon o & yatta. & & \\
top-subj & obj2 & book obj & give-past & & \\
\multicolumn{6}{l}{{\it ``Hanako gave Taroo a book.''}}\\
\multicolumn{6}{l}{EMPATHY=SUBJ=HANAKO} \\
\end{tabular}
\normalsize
The use of deictic verbs such as {\it kuru\/} (`come'), and {\it
iku\/} (`go') also indicate speaker's perspective.
Kuno calls a verb that is sensitive to the speaker's perspective an
{\sc Empathy-loaded} verb, and defines {\sc Empathy locus} as the
argument position whose referent the speaker is identifying
with\footnote{The speaker does not necessarily take his/her own
perspective to describe an event in which s/he is involved.}.
Any Japanese verb can be made into an empathy-loaded verb by using an
empathy-loaded verb as an auxiliary, which is suffixed onto the main
verb stem. The complex predicate made by this operation inherits the
empathy-locus of the suffixed verb. The {\em kureru} form of (`give')
can be used as a suffix, to mark {\sc obj} or {\sc obj2} as the
empathy-locus, as can the deictic verb {\em kuru\/} (`come') The use of
the suffix {\em kureta} is shown in (3).
\noindent (3)\\
\footnotesize
\begin{tabular}{llllll}
Hanako wa & Taroo ni & hon o & yonde-kureta. & & \\
& & book & read-gave & & \\
\multicolumn{6}{l}{{\it ``Hanako gave Taroo a favor of reading a book.''}}\\
\multicolumn{6}{l}{EMPATHY=OBJ2=TAROO} \\
\end{tabular}
\normalsize
The suffixation of verbs
such as {\em iku\/} (`go') and the {\em yaru} form of (`give'), mark
{\sc subject} as the empathy-locus, e.g. {\em itta} in (4).
\noindent (4)\\
\footnotesize
\begin{tabular}{llllll}
Hanako wa & Taroo o & tazunete-itta. & & & \\
& & visit-went & & & \\
\multicolumn{6}{l}{{\it ``Hanako went to visit Taroo.''}}\\
\multicolumn{6}{l}{EMPATHY=SUBJ=HANAKO} \\
\end{tabular}
\normalsize
The relevance of speaker's empathy to centering is that a discourse entity
realized as the empathy-locus is more salient, so that the empathy-locus
position is ranked higher on the $\rm{Cf}$. Therefore, we use a ranking
for the $\rm{Cf}$ in
Japanese that incorporates {\sc Empathy} as follows:
\begin{quote}
{\bf Cf Ranking for Japanese} \\
{\sc topic} $>$ {\sc empathy} $>$ {\sc subj} $>$ {\sc obj2} $>$ {\sc obj}
\end{quote}
This ranking is a slight variation of that proposed by
Kameyama\cite{Kameyama86a}. The centering algorithm works by taking the
arguments of the verb and ordering them according to the Cf ranking
for Japanese given above. In the cases where there are zero pronouns,
there will be multiple possibilities for their interpretation and this
will result in there being a priori several possible Cf
lists\footnote{A discourse entity can simultaneously fulfill multiple
roles. The entity is ranked according to the highest ranked role.}.
These Cf lists are filtered according to the centering rules and
constraints in section
\ref{cent-form}. If there are still multiple possibilities, then the
ordering on transitions applies, and {\sc continuing} interpretations
are preferred.
Many cases of the preference for one interpretation over another
follow directly from the distinction between {\sc continuing} and {\sc
retaining}.
\noindent (5) \\
\footnotesize
\begin{tabular}{llllll}
$\rm{U_{n}}$: & & & & & \\
Taroo wa & paatii ni & syootai-sareta. & & &\\
& party to & invited-was & & &\\
\multicolumn{5}{l}{{\it ``Taroo was invited to the party.''}}&\\
\end{tabular}
\begin{tabular}{|llllll|}
\hline
{\bf Cb:} & TAROO & & & & \\
{\bf Cf:} & [TAROO] & & & & \\ \hline
\end{tabular}
\begin{tabular}{llllll}
$\rm{U_{n+1}}$: & & & & & \\
0 & Hanako o & totemo & kiniitta. & & \\
& & very-much & was-fond-of & & \\
\multicolumn{5}{l}{{\it ``He liked Hanako very much.''}} &\\
\end{tabular}
\begin{tabular}{|llllll|}
\hline
{\bf Cb:} & TAROO & & & & \\
{\bf Cf:} & [TAROO, & HANAKO] & & & \\
& subj & obj & & & \\ \hline
\end{tabular}
\begin{tabular}{llllll}
$\rm{U_{n+2}}$: & & & & & \\
Kinoo & 0 & 0 & eiga ni & sasotta rasii.& \\
yesterday & & & movie to & invite seems & \\
\multicolumn{5}{l}{{\it ``Seemingly he invited her to a movie.''}} &\\
\end{tabular}
\begin{tabular}{|llllll|}
\hline
{\bf Cb:} & TAROO & & & & \\
{\bf Cf1:} & [TAROO, & HANAKO] & CONTINUING & & \\
& subj & obj & & & \\
{\bf Cf2:} & [HANAKO, & TAROO] & RETAINING & & \\
& subj & obj & & & \\ \hline
\end{tabular}
\normalsize
When the centering algorithm applies in (5) to $\rm{U_{n+2}}$,
constraint 3 says the $\rm{Cb}$ must be the highest ranked element of
Cf($\rm{U_{n+1}}$) realized in $\rm{U_{n+2}}$. Because there are 2
zeros in $\rm{U_{n+2}}$, TAROO must be realized and therefore must be
the $\rm{Cb}$. The only {\sc continuing} interpretation available,
{\em Taroo invited Hanako ...}, corresponds to the forward centers
list Cf1. The fact that the preferred interpretation
is the one in which the {\sc subject} zero pronoun
takes a {\sc subject} antecedent is epiphenomenal.
Example (6) demonstrates the effect of speaker's empathy on
the salience of discourse entities.
\noindent (6) \\
\nopagebreak
\footnotesize
\begin{tabular}{llllll}
$\rm{U_{n}}$: & & & & & \\
Hanako wa & tosyokan de & benkyoositeita. & & &\\
& library in & studying-was & & &\\
\multicolumn{5}{l}{{\it ``Hanako was studying in the library.''}} &\\
\end{tabular}
\begin{tabular}{|llllll|}
\hline
{\bf Cb:} & HANAKO & & & & \\
{\bf Cf:} & [HANAKO] & & & & \\ \hline
\end{tabular}
\footnotesize
\begin{tabular}{llllll}
$\rm{U_{n+1}}$: & & & & & \\
Taroo ga & Hanako o & tetudatte-kureta. & & & \\
& & help-gave & & & \\
\multicolumn{5}{l}{{\it ``Taroo gave Hanako a favor in helping her.''}} &\\
\end{tabular}
\begin{tabular}{|llllll|}
\hline
{\bf Cb:} & [HANAKO] & & & & \\
{\bf Cf:} & [HANAKO, & TAROO] & & & \\
& empathy & subj & & & \\ \hline
\end{tabular}
\begin{tabular}{llllll}
$\rm{U_{n+2}}$: & & & & & \\
Tugi no hi & 0 & 0 & eiga ni & sasotta. & \\
next of day &SUBJ & OBJ & movie to & invited & \\
\multicolumn{5}{l}{{\it ``Next day she invited him to a movie.''}}& \\
\end{tabular}
\begin{tabular}{|llllll|}
\hline
{\bf Cb:} & HANAKO & & & & \\
{\bf Cf:} & [HANAKO, & TAROO] & CONTINUING & & \\
& subj & obj & & & \\ \hline
\end{tabular}
\normalsize
In (6), HANAKO is the most highly ranked entity from $\rm{U_{n+1}}$
realized in $\rm{U_{n+2}}$ , and therefore must be the $\rm{Cb}$. The
preferred interpretation will therefore be the {\em she invited
him...} one that results from the more highly ranked {\sc continuing}
transition, in which HANAKO is the preferred center ($\rm{Cp}$).
The centering algorithm can also be applied successfully to
intrasentential anaphora, by treating the subordinate clause as though
it were a separate utterance for the purposes of pronoun
interpretation. Consider:
\noindent (7) \\
\footnotesize
\begin{tabular}{llllll}
Taroo wa & Kim ni &[0 0 & bengosuru] &koto o &hanasita. \\
& & & defend & comp & told \\
\multicolumn{6}{l}{{\it``Taroo told Kim that he would defend her''}}\\
\end{tabular}
\begin{tabular}{|llllll|}
\hline
{\bf Cb:} & TAROO & & & & \\
{\bf Cf1:} & [TAROO, & KIM] & CONTINUING & & \\
& subj/top & obj2 & & & \\
{\bf Cf2:} & [HANAKO, & KIM] & RETAINING & & \\
& subj/top & obj2 & & & \\ \hline
\end{tabular}
\normalsize
The {\sc continuing} interpretation, {\em Taroo told Kim that he
would defend her}, is preferred to the {\sc retaining} interpretation,
{\em Taroo told Kim that she would defend him}.
\section{Topic ambiguity}
The centering process reduces but does not necessarily eliminate
semantic ambiguity in
Japanese discourse. Within a loosely defined context, a native
speaker's intuitions sometimes
still allow for more than one equally preferred interpretation of an utterance.
\subsection{Center Establishment}
In the ``Introduce'' example shown in (8) below, ambiguity
arises from the combined facts that the $\rm{Cb}$ of $\rm{U_{1}}$ is
neutral (undefined),
and there are more entities on the Cf list of $\rm{U_{1}}$ than there are zero
pronouns in $\rm{U_{2}}$.
\noindent (8)\\
\nopagebreak
\footnotesize
\begin{tabular}{llllll}
$\rm{U_{1}}$: & & & & & \\
Lyn-ga & Masayo-ni & Sharon-o & shookaisita & & \\
SUBJ & OBJ2 & OBJ & introduced & & \\
\multicolumn{6}{l}{\it {``Lyn introduced Sharon to Masayo.''}}\\
\end{tabular}
\begin{tabular}{|llllll|}
\hline
{\bf Cb:} & [?] & & & & \\
{\bf Cf:} & [LYN, & MASAYO, & SHARON] & & \\
& subj & obj2 & obj & & \\ \hline
\end{tabular}
\begin{tabular}{llllll}
$\rm{U_{2}}$: & & & & & \\
0 & 0 & kiniitteiru & & & \\
\multicolumn{6}{l}{\it {``Lyn likes Masayo''} (Cf1a)}\\
\multicolumn{6}{l}{\it {``Lyn likes Sharon''} (Cf1b)}\\
\multicolumn{6}{l}{\it {``Masayo likes Sharon''} (Cf2)}\\
\end{tabular}
\begin{tabular}{|llllll|}
\hline
{\bf Cb1:} & LYN & & & & \\
{\bf Cb2:} & MASAYO & & & & \\
{\bf Cf1a:} & [LYN, & MASAYO] & & & \\
& subj & obj & & & \\
{\bf Cf1b:} & [LYN, & SHARON] & & & \\
& subj & obj & & &\\
{\bf Cf2:} & [MASAYO, & SHARON] & & & \\
& subj & obj & & &\\ \hline
\end{tabular}
\normalsize
All three of these readings of $\rm{U_{2}}$ are equally preferred {\sc
continuations}. To explain this fact, we posit that the $\rm{Cb}$ of
an initial utterance $\rm{U_{n}}$ may be treated as a variable,
indicated by [?], which can be equated with whatever $\rm{Cb}$ is
assigned to the subsequent utterance $\rm{U_{n+1}}$\footnote{Future
work will discuss center
establishment in more detail, as well as other
interactions, e.g., the effect of {\it wa} marking.}. For example,
because there are 2 zeros in $\rm{U_{2}}$ of (8) and there are 3
entities available to fill these positions, constraint 3 implies that
SHARON (the lowest ranked entity) can never be the $\rm{Cb}$, since it
will never be the most highly ranked element of $\rm{Cf(U_1)}$
realized in $\rm{U_2}$. Therefore, whenever LYN is realized, the {\sc
continuation} interpretation will place LYN in subject position, thus
explaining the first two readings of $\rm{U_2}$. The third reading is
available because no $\rm{Cb}$ has yet been established for
$\rm{U_{1}}$, so that a {\sc continuation} does not require the
realization of LYN in $\rm{U_{2}}$. Notice that any reading that
assigns SHARON to the subject position or LYN to a non-subject
position would produce a {\sc retention}.
\subsection{Zero Topics}
Another class of ambiguities can result from the optional assignment of
{\sc topic} to
a zero pronoun. We propose a topic assignment rule:
\begin{quote}
{\bf Zero Topic Assignment} \\
When no {\sc continuation} transition is available, and a zero pronoun
in $\rm{U_{m}}$ represents an entity that was the $\rm{Cb(U_{m-1})}$
and if no other entity in $\rm{U_{m}}$ is overtly marked as the {\sc
topic}, that zero may be interpreted as the {\sc topic} of
$\rm{U_{m}}$.
\end{quote}
This fact, which has been overlooked in previous treatments of zero
pronouns in Japanese, explains the interesting contrast between the
two discourse segments in examples (9) and (10) below. Assume in
(9) and (10) that TAROO and HANAKO have already been under
discussion:\footnote{Due to lack of space, we can not discuss the
interaction of center establishment with zero topic assignment here.}
\noindent (9)\\
\footnotesize
\begin{tabular}{llllll}
$\rm{U_{n}}$: & & & & & \\
Taroo wa & kooen o & sanpo-siteita & & & \\
SUBJ & park & walk-around & & & \\
\multicolumn{6}{l}{\it {``Taroo was walking around the park''}}\\
\end{tabular}
\begin{tabular}{|llllll|}
\hline
{\bf Cb:} & TAROO & & & & \\
{\bf Cf:} & [TAROO, & PARK] & & & \\
& subj & obj & & & \\ \hline
\end{tabular}
\begin{tabular}{llllll}
$\rm{U_{n+1}}$: & & & & & \\
Hanako ga & 0 & yatto & mituketa & & \\
SUBJ & & finally & found & & \\
\multicolumn{6}{l}{\it {``Hanako finally found (him).''}}\\
\end{tabular}
\begin{tabular}{|llllll|}
\hline
{\bf Cb:} & TAROO & & & & \\
{\bf Cf1:} & [TAROO, & HANAKO] & (C) & &\\
& topic/obj & subj & & & \\
{\bf Cf2:} & [HANAKO, & TAROO] & (R) & & \\
& subj & obj & & & \\ \hline
\end{tabular}
\begin{tabular}{llllll}
$\rm{U_{n+2}}$: & & & & & \\
0 & 0 & yotei o & setumeisita & & \\
SUBJ& OBJ & schedule & explained & & \\
\multicolumn{6}{l}{\it{He explained the schedule to her.} (Cf1)}\\
\multicolumn{6}{l}{\it{She explained the schedule to him.} (Cf2)}\
\end{tabular}
\begin{tabular}{|llllll|}
\hline
{\bf Cb1:} & TAROO & & & & \\
{\bf Cb2:} & HANAKO & & & & \\
{\bf Cf1:} & [TAROO, & HANAKO] & (C) & & \\
& subj & obj & & &\\
{\bf Cf2:} & [HANAKO, & TAROO] & (S-1) & & \\
& subj & obj & & &\\ \hline
\end{tabular}
\normalsize
In (9), there are actually two possible Cf lists in $\rm{U_{n+1}}$;
Cf2, which is the only list possible without topic ambiguity,
represents a {\sc retention} (R) rather than a {\sc continuation} (C),
thus triggering zero topic assignment. The utterance $\rm{U_{n+1}}$,
actually has the same meaning for both Cf lists. The ambiguity in
$\rm{U_{n+2}}$ is caused by the fact that the hearer simultaneously
entertains both of the $\rm{Cf(U_{n+1})}$. The availability of zero
topic assignment means that TAROO can be the $\rm{Cp}$ even when TAROO
is realized as the topic/object. The {\sc shift-1} interpretation
results from the algorithm's application to Cf2 of $\rm{U_{n+1}}$. We
can test to see if topic ambiguity is actually the discourse
phenomenon at work here by contrasting (9) with its minimal pair (10),
in which overt topic marking in $\rm{U_{n+1}}$ rules out topic ambiguity.
\noindent (10)\\
\footnotesize
\begin{tabular}{llllll}
$\rm{U_{n}}$: & & & & & \\
Taroo wa & kooen o & sanpo-siteita & & & \\
SUBJ & park & walk-around & & & \\
\multicolumn{6}{l}{\it {``Taroo was walking around the park''}}\\
\end{tabular}
\begin{tabular}{|llllll|}
\hline
{\bf Cb:} & TAROO & & & & \\
{\bf Cf:} & [TAROO, & PARK] & & & \\
& subj & obj & & & \\ \hline
\end{tabular}
\begin{tabular}{llllll}
$\rm{U_{n+1}}$: & & & & & \\
Hanako-wa & 0 & yatto & mituketa & & \\
TOP/SUBJ & & finally & found & & \\
\multicolumn{6}{l}{\it {``Hanako finally found (him).''}}\\
\end{tabular}
\begin{tabular}{|llllll|}
\hline
{\bf Cb:} & TAROO & & & & \\
{\bf Cf:} & [HANAKO, & TAROO] & (R) & & \\
& top/subj & obj & & & \\ \hline
\end{tabular}
\begin{tabular}{llllll}
$\rm{U_{n+2}}$: & & & & & \\
0 & 0 & yotei-o & setumeisita & & \\
& & schedule & explained & & \\
\multicolumn{6}{l}{\it {``She explained the schedule to him''}}\\
\end{tabular}
\begin{tabular}{|llllll|}
\hline
{\bf Cb:} & HANAKO & & & & \\
{\bf Cf:} & [HANAKO, & TAROO] &(SHIFT1) & & \\
& subj & obj & & & \\ \hline
\end{tabular}
\normalsize
In (10) the only Cf possible for $\rm{U_{n+1}}$ is the {\sc retention}
in the parallel utterance in (9). Given that there are 2 zero
pronouns in $\rm{U_{n+2}}$, constraint 3 forces a shift. The {\em
Hanako explained ...} interpretation is preferred because it is the
more highly ranked {\sc shift-1} transition. If {\sc hanako} could
represent a {\sc topic-obj} there would be another equally ranked {\sc
shift-1} interpretation. However, HANAKO can not be a zero topic
because it was not the $\rm{Cb}$ of the previous utterance.
\section{Discussion}
We have demonstrated a computational treatment of the resolution of
zero pronouns in Japanese. Kameyama proposed an analysis of Japanese
zero pronouns that used centering, but did not distinguish between
{\sc continuing} and {\sc retaining}, and thus required an extra
mechanism, i.e. property-sharing\cite{Kameyama85}. Our examples (5), (6)
and (7) show that property-sharing is an unnecessary stipulation. In
addition, there are a number of cases in which property-sharing just
doesn't work. Our ``introduce'' example (8) illustrates that it is not
essential for a zero pronoun to share a grammatical function property
with its antecedent. In fact property-sharing would falsely predict that the
{\em Masayo likes Sharon} interpretation of (8) $\rm{U_2}$ is not
possible, as well as falsely predicting the ungrammaticality of
examples like (11) below.
\noindent(11)\\
\footnotesize
\begin{tabular}{llllll}
$\rm{U_{n}}$: & & & & & \\
& Hanako wa & repooto o & kaita. & & \\
& & report & wrote & & \\
& \multicolumn{5}{l}{\it {``Hanako wrote a report''}}\\
& & & & & \\
$\rm{U_{n+1}}$: & & & & & \\
& $0_{i}$ & Taroo ni & aini-itta. & & \\
& & & to see-went & & \\
& \multicolumn{5}{l}{\it {``She went to see Taroo''}}\\
& \multicolumn{5}{l}{$0_{i}$ = Hanako [SUB EMPATHY]}\\
& & & & & \\
$\rm{U_{n+2}}$: & & & & & \\
& Taroo wa & $0_{i}$ & kibisiku hihansita. & & \\
& & & severely criticized & & \\
& \multicolumn{5}{l}{\it {``Taroo severely criticized her.''}}\\
& \multicolumn{5}{l}{$0_{i}$ = Hanako [nonSUB nonEMPATHY]}\\
\end{tabular}
\normalsize
Property-sharing requires that in $\rm{U_{n+2}}$, $i \neq $ HANAKO,
since the zero carries the properties ({\sc subj},
{\sc empathy}) in $\rm{U_{n+1}}$, but has
the properties ({\sc nonsubj},{\sc nonempathy}) in
$\rm{U_{n+2}}$\footnote{Kameyama called the Empathy property IDENT.}.
But in fact $\rm{U_{n+2}}$ is perfectly acceptable under the intended
reading of {\em Taroo severely criticized Hanako}. Nothing special
needs to be said about these to get the correct interpretation using
the centering algorithm.
We have also proposed a notion of topic ambiguity, which arises from
the fact that the grammatical function of unexpressed zero arguments
is indeterminate. The application of zero topic assignment also
depends on the centering theory distinction between {\sc continuing}
and {\sc retaining}. In addition, the centering construct of
backward-looking center, $\rm{Cb}$, gives us a computational way of
determining when a zero pronoun may be assigned {\sc Topic}. Topic
ambiguity has been ignored in previous analyses, but it explains the
availability of interpretations that previous accounts would predict
as ungrammatical.
This analysis has implications for the design of language-independent
discourse processing modules. We claim that the syntactic factors that
affect the ranking of the items on the forward center list, $\rm{Cf}$,
will vary from language to language. The ordering for Japanese
incorporates {\sc topic} and {\sc empathy} into the $\rm{Cf}$ ranking,
which is a single parameter of the centering algorithm. In every
other respect, the rules and constraints of the centering framework
that the centering algorithm implements remain invariant.
\section{Acknowledgements}
The authors would like to thank Aravind Joshi, Carl Pollard, and Ellen
Prince for their comments and support. This paper has also benefited
from suggestions by Megan Moser, Peter Sells, Enric Vallduv\'{\i},
Bonnie Webber and Steve Whittaker. This research was partially funded
by ARO grants DAAG29-84-K-0061 and DAAL03-89-C0031PRI, DARPA grant
N00014-85-K0018, and NSF grant MCS-82-19196 at the University of
Pennsylvania, and by Hewlett-Packard Laboratories.
| proofpile-arXiv_065-457 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
The magnetic moments of the stable baryons are very well measured, and
their theoretical calculation gives a sensitive test of our understanding
of baryon structure in quantum chromodynamics (QCD). Although
the basic pattern and approximate magnitudes of the
moments can be explained using the nonrelativistic
quark model, the deviation of the moments from the quark-model pattern
has not been explained dynamically despite many attempts. We discuss that
problem here from the point of view of nonperturbative QCD. In particular,
we have analyzed the theory of the moments in the approximate QCD setting
developed in the work of Brambilla {\em et al.} \cite{brambilla}, who used
a Wilson-loop approach to study the $qqq$ bound state
problem. By modifying their derivation of the $qqq$ interaction and wave
equation, we have derived expressions for the baryon moments in
terms of the underlying quark moments, including the first corrections
associated with the binding of the quarks in baryons. Our results hold in
the ``quenched'' approximation in which there are no internal quark loops
embedded in the world sheet swept out by the Wilson lines joining the
valence quarks, and no pairs associated with the valence lines.
The corrections to the moments are of relative order
$\langle V\rangle/m_q$, where $V$ is a typical component of the binding
potential and $m_q$ is a constituent quark mass, and
are potentially large enough to explain the deviations of the measured
moments from the quark-model values.
To test the theory, we
have constructed variational wave functions for the baryons using the
interactions derived by Brambilla {\em et al.} \cite{brambilla}, including
all spin and orbital configurations possible for $J^P=\frac{1}{2}^+$ and
internal and total orbital angular momenta $L\leq 2$, and used them to calculate the moments. The effects of excited orbital
contributions to the moments are negligible, as expected.
The new contributions tend to
cancel, and the remnants do not have the correct pattern
to explain the discrepancies
between theory and experiment.
The approximations underlying the
Wilson-line analysis are known. Since of these, only the quenched
approximation has a direct effect on the spin dependent
terms with which we are dealing, we
conclude that the use of the quenched approximation is
responsible for the deviations of theory from experiment, and
that the moment problem
provides a sensitive test of this standard approximation in lattice
and analytic QCD.
In a world-sheet picture, the inclusion of internal quark loops which describe meson emission and absorption by the baryon would allow
the introduction of internal orbital angular momentum and spin, and would
affect the moments. We are now investigating meson-loop
contributions using approximations suggested by the world-sheet picture
and chiral pertubation theory. In the following sections, we sketch our
derivations and calculations, and justify our conclusions. A more detailed
discussion will be given elsewhere.
\section{Baryon moments in QCD}
\subsection{The problem}
The simple, nonrelativistic quark model gives a qualitatively good
description of the baryon moments. Under the assumption that each baryon
is composed of three valence or constituent quarks in a state with all
internal orbital angular momenta equal to zero, the moments are given
by expectation values of the spin moment operators
\begin{equation}
\mbox{\boldmath$\mu$}_{\rm QM}=\sum_{\rm q}\,\mu_{\rm q}
\mbox{\boldmath$\sigma$}_{\rm q},
\label{eq:mu}
\end{equation}
leading to the standard expressions
\begin{equation}
\mu_{\rm p}=\frac{1}{3}(4\mu_{\rm u}-\mu_{\rm d}),\,\ldots,\quad \mu_{\rm q}=
\frac{e_{\rm q}}{2m_{\rm q}}\,. \label{eq:mu_quark}
\end{equation}
The masses in the quark moments are clearly effective masses, and can be
treated as free parameters in attempting to fit the data. A
least-squares fit to the measured octet moments,
taken with equal weights, is shown in Table 1. The moments are given in nuclear magnetons (nm).
The pattern of the signs of the quark model moments agrees with observation,
while the root-mean-square deviation of
theory from experiment is 0.14 nuclear
magnetons, or about 9\% of the average magnitudes of the moments.
Agreement at this level can
be regarded as an outstanding success of the quark model, but the deviations
also give a very sensitive test of baryon structure: there is presently
no completely successful first-principles theory of the moments.
\bigskip
\begin{center}
Table 1: Quark model fit to the magnetic moments in the baryon octet.
All moments are given in nuclear magnetons.
\medskip
\begin{tabular}{|c|c|c|r|}
\hline
Baryon & Experiment & Quark Model & $\Delta\mu$\\
\hline
$p$ & 2.793 & 2.728 & 0.065 \\
$n$ & -1.913 & -1.819 & -0.094 \\
$\Sigma^+$ & $2.458\pm 0.010$ & 2.639 & -0.181 \\
$\Sigma^-$ & $-1.160\pm 0.025$ & -0.999 & -0.161 \\
$|\Sigma^0\rightarrow\Lambda\gamma|$ & $1.61\pm 0.08$ & 1.575 & -0.03 \\
$\Lambda$ & $-0.613\pm 0.004$ & -0.642 & 0.029 \\
$\Xi^0$ & $-1.250\pm 0.014$ & -1.462 & 0.212 \\
$\Xi^-$ & $-0.651\pm 0.025$ & -0.553 & -0.098 \\
\hline
\end{tabular}
\end{center}
\medskip
\subsection{The Wilson-loop approach to the baryon moments}
Our approach to the baryon moment problem is based on the work of
Brambilla {\em et al.} \cite{brambilla}, who derived the interaction
potential and wave equation for the valence quarks in a baryon
from QCD using a Wilson-line construction. The basic idea is to construct a
Green's function for the propagation of a gauge-invariant combination of
quarks joined by path ordered Wilson-line factors
\begin{equation}
U=P\exp ig\!\int\! A_g\cdot dx, \label{wilsonline}
\end{equation}
where $A_g$ is the color gauge field. The Wilson lines sweep out a
three-sheeted world sheet of the form shown in Fig.\ \ref{fig:worldsheet}
as the quarks move from their initial to their final configurations.
By making an expansion in powers of $1/m_q$ and considering only forward
propagation of the quarks in time using the Foldy-Wouthuysen approximation,
Brambilla {\em et al.}
are able to derive a Hamiltonian and Schr\"{o}dinger equation for the
quarks, with an interaction which involves an average over the gauge field.
That average is performed using the minimal surface approximation in which
fluctuations in the world sheet are ignored,
and the geometry is chosen to minimize the total area of the world sheet
{\em per} Wilson, subject to the motion of the quarks. The short-distance QCD
interactions are taken into account explicitly.
\begin{center}
\begin{figure}[t]
\psfig{figure=worldsheetfig.eps,height=1.9in}
\caption{World sheet picture for the structure of a baryon.
\label{fig:worldsheet}}
\end{figure}
\end{center}
The result of this construction is an
effective Hamiltonian to be used in a semirelativistic Schr\"{o}dinger
equation $H\Psi=e\Psi$,
\begin{eqnarray}
H &=& \sum_i\sqrt{p_i^2+m_i^2}+\sigma(r_1+r_2+r_3)-\sum_{i<j}\frac{2}{3}
\frac{\alpha_{\rm s}}{r_{ij}} \nonumber \\
&&-\frac{1}{2m_1^2}\frac{\sigma}{r_1}{\bf s}_1
\cdot ({\bf r}_1\times{\bf p}_1)
+\frac{1}{3m_1^2}{\bf s}_1\cdot[({\bf r}_{12}\times{\bf p}_1)
\frac{\alpha_{\rm s}}{r_{12}^3} + ({\bf r}_{13}\times{\bf p}_1)
\frac{\alpha_{\rm s}}{r_{13}^3}] \nonumber\\
&&-\frac{2}{3}\frac{1}{m_1m_2}\frac{\alpha_{\rm s}}{r_{12}^3}
{\bf s}_1\cdot{\bf r_{12}}\times{\bf p}_2
-\frac{2}{3}\frac{1}{m_1m_3}\frac{\alpha_{\rm s}}{r_{13}^3}
{\bf s}_1\cdot{\bf r_{13}}\times{\bf p}_3+\cdots.
\label{hamiltonian}
\end{eqnarray}
Here ${\bf r}_{ij}={\bf x}_i-{\bf x}_j$ is the
separation of quarks $i$ and $j$, $r_i$ is the distance of quark $i$ from
point at which the sum $r_1+r_2+r_3$ is minimized, and ${\bf p}_i$ and
${\bf s}_i$ are the momentum and spin operators for quark $i$.
The parameter $\sigma$
is a ``string tension'' which specifies the strength of the long range
confining interaction, and $\alpha_{\rm s}$ is the strong coupling.
The terms hidden in the ellipsis include tensor and spin-spin
interactions which will not play a role in the
analysis of corrections to the moment operator, and the terms that
result from permutations of the particle
labels. The full Hamiltonian is given in \cite{brambilla}.
This Hamiltonian, including the terms omitted here,
gives a good description of the
baryon spectrum as shown by Carlson, Kogut, and Pandharipande \cite{kogut}
and by Capstick and Isgur \cite{isgur}, who proposed it on the basis of
reasonable physical arguments, but did not give formal derivations from
QCD.
The presence of the quark momenta ${\bf p}_i$ in the Thomas-type spin-dependent
interactions in Eq.\ (\ref{hamiltonian}) suggests that new contributions to
the magnetic moment operator could arise in a complete theory through
the minimal substitution
\begin{equation}
{\bf p}_i\rightarrow {\bf p}_i-e_i{\bf A}_{\rm em}(x_i), \label{minimalsub}
\end{equation}
with ${\bf A}_{\rm em}(x_i)$ the electromagnetic
vector potential associated with an
external magnetic field. However, this point is obscured in the work
of Brambilla {\em at al.} by the transformations they make to express the
equations for the Green's function in Wilson-loop form. We have therefore
repeated their derivation of the valence-quark Hamiltonian in Eq.\ (
\ref{hamiltonian}), replacing the SU(3)$_c$ color gauge field $gA_g$ by the
full SU(3)$_c\times$U(1)$_{\em em}$ gauge field $gA_g+e_qA_{\rm em}$
at the beginning.
By then reorganizing the calculation of the three-quark Hamiltonian
to keep $e_qA_{\rm em}$ explicit
throughout, and then expanding to first order in $A_{\rm em}$ with
\begin{equation}
{\bf A}_{\rm em}=\frac{1}{2}{\bf B}\times{\bf x}_q,
\quad {\bf B}={\rm constant,} \label{magneticpotential}
\end{equation}
we can identify the modified magnetic moment operator through the relation
\begin{equation}
\Delta H=-\mbox{\boldmath$\mu$}\cdot{\bf B}. \label{deltaH}
\end{equation}
The new moment operator, $\mbox{\boldmath$\mu$}=\mbox{\boldmath$\mu$}_{\rm
QM}+\Delta\mbox{\boldmath$\mu$}$, involves the leading corrections to the
quark-model operator associated with the binding interactions.
$\Delta\mbox{\boldmath$\mu$}$ {\em can}, in fact, be read off from the
terms in Eq.\ (\ref{hamiltonian}) which depend on both the quark spins
and momenta by making the minimal substitution in Eq.\ (\ref{minimalsub}).
For example, the term which depends on ${\bf s}_1\cdot{\bf r}_{12}\times
{\bf p}_1$ gives an extra contribution
\begin{equation}
\frac{e_1}{6m_1^2}{\bf x}_1\times({\bf s}_1\times{\bf r}_{12})\frac{\alpha_
{\rm s}}{r_{12}^3} \label{deltamu1}
\end{equation}
to $\mbox{\boldmath$\mu$}_1$. There are also possible orbital contributions
to the moments because the Hamiltonian mixes states with nonzero orbital
angular momenta with the ground state. These have the standard form to the
accuracy we need.
The baryon moments are now given by expressions of the form
\begin{equation}
\mu=\sum_{i=1}^3\mu_i\langle\sigma_{i,z}\rangle(1+\delta_i)+\sum_{i=1}^3
\mu_i\langle L_{i,z}\rangle, \label{mufinal}
\end{equation}
where we have quantized along $\bf B$, taken along the $z$ axis. The
expectation values are to be calculated in the baryon ground states. The
correction terms $\delta_i$ from the new operators are given for the $L=0$
baryons other than the $\Lambda$ by
\begin{eqnarray}
\delta_i &=& \frac{3\epsilon_0+\epsilon_1}{2m_1}+\frac{e_3}{e_1}
\frac{\epsilon_2}{m_3}-\frac{\Delta_0+\Delta_1}{2m_1},\quad i=1,2,\nonumber\\
\delta_3 &=& \frac{\epsilon_2}{m_3}+\frac{2e_1}{e_3}\frac{\epsilon_1}{m_1}
-\frac{\Delta_2}{m_3},\label{deltas}
\end{eqnarray}
where the $\epsilon$'s and $\Delta$'s are ground state radial matrix elements,
\begin{eqnarray}
\epsilon_0=\langle\frac{2}{3}\frac{\alpha_{\rm s}}{6r_{12}}\rangle,\quad
& \displaystyle \epsilon_1=\langle\frac{2}{3}\frac{\alpha_{\rm s}
{\bf r}_{23}\cdot{\bf z}_2}{3r_{23}^3}\rangle,\quad
& \epsilon_2= \langle\frac{2}{3}\frac{\alpha_{\rm s}{\bf r}_{31}\cdot
{\bf z}_3}{3r_{31}^3}\rangle \nonumber\\
\Delta_0=\langle\frac{\sigma r_{12}}{12}\rangle,\quad & \displaystyle
\Delta_1=\langle\frac{\sigma {\bf r}_{23}\cdot{\bf z}_2}{6r_{23}}\rangle,\quad
& \Delta_2=\langle\frac{\sigma {\bf r}_{31}\cdot{\bf z}_3}{6r_{31}}\rangle.
\label{matrixelements}
\end{eqnarray}
The identical quarks in these baryons are labelled 1 and 2, the unlike quark,
3. In writing these results, we have made the approximation $r_1+r_2+r_3\approx
\frac{1}{2}(r_{12}+r_{23}+r_{31})$, known to be reasonably accurate for the
ground state baryons \cite{kogut}, and used the corresponding Thomas spin
interaction. The result for the $\Lambda$ is similar.
\subsection{Tests of the model}
Rough estimates of the matrix elements above suggest that $\epsilon_i/m_l
\approx 0.05$ and that $\Delta_i/m_l\approx 0.3$ for the
light-quark masses used in Refs.\ 2 and 3. Binding effects are therefore
potentially large, and are different for different quarks and baryons. To
obtain a quantitative test of these effects and reliable estimates of the
orbital contributions to the moments, expected to be small, we have
performed a detailed analysis of the baryon wavefunctions using the
Hamiltonian given in \cite{brambilla}. We use Jacobi-type internal coordinates
\mbox{\boldmath $\rho, \lambda$}, with $\mbox{\boldmath $\rho$}=
{\bf x}_1-{\bf x}_2$, and $\mbox{\boldmath $\lambda$}={\bf R}_{12}-{\bf x}_3$,
where ${\bf R}_{ij}$ is the coordinate of the center of mass of quarks $i$
and $j$. The most general $j=\frac{1}{2}^+$ baryon wave function for
quarks 1 and 2 identical and orbital angular momenta $L_\rho, L_\lambda, L
\leq 2$ has the form
\begin{eqnarray}
\psi_{\frac{1}{2},m}&=&\left[\right.a_0\psi_a{\bf 1}
+ib_0\psi_b\,(\mbox{\boldmath$\sigma$}_1-\mbox{\boldmath$\sigma$}_2)
\cdot\mbox{\boldmath$\rho$}\times\mbox{\boldmath$\lambda$}
+c_0\psi_c\,t_{12}(\mbox{\boldmath$\rho$})\nonumber\\
&&+d_0\psi_d\,t_{12}(\mbox{\boldmath$\lambda$})
+e_0\psi_d\,(\mbox{\boldmath$\sigma$}_1-\mbox{\boldmath$\sigma$}_2)\cdot
\mbox{\boldmath$\sigma$}_3\,\mbox{\boldmath$\rho$}\cdot
\mbox{\boldmath$\lambda$}\left.\right]\chi_{\raisebox{-.5ex}
{$\scriptstyle\frac{1}{2},m$}}^{S_{12}=1}. \label{wavefunc}
\end{eqnarray}
Here $t_{12}$ is the usual tensor operator
\begin{equation}
t_{12}({\bf x})=3\mbox{\boldmath$\sigma$}_1\cdot{\bf x}\mbox{\boldmath$\sigma$}
_2\cdot{\bf x}-\mbox{\boldmath$\sigma$}_1\cdot\mbox{\boldmath$\sigma$}_2
{\bf x}^2,
\end{equation}
and $\chi_{\raisebox{-.5ex}
{$\scriptstyle\frac{1}{2},m$}}^{S_{12}=1}$ is the standard three-particle
spinor for $S_{12}=1,\,j=\frac{1}{2},\, j_3=m$. The scalar functions $\psi_i=\psi_i(\rho^2,\lambda^2)$ are normalized together with
the accompanying spin operators. The constant coefficients $a_0,\ldots,e_0$,
normalized to unity, give the fractions of the various component
states in $\psi$. With this form of the wave function,
we can use trace methods for the spins to calculate such quantities as
$\langle H\rangle$ and $\langle\mbox{\boldmath$\mu$}\rangle$.
We have made variational calculations of the energies and approximate
wave functions of the ground state baryons and their first excited states
using the Hamiltonian in \cite{brambilla}. The information on the excited
states
allows us to calculate the coefficients $b_0,\ldots, e_0$ perturbatively.
While the Hamiltonian does mix orbitally excited states into the
$L_\rho=L_\lambda=L=0$ quark-model ground state,
the coefficients are very small, ranging from essentially zero to about
0.02 depending on the baryon. Because these coefficients only
appear quadratically in the baryon moments, the orbital contributions to the
moments are completely negligible.
The radial matrix elements $\epsilon_i$ and $\Delta_i$ defined in Eq.\
(\ref{matrixelements}) are significant, with the $\epsilon$'s ranging
from 10 to 20 MeV, and the $\Delta$'s from 40 to 70 MeV. A
new fit to the moment data using the expressions in Eqs.\ (\ref{mufinal})
and (\ref{deltas}), with the quark masses allowed to vary, gives a small
improvement in the fitted moments, with the root-mean-square deviation
from the measured moments decreasing from 0.14\,nm for the quark model to
0.10\,nm. A secondary effect of the corrections is to change the fitted
quark masses or moments significantly, with the effective quark masses
decreasing, or the moments increasing.
We have concluded from this exercise first, that the finer details of baryon
structure are not yet described correctly in the present QCD-based model,
and second, that the baryon magnetic moments provide a very sensitive
check on the theory.
\section{Possible improvements in the theory}
We begin the discussion of possible improvements in the theory of the
moments by recalling the
approximations used in the construction given by Brambilla
{\em et al.} \cite {brambilla} and in our derivation of the
moment operators:
\newcounter{approx}
\begin{list}{(\roman{approx})}{\usecounter{approx}}
\item
the $1/m_q$ expansion;
\item
the minimal surface approximation;
\item
forward propagation of the quarks in time;
\item
and the quenched approximation.
\end{list}
Of these approximations, only (iii) and (iv) are likely to affect the
moments significantly.
The $1/m_q$ expansion is to be interpreted as an expansion in
constituent quark masses. It can be resummed in the kinetic terms in the
Hamiltonian as in Eq.\ (\ref{hamiltonian}),
and leads elsewhere to the appearance of effective
inverse masses $1/m_i$ which need not be the same as the kinematic masses,
but represent averages of quantities such as $1/E_i=1/\sqrt{p_i^2+m_i^2}$.
Since there is no new spin dependence involved, and
we have treated the quark masses as free parameters in fitting the
moments, relativistic corrections of this sort are unlikely to change
our results.
The average over the color gauge field $A_g$ in the
minimal surface approximation
neglects fluctuations, and minimizes the surface energy of the
world sheet to obtain the approximate Hamiltonian. Since the color fields do
not carry charge, they do not contribute internal currents in the baryon,
and an improved
treatment of the averaging would presumably not change the moment operator
directly. While it could change the functional form of the Hamiltonian
somewhat, the quantitative success of the model in describing
baryon spectra suggests that the changes would not be large, and their effect
on the moments through the $\epsilon$ and $\Delta$ parameters would be minimal.
In contrast, the remaining two approximations affect the moment operator
directly. Internal quark loops can contribute circulating currents in the
baryon. These are omitted in the quenched approximation. In addition, the
forward propagation of the quarks in time inherent in the use of the
Foldy-Wouthuysen approximation eliminates quark pair effects connected
with the valence lines in Fig. 1. We believe that this general quenched
picture underlies the difficulties with the model, and that it will be
necessary to include pair effects to obtain a precise dynamical description of
the moments.
One effect of internal quark loops can be seen in Fig.\ \ref{fig:mesonloop}.
In this figure, we suppose that a quark loop is embedded in the minimal
world sheet of the baryon. The effect is to give a meson state and a new
baryon in a world-sheet picture of meson emission and absorption.
\begin{figure}[h]
\psfig{figure=mesonloopfig.eps,height=1.3in}
\caption{Quark loops embedded in the world sheet give meson-baryon states.
\label{fig:mesonloop}}
\end{figure}
\noindent
An external magnetic field interacting with the system will see a meson
current as well as the baryon current, and the moment will be modified.
Meson currents were invoked in the past in attempts to explain the {\em full}\/
anomalous moments of the nucleons. Here we are only concerned with
presumably small corrections to the quark-model moments.
One approach to the calculation of meson loop effects is through
chiral perturbation theory. This has a long history, and has
been studied recently in the context of baryon moments by a number of
authors \cite{jenkins,luty,bos}. The relevant diagrams
for the interaction of the baryons with the pseudo Goldstone bosons
of the chiral theory are shown
in Fig.\ \ref{fig:goldstone}.
\begin{figure}[h]
\psfig{figure=goldstonefig.eps,height=1.2in}
\caption{Goldstone-boson diagrams which contribution to the magnetic moments.
\label{fig:goldstone}}
\end{figure}
The work of Jenkins {\em et al.} \cite{jenkins}
uses heavy baryon chiral perturbation theory \cite{manohar}, and studies the
effect on the moments of the ``nonanalytic terms'' in the symmetry breaking
parameter $m_{\rm s}$. The analytic
terms in $m_{\rm s}$ are ambiguous because the inevitable
appearance of new couplings at each order in the chiral expansion,
and are ignored. The results of Jenkins {\em et al.}
are not encouraging as they stand, but we have found errors in some
of the coupling factors given in the published paper. Luty
{\em et al.} \cite{luty}
concentrate on a simultaneous expansion in $1/N_{\rm c}$ and $m_{\rm s}$,
and obtain interesting sum rules for the
moments but little dynamical information. Finally, Bos {\em et al.} \cite{bos}
consider the moments from the point of view of flavor
SU(3) breaking in the baryon octet
using a different chiral counting than that usually used, and obtain
a very successful parametrization for the moments. However, this model
is again nondynamical, and has seven parameters to
describe eight measured moments.
In fact, a fundamental problem with the chiral expansion from our perspective is that it simply parametrizes the moments with an
expansion consistent with QCD, but does not control the higher order terms
in what appears to be a slowly convergent series.
We are presently investigating meson loop effects using a somewhat
different approach suggested by the world-sheet picture. The baryon appears in
this picture as an extended object which must absorb the recoil momentum
in the emission of a meson. We would therefore expect wave function effects
(form factors) to be important for high meson momenta, and to supply a natural
cutoff for loop graphs. The wave function appears naturally when the
process is viewed using ``old fashioned'' instead of Feynman perturbation
theory, as indicated in Fig.\ \ref{fig:wavefunctionvertex}: energy
\begin{figure}
\psfig{figure=wavefunctionvertex.eps,height=1.4in}
\caption{(a) Appearance of Bethe-Salpeter wave functions in the diagrams.
(b) Schematic reminder of the distributed nature of the meson-baryon vertex.
\label{fig:wavefunctionvertex}}
\end{figure}
denominators and vertex functions combine to give the $B'M$ component of
the wave function for a baryon $B$, or, with different time ordering, the
$BM$ component of the wave function of $B'$. Since the wave functions are
expected to be fairly soft, with characteristic momenta below the
chiral cutoff of $\sim 1$ GeV, they can be expected to combine a number of
terms in the chiral expansion in a way that is dynamically accessible
through approximate models derived from QCD such as that of Brambilla
{\em et al.} \cite{brambilla}. The non-point character of the meson-baryon vertex in spacetime indicated in Fig.\ \ref{fig:wavefunctionvertex}
contrasts sharply with the point vertex used in chiral
perturbation theory, Fig.\ \ref{fig:goldstone}, and is also likely to play
a role. We have reached the same conclusions by
studying the exact sideways dispersion relations for the moments given by
Bincer \cite{bincer}. The problem there is in extracting the quark-model
moments.
We note finally that a different aspect of symmetry breaking, the
suppression of strange-quark loops through mass effects, appears as a
natural possibility in a world-sheet picture. Note that this is {\em not}
the same as the suppression of kaon loops considered by other authors; see,
e.g., \cite{jenkins} and the references given there.
\section{Conclusions}
On the basis of the work sketched above, we have concluded that the magnetic
moments of the baryons give a sensitive test of baryon structure and of
approximations in QCD. In particular, the quenched approximation, while reasonably successful when used in the calculation of
baryon and meson spectra, appears to fail for the moments. The accurate
calculation of the moments by lattice methods will presumably involve
going beyond that approximation, and will provide a precision test of
the methods used.
We find also that the world-sheet picture of baryon structure gives useful
insights into the moments problem, and provides a new point of view which
could be developed further. Problems which need further study within this
approach to QCD include the following:
\begin{list}{(\roman{approx})}{\usecounter{approx}}
\item
establishing the connection to, and the relevance of, the chiral limit;
\item
the development of methods to incorporate loop effects which build in
the extended structure of the baryons;
\item
and the possible usefulness of string theory methods in the calculation
of loop effects.
\end{list}
\section*{Acknowledgments}
The authors have benefited from conversations with Dr.\ Nora Brambilla,
and appreciate her interest in this work, and her organization
of the Como Conference.
This research was supported in part by the U.S. Department of Energy under
Grant No.\ DE-FG02-95ER40896, and in part by the University of Wisconsin
Graduate School with funds from the Wisconsin Alumni Research Foundation.
\section*{References}
| proofpile-arXiv_065-458 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
There is a wide class of integrable models which describe the
flows between two different models of Conformal Field Theory (CFT)
in UV and IR regions \cite{zam}. To describe these models the factorized
scattering of massless particles was proposed in \cite{zz1}. In spite
of general difficulties arising from the fact that the scattering
of massless particles is not properly defined in 2D the application
of this method provided very good results for several particular models.
Certainly, the physical results are related to possibility of
extracting the off-shell information from these S-matrices.
Similarly to the massive case there are two ways to do that.
The first way is TBA approach. The TBA equations can be written for
the massless scattering which allow to calculate the effective
central charge and to show that the latter really interpolates between
corresponding UV and IR values \cite{zz1}.
The second way consists in generalization proposed in \cite{muss}
of the form factor bootstrap
approach (which is originally formulated for the massive models \cite{book})
to the massless flows. One must distinguish between the form factor bootstrap
in massive and massless cases. In the massive case the form factor
bootstrap stays on the solid ground because the space of states
is well defined as the Fock space of particles. In the massless case
this very definition is doubtful and one has to consider the
form factor bootstrap rather as intuitive than rigorous
procedure. Indeed for certain operators (as, for example,
the order-disorder operators for the flow from tricritical to
critical Ising model) the straightforward application of the
method leads to divergent series for the Green functions \cite{muss}.
However, the situation can be improved even for those operators,
furthermore there are operators for which the series converge
providing spectacular examples of interpolation between UV and IR
limits \cite{muss}. So, the form factor bootstrap is a method
which works in massless case in spite of all the problems of
general character.
It must be noticed, however, that the complete construction
of all the form factors is not given in the paper \cite{muss}
even for the simplest model describing the
flow from tricritical to
critical Ising model
considered there. In the present paper
we shall give the complete construction for more complicated model:
the Principal Chiral Field with
Wess-Zumino-Novikov-Witten term.
We shall explain briefly that our construction can be generalized
to a wide class of models including the one considered in \cite{muss}.
\section{Formulation of the problem.}
On of the most beautiful examples of the massless flows for
which the S-matrix is known is given by the Principal Chiral Field with
Wess-Zumino-Novikov-Witten (WZNW) term on level $1$ (PCM${}_1$):
\begin{eqnarray}
S={1\over 2\lambda ^2}\int tr(g^{-1}\partial_{\mu}g)
(g^{-1}\partial_{\mu}g) d^2x\ +\ i\Gamma (g)
\nonumber
\end{eqnarray}
where the WZNW term $\Gamma (g)$ is defined by means of continuation
of $g$ to 3D manifold $B$ for which the 2D space-time is the boundary:
\begin{eqnarray}
\Gamma (g)=\int\limits _{B} \epsilon ^{\mu \nu \lambda}
tr(g^{-1}\partial_{\mu}g)
(g^{-1}\partial_{\nu}g)(g^{-1}\partial_{\lambda}g)d^3x
\nonumber
\end{eqnarray}
In the UV limit the pure PCM action dominates the central charge
being equal to 3. In the IR region the flow is attracted
by the fixed point $\lambda ^2=8\pi$ which corresponds to the conformal
WZNW model with the central charge equal to 1. It is explained in \cite{zz1}
that the flow arrives at the IR point along the direction
defined by the irrelevant operator $\overline{T}T$ composed of the right
and left components of the energy-momentum tensor.
In the IR region the theory is conformal, so, two charialities
essentially decouples. One describes corresponding left and right
level-1 WZNW models in terms of massless particles.
This is exactly left-left and right-right scattering which
seems to be doubtful in 2D. The prescription of the
paper \cite{zz1} for the definition of this scattering can be
understood in the following way.
We know that the level-1 WZNW model coincides with the UV limit of
the massive $SU(2)$-invariant Thirring model.
The local operators for the latter model are defined
via their form factors \cite{book}.
For the operators
chiral in the limit (as chiral components of the energy-momentum
tensor) the limiting values of the correlation functions
are obtained by replacing the massive dispersion
by the massless ones.
On the other hand these
limiting correlation functions
coincide with the conformal ones. So, we get the representation
of the conformal correlation functions for chiral operators in terms of
the form factor series with massless particles.
The form factors are defined through the S-matrix for the original massive
theory which is considered now as S-matrix of massless
particles.
It is proposed to
use this representations
of the correlators as a starting
point of the description of massless flows.
More precisely, left
and right particles are parametrized by the rapidities
$\beta$ and $\beta'$ such that the energy-momentum are respectively
$$e=-p=Me^{-\beta},\qquad e=p=Me^{\beta'} $$
where $M$ is the mass scale which can be chosen arbitrary
on this stage. The theory possesses $SU(2)_L\otimes SU(2)_R$
symmetry, the left (right) movers are doublets with respect
to $SU(2)_L$ ($SU(2)_R$). The factorizable S-matrices which
describe the left and right CFT are given by
\begin{eqnarray}
S_{LL}(\beta _1,\beta_2)=S^Y(\beta _1-\beta_2),\qquad
S_{RR}(\beta _1',\beta_2')=S^Y(\beta _1'-\beta_2')
\nonumber
\end{eqnarray}
where $S^Y(\beta)$ is the Yangian S-matrix for the scattering
of spin-${1\over 2}$ particles \cite{zz}:
\begin{eqnarray}
S^Y(\beta)={\Gamma ({1\over2}+{\beta\over 2\pi i})\Gamma (-{\beta\over 2\pi i})\over
\Gamma ({1\over2}-{\beta\over 2\pi i})\Gamma ({\beta\over 2\pi i}) }
\({\beta I-\pi i P\over \beta-\pi i}\)
\nonumber \end{eqnarray}
where $I$ and $P$ are respectively unit and permutation
operators acting in the tensor product of two 2-dimensional
isotopic spaces.
The crucial point is in introducing the non-trivial left-right and right-left
S-matrices. Contrary to $S_{LL}$ and $S_{RR}$ whose definition is rather formal
the S-matrices $S_{LR}$ and $S_{RL}$
allow quite rigorous interpretation.
For the PCM${}_1$ the proposal of \cite{zz1} is
$$S_{RL}(\beta'-\beta)={1\over S_{LR}(\beta-\beta')}=U(\beta'-\beta),\qquad
U(\beta)=\tanh{1\over2}\(\beta -{\pi i\over 2}\) $$
The scale normalization $M$ is fixed by the requirement that
the zero of this S-matrix is situated exactly
at $\beta ={\pi i\over 2}$.
It is quite amusing that the IR limit corresponds to
$\beta-\beta '=\log \Lambda$, $\Lambda\to\infty$, indeed in this
limit the $s$-variable goes to zero. This fact is very
interesting because in the massive case infinite rapidities are
always related to UV behaviour of the form factors which has
been investigated in several cases in \cite{book}, so, we can
use the familiar methods for solving absolutely different problems.
Let us describe the form factor bootstrap approach to
massless flows as it is formulated in \cite{muss}.
Consider the matrix element of certain local operator
$\CO$ taken between the vacuum and the state containing
The left and right particles with rapidities
$\beta_1,\cdots,\beta_l$ and $\beta_1',\cdots,\beta_k'$ respectively:
\begin{eqnarray}
f_{\CO}(\beta_1,\cdots,\beta_l\ |\ \beta_1',\cdots,\beta_k')
\label{Fb}
\end{eqnarray}
It is very convenient to collect all the rapidities together
into the set $\theta _1,\cdots,\theta _{k+l}=$
$\beta_1,\cdots,\beta_l,\beta_1',\cdots,\beta_k'$ and to introduce index
$a_i=L,R$ which distinguish the left and right particles.
The first requirement of the form factors is that
of symmetry:
\begin{eqnarray}
f_{\CO}(\cdots ,\theta _i,\theta _{i+1},\cdots)_{\cdots,a_i,a_{i+1},\cdots}
S(\theta _i-\theta _{i+1})_{a_i,a_{i+1}} =
f_{\CO}(\cdots ,\theta _{i+1},\theta _i,\cdots)_{\cdots,a_{i+1},a_i,\cdots}
\label{a1}
\end{eqnarray}
If $a_i\ne a_j$ this equation has to be considered as definition
which allows to construct the form factor with arbitrary placed
left and right particles starting from (\ref{Fb}).
The second requirement is
\begin{eqnarray}
&&f_{\CO}(\theta _1,\cdots ,\theta _{k+l-1},\theta _{k+l}+2\pi i)
_{a_1,\cdots ,a_{k+l-1},a_{k+l}} =
f_{\CO}(\theta _{k+l},\theta _1,\cdots ,\theta _{k+l-1})
_{a_{k+l},a_1,\cdots ,a_{k+l-1}}= \nonumber \\&&=
f_{\CO}(\theta _1,\cdots ,\theta _{k+l-1},\theta _{k+l})
_{a_1,\cdots ,a_{k+l-1},a_{k+l}}
S_{a_{k+l-1},a_{k+l}}(\theta _{k+l-1}-\theta _{k+l})
\cdots
S_{a_{1},a_{k+l}}(\theta _{1}-\theta _{k+l})
\label{a2}
\end{eqnarray}
Since we do not have bound states in the theory the form factor
$f_{\CO}(\theta _1,\cdots ,\theta _{k+l})$ is supposed to be
a meromorphic function of $\theta _{k+l} $ in the strip
$0<\theta _{k+l} <2\pi$ whose only singularities are the
simple poles at the points $\theta _{k+l} =\theta _{j} +\pi i$.
It is important that these singularities appear only in
left-left and right-right chanels. The residue at
$\theta _{k+l} =\theta _{k+l-1} +\pi i $
is given
by
\begin{eqnarray}
&&2\pi i\ res
f_{\CO}(\theta _1,\cdots ,\theta _{k+l-2},\theta _{k+l-1},\theta _{k+l})
_{a_1,\cdots ,a_{k+l-2} ,a_{k+l-1},a_{k+l}}=
\nonumber \\&&\hskip 1cm =
\delta _{a_{k+l-1},a_{k+l}}f_{\CO}(\theta _1,\cdots ,\theta _{k+l-2})
_{a_1,\cdots ,a_{k+l-2}}\otimes c_{k+l-1,k+l}
\nonumber \\&&\hskip 1cm
\times
\(1\ -\ S_{a_{k+l-1},a_{1}}(\theta _{k+l-1}-\theta _{1})\cdots
S_{a_{k+l-1},a_{k+l-2}}(\theta _{k+l-1}-\theta _{k+l-2})\)
\label{a3}
\end{eqnarray}
here $c_{k+l-1,k+l}$ is a vector in the tensor product
of two isotopic spaces constructed from the charge conjugation
matrix, in our case it is the singlet vector in the tensor
product of two spin-${1\over 2}$ representations of $SU(2)$.
These requirements on the massless form factors do not differ too
much from the form factor axioms of \cite{book}. However, the
physical situation is quite different and the solutions to these
equation can not be found in \cite{book}.
\section{Form factors of the energy-momentum tensor.}
It the present paper we are going to construct the form factors
of the trace of energy-momentum tensor ($\Theta$) for PCM${}_1$.
Our methods are applicable to other operators, but we are
considering this particular one because of its nice properties
and physical importance.
Since the symmetry under $SU(2)_L\otimes SU(2)_R$ is not broken by
the perturbation the form factors have to be singlets with
respect to both isotopic groups. That is why $l=2n$ and $k=2m$.
The form factors of $\Theta$ satisfy general conditions (\ref{a1},
\ref{a2}, \ref{a3})
and additional requirements following from the fact that we consider
this particular operator.
1. The energy-momentum conservation implies that
\begin{eqnarray}
f_{\Theta}(\beta_1,\cdots,\beta_{2n}\ |\ \beta_1',\cdots,\beta_{2m}')=
(\sum e^{-\beta _j})(\sum e^{\beta _j '})
f(\beta_1,\cdots,\beta_{2n}\ |\ \beta_1',\cdots,\beta_{2m}')
\label{c1}
\end{eqnarray}
where for $n>1$ and $m>1$ the function $f$ does not other
singularities that those of $f_{\Theta}$, for $n=1$($m=1$)
it has simple poles at $\beta _2=\beta _1 +\pi i$ ($\beta _2 '=\beta _1 '+\pi i$)
which are cancelled by $e^{-\beta _1 } +e^{-\beta _2 }$
($e^{\beta _1' } +e^{\beta _2 '}$ ).
2. The lowest form factor of $\Theta$ is that corresponding to
2+2 particles. However by the conservation law we can construct
from $f_{\Theta}$ the form factors of the left and right components of
the energy momentum tensor $T$ and $\overline{T}$ whose lowest
form factors are of the type $2n+0$ and $0+2m$ respectively.
These lowest form factors must coincide with the form
factors of pure $k=1$ WZNW model i.e. with those of
$SU(2)$-invariant Thirring model. One easily finds that it implies:
\begin{eqnarray}
&&2\pi i\ res _{\beta_2'=\beta _1'+\pi i}\sum e^{-\beta _j}\
f(\beta_1,\cdots,\beta_{2n}\ |\ \beta_1',\beta_{2}') = \nonumber \\&&=
\widehat{f}_{T}(\beta_1,\cdots,\beta_{2n})\otimes c_{1',2'} \(1-
S_{LR}(\beta _1'-\beta_1)\cdots S_{LR}(\beta _1'-\beta_1)\) , \nonumber \\ &&
2\pi i \ res _{\beta_2=\beta _1+\pi i}\sum e^{\beta _j'}\
f(\beta_1,\beta_{2}\ |\ \beta_1',\cdots,\beta_{2m}')= \nonumber \\&&=
\widehat{f}_{\overline{T}}(\beta_1',\cdots,\beta_{2m}')
\otimes c_{1',2'} \(1-
S_{RL}(\beta _1-\beta_1')\cdots S_{RL}(\beta _1-\beta_1')\) \label{c2}
\end{eqnarray}
where $\widehat{f}_{T}$ and $\widehat{f}_{\overline{T}}$ are the form factors
of left and right components
of the energy-momentum tensor for the $SU(2)$ -Thirring model \cite{book}.
3. The IR limit corresponds to $\beta _i-\beta _j '\simeq\log\Lambda$ and
$\Lambda\to\infty$. In this limit one has to reproduce the
operator $T\overline{T}$ which defines the direction of the flow in the IR
region. So, we must have
\begin{eqnarray}
f_{\Theta}(\beta_1+\log\Lambda,\cdots,\beta_{2n}+\log\Lambda|\ \beta_1',\cdots,\beta_{2m}')
\to
(M\Lambda) ^{-2 }
\widehat{f}_{T}(\beta_1,\cdots,\beta_{2n})
\widehat{f}_{\overline{T}}(\beta_1',\cdots,\beta_{2m}')
\label{c3}
\end{eqnarray}
Let us try to satisfy all this requirement. The simple form of the
left-right S-matrix allows to exclude it from the equations (\ref{a1},\ref{a2}).
Consider the function $g$ defined as follows:
\begin{eqnarray}
f_{\Theta}(\beta_1,\cdots,\beta_{2n}\ |\ \beta_1',\cdots,\beta_{2m}')
=\prod\psi(\beta_i,\beta_j')g(\beta_1,\cdots,\beta_{2n}\ |\ \beta_1',\cdots,\beta_{2m}')
\label{fg}
\end{eqnarray}
where
$$\psi(\beta,\beta ')=
2^{-{3\over 4}}
\exp \(
-{1\over 4}(\beta+\beta ')
-\int _0^{\infty}
{2\sin ^2{1\over 2}(\beta-\beta ' +\pi i)k+\sinh ^2 {\pi k\over 2}
\over 2k \sinh \pi k \cosh {\pi k\over 2}}dk\)$$
The function $\psi (\beta,\beta ')$ satisfies the equations
\begin{eqnarray}
&&\psi (\beta,\beta '+2\pi i)=\psi (\beta,\beta ')S_{RL}(\beta '-\beta),\quad
\psi (\beta+2\pi i,\beta ')=\psi (\beta,\beta ')S_{LR}(\beta -\beta ') \nonumber \\ &&
\psi (\beta,\beta '+\pi i) \psi (\beta,\beta ')={1\over e^{\beta}-i e^{\beta '}},
\quad \psi (\beta +\pi i,\beta ') \psi (\beta,\beta ')={1\over ie^{\beta}- e^{\beta '}}
\label{psi}
\end{eqnarray}
It is clear that the equation (\ref{a2})
rewritten in terms of $g$
does not contain the left-right
S-matrices which means that the function $g$ must be of the form
\begin{eqnarray}
g(\beta_1,\cdots,\beta_{2n}\ |\ \beta_1',\cdots,\beta_{2m}')=\sum\limits _{K,L}
c _{K,L}(\beta_1,\cdots,\beta_{2n}\ |\ \beta_1',\cdots,\beta_{2m}')
\widehat{f}^K(\beta_1,\cdots,\beta_{2n})\widehat{f}^L (\beta_1',\cdots,\beta_{2m}')
\label{xx}
\end{eqnarray}
where
$\widehat{f}^K(\beta_1,\cdots,\beta_{2n})$ and
$\widehat{f}^L (\beta_1',\cdots,\beta_{2m}')$ are different singlet solutions
(counted by $K$ and $L$ whose nature will be explained later)
of the equations
\begin{eqnarray}
&&\widehat{f}^K(\cdots,\beta_i,\beta_{i+1},\cdots)S^Y(\beta_i-\beta_{i+1} )=
\widehat{f}^K(\cdots,\beta_{i+1},\beta_i,\cdots), \nonumber \\ &&
\widehat{f}^K(\beta_1,\cdots,\beta_{2n-1},\beta_{2n}+2\pi i)=
\widehat{f}^K(\beta_{2n},\beta_1,\cdots,\beta_{2n-1}), \label{leq}\\ &&
\widehat{f}^L(\cdots,\beta_i',\beta_{i+1}',\cdots)S^Y(\beta_i'-\beta_{i+1}' )=
\widehat{f}^L(\cdots,\beta_{i+1}',\beta_i',\cdots), \nonumber \\ &&
\widehat{f}^L(\beta_1',\cdots,\beta_{2m-1}',\beta_{2m}'+2\pi i)=
\widehat{f}^L(\beta_{2m}',\beta_1',\cdots,\beta_{2m-1}'), \nonumber
\end{eqnarray}
The functions $c _{K,L}(\beta_1,\cdots,\beta_{2n}\ |\ \beta_1',\cdots,\beta_{2m}') $
are quasiconstants: $2\pi i$-periodical symmetrical with
respect to $ \beta_1,\cdots,\beta_{2n} $ and $ \beta_1',\cdots,\beta_{2m}'$
functions with possible singularities only at $\beta_i,\beta_i'=\pm\infty$.
The equations for left and right parts are the same, so, let us concentrate
for the moment only on the left one.
It is well known \cite{book,count} that the solutions to the equations
(\ref{leq}) are counted by the functions
$K(A_1,\cdots , $ $A_{n-1}|
B_1,\cdots ,B_{2n})$ which are antisymmetrical polynomials
of $A_1,\cdots ,A_{n-1}$ such that $1\le deg_{A_i}(K)\le 2n-1$,
$\forall i$
and symmetrical Laurent polynomials of $B_1,\cdots ,B_{2n} $. The solutions are
given by the formula
\begin{eqnarray}
&&\widehat{f}^K(\beta_1,\cdots ,\beta_{2n})=d^n
\exp\({n\over 2}\sum \beta _j\)\prod\limits _{i,j}\zeta(\beta _i-\beta _j)
\label{int} \\&& \times\int\limits _{-\infty}^{\infty}d\alpha _1\cdots
\int\limits _{-\infty}^{\infty}d\alpha _{n-1}
\prod\limits_{i,j}
\widetilde{\varphi}(\alpha _i,\beta _j)
\langle\Delta _n^{(0)}\rangle _n
(\alpha _1,\cdots ,\alpha _{n-1} |\beta _{1},\cdots ,\beta _{2n} )
K(e^{\alpha _1},\cdots ,e^{\alpha _{n-1}} |e^{\beta _{1}},\cdots ,e^{\beta _{2n}})
\nonumber
\end{eqnarray}
where
$$
\widetilde{\varphi}(\alpha _i,\beta _j) =e^{-{1\over 2}(\alpha +\beta)}
\Gamma \({1\over 4}+{\alpha -\beta\over 2\pi i} \)
\Gamma \({1\over 4}-{\alpha -\beta\over 2\pi i} \)
$$
We do not give here the formulae for
$\langle\Delta _n^{(0)}\rangle _n
(\alpha _1,\cdots ,\alpha _{n-1} |\beta _{1},\cdots ,\beta _{2n} ) $ which is a rational
function of all variables with values in the tensor product of
the isotopic spaces, for $\zeta (\beta)$ which is certain transcendental
function
and for the constant $d$:
these formulae can be found in the
book \cite{book} (Chapter 7).
It has to be emphasized that the integral (\ref{int})
vanishes for two kinds of function $K$ \cite{count,bbs2}:
\begin{eqnarray}
&&K(A_1,\cdots ,A_{n-1}|B_1,\cdots ,B_{2n})=
\sum\limits _{k=1}^{n-1}(-1)^k (P(A_k)-P(-A_k))
K'(A_1,\cdots ,\widehat{A_k},\cdots ,A_{n-1}|B_1,\cdots ,B_{2n}) , \nonumber \\&&
K(A_1,\cdots ,A_{n-1}|B_1,\cdots ,B_{2n})=
\sum\limits _{k<l}(-1)^{k+l} C(A_k,A_l)
K''(A_1,\cdots ,\widehat{A_k},\cdots ,\widehat{A_l}\cdots ,A_{n-1}
|B_1,\cdots ,B_{2n})\quad \quad \label{zero}
\end{eqnarray}
where $K'$,$K''$ are some polynomials of the less number of variables
with the same properties as $K$,
$P(A)=\prod\limits _j (A_k+iB_j)$ and
$$
C(A_1,A_2 )={1\over A_1A_2}\left\{ {A_1-A_2\over A_1+A_2 }
(P(A_1)P(A_2)-P(-A_1)P(-A_2))
+
(P(-A_1)P(A_2)-P(A_1)P(-A_2))\right\} \label{C}
$$
So, the polynomials $K$ are defined modulo the polynomials
of the kind (\ref{zero}) , the fact that has been used in \cite{count}
to show that we have correct number of solutions to (\ref{leq}).
Combining (\ref{fg}),(\ref{xx}) and (\ref{int}) we find that
the from factors satisfying (\ref{a1}) and (\ref{a2}) are of the form
\begin{eqnarray}
&&f_{\Theta}(\beta_1,\cdots,\beta_{2n}\ |\ \beta_1',\cdots,\beta_{2m}')
=M^2\prod\psi(\beta_i,\beta_j')
\prod\limits _{i,j}\zeta(\beta _i-\beta _j)
\prod\limits _{i,j}\zeta(\beta _i'-\beta _j')
\nonumber \\ &&\times
\int\limits _{-\infty}^{\infty}d\alpha _1\cdots
\int\limits _{-\infty}^{\infty}d\alpha _{n-1}
\int\limits _{-\infty}^{\infty}d\alpha '_1\cdots
\int\limits _{-\infty}^{\infty}d\alpha '_{m-1}
\prod\limits _{i=1}^{n-1}\prod\limits_{j=1}^{2n} \widetilde{\varphi}(\alpha _i,\beta _j)
\prod\limits _{i=1}^{m-1}\prod\limits_{j=1}^{2m} \widetilde{\varphi}(\alpha '_i,\beta '_j)
\nonumber \\ &&\times
\langle\Delta _n^{(0)}\rangle _n
(\alpha _1,\cdots ,\alpha _{n-1} |\beta _{1},\cdots ,\beta _{2n} )
\langle\Delta _n^{(0)}\rangle _n
(\alpha' _1,\cdots ,\alpha _{m-1}' |\beta _{1}',\cdots ,\beta _{2m}' )
\nonumber \\ &&\times
M_{n,m}(e^{\alpha _1},\cdots ,e^{\alpha _{n-1}} |
e^{\alpha _1'},\cdots ,e^{\alpha _{m-1}'}|
e^{\beta _{1}},\cdots ,e^{\beta _{2n}}|
e^{\beta _{1}'},\cdots ,e^{\beta _{2m}'})
\label{ff}
\end{eqnarray}
where
$ M_{n,m}(A _1,\cdots ,A _{n-1} |
A _1',\cdots ,A _{m-1}'|
B _{1},\cdots ,B _{2n}|
B _{1}',\cdots ,B _{2m}') $
is an antisymmetrical polynomial of $ A _1,$ $\cdots ,A _{n-1} $
($A _1',\cdots ,A _{m-1}'$) whose degree with respect to every
variable is from $1$ to $2n-1$ (from $1$ to $2m-1$) and
symmetrical Laurent polynomial of $B _{1},\cdots ,B _{2n}$
($B _{1}',\cdots ,B _{2m}'$).
Now we have to satisfy the rest of our requirements on the form
factors. In the paper \cite{count} there is a general prescription
for handling the residue condition (\ref{a3}) for the integrals of the
form (\ref{int}). Applying this prescription to our situation and
using the equations (\ref{psi}) one finds that the residue condition
(\ref{a3}) is satisfied if and only if the function $M_{n,m}$ possesses the
properties:
\newline
First,
\begin{eqnarray}
&&M_{n,m}(A _1,\cdots ,A _{n-1} |A _1',\cdots ,A _{m-1}'|
B _{1},\cdots ,B _{2n-2},B,-B|B _{1}',\cdots ,B _{2m}')= \nonumber \\&&=
\sum\limits _{k=1}^{n-1}(-1)^k\prod\limits _{p\ne k}(A_p^2+B^2)
M_{n-1,m}^k(A _1,\cdots ,A _{n-1} |A _1',\cdots ,A _{m-1}'|
B _{1},\cdots ,B _{2n-2}|B|B _{1}',\cdots ,B _{2m}') \nonumber \\ &&
M_{n,m}(A _1,\cdots ,A _{n-1} |A _1',\cdots ,A _{m-1}'|
B _{1},\cdots ,B _{2n}|B _{1}',\cdots ,B _{2m-2}',B',-B')= \label{rel1}\\&&=
\sum\limits _{k=1}^{m-1}(-1)^k\prod\limits _{p\ne k}((A_p')^2+(B')^2)
M_{n,m-1}^k(A _1,\cdots ,A _{n-1} |A _1',\cdots ,A _{m-1}'|
B _{1},\cdots ,B _{2n}|B _{1}',\cdots ,B _{2m-2}'|B')
\nonumber
\end{eqnarray}
where $M_{n-1,m}^k $ and $M_{n,m-1}^k$
are some {\it polynomials} in $A_i$ and
$A_i'$.
\newline
Second,
\begin{eqnarray}
&&M_{n-1,m}^k(A _1,\cdots ,A _{k-1},\pm iB ,A _{k-1},\cdots ,A _{n-1}
|A _1',\cdots ,A _{m-1}'|
B _{1},\cdots ,B _{2n-2}|B|B _{1}',\cdots ,B _{2m}')= \nonumber \\&& =
\pm B\prod\limits _{j=1}^{2m}(B\mp iB'_j)
M_{n-1,m}(A _1,\cdots ,A _{k-1},A _{k-1},\cdots ,A _{n-1}
|A _1',\cdots ,A _{m-1}'|
B _{1},\cdots ,B _{2n-2}|B _{1}',\cdots ,B _{2m}'), \nonumber \\&&
M_{n,m-1}^k(A _1,\cdots ,A _{n-1}
|A _1',\cdots ,A _{k-1}',\pm iB' ,A _{k-1}',\cdots ,A _{m-1}'|
B _{1},\cdots ,B _{2n}|B _{1}',\cdots ,B _{2m-2}'|B')=
\label{rel2} \\ &&=
\pm B'\prod\limits _{j=1}^{2m}(B'\pm iB_j)
M_{n,m-1}(A _1,\cdots ,A _{n-1}
|A _1',\cdots ,A _{k-1}',A _{k-1}',\cdots ,A _{m-1}'|
B _{1},\cdots ,B _{2n}|B _{1}',\cdots ,B _{2m-2}') \nonumber
\end{eqnarray}
These equations are necessary and sufficient for the formula
(\ref{ff}) to define form factors of a local operator. Certainly,
they have infinitely many solutions. We shall give only one of these solutions
corresponding to the operator $\Theta$. Let us introduce the notations
for the sets of integers:
$S=\{1,\cdots ,2n\}$, $S'=\{1,\cdots ,2m\}$ then
\begin{eqnarray}
&&M_{n,m}(A _1,\cdots ,A _{n-1} |A _1',\cdots ,A _{m-1}'|
B _{1},\cdots ,B _{2n}|B _{1}',\cdots ,B _{2m}') =
\prod\limits _{i<j}(A_i-A_j)\prod\limits _{i<j}(A_i'-A_j')
\nonumber \\&& \times \prod\limits _{j=1}^{2n} B_j^{-1}
\prod\limits _{i=1}^{n-1} A_i^2
\prod\limits _{i=1}^{m-1} A_i '
\sum\limits _{{T\subset S\atop\# T=n-1}}
\sum\limits _{{T'\subset S'\atop\# T'=m-1}}
\prod\limits _{j\in T} B_j
\prod\limits _{i=1}^{n-1}
\prod\limits _{j\in T}(A_i+iB_j)
\prod\limits _{i=1}^{m-1}
\prod\limits _{j\in T'}(A_i'+iB_j') \nonumber \\&&
\times
\prod\limits _{{i,j\in \overline{T}\atop i<j}}(B_i+B_j)
\prod\limits _{{i,j\in \overline{T}'\atop i<j}}(B_i'+B_j')
\prod\limits _{{i\in T\atop j\in \overline{T}}}{1\over B_i-B_j}
\prod\limits _{{i\in T'\atop j\in \overline{T}'}}{1\over B_i'-B_j'} \nonumber \\&& \times
\prod\limits _{{i\in T\atop j\in T'}}(B_i+iB_j')
\prod\limits _{{i\in \overline{T}\atop j\in \overline{T}'}}(B_i-iB_j')
\ X_{T,T'}(B_1,\cdots ,B_{2n}|B_1',\cdots ,B_{2m}')
\label{poly}
\end{eqnarray}
where $\overline{T}=S\backslash T$, $\overline{T'}=S'\backslash T'$,
\begin{eqnarray}
X_{T,T'}(B_1,\cdots ,B_{2n}|B_1',\cdots ,B_{2m}')=
\sum\limits _{i_1,i_2\in\overline{T}}
\prod\limits _{p=1}^2\(
{\prod\limits _{j\in T}(B_{i_p}+B_j)
\over \prod\limits _{j\in \overline{T}\backslash \{i_1,i_2\}}(B_{i_p}-B_j)}
{\prod\limits _{j\in T'}(B_{i_p}+iB_j')
\over \prod\limits _{j\in \overline{T}'}(B_{i_p}-iB_j')} \)
\nonumber
\end{eqnarray}
The polynomial $X_{T,T'}(B_1,\cdots ,B_{2n}|B_1',\cdots ,B_{2m}') $
is in fact quite symmetric with respect to replacement
$B\leftrightarrow B'$.
Let us show that $M_{n,m}$ satisfies all necessary requirements.
The relations (\ref{rel1},\ref{rel2}) are easily checked using the
formula
$$\prod\limits _{i=1}^{n-1}
\prod\limits _{j=1}^{n-1}(A_i+iB_j) \prod\limits _{i<j}(A_i-A_j)=
\prod\limits _{i=1}^{n-1}
\prod\limits _{j=1}^{n-1}(A_i^2+B_j^2)
\prod\limits _{i<j}^{n-1}{1\over B_i-B_j}\ det\({1\over A_i-iB_j}\) $$
So, $M_{n,m}$ really defines a local operators. We have to
show that the additional conditions formulated at the
beginning of this section are satisfied in order to show
that this local operator is indeed the trace of the energy-momentum
tensor.
Obviously, $M_{n,m}$ is a homogeneous function of all its
variables ($A,B,A',B'$) of total degree $(m+n)^2-2m-2n$.
Considering the formula (\ref{ff}) one realizes that this fact
provides that the operator defined by $M_{n,m}$ is Lorentz
scalar i.e. its form factors are invariant under simultaneous
shift of all the rapidities.
Let us consider now the conditions 1-3 formulated earlier.
We start form the condition 3. One finds that
$$\psi (\beta +\log \Lambda ,\beta ')\to \Lambda ^{-{1\over 2}} e^{-{1\over 2}\beta} $$
The integrals with respect to $\alpha _i$ in (\ref{ff}) are
concentrated near the points $\beta _j$, so when $\beta_j$ become of
order $\log \Lambda $ the integration variables $\alpha _i$ must be of the
same order. One finds that when $\log \Lambda \to \infty$
\begin{eqnarray}
&&M_{n,m}(\Lambda A _1,\cdots ,\Lambda A _{n-1} |A _1',\cdots ,A _{m-1}'|
\Lambda B _{1},\cdots ,\Lambda B _{2n}|B _{1}',\cdots ,B _{2m}') \to \nonumber \\
&&\to
\Lambda ^{2mn+n^2-2n-2}
\prod\limits _{j=1}^{2n} B_j^{2m-1}
\sum\limits _{j=1}^{2n} B_j^{-1}
\prod\limits _{i=1}^{n-1} A_i^3
\prod\limits _{i<j}(A_i^2-A_j^2)
\sum\limits _{j=1}^{2m} B_j '
\prod\limits _{i=1}^{m-1} A_i '
\prod\limits _{i<j}(A_i'^2-A_j'^2)
\nonumber
\end{eqnarray}
This formula is equivalent to (\ref{c3}) because
the form factors of the energy-momentum tensor of $SU(2)$-invariant
Thirring model $\widehat{f}_{T}(\beta_1,\cdots,\beta_{2n})$
$\widehat{f}_{\overline{T}}(\beta_1',\cdots,\beta_{2m}') $ are given by the formulae
on the type (\ref{int}) with the polynomial $K$ equal respectively to
\begin{eqnarray}
M^2\prod\limits _{j=1}^{2n} B_j^{-1}
\sum\limits _{j=1}^{2n} B_j^{-1}\prod\limits _{i=1}^{n-1} A_i^3
\prod\limits _{i<j}(A_i^2-A_j^2)
\qquad and
\qquad M^2\sum\limits _{j=1}^{2m} B_j '
\prod\limits _{i=1}^{m-1} A_i '
\prod\limits _{i<j}(A_i'^2-A_j'^2)
\label{T}
\end{eqnarray}
Let us consider the condition (\ref{c2}).
One finds that
\begin{eqnarray}
&&\left.
{1\over B _{1}' +B _{2}' }
M_{n,1}(A _1,\cdots ,A _{n-1} |\emptyset|
B _{1},\cdots ,B _{2n}|B _{1}' ,B _{2}') \right| _{B _{2}' =-B _{1}'}=
\nonumber \\&&\hskip 1cm =
\prod\limits _{j=1}^{2n} B_j^{-1}
\prod\limits _{i=1}^{n-1} A_i^3
\prod\limits _{i<j}(A_i^2-A_j^2)
\(\prod (B_1'+iB_j )-\prod (B_1'-iB_j )\)
\nonumber
\end{eqnarray}
which together with (\ref{T}) gives the first equation from (\ref{c2}),
the second relation is proven similarly.
The condition (\ref{c1}) is the most complicated to prove.
Naively it has to be equivalent to the fact that $M_{n,m} $
is divisible by $\sum B_j^{-1}$ and $\sum B_j'$, but that is not
the case: the function $M_{n,m} $ has to be substituted into
the integral hence it is defined modulo the functions of
the type (\ref{zero}) (and similar functions of $A_i'$). Thus
the divisibility has to be proven modulo these null-polynomials.
We have checked this fact for many particular examples, but still
we lack a general proof. However, the calculations in particular
cases go so nicely that we have no doubt that the relation (\ref{c1})
is satisfied generally.
\section{Some generalizations.}
The model considered in this paper provides a special
case of wide class of massless flows. Consider the massless
flow \cite{sot} between the UV
coset model $su(2)_{k+1}\otimes su(2)_k/su(2)_{2k+1}$ and
the IR coset model $su(2)_{k}\otimes su(2)_1/su(2)_{k+1}$,
the latter model is nothing but the minimal
model $M_{k+2}$. This flow is defined in UV by the relevant
operator of dimension
$1-2/(2k+3) $,
it arrives at IR region along $T\overline{T}$.
The massless S-matrices for these flows are written in terms of
RSOS restriction of the sine-Gordon (SG) S-matrix $S^{\xi}(\beta)$
($\xi$ is SG coupling constant defined as in \cite{book}).
Namely \cite{bl},
$$S_{LL}(\beta _1,\beta_2)=S^{\pi(k+2)}_{RSOS}(\beta _1-\beta_2),\qquad
S_{RR}(\beta _1',\beta_2')=S^{\pi(k+2)}_{RSOS}(\beta _1'-\beta_2') $$
The left-right S-matrix is independent of $k$, it is the same as above.
When $k=\infty$ the model coincides with PCM${}_1$. Another
extreme case is $k=1$ when the model describes the flow
between tricritical and critical Ising models. It is well known
that the RSOS-restriction for $\xi =3\pi$ effectively
reduces soliton to one-component particle with free scattering:
$$S^{\pi(k+2)}_{RSOS}(\beta)=-1$$
The results of this paper allow straightforward generalization to
these flows. One has to replace the formulae of the type (\ref{int}) by
their SG analogues. This does not disturb the function
$M_{n,m}$ because the way of counting solution to the equation
of the type (\ref{leq}) in SG case does not depend on the
coupling constant as well as all the equations on $M_{n,m}$.
So, the form factors are defined by (\ref{ff}) where one has to
replace the
functions $\zeta$, $\varphi$, $\langle \Delta ^{(0)}\rangle$
by their SG-analogues and to take RSOS restriction.
Let us see how it works in the case $k=1$. For generic
coupling constant the
formulae (\ref{zero}) present the only reason for vanishing the
integrals of the type
(\ref{int}). However when $ \xi=3\pi$ and RSOS restriction is
taken the integral does not vanish only for the antisymmetrical
polynomial $K(A_1,\cdots ,A_{n-1})$ of very special kind:
$$ K(A_1,\cdots ,A_{n-1}) =\prod A_i^2\prod\limits _{i<j}(A_i^2-A_j^2)$$
The value of the integral for this kind of polynomial
(taking in account the functions $\zeta$ also) is
$$\prod\limits _{i<j}\tanh {1\over 2}(\beta_i-\beta _j)
\exp\({1\over 2}\sum \beta_j \)$$
Consider the formula (\ref{poly}). We have to take the functions
$$
\prod\limits _{i<j}(A_i-A_j)
\prod\limits _{i=1}^{n-1} A_i^2
\prod\limits _{i=1}^{n-1}
\prod\limits _{j\in T}(A_i+iB_j)
\quad and\quad
\prod\limits _{i<j}(A_i'-A_j')
\prod\limits _{i=1}^{m-1} A_i '
\prod\limits _{i=1}^{m-1}
\prod\limits _{j\in T'}(A_i'+iB_j'),
$$
to decompose them with respect to antisymmetrical polynomials
of $A_i$ and $A'_i$ corresponding to different partitions
and to find the coefficients with which enter
the polynomials $\prod A_i^2\prod\limits _{i<j}(A_i^2-A_j^2)$ and
$\prod (A'_i)^2\prod\limits _{i<j}((A'_i)^2-(A'_j)^2)$. These coefficients are
$$
\prod\limits _{j\in T}B_j
\prod\limits _{{i,j\in T\atop i<j}}(B_i+B_j)
\quad and \quad
\prod\limits _{{i,j\in T'\atop i<j}}(B_i'+B_j') $$
Thus we find the following formula for the form factors of $\Theta$ for
this model
\begin{eqnarray}
&&f_{\Theta}(\beta_1,\cdots,\beta_{2n}\ |\ \beta_1',\cdots,\beta_{2m}')
=M^2 \prod\psi(\beta_i,\beta_j')
\prod\limits _{i,j}\tanh {1\over 2}(\beta_i-\beta _j)
\prod\limits _{i,j}\tanh {1\over 2}(\beta_i-\beta _j)
\nonumber \\ &&\times
\exp\({1\over 2}\sum \beta_j+{1\over 2}\sum \beta_j' \)
Q_{n,m}
(e^{\beta _{1}},\cdots ,e^{\beta _{2n}}|
e^{\beta _{1}'},\cdots ,e^{\beta _{2m}'})
\nonumber
\end{eqnarray}
where
\begin{eqnarray}
&&Q_{n,m}(
B _{1},\cdots ,B _{2n}|B _{1}',\cdots ,B _{2m}') =
\nonumber \\&& = \prod\limits _{j=1}^{2n} B_j^{-1}
\sum\limits _{{T\subset S\atop\# T=n-1}}
\sum\limits _{{T'\subset S'\atop\# T'=m-1}}
\prod\limits _{j\in T} B_j^2
\prod\limits _{{i,j\in T\atop i<j}}(B_i+B_j)
\prod\limits _{{i,j\in T'\atop i<j}}(B_i'+B_j')
\nonumber \\&&
\times
\prod\limits _{{i,j\in \overline{T}\atop i<j}}(B_i+B_j)
\prod\limits _{{i,j\in \overline{T}'\atop i<j}}(B_i'+B_j')
\prod\limits _{{i\in T\atop j\in \overline{T}}}{1\over B_i-B_j}
\prod\limits _{{i\in T'\atop j\in \overline{T}'}}{1\over B_i'-B_j'} \nonumber \\&& \times
\prod\limits _{{i\in T\atop j\in T'}}(B_i+iB_j')
\prod\limits _{{i\in \overline{T}\atop j\in \overline{T}'}}(B_i-iB_j')
X_{T,T'}(B_1,\cdots ,B_{2n}|B_1',\cdots ,B_{2m}')
\label{kon}
\end{eqnarray}
one can write a formula for this polynomial in determinant form,
but we think that (\ref{kon}) shows quite transparently how all the
required properties of this polynomial \cite{muss} are satisfied.
| proofpile-arXiv_065-459 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{INTRODUCTION}
\begin{figure}
\psfig{file=snfig.ps,height=9.53cm,rheight=8.3cm}
\caption{\label{snfig}\small Direct-capture cross sections at 30
keV for different Sn isotopes.
Levels and masses are calculated with models by Sharma et
al.~\protect{\cite{sharma}} (triangles),
M\"oller et al.~\protect{\cite{moeller}} (dots), and Dobaczewski et
al.~\protect{\cite{doba}} (squares). The lines are drawn to guide the eye.}
\end{figure}
\begin{figure}
\psfig{file=nic96figc.ps,height=9.53cm,rheight=8.3cm}
\caption{\label{states}\small Dependence of level energies on mass number for
the even-odd isotopes $^{125-135}$Sn in the RMFT model~\protect{\cite{sharma}}
(right) and for the isotopes $^{125-145}$Sn in the HFB
model~\protect{\cite{doba}} (left).
Shown are the 1/2$^-$ state (open circles), the
3/2$^-$ state (triangles) and the calculated neutron separation energy (full
circles). The lines are drawn to guide the eye. Note the different range in
mass numbers in the two plots.}
\end{figure}
Explosive nuclear burning in astrophysical environments produces
unstable nuclei which again can be targets for subsequent reactions.
Most of these nuclei are not accessible in terrestrial labs or not fully
explored by experiments, yet. For the majority of unstable nuclei the
statistical model (Hauser-Feshbach) can be used to determine the cross
sections. However, for nuclei close to the dripline the level density
becomes too low to apply the statistical model~\cite{tommy}
and contributions of a direct
interaction mechanism (DI) may dominate the cross sections.
The DI requires the detailed knowledge of energy levels (excitation energies,
spins, parities), contrary to the statistical model which averages over
resonances. Lacking experimental data, this information has to be extracted
from microscopic nuclear-structure models.
We compare the results for direct neutron capture (calculated in the optical
model~\cite{kim}) on the even-even isotopes
$^{124-145}$Sn with energy
levels,
masses, and nuclear density distributions
taken from different nuclear-structure models.
The utilized structure models were a Hartree-Fock-Bogoliubov model
(HFB) with SkP force~\cite{doba}, a Relativistic Mean Field Theory (RMFT)
with the
parameter set NLSH~\cite{sharma} and a Shell Model based on folded-Yukawa
wave functions (FYSM)~\cite{moeller}.
A similar study has already been performed for neutron-rich Pb
isotopes~\cite{tom}.
\section{METHOD}
The cross sections were calculated in the optical model for direct
capture~\cite{kim}, utilizing optical potentials derived by the folding
procedure~\cite{satch}. In the folding approach the nuclear target density
is folded with an energy- and density-dependent effective nucleon-nucleon
interaction in order to obtain the potentials for the bound and
scattering states. Only one open parameter $\lambda$ remains which accounts
for the effects of antisymmetrization and is close to unity.
The densities required for the determination of the
folding potentials were consistently calculated from the wave functions
of the respective nuclear-structure model.
For the bound states the strength parameter $\lambda$ was fixed by the condition
to reproduce the given binding energy of the captured neutron. The
value of $\lambda$ for the scattering potential was adjusted to yield the
same value of 425 MeV fm$^3$~\cite{werner} for the volume integral as
determined from the
experimental scattering data on stable Sn isotopes~\cite{mug,cinda}.
In order to be able to directly
compare the different models, all nuclei were assumed to be spherical and
the spectroscopic factors were set to 1.
\section{RESULTS AND DISCUSSION}
The results of the calculations for projectiles at $E_{\mathrm{c.m.}}=30$ keV
are summarized in Fig.~\ref{snfig}. For each model we calculated the
capture cross section only up to the r-process path. The most extreme
location of the
path (farthest away from the line of stability) is determined by neutron
separation energies $E_{\mathrm{n}} \approx 2$ MeV~\cite{fkt}. Depending on
the microscopic model, the path will then be located at higher or lower
mass numbers $A$. In the case of RMFT and FYSM it will go through $A\approx$
132--134, for HFB the path will be shifted to
considerably higher mass numbers $A\approx$ 142--144. (The neutron dripline
is also shifted to higher masses in the latter model.)
Similar effects as seen in the behavior of the Pb cross sections~\cite{tom}
can also be found for the Sn cross sections. The cross section can vary by
order of magnitudes when going from one isotope to the next and also differ
vastly between the different microscopic models.
As the capture to low-spin states ($J$=1/2, 3/2) accounts for the largest
contributions to the cross section, the results are very sensitive to the
presence of bound low-spin states. Since the microscopic models not only
yield different masses (i.e.\ neutron separation energies) but also exhibit
different behaviors of the level energies with changing mass,
``jumps'' and ``gaps'' can be seen with some models (RMFT, FYSM), whereas
others (HFB)
result in a smoother behavior of the capture cross sections in an
isotopic chain.
This is illustrated in Fig.~\ref{states}, which shows the neutron
separation energy and the excitation energy of the 1/2$^-$ and 3/2$^-$
states in RMFT and HFB. As long as both states are unbound in the RMFT, the
cross sections remain low and only jump to higher values when those
states become bound at the shell closure. As at least the 3/2$^-$ level is
always bound in HFB, the cross sections show a smoother behavior.
The variation in the FYSM cross sections can be explained in a similar
way.
\section{CONCLUSION}
With this work we have underlined that the calculation of purely
theoretical direct capture cross sections far from stability still contains a
large error, even when using most recent nuclear-structure models.
In the previously discussed case of Pb isotopes~\cite{tom}, the r-process path
contains nuclei in or at the border of a region expected to be deformed,
leading to
higher level densities and thus favoring the compound nucleus mechanism.
This is not true for neutron-rich isotopes in the Sn region, especially
around the neutron magic number $N=82$ where the level density becomes too
low for the statistical model. Therefore the neutron capture cross sections
have to be calculated using input from nuclear-structure models and will
be subject to the quoted uncertainties, even when the different models
yield similar values for other nuclear properties, such as masses.
Similar problems may be encountered on the proton-rich side when predicting
proton capture cross sections close to the proton dripline.
| proofpile-arXiv_065-460 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
{\Large 1. Introduction}\\
\noindent
One of the most precise and from the theoretical point unambiguous
determinations of the strong coupling constant $\alpha_s$ has been
obtained from the hadronic decay rate of the $Z$ boson. Ingredients are
the large event rate which allows to reduce the statistical
uncertainty in $\alpha_s$ down to $0.003$, the precise calibration
through the accurate measurement of the luminosity or the leptonic
rate \cite{Blondel} and, last but not least, the elaborate theoretical
calculations \cite{YellowReport,CKKRep}.
This high energy determination of $\alpha_s$
should and could be complemented by a similar measurement of $R_{\rm
had}$ in the energy region just below the $B$ meson threshold: large
event rates are available at the $e^+ e^-$ storage ring CESR and an
experimental analysis is currently underway \cite{DaveBessonpriv}.
Theoretical predictions for $R_{\rm had}$ below the $B{\overline B}$
threshold have been presented earlier \cite{rhad}. These include
terms up to order $\alpha_s^3$ for the massless limit and for $m_c^2/s$
corrections as well, terms of order $\alpha_s^2$ for the quartic mass
terms $(m_c^2/s)^2$, terms of order $\alpha_s (m_c^2/s)^3$ and
contributions from virtual bottom quarks up to order $\alpha_s^2 (s/m_b^2)$.
Recently it has been demonstrated that this expansion in $(m_c^2/s)$
provides an excellent approximation not only around $\sqrt{s} =
10$~GeV, but also down to 6 or even 5 GeV.
\par
The forthcoming experimental analysis necessitates on the one hand a
thorough understanding of the background, e.g. from two photon events
or the feedthrough from $\tau$ pairs into the hadronic cross section,
and, on the other hand, a realistic prediction for the annihilation
channel. The former is outside the scope of this paper, the latter is
the subject of this work. It requires to
incorporate the effect of the running QED coupling constant and
initial state radiation. The photon vacuum polarization leads to an
increase of the cross section by about 7--8\%. Magnitude and even sign
of initial state corrections depend on the experimental procedure,
in particular on the minimal mass of the hadronic system accepted
for the event sample. Another
important issue is the treatment of the tails of the $\Upsilon$
resonances, which
contribute coherently through mixing with the photon and
incoherently through their radiative tails. In order to study these
effects and the associated theoretical uncertainties separately
the corresponding analysis for the reaction $e^+ e^- \to \tau^+ \tau^-$
is presented in section 2. These results may also serve as an
independent experimental
calibration of the cross section. The corresponding predictions for
the hadronic cross section are discussed in section 3. In section 4
we comment on the accuracy of the presented approach and section 5
contains our summary and conclusions.
\vskip 1 cm
\noindent
{\Large 2. Lepton Pair Production}\\
\noindent
To calibrate the predictions for the hadron production cross section
it seems appropriate and useful to calculate in a first step the cross
section for lepton pair production. To arrive at a reliable
prediction, initial and final state radiation, the effect of vacuum
polarization and the influence of the nearby $\Upsilon$ resonances must
be included. In this section results are presented for $\tau$ pair production.
\vskip 0.5 cm
\noindent
{\bf (a) Initial state radiation}
\noindent
The most important correction to the total cross section is introduced
by initial state radiation. It leads to a reduction of the invariant
mass of the produced lepton pair or hadronic system and, for fairly
loose cuts, to a significant enhancement of the cross section, albeit
with events of significantly lower invariant mass of the system of
interest. For the precise determination of $R_{\rm had}$ discussed below
it is advisable to exclude the bulk of these low mass events. This reduces the
size of the correction and at the same time the dependence on the
input for $R_{\rm had}$ from the lower energy region.
The treatment of initial state radiation has advanced significantly as
a consequence of the detailed calculations performed for the analysis
of the $Z$ line shape. The result for the inclusive cross section can
be written in the form
\begin{equation}
\sigma(s) = \int_{z_0}^1 {\rm d}z\,\sigma_0(sz)\,G(z)\,.
\label{eqisrint}
\end{equation}
The cross section including photon vacuum polarization (running $\alpha$)
is denoted by $\sigma_0$. The invariant mass of the produced fermion
pair is given by $s\,z$, where
\begin{equation}
\frac{m_{\rm min}^2}{s} \leq z_0 \leq z \leq 1
\label{eqisrborder}
\end{equation}
and $m_{\rm min} = 2 m_\ell$ for lepton pairs and $m_{\rm min} = 2 m_\pi$
for hadron production. In the cases of interest for this paper $z_0$
will have to be adopted to the experimental setup. Typically it is
significantly larger than the theoretically allowed minimal value.
For $E_{\rm cm} = 10.52$~GeV a cut in $z$ around $0.25$ corresponding
to roughly $5$~GeV in the minimal mass will in the case of hadronic
final states exclude the charmonium resonance region and the charm
threshold region as well, thus limiting the hadron analysis to truely
multihadronic final states.
The complete radiator function $G(z)$ up to order $\alpha^2$ has been
calculated in \cite{Burgers,vanNeerven}. The resummation of leading
logarithms is discussed in \cite{YRBerends} (see also \cite{Kuraev}).
For the present purpose an approximation is adequate which is exact in
order $\alpha$ and which includes the dominant terms of order
$\alpha^2$ plus leading logarithms. For the radiator function $G$ we
thus take \cite{JadachWard}
\begin{equation}
G(z) = \beta (1-z)^{\beta-1}\,{\rm e}^{\delta_{yfs}}\,F\,\left(
\delta_C^{V+S} + \delta_C^H \right)\,,
\label{eqGc}
\end{equation}
with
\begin{eqnarray}
\beta &=& \frac{2\alpha}{\pi}(L-1)\,,\nonumber\\
L &=& \ln\frac{s}{m_e^2}\,,\nonumber\\
\delta_{yfs} &=& \frac{\alpha}{\pi} \left( \frac{L}{2} - 1 + 2\zeta(2)
\right)\,,\nonumber\\
\delta_c^{V+S} &=& 1+\frac{\alpha}{\pi}(L-1)+\frac{1}{2}
\left(\frac{\alpha}{\pi}\right)^2 L^2\,,\nonumber\\
\delta_C^H &=& -\frac{1-z^2}{2}+\frac{\alpha}{\pi}L\left[-\frac{1}{4}
\left(1+3z^2\right)\ln z -1+z\right]\,,\nonumber\\
F &=& \frac{{\rm e}^{-\beta\gamma_E}}{\Gamma(1+\beta)}\,,\nonumber
\end{eqnarray}
where $\gamma_E = 0.5772\ldots$ is Euler's constant.
Eq.~(\ref{eqGc}) is suited for quick numerical integration.
The difference to the initial state radiation convolution using
the complete order $\alpha^2$ result (eq~(3.12) of \cite{YRBerends})
is below one permille.
The predictions for the reference energy of $10.52$~GeV and a
variety of cuts $z_0$ are listed in Table~\ref{table1}.
\begin{table}[htb]
\caption{Dependence of the cross section $e^+ e^- \to \tau^+ \tau^-$
on the cutoff $z_0$ for $E_1 = 10.52$~GeV.}
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|} \hline
$z_0$
& 0.114 & 0.20 & 0.25 & 0.5 & 0.8 & 0.9 & 0.95
\\ \hline
$m_{\rm cut}\ [GeV]$
& $2m_{\tau}$ & 4.705 & 5.260 & 7.439 & 9.409 & 9.980 & 10.254
\\ \hline\hline
$\sigma\ [nb]$
& 0.9269 & 0.9101 & 0.8998 & 0.8543 & 0.7831 & 0.7359 & 0.6922
\\ \hline
$\sigma/\sigma_{pt}$
& 1.1822 & 1.1608 & 1.1477 & 1.0896 & 0.9988 & 0.9386 & 0.8829
\\ \hline
\end{tabular}
\end{center}
\label{table1}
\end{table}
\begin{figure}[htb]
\begin{center}
\leavevmode
\epsfxsize=17.cm
\epsffile[70 290 510 530]{taudiff.ps}
\vskip -3mm
\caption[]{\label{fig1ab} {\em (a) The differential cross section
${\rm d}\sigma(e^+ e^- \to \tau^+ \tau^-)/{\rm d}z$ and (b) the
cutoff dependent cross section $\sigma(z > z_0)$ in nb for
the centre of mass energy $E_1 = 10.52$~GeV as functions of $z$ and
$z_0$, respectively.}}
\end{center}
\end{figure}
The differential distribution ${\rm d}\sigma/{\rm d}z$ and the
cutoff dependent cross section $\sigma(z > z_0)$ are displayed
in Fig.~\ref{fig1ab}a, b, respectively, as functions of
$z$, $z_0$. For the cross section before convolution the corrections from
the vacuum polarization, the tails of the $\Upsilon$ resonances and
final state radiation are included.
Initial state radiation of lepton pairs and of hadrons has been
treated in \cite{vanNeerven,KKKS}. These could easily be incorporated
in the present formalism. The resulting corrections are at the
permille level and will be ignored for the moment.
Corrections from initial state radiation are evidently huge for loose
or extremely tight cutoffs, with opposite sign. The strong cutoff
dependence of the result is apparent from Table \ref{table1} and
Fig.~\ref{fig1ab} as well. The result is relatively stable against
variations of $z_0$ for values of $z_0$ below 0.75. At the
same time the sensitivity to the input for $\sigma_0$ from smaller
$s' = s\,z$ is eliminated for this choice.
From this viewpoint a value around 0.7 to 0.8 seems optimal for
the experimental determination of $R_{\rm had}$. On the other
hand, if one allows to use input from smaller $s'$ from other experiments
or from the measured events with photons from initial state radiation,
then also lower values of $z_0$ down to about 0.25 are equally
acceptable.
The distribution ${\rm d}\sigma/{\rm d}z$ is flat in the
region $0.25 < z < 0.75$ and numerically small. An experimental
analysis based on a cutoff $z_0 = s'/s$ with a reasonably symmetric
resolution in $s'$ is thus insensitive towards the details of the
resolution function. In fact, even an uncontrolled shift of the
central value by $\Delta z_0 = 0.05$ would lead to a 1\% deviation
of the cross section only.
\noindent
{\bf (b) Final state radiation}
\noindent
Initial state radiation is strongly enhanced, a consequence of the
inherently large logarithm in the correction
function (cf. eq.~(\ref{eqGc})). Final state radiation with loose
cuts, in
contrast, is typically of order $\alpha/\pi$ without a large logarithm.
In the massless limit and without a cutoff on the invariant fermion
pair mass the correction factor $(1 + 3/4\,\alpha/\pi)$ amounts
to two permille only. It depends, however, on the lepton mass
and the cutoff. For the relatively loose cuts with $z_0$ between
0.25 and 0.8 one may well use the totally inclusive correction
function $r^{(1)}$ defined by
\begin{equation}
R_{f\bar f}\, := \, \frac{\sigma(e^+e^-\to\gamma^*\to f\bar f\ldots)}
{\sigma_{pt}}
\quad = \quad r^{(0)} +
\,\bigg(\frac{\alpha}{\pi}\bigg)\,r^{(1)} + \ldots\,,
\label{eqrdef}
\end{equation}
with $\ \sigma_{pt} = 4\pi\alpha^2/3s\ $
and
\begin{eqnarray}
r^{(0)} & = & \frac{\beta}{2}\,(3-\beta^2)\,,\\[2mm]
r^{(1)} & = &
\frac{\left( 3 - {\beta^2} \right) \,\left( 1 + {\beta^2} \right) }{2
}\,\bigg[\, 2\,\mbox{Li}_2(p) + \mbox{Li}_2({p^2}) +
\ln p\,\Big( 2\,\ln(1 - p) + \ln(1 + p) \Big)
\,\bigg] \,\nonumber\,\\
& & \mbox{} -
\beta\,( 3 - {\beta^2} ) \,
\Big( 2\,\ln(1 - p) + \ln(1 + p) \Big) -
\frac{\left( 1 - \beta \right) \,
\left( 33 - 39\,\beta - 17\,{\beta^2} + 7\,{\beta^3} \right) }{16}\,
\ln p\,\nonumber\,\\
& & \mbox{} +
\frac{3\,\beta\,\left( 5 - 3\,{\beta^2} \right) }{8}
\,,
\end{eqnarray}
where
\begin{eqnarray}
p \, = \, \frac{1-\beta}{1+\beta}\,,\qquad\,
\beta \, = \, \sqrt{1-4m_f^2/s}\,.
\end{eqnarray}
In the case of hadron production, the main motivation
of this investigation, photons from final state radiation will
in general anyhow be included in the hadronic invariant mass.
\vskip 0.5 cm
\noindent
{\bf (c) Leptonic and hadronic vacuum polarization}
\noindent
The leptonic vacuum polarization in one loop approximation is given
by
\begin{equation}
\widehat\Pi_{\gamma\gamma}(s) :=
\Pi_{\gamma\gamma}(s) - \Pi_{\gamma\gamma}(0) =
\frac{\alpha}{3\pi}\,\sum_f N_c \, Q_f^2 \, P(s, m_f)\,,
\label{eqvacpollepdef}
\end{equation}
with
\begin{equation}
P(s, m_f) = \frac{1}{3} - \left(1+\frac{2m_f^2}{s}\right)\,
\left(2+\beta\ln\frac{\beta-1}{\beta+1}\right)\,.
\label{eqvacpollepp}
\end{equation}
For $|s| \gg m_{\ell}^2$ the function $P(s, m_f)$ is well approximated by
\begin{equation}
P_{\rm asymp.}(s, m_f) = -\frac{5}{3} + \ln\bigg(-\frac{s}{m_f^2}
+ i \,\epsilon\bigg)\,.
\label{eqvacpolleppappr}
\end{equation}
The hadronic vacuum polarization is obtained from it's imaginary part
via a dispersion relation
\begin{equation}
{\rm Re}\widehat\Pi_{\rm had}(q^2) = \frac{\alpha q^2}{3\pi}
\, {\rm P} \int_{m_{\pi}^2}^{\infty} \frac{R_{\rm had}(s')}{s'(s'-q^2)}
\, {\rm d}s'\,.
\label{eqvacpolhad}
\end{equation}
To avoid complications that arise from the numerical evaluation of
the dispersion integral over the data in the timelike region, the
integral is evaluated at the corresponding value of $s$ in the
spacelike region.
We use parametrisations from the evaluation of eq.~(\ref{eqvacpolhad})
in the spacelike region as provided by \cite{jegerl,burkhardt}.
The results for the
complete cross sections calculated with the different parametrisations
from \cite{jegerl} and \cite{burkhardt} agree to better
than $10^{-4}$ in the region of interest. The error from the
identification of spacelike and
timelike $\widehat\Pi_{\rm had}(q^2)$ should be
small except for
the $b$ quark contribution, where the threshold is fairly close to the
$q^2$ values of interest. Therefore we subtract the perturbative
$b$ quark contribution evaluated for spacelike $q^2$ and add the
corresponding value for timelike $q^2 = s$. We have checked that this
ansatz is in excellent agreement with the numerical
evaluation\footnote{We thank H.~Burkhardt for providing the numerical
evaluation of (\ref{eqvacpolhad}) required for this comparison.} of
(\ref{eqvacpolhad}) in the timelike region.
Effects due to $\Upsilon$ resonances are described in detail below.
The running $\alpha$ is then obtained from the real part of $\widehat\Pi$
through
\begin{equation}
\alpha(s) = \frac{\alpha}{1-{\rm Re}\widehat\Pi(s)}\,.
\label{eqalpharunning}
\end{equation}
The relative shift in $\alpha(s)$ from the
hadronic plus leptonic vacuum polarization is shown in
Fig.~\ref{figalpha} as a function of $\sqrt{s}$. The size
of the individual contributions is
listed in Table~\ref{table2} for the reference energy
$E_1=10.52$ GeV and for a few selected lower energies.
The uncertainty in the cross section from our treatment of the
hadronic vacuum polarization in the timelike region is estimated
to be below two permille.
\begin{table}[htb]
\caption{Individual contributions to ${\rm Re}\widehat\Pi(s) \cdot 10^2$.}
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|} \hline
$\sqrt{s}/$GeV & $2m_{\tau}$ & 5 & 7 & 9 & 10.52 \\ \hline
$e$ & 1.241 & 1.294 & 1.346 & 1.385 & 1.409 \\
$\mu$ & 0.415 & 0.468 & 0.520 & 0.559 & 0.583 \\
$\tau$ & -0.207 & -0.049 & 0.047 & 0.101 & 0.132 \\
had & 0.923 & 1.096 & 1.269 & 1.379 & 1.456 \\ \hline
$1/\alpha(s)$ & 133.79 & 133.19 & 132.68 & 132.34 & 132.13 \\ \hline
\end{tabular}
\end{center}
\label{table2}
\end{table}
\begin{figure}[htb]
\begin{center}
\leavevmode
\epsfxsize=12.cm
\epsffile[110 280 460 560]{alpha.ps}
\vskip -3mm
\caption[]{\label{figalpha} {\em The relative shift in $\alpha(s)$ as
a function of $\sqrt{s}$ from the hadronic plus leptonic vacuum
polarization as described in the text.}}
\end{center}
\end{figure}
\vskip 0.5 cm
\noindent
{\bf (d) Narrow $\Upsilon$ resonances}
\begin{table}[htb]
\caption{Relative contributions from the $\Upsilon$ resonances to
the leptonic cross section $\sigma(e^+ e^- \to \tau^+ \tau^-)$
for the energies $E_1 = 10.52$~GeV and $E_2 = 9.98$~GeV and two
different values of the cutoff $m_{\rm min}$. Interference
terms and radiative tails are listed separately. Also given are the
continuum contributions and the resulting predictions for the total
cross section.}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline
&$\Upsilon(1)$&$\Upsilon(2)$&$\Upsilon(3)$&
$\Upsilon(4)$&$\Upsilon(5)$&$\Upsilon(6)$\\ \hline\hline
$M\ [GeV]$ &
$9.460$ & $10.023$ & $10.355$ & $10.580$ & $10.865$ & $11.019$ \\ \hline
$\Gamma_e\ [keV]$ &
$1.32$ & $0.576$ & $0.476$ & $0.24$ & $0.31$ & $0.13$ \\ \hline
$\Gamma_{\rm tot}\ [MeV]$ &
$0.0525$ & $0.044$ & $0.0263$ & $23.8$ & $110$ & $79$ \\ \hline\hline
\multicolumn{7}{|l|}{$E_1 = 10.52$ GeV, $m_{\rm min}=2m_{\tau}=3.554$ GeV:}
\\ \hline
Continuum: & \multicolumn{6}{c|}{$0.9256$ nb} \\ \hline
Interf.:
& $2.7\cdot 10^{-4}$ & $2.9\cdot 10^{-4}$ & $7.5\cdot 10^{-4}$
& $-1.04\cdot 10^{-3}$ & $-3.0\cdot 10^{-4}$ & $-9.7\cdot 10^{-5}$ \\ \hline
$\sum$ interf.: & \multicolumn{6}{c|}{$-1.3\cdot 10^{-4}$} \\ \hline
Rad.~tails~:
& $5.5\cdot 10^{-4}$ & $2.5\cdot 10^{-4}$ & $8.1\cdot 10^{-4}$ &
-- & -- & -- \\ \hline
$\sum$ rad. tails: & \multicolumn{6}{c|}{$1.61 \cdot 10^{-3}$} \\ \hline
Total: & \multicolumn{6}{c|}{$0.9269$ nb} \\ \hline\hline
\multicolumn{7}{|l|}{$E_1 = 10.52$ GeV, $m_{\rm min}=5$ GeV:} \\ \hline
Continuum: & \multicolumn{6}{c|}{$0.9033$ nb} \\ \hline
Interf.:
& $2.9\cdot 10^{-4}$ & $3.1\cdot 10^{-4}$ & $7.7\cdot 10^{-4}$
& $-1.07\cdot 10^{-3}$ & $-3.1\cdot 10^{-4}$ & $-9.8\cdot 10^{-5}$ \\ \hline
$\sum$ interf.: & \multicolumn{6}{c|}{$-9.7\cdot 10^{-5}$} \\ \hline
Rad.~tails~:
& $5.7\cdot 10^{-4}$ & $2.6\cdot 10^{-4}$ & $8.3\cdot 10^{-4}$ &
-- & -- & -- \\ \hline
$\sum$ rad. tails: & \multicolumn{6}{c|}{$1.65 \cdot 10^{-3}$} \\ \hline
Total: & \multicolumn{6}{c|}{$0.9047$ nb} \\ \hline\hline
\multicolumn{7}{|l|}{$E_2 = 9.98$ GeV, $m_{\rm min}=2m_{\tau}=3.554$ GeV:}
\\ \hline
Continuum: & \multicolumn{6}{c|}{$1.0200$ nb} \\ \hline
Interf.:
& $6.4\cdot 10^{-4}$ & $-3.51\cdot 10^{-3}$ & $-4.5\cdot 10^{-4}$ &
$-1.6\cdot 10^{-4}$ & $-1.6\cdot 10^{-4}$ & $-5.9\cdot 10^{-5}$ \\ \hline
$\sum$ interf.: & \multicolumn{6}{c|}{$-3.70\cdot 10^{-3}$} \\ \hline
Rad.~tails:
& $1.08\cdot 10^{-3}$ & -- & -- & -- & -- & -- \\ \hline
$\sum$ rad. tails: & \multicolumn{6}{c|}{$1.08\cdot 10^{-3}$} \\ \hline
Total: & \multicolumn{6}{c|}{$1.0173$ nb} \\ \hline\hline
\multicolumn{7}{|l|}{$E_2 = 9.98$ GeV, $m_{\rm min}=5$ GeV:} \\ \hline
Continuum: & \multicolumn{6}{c|}{$0.9947$ nb} \\ \hline
Interf.:
& $6.8\cdot 10^{-4}$ & $-3.60\cdot 10^{-3}$ & $-4.6\cdot 10^{-4}$ &
$-1.6\cdot 10^{-4}$ & $-1.6\cdot 10^{-4}$ & $-5.8\cdot 10^{-5}$ \\ \hline
$\sum$ interf.: & \multicolumn{6}{c|}{$-3.75\cdot 10^{-3}$} \\ \hline
Rad.~tails:
& $1.11\cdot 10^{-3}$ & -- & -- & -- & -- & -- \\ \hline
$\sum$ rad. tails: & \multicolumn{6}{c|}{$1.11\cdot 10^{-3}$} \\ \hline
Total: & \multicolumn{6}{c|}{$0.9920$ nb} \\ \hline
\end{tabular}
\end{center}
\label{table3}
\end{table}
\noindent
Up to this point we have not included the contributions from the
$\Upsilon$ resonances. As a consequence of their close proximity
their effect will be enhanced and thus should
be discussed separately. The Breit--Wigner amplitudes from the
$\Upsilon$ resonances above and below the $B{\overline B}$ threshold
interfere with the virtual photon amplitude and thus enhance or
decrease the cross section by
\begin{equation}
\delta\sigma_{\rm int}=
\frac{6\,\Gamma_e}{\alpha(s)\,s} \,
\frac{M^3(s-M^2)}{(s-M^2)^2+\Gamma^2 M^2}\,\sigma_0\,.
\label{eqbwint}
\end{equation}
The individual contributions are listed in Table~\ref{table3} for the
two cms energies $E_1 = 10.52$~GeV and $E_2 = 9.98$~GeV after the
convolution with initial state radiation. For $E_1$ these interference
terms happen to cancel to a large extent, leaving a minute correction
with a relative magnitude around $10^{-4}$. For $E_2$, however, due to
the close proximity of $\Upsilon(2S)$ one receives a non--negligible
negative contribution of about $-0.4$\%. This effect could become
relevant in precision tests.
\par
Also the radiative tails of the resonances with $M < \sqrt{s}$ have to
be taken into consideration. These can be easily calculated with the
radiator function eq.~(\ref{eqGc}), taking either the full
Breit--Wigner resonance
\begin{equation}
\delta\sigma_{\rm R} =
\left(\frac{3\,\Gamma_e\,M}{\alpha(s)\,s}\right)^2
\frac{M^4}{(s-M^2)^2+\Gamma^2 M^2}\,\sigma_0
\label{eqbwtailfull}
\end{equation}
or the narrow width approximation
\begin{equation}
\delta\sigma_{\rm R} \Big|_{\rm NW} =
\frac{9\,\Gamma_e^2}{\alpha(s)^2} \frac{M}{\Gamma} \,\pi\, \delta(s-M^2)
\,\sigma_0\,.
\label{eqbwtailnarroww}
\end{equation}
Both formulae lead to nearly identical results. The individual
contributions are also listed in Table~\ref{table3}.
In total they raise the lepton pair cross section by about 0.16\% or 0.11\%
for $E_1$ and $E_2$, respectively.
At this point a brief comment ought to be made concerning the
treatment of $\Upsilon(4S)$. The approximation of neglecting the
energy dependence of the width of the resonance
is justified even for $E_1$, since the nominal width $\Gamma = 24$~MeV is
significantly smaller than the difference between the cms energy $E_1$
and the mass of the resonance: $\Delta E = 60$~MeV.
\pagebreak
\noindent
{\Large 3. The Total Cross Section for Hadron Production}\\
\noindent
As stated already in the introduction the emphasis of this work is on
the energy just below the bottom meson threshold. Virtual $b$ quark
loops can be easily taken into account, and in fact it has been
demonstrated that the hard mass expansion works surprisingly well
for these diagrams --- not only far below but even close to the
nominal $b$ quark threshold. On the other hand this energy is
sufficiently far above the open charm threshold such that charm
quark mass effects can be included through an expansion in powers
of $m_c$, if quadratic and quartic terms are incorporated. With
this method the production cross section can be predicted reliably
not only in the high energy region but even relatively close to
threshold through an expansion in $m^2/s$.
This approach was suggested originally in \cite{quart,rhad}. The
calculation of $R_{\rm had}$ to second order, including the full mass
dependence \cite{hkt1} has demonstrated the nearly perfect agreement
between approximate and exact result for the coefficient of the
$\alpha_s^2$ term for $E_{\rm cm}$ above $4m$.
In fact, even for $E_{\rm cm}$ around $3m$, which is around the lowest
advisable value of the cutoff $m_{\rm min}=5$ GeV, the deviation of the
$\alpha_s^2$ coefficient from the complete result leads to a
difference of less than five percent in the rate. For the energy
around 10 GeV this region contributes through the radiative tail
only, and the approximation is thus adequate throughout. In total
we thus use the following individual contributions
\begin{equation}
R = R_{\rm NS} + R_{\rm S} + \delta R_{m_b} + \delta R_{m_c} +
\delta_{\rm QED}\,,
\label{eqRhad}
\end{equation}
where
\begin{eqnarray}
R_{\rm NS} & = & \sum_{f = u,d,s,c} 3 \, Q_f^2
\left[
1 + \frac{\alpha_s}{\pi}
+ 1.5245
\left(\frac{\alpha_s}{\pi}\right)^2
-11.52033
\left(\frac{\alpha_s}{\pi}\right)^3\,
\right]
\,,\nonumber\\
R_{\rm S} & = &
-\left(\frac{\alpha_s}{\pi}\right)^3
\Big(\sum_{u,d,s,c} Q_f\Big)^2 \, 1.239 \ = \
-0.55091
\left(\frac{\alpha_s}{\pi}\right)^3
\,,\nonumber\\
\delta R_{m_b} & = & \sum_{f = u,d,s,c} 3 \, Q_f^2
\left(\frac{\alpha_s}{\pi}\right)^2 \frac{s}{{\overline m}_b^2}
\left[
\frac{44}{675} + \frac{2}{135} \log \frac{{\overline m}_b^2}{s}
\right]
\,,\nonumber\\
\delta R_{m_c} & = &
3\,Q_c^2 \,12\,\frac{m_c^2}{s} \frac{\alpha_s}{\pi}
\left[
1
+
9.097
\frac{\alpha_s}{\pi}
+
53.453
\left(\frac{\alpha_s}{\pi}\right)^2
\right]
-
3 \sum_{f=u,d,s,c}Q_f^2\frac{m_c^2}{s}
\left(\frac{\alpha_s}{\pi}\right)^3
6.476
\nonumber\\
& & +
3\,Q_c^2 \frac{m_c^4}{s^2}
\left[
-6
-22
\frac{\alpha_s}{\pi}
+
\left(
141.329 - \frac{25}{6}\ln\frac{m_c^2}{s}
\right)
\left(\frac{\alpha_s}{\pi}\right)^2
\right]
\nonumber\\
& &+3 \sum_{f=u,d,s,c} Q_f^2
\frac{m_c^4}{s^2}
\left(\frac{\alpha_s}{\pi}\right)^2
\left[
-0.4749
- \ln\frac{m_c^2}{s}
\right]
-3\,Q_c^2 \frac{m_c^6}{s^3}
\left[
8
+\frac{16}{27}
\frac{\alpha_s}{\pi}
\left(
6\ln\frac{m_c^2}{s} + 155
\right)
\right]\,,
\nonumber\\
\delta_{\rm QED} & = &
\sum_{f=u,d,s,c} 3\,Q_f^4 \frac{\alpha}{\pi} \frac{3}{4}
\,.
\nonumber
\end{eqnarray}
The formulae are evaluated for $n_f=4$ with $\alpha_s$ and the charm
quark mass interpreted accordingly. For the massless case and
the $m^2/s$ terms the results are available in third order, for
the quartic and $m^6/s^3$ terms in second and first order
$\alpha_s$, respectively. Tables which list the numerical values of
the running quark masses and the magnitude of the individual
contributions can be found in \cite{rhad,CKKRep} together with the
details of the matching between the theories with $n_F = 4$ and $5$.
\begin{figure}[htb]
\begin{center}
\leavevmode
\epsfxsize=17.cm
\epsffile[60 280 520 540]{rhad.ps}
\vskip -3mm
\caption[]{\label{fig3ab} {\em (a) $R_{\rm had}(s)$ as defined in
eq.~(\ref{eqRhad}), and (b) only the contributions from the light
($u$, $d$ and $s$) quark currents for different values of $\alpha_s$.}}
\end{center}
\end{figure}
The predictions for $R_{\rm had}$ as a function of $E_{\rm cm}$ are shown
in Fig.~\ref{fig3ab} for three values of the strong coupling constant,
$\alpha_s(M_Z^2) = 0.115$, $0.120$ and $0.125$.
Fig.~\ref{fig3ab}a displays the contributions induced by the light plus
charm quark currents (as defined in eq.~(\ref{eqRhad})),
whereas in Fig.~\ref{fig3ab}b only the light ($u$, $d$ and $s$) quark
current contributions are shown. Note, that this figure by definition
does not contain
QED--corrections from initial state radiation and vacuum polarization,
but includes the tiny singlet terms, which cannot be attributed to one
individual quark species and the final state QED--corrections
$\delta_{\rm QED}$.
The quark masses are chosen to be
${\overline m}_c({\overline m}_c) = 1.24$~GeV ($n_f=4$)
and ${\overline m}_b({\overline m}_b) = 4.1$~GeV ($n_f=5$)
corresponding to pole masses of $1.6$~GeV and $4.7$~GeV,
respectively, if $\alpha_s^{(n_f=5)}(M_Z)=0.120$ is adopted.
A variation of the charm quark mass around the default value
by $+300$~MeV/$-300$~MeV changes
$R_{\rm had}$ only by $+0.006/-0.004$ for $\sqrt{s} = 10.52$ GeV and by
$+0.031/-0.028$ for $\sqrt{s} = 5$ GeV.
Throughout this paper the QCD results are interpreted in fixed order
$\alpha_s^3$ without any attempt to improve the formulae through the
inclusion of guesses for higher order coefficients. An estimate of the
scale dependence is easily obtained through the evaluation of a
variant of eq.~(\ref{eqRhad}) where $R_{\rm NS}$ is calculated for a general
t'Hooft scale $\mu^2$. Adopting $\alpha_s(10.5\ {\rm GeV}) = 0.177$
corresponding to $\alpha_s(M_Z) = 0.12$ and varying $\mu^2$ between
$s/4$ and $4 s$ the predicted value of $R$ varies by -2 and +0.2
permille. This is well below the anticipated experimental precision.
Alternatively we may include in eq.~(\ref{eqRhad}) an $\alpha_s^4$
term with the coefficient based on a recent estimate
in \cite{Kataev}. This would lead to a decrease in $R$ by 1.7
permille, again far below the forseeable experimental accuracy.
Let us now discuss the impact of initial state radiation, the running
$\alpha$ and the $\Upsilon$ resonances on the hadronic cross section,
as observed at around $10$~GeV under realistic experimental
conditions. Lower energies contribute again through initial state
radiation (eq.~(\ref{eqisrint})). If we restrict the cutoff $z_0$ to a
value of $0.25$ which corresponds to a cut on the mass of the hadronic
system of around $5$~GeV, eq.~(\ref{eqRhad}) for $R_{\rm had}$ can be
applied also for $\sigma_0(z s')$ which appears in the integrand of
(\ref{eqisrint}). At the same time this cutoff excludes the region of
the broad charmonium resonances which have not been well explored up
today. A cutoff around $z_0 = 0.7$ reduces the initial state radiation
corrections further and eliminates contributions from
the lower energy range completely.
The vacuum polarization of the virtual photon has been discussed
before for the case of the $\tau$ lepton production and is identical
for the hadronic cross section.
\begin{table}[htb]
\caption{Relative contributions from the $\Upsilon$ resonances to
the hadronic cross section $\sigma(e^+ e^- \to {\rm hadrons})$
for the energies $E_1 = 10.52$~GeV and $E_2 = 9.98$~GeV and
the two different values of the cutoff $m_{\rm min} = 5$~GeV and 9~GeV.
Interference terms and radiative tails are listed separately.
Also given are the continuum contributions and the resulting
predictions for the total cross section.}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline
&$\Upsilon(1)$&$\Upsilon(2)$&$\Upsilon(3)$&
$\Upsilon(4)$&$\Upsilon(5)$&$\Upsilon(6)$\\ \hline\hline
\multicolumn{7}{|l|}{$E_1 = 10.52$ GeV, $m_{\rm min}=5$ GeV:} \\ \hline
Continuum: & \multicolumn{6}{c|}{$3.2228$ nb} \\ \hline
Interf.:
& $2.9\cdot 10^{-4}$ & $3.1\cdot 10^{-4}$ & $7.7\cdot 10^{-4}$ &
$-1.06\cdot 10^{-3}$ & $-3.1\cdot 10^{-4}$ & $-9.7\cdot 10^{-5}$ \\ \hline
$\sum$ interf.: & \multicolumn{6}{c|}{$-1.0\cdot 10^{-4}$} \\ \hline
Rad.~tails:
& $5.88\cdot 10^{-3}$ & $5.38\cdot 10^{-3}$ & $1.212\cdot 10^{-2}$ &
-- & -- & -- \\ \hline
$\sum$ rad. tails: & \multicolumn{6}{c|}{$2.343\cdot 10^{-2}$} \\ \hline
Total: & \multicolumn{6}{c|}{$3.2980$ nb} \\ \hline\hline
\multicolumn{7}{|l|}{$E_1 = 10.52$ GeV, $m_{\rm min}=9$ GeV:} \\ \hline
Continuum: & \multicolumn{6}{c|}{$2.8515$ nb} \\ \hline
Interf.:
& $4.0\cdot 10^{-4}$ & $3.7\cdot 10^{-4}$ & $8.9\cdot 10^{-4}$ &
$-1.19\cdot 10^{-3}$ & $-3.3\cdot 10^{-4}$ & $-1.1\cdot 10^{-4}$ \\ \hline
$\sum$ interf.: & \multicolumn{6}{c|}{$3.9\cdot 10^{-5}$} \\ \hline
Rad.~tails:
& $6.64\cdot 10^{-3}$ & $6.08\cdot 10^{-3}$ & $1.370\cdot 10^{-2}$ &
$5\cdot 10^{-5}$ & -- & -- \\ \hline
$\sum$ rad. tails: & \multicolumn{6}{c|}{$2.648\cdot 10^{-2}$} \\ \hline
Total: & \multicolumn{6}{c|}{$2.9272$ nb} \\ \hline\hline
\multicolumn{7}{|l|}{$E_2 = 9.98$ GeV, $m_{\rm min}=5$ GeV:} \\ \hline
Continuum: & \multicolumn{6}{c|}{$3.5568$ nb} \\ \hline
Interf.:
& $6.7\cdot 10^{-4}$ & $-3.58\cdot 10^{-3}$ & $-4.6\cdot 10^{-4}$ &
$-1.6\cdot 10^{-4}$ & $-1.6\cdot 10^{-4}$ & $-5.8\cdot 10^{-5}$ \\ \hline
$\sum$ interf.: & \multicolumn{6}{c|}{$-3.74\cdot 10^{-3}$} \\ \hline
Rad.~tails:
& $1.145\cdot 10^{-2}$ & $9\cdot 10^{-5}$ & -- & -- & -- & -- \\ \hline
$\sum$ rad. tails: & \multicolumn{6}{c|}{$1.154\cdot 10^{-2}$} \\ \hline
Total: & \multicolumn{6}{c|}{$3.5845$ nb} \\ \hline\hline
\multicolumn{7}{|l|}{$E_2 = 9.98$ GeV, $m_{\rm min}=9$ GeV:} \\ \hline
Continuum: & \multicolumn{6}{c|}{$3.0666$ nb} \\ \hline
Interf.:
& $8.8\cdot 10^{-4}$ & $-4.1\cdot 10^{-3}$ & $-5.0\cdot 10^{-4}$ &
$-1.7\cdot 10^{-4}$ & $-1.6\cdot 10^{-4}$ & $-6.1\cdot 10^{-5}$ \\ \hline
$\sum$ interf.: & \multicolumn{6}{c|}{$-4.14\cdot 10^{-3}$} \\ \hline
Rad.~tails:
& $1.328\cdot 10^{-2}$ & $1.04\cdot 10^{-4}$ & -- & -- & -- & -- \\ \hline
$\sum$ rad. tails: & \multicolumn{6}{c|}{$1.338\cdot 10^{-2}$} \\ \hline
Total: & \multicolumn{6}{c|}{$3.0949$ nb} \\ \hline
\end{tabular}
\end{center}
\label{table4}
\end{table}
\begin{figure}[htb]
\begin{center}
\leavevmode
\epsfxsize=12.cm
\epsffile[120 280 460 560]{haddiff.ps}
\vskip -3mm
\caption[]{\label{fig4} {\em Dependence of the hadronic cross section
$\sigma(e^+ e^- \to {\rm hadrons})$ on the cutoff in the minimal
invariant mass of the hadronic system, $m_{\rm min}$, for the two
energies $E_1 = 10.52$ GeV, $E_2 = 9.98$ GeV and $\alpha_s(M_Z) =
0.120$.}}
\end{center}
\end{figure}
The interference
between the $\Upsilon$ resonances and the virtual photon is completely
analogous to the leptonic case and leads to the same relative
corrections. However, the contributions from the radiative tails of
the Breit--Wigner amplitudes are distinctively different. Without
initial state radiation the tail from one resonance is given by
\begin{equation}
\sigma_{\rm had, R} =
\frac{12\,\pi\,\Gamma_e\,\Gamma_{\rm had}\,M^4}{s^3}
\frac{M^2}{(s-M^2)^2+\Gamma^2 M^2}\,.
\label{eqtailhad}
\end{equation}
For the $\Upsilon(4,\,5,\,6)$ resonances an energy dependent width
$\Gamma(s)$ has to be used in (\ref{eqtailhad}), which varies rapidly
in the region of interest. For $s$ below the $B{\overline B}$ threshold this
leads to a drastic suppression of these contributions.
In the narrow width approximation eq.~(\ref{eqtailhad}) implies
\begin{equation}
\sigma_{\rm had, R} \Big|_{\rm NW} =
\frac{12\,\pi^2\,\Gamma_e\,\Gamma_{\rm had}}{s}
\frac{M}{\Gamma}\,\delta(s-M^2) \,.
\label{eqtailhadnw}
\end{equation}
The contributions of interference terms and radiative tails from the
individual resonances to the hadronic cross section are listed in
Table~\ref{table4} for the energies $E_1 = 10.52$ GeV and $E_2 = 9.98$
GeV and $\alpha_s(M_Z^2) = 0.120$. Also given are the continuum
contribution and the sum. The cutoffs $m_{\rm min} = 5$~GeV and 9~GeV are
adopted. These correspond to
the minimal and maximal values recommended for a precision measurement
of $R_{\rm had}$. The relative strength of the interference terms is
identical for hadronic and leptonic final states. Again one observes
the accidental cancellation at $10.52$~GeV and the dominance of the
(negative) $\Upsilon(2)$ contribution at the energy $E_2 = 9.98$~GeV just
below this resonance. Compared to the leptonic cross section
(Table~\ref{table3}) the radiative tails are significantly more
prominent --- a consequence
of the relatively large direct, non--QED--mediated hadronic decay rates of
the $\Upsilon$ resonances. This leads to a positive correction of 2.3\%
and 1.2\% for $E_1$ and $E_2$, if $m_{\rm min} = 5$~GeV and to 2.7\%
and 1.3\% if $m_{\rm min} = 9$~GeV, respectively.
The difference in the strength of the $\Upsilon$
tails leads to an apparent variation of $R_{\rm had}$ around 1.3\%.
The cutoff dependence of the cross section is illustrated
in Fig.~\ref{fig4}.
The strong dependence of the cross section for tight cuts is again
clearly visible, suggesting a cut between about 5 and 9 GeV for $E_1$
and 5 and 8.5 GeV for $E_2$.
\vskip 1 cm
\noindent
{\Large 4. Uncertainties}\\
\noindent
From the comparison of different radiator functions for initial state
radiation the theoretical uncertainty from this source can be estimated
to be below one permille. The error induced through the present
simplified treatment of the hadronic vacuum polarization in the timelike
region is estimated around two permille and could easily be reduced even
further, if required. The combined theoretical uncertainty from these and
other effects is generously estimated below five permille. In addition
photons from initial state radiation might be included in the
invariant mass of the hadronic system or, conversely, photons from final
state radiation may escape the detection. This ${\cal O}(\alpha)$ effect
can only be evaluated for the concrete experimental analysis with the
help of a Monte Carlo simulation.
These theoretical uncertainties are significantly below the expected
experimental error of roughly two percent, which is dominated by
systematical uncertainties \cite{DaveBessonpriv}.
\vskip 1 cm
\noindent
{\Large 5. Summary and Conclusions}\\
\noindent
Precise predictions have been presented for the total cross section in
the energy region explored presently by the CLEO experiment and at
a future $B$ meson factory. The present sample of nearly one million
hadronic events allows for a small statistical error. These
measurements will determine the value for $\alpha_s$ under
particularly clean conditions similar to the $Z$ line shape
measurements but at a different energy.
When compared to the experimental results from $Z$ decays, a
determination of $R$ with a precision of 2.5\% would evidently
demonstrate the running of $\alpha_s$ between 90 and 10 GeV. A precision
of 0.3\% would be competitive with the $\alpha_s$ measurements from the
$Z$ line shape which are based on the combined results of all four LEP
experiments.
\vskip 1 cm
\noindent
{\Large Acknowledgements}\\
\noindent
We thank Fred Jegerlehner and Helmut Burkhardt for providing their
programs and Fred Jegerlehner for important discussions concerning
the hadronic vacuum polarization. The interest of Dave Besson in this
study was essential for its completion.
TT thanks the UK Particle Physics and Astronomy Research Council and the Royal
Society for support.
This work was supported by BMFT under Contract 057KA92P(0),
and INTAS under Contract INTAS-93-0744.
\def\app#1#2#3{{\it Act. Phys. Pol. }{\bf B #1} (#2) #3}
\def\apa#1#2#3{{\it Act. Phys. Austr.}{\bf #1} (#2) #3}
\defProc. LHC Workshop, CERN 90-10{Proc. LHC Workshop, CERN 90-10}
\def\npb#1#2#3{{\it Nucl. Phys. }{\bf B #1} (#2) #3}
\def\plb#1#2#3{{\it Phys. Lett. }{\bf B #1} (#2) #3}
\def\prd#1#2#3{{\it Phys. Rev. }{\bf D #1} (#2) #3}
\def\pR#1#2#3{{\it Phys. Rev. }{\bf #1} (#2) #3}
\def\prl#1#2#3{{\it Phys. Rev. Lett. }{\bf #1} (#2) #3}
\def\prc#1#2#3{{\it Phys. Reports }{\bf #1} (#2) #3}
\def\cpc#1#2#3{{\it Comp. Phys. Commun. }{\bf #1} (#2) #3}
\def\nim#1#2#3{{\it Nucl. Inst. Meth. }{\bf #1} (#2) #3}
\def\pr#1#2#3{{\it Phys. Reports }{\bf #1} (#2) #3}
\def\sovnp#1#2#3{{\it Sov. J. Nucl. Phys. }{\bf #1} (#2) #3}
\def\jl#1#2#3{{\it JETP Lett. }{\bf #1} (#2) #3}
\def\jet#1#2#3{{\it JETP Lett. }{\bf #1} (#2) #3}
\def\zpc#1#2#3{{\it Z. Phys. }{\bf C #1} (#2) #3}
\def\ptp#1#2#3{{\it Prog.~Theor.~Phys.~}{\bf #1} (#2) #3}
\def\nca#1#2#3{{\it Nouvo~Cim.~}{\bf #1A} (#2) #3}
| proofpile-arXiv_065-461 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{$D$-branes and the WZNW path integral.}
The standard WZNW action on a group manifold $R$
reads
\begin{equation} S(r)\equiv {1\over 4\pi}\int d\xi^+ d\xi^-\langle \partial_+ r~r^{-1},
\partial_- r~r^{-1}\rangle +{1\over 24\pi}
\int d^{-1}\langle dr~r^{-1},[dr~r^{-1},
dr~r^{-1}]\rangle .\end{equation}
Here $\xi^{\pm}$ are the standard lightcone variables on the world-sheet
\begin{equation} \xi^{\pm}\equiv{1\over 2}(\tau \pm \sigma),\quad \partial_{\pm}
\equiv \partial_{\tau}\pm\partial_{\sigma}\end{equation}
and $\langle.,.\rangle$ denotes a non-degenerate invariant bilinear form on the
Lie algebra ${\cal R}$ of $R$.
The second term in the WZNW action is commonly referred to as the
WZNW term and it provide the action
with the antisymmetric tensor part. It is well-known that
this antisymmetric tensor $B$ of the WZNW background
is not globally defined (for compact groups) because the
WZNW form $\Omega$ is a non-trivial cocycle in the third de Rham cohomology
$H^3(R)$
of the group manifold $R$. Inspite of this, the classical
WZNW theory is well defined for the case of closed strings.
The reason is simple: Consider an evolving loop
which sweeps out a cylindrical world-sheet $g(\sigma,\tau)$
on the group manifold.
The variational problem requires fixing
of the initial and final position of the loop and slightly varying
the position of
the cylinder between: $r(\sigma,\tau)\to r(\sigma,\tau)+\delta r(\sigma,\tau);
~\delta r\vert_{initial,final}=0$.
The antisymmetric tensor part of the variation of the
action can be thus written as
\begin{equation} \int (r+\delta r)^*B-\int r^*B = \oint dB=\oint
\Omega.\end{equation}
The integral $\oint$ is taken over the volume interpolating between
the world-sheets $r$ and $r+
\delta r$ and * means the pull-back of the map. We conclude that the
variation of the action does indeed depend
only on the WZNW three-from $\Omega$ and not on a choice of its potential $B$.
Note that the interpolating volume is given unambiguously because the
variation of the action is infinitesimal.
A well known additional topological problem may occur if we wish
to define a path integral for the WZNW theory of closed strings \cite{Wit}:
Consider a set of fixed loops in $R$ and all world-sheets interpolating
among them. We wish to evaluate the WZNW action $S$ of every world-sheet $s$,
form an expression $\exp{iS}$ and sum up it over all interpolating world-sheets
of arbitratry topology. Suppose
we choose some reference interpolating world-sheet $s_{ref}$ and calculate
its WZNW
action
$S_{ref}$ for some choice of the potential $B$. The action $S$ of any other
world-sheet $s$ can be computed in the same way. It is tempting to conclude
that the difference $S-S_{ref}$ does not depend on the choice of the potential
$B$. Indeed, by using the same argument as in the variational problem, we
easily see that the difference of the integral of $B$ over the
both world-sheets
is given solely in terms of the integral $\oint \Omega$ over the three-surface
which interpolates between the world-sheets\footnote{We shall alway assume that
the group manifold in question is simply connected. By Hurewicz isomorphism
and the fact the second homotopy group of any Lie group
vanishes we thus have that the second cohomology of the simply connected
group manifold
vanishes. This means that the interpolating three-surface always exists.}.
But now the two world-sheets
do not differ
only infinitesimaly! It therefore seems that the interpolating three-surface
is
not given unambiguously. The way to get out of the trouble lies in comparing
the quantity $S-S_{ref}$ for two non-homotopical three-surfaces interpolating
between $s$ and $s_{ref}$. This difference is obviously given in terms of the
integral $\oint \Omega$ over a three-cycle obtained by taking the difference of
(or the sum of oppositely oriented) non-homotopical
three-surfaces interpolating between
$s$ and $s_{ref}$. Fortunately, the WZNW three-form $\Omega$ is an integer-valued
cocycle \cite{Wit} in $H_3(R)$ hence it is enough to normalize action $S$
properly in order to ensure that the quantities $S-S_{ref}$ differ by
a term $2\pi k, k\in Z$
for any two interpolating three-surfaces. These $2\pi k$ terms do not
contribute to the path integral and, moreover, a dependence on the reference
surface
$s_{ref}$ results only in an unobservable change of the total phase
of the path integral. We finish this little review
by concluding that the WZNW path integral is well
defined for the case of the interacting closed strings.
Consider now a $D$-branes configuration in the group target $R$. By this we
simply mean that there are two given submanifolds $D_i$ and $D_f$ of $R$
and open strings propagate on $R$ in such a way that their end-points
$i$ and $f$ stick on the $D$-branes $D_i$ and $D_f$, respectively.
We define the WZNW theory for this $D$-branes configuration
by choosing
two-forms $\alpha_i$ and $\alpha_f$, living respectively on $D_i$ and $D_f$
such that
\begin{equation} d\alpha_{i(f)}= \Omega\vert_{D_i(D_f)} .\end{equation}
In words: the exterior derivative of $\alpha_{i(f)}$ has to be equal to the
restriction of the WZNW three form $\Omega$ to the $D$-brane $D_{i(f)}$.
The construction of the $WZNW$ theory based on the triplet $(\Omega,\alpha_i,\alpha_f)$
goes as follows:
Pick up an open string $r(\sigma,\tau)$
with the topology of an open strip. The variational
problem requires fixing of the initial and the final positions
of the string on the target. Consider now such a variation
$\delta r(\sigma,\tau),
\delta r(\sigma,\tau_{i,f})=0$.
The both original open strip and its variation form together a closed
strip (a `diadem'), whose edges lie on the opposite $D$-branes. We
can define the variation $\delta S_{WZNW}$ of the WZNW term of the
WZNW action
by choosing an interpolating
surface $\Sigma_{i(f)}\subset D_{i(f)}$ between the edges of the original and
the varied strip. This variation then reads
\begin{equation} \delta S_{WZNW}=
\oint \Omega -\int_{\Sigma_i} \alpha_i - \int_{\Sigma_f} \alpha_f,\end{equation}
where the $\oint \Omega$ is taken over the volume of the figure enclosed by
$\Sigma_i$, $\Sigma_f$, the original strip and its variation.
Note that this variation does not depend on the choice of the interpolating
surface $\Sigma_{i(f)}$ because $d\alpha=\Omega\vert_D$ and all infinitesimal
interpolating surfaces are mutually homotopic. Hence we conclude, that
the classical WZNW theory of open strings with end-points on the $D$-branes
is well defined in terms of the triplet $(\Omega,\alpha_i,\alpha_f)$.
The reader may wish to have a more concrete idea of how to compute
the WZNW action of a single strip.
For a particular choice of the potential $B$ ($dB=\Omega$)
the combination $\alpha-B$ on the $D$-brane is a closed form, hence, at least
locally, it has a potential $A$ on $D$. The WZNW action $S$ for an open string
configuration $r(\sigma,\tau)$ which sweeps out a two-surface $s$ in the
target $R$
and
respects the $D$-branes boundary conditions
can now be written as follows
\begin{equation} 4\pi S(r)=\int \langle \partial_+ r~r^{-1},
\partial_- r~r^{-1}\rangle + \int_s B +\int_{\delta s\cap D} A.
\end{equation}
Upon a change of
\begin{equation} B\to B+d\lambda ,\end{equation}
$A$ has to be replaced
by
\begin{equation} A-\lambda\vert_D . \end{equation}
We may intepret the $A$-term of the action (6) as if there were
equal and opposite charges on the end-points of the string
which feel the electromagnetic fields $A_i$ and $A_f$ on the $D$-branes.
This interpretation does not have an invariant meaning, however, because
of the `gauge invariance' (7) and (8). Moreover it holds
only locally. We stress that the {\it global}
invariant
description of the WZNW model for $D$-branes configuration is given
in terms of the triplet $(\Omega,\alpha_i,\alpha_f)$. We remark that in general
there is no natural {\it closed} two-form living on the $D$-branes. This
is true only in the case if the restriction of the WZNW three-form $\Omega$
on the $D$-brane vanishes. Note also that if the $D$-brane is as many
dimensional as the whole group
target
$R$ is, then the form $\alpha$ is nothing but some concrete choice of the
potential
$B$ which, however, may be different for the different end-points
of the string.
At the presence of the $D$-branes and open strings, the discussion of the
string path integral is more involved as before.
The group manifold will be always taken as
simply connected and, for a while,
we consider the case where also the $D$-branes are connected and
simply connected.
Now draw a general string
diagram respecting the $D$-branes configurations. It is an interpolating
world-sheet between a set of fixed open segments with end-points
located on the $D$-branes
and a fixed set of loops on the target $R$. Much as before, we can choose
some reference interpolating world-sheet $s_{ref}$ and calculate
its WZNW part of the
action $S_{ref}$ for some choice of $B$ and $A$ according to the
formula (6). Now we can take any other interpolating world-sheet $s$ and
calculate
its action $S$ in the same way. As in the
case of the variational principle, the quantity $S-S_{ref}$ does not depend
on the particular choice of $B$ and $A$ but only on the invariant
globally defined triplet $(\Omega,\alpha_i,\alpha_f)$. The reason for this is the
following: the union of the intersections $(\partial s_{ref}\cap D_{i(f)})
\cup (\partial s\cap D_{i(f)})$ is a contractible cycle in $D_{i(f)}$,
hence it is a boundary of
some two-surface $\Sigma_{i(f)}$. Now the union $s\cup s_{ref}\cup\Sigma_i\cup
\Sigma_f$ is a two-boundary of some interpolating three-surface
in the group manifold, because the second
cohomology of the group manifold vanishes by assumption. Then
the antisymmetric tensor (the WZNW term) part of $S-S_{ref}$
is defined by (5) where $\oint$ is taken over the interpolating
three-surface.
There occurs the same
problem as for the closed strings, namely, the interpolating three-surfaces
between $s$ and $s_{ref}$ do not have to be homotopically
equivalent. This means that the quantity $S-S_{ref}$ may depend on the
homotopy of the chosen interpolating three-surface. But if the ambiguity
in $S-S_{ref}$ is only of the form $2\pi k, k\in Z$ then the
term $\exp{i(S-S_{ref})}$ is unambiguous and the path integral
is well defined.
It is not difficult to find a cohomological formulation
of the condition of the integer-valued
ambiguity. All what we need is the notion of the
relative singular homology $H_*(R,D_i\cup D_f)$
of the manifold $R$ with respect to its submanifolds
$D_i$ and $D_f$ (with real coefficients).
The relative chains are the elements of the vector
space of the standard
chains in $R$ factorized by its subspace of all chains lying in $D_i\cup D_f$.
The operation of taking the boundary is the standard one. The corresponding
homology is the relative singular homology $H_*(R,D_i\cup D_f)$.
The triplet $(\Omega,\alpha_i,\alpha_f)$ can act on a relative cycle $\gamma$ by
the following prescription
\begin{equation} \langle (\Omega,\alpha_i,\alpha_f),\gamma\rangle\equiv \int_{\gamma}\Omega
-\int_{D_i\cap\partial\gamma}\alpha_i -\int_{D_f\cap\partial\gamma}\alpha_f.\end{equation}
If the cycle $\gamma$ is itself a boundary then the pairing vanishes
because $\Omega$ is closed. Hence
our triplet $(\Omega,\alpha_1,\alpha_f)$ is an element (cocycle)
of the relative singular cohomology $H^*(R,D_i\cup D_f)$
because it vanishes
on the boundary of any relative chain.
Now we may conclude that if the cocycle $(\Omega,\alpha_1,\alpha_2)$
is integer-valued\footnote{The precise statement is as follows: The
cocycle
$(\Omega,\alpha_1,\alpha_2)$ is integer-valued, if it lies
in the image of the natural map from
the singular cohomology with integer coefficients to the singular
cohomology with real coefficients.} the WZNW path integral is well-defined.
Indeed, if we choose two non-homotopical three-surfaces interpolating
between the world-sheets $s$ and $s_{ref}$ their oriented sum is
a closed cycle in the relative singular homology and its pairing (9)
with the triplet $(\Omega,\alpha_1,\alpha_2)$ is integer-valued.
It turns out that we can extend our discussion to the case of connected
but not necessarily simply connected $D$-branes.
The main problem to be addressed is the fact that now the
union of the intersections
$(\partial s_{ref}\cap D_{i(f)})
\cup (\partial s\cap D_{i(f)})$ is not necessarily a contractible cycle in
$D_{i(f)}$
(which means that $s\cup s_{ref}$ is a relative two-cycle
but not a relative two-boundary).
Thus the two-surface $\Sigma_{i(f)}$ does not have to exist and we cannot
in general use the formula (5) in order to determine $\exp{i(S-S_{ref})}$.
It may seem that we may take some reference world-sheet
for each homotopy class of the one-chain $\partial s\cap D_{i(f)}$ and assign
it an arbitrary reference phase. But there is still a consistency condition
that under summing of the relative two-cycles (unions of $s$ and $s_{ref}$)
the phases $\exp{iS}$ should be additive!
Recall that we can
unambiguously assign the $\exp{iS}$ to every relative two-boundary
in such a way that this mapping is homomorphism $f$ from the group $B$ of
relative two-boundaries (with integer coefficients) into the group $U$
of complex units (phases). The consistency condition means
that there should exist an extension $\tilde f: Z\to U$
of this
homomorphism defined on the group $Z$ of all relative two-cycles. We now
prove that such
an extension always exists because $U$ is the divisible group (this means
that the equation $nx=a, a\in U, n\in {\bf N}$ has always a solution $x\in U$).
Consider the group $H_f=Z+U/\{b-f(b),b\in B\}$. We have an exact sequence
\begin{equation} 0\to U\to H_f \to H\to 0,\end{equation}
where $H\equiv H_2(R, D_i\cup D_f)=Z/B$ and all homomorphisms are naturally
defined. Suppose now that we do have an extension $\tilde f:Z\to U$ of the map
$f:B\to U$. Such an extension enables us to write
\begin{equation} H_f= H+U.\end{equation}
In words: $H_f$ is a direct sum of $H$ and $U$. Indeed, for
$z+c, z\in Z,c\in U$
we have
\begin{equation} z+c = (z- \tilde f(z)) +(0 +c+\tilde f(z)).\end{equation}
Evidently, the first term on the right hand side is from $Z$ and the second
from $U$. The decomposition (12) is consistent with the factorization
by $\{b-f(b), b\in B\}$ because $\tilde f$ is the homomorphism.
The converse is also true: if we can write $H_f$ as the direct sum
$H+U$ then there exists an extension $\tilde f:Z\to U$ which is a homomorphism.
Indeed, consider $z\in Z$ and embed it naturally into $H_f$ i.e.
$z\to z+0\in H_f$. $z+0$ can be decomposed as $y+g, y\in H, g\in U$
by assumption, hence we obtain a natural homomorphism from $Z$ into $U$:
$z\to g$. This homomorphism is the extension of $f$ which we look for.
Summarizing, if we prove that $H_f$ is the direct sum of $H$ and $U$,
we are guaranteed that the extension $\tilde f:Z\to U$ always exists. But it is
easy to prove this, by using the well-known result from the homological
algebra that every extension of an (Abelian) group G
by a divisible group $X$ is necessarily
the direct sum of $G$ and $X$. In our case, we know from
the exact sequence (10) that $H_f$ is the extension of $H$ by $U$. Therefore
$H_f=H+U$, what was to be proved.
\noindent {\it Notes}:
\noindent 1.
We have a certain freedom in writing $H_f$ as a direct sum of $H$ and $U$
which is
described by the group of homomorphisms $Hom(H,U)$. The easiest way to see
it is by noting that if we have an extension $\tilde f:Z\to U$ it can be modified
by adding to it any homomorphism which vanishes on $b\in B$. Any
such homomorphism is obviously from $Hom(H,U)$. The modified $\tilde f$ then
gives another partition of $H_f$ into the direct sum of $H$ and $U$.
\noindent 2. It may be instructive to relate the group $H$ of the relative
two-cycles with the fundamental groups $\pi_1$ of the $D$-branes.
We have a natural exact sequence
\begin{equation} 0=H_2(R)\to H_2(R,D_i\cup D_f)\to H_1(D_i)+H_1( D_f)\to 0=H_1(R).\end{equation}
Hence
\begin{equation} H=H_1(D_i) +H_1(D_f)\end{equation}
and
\begin{equation} H_1(D_{i(f)})=\pi_1(D_{i(f)})/[\pi_1(D_{i(f)}),\pi_1(D_{i(f)})].\end{equation}
The last equality is the Hurewicz isomorphism which holds due to the assumption
that the $D$-branes are connected.
\section{PL symmetries of WZNW models}
For the description of the PL $T$-duality, we need the
crucial concept
of the Drinfeld double, which is simply a Lie group $D$ such that
its Lie algebra ${\cal D}$ (viewed as a vector space)
can be decomposed as the direct sum of two subalgebras, ${\cal G}$ and $\ti{\cal G}$,
maximally isotropic with
respect to a non-degenerate invariant bilinear form on ${\cal D}$ \cite{D}.
It is often convenient to identify the dual linear space to ${\cal G}$ ($\ti{\cal G}$)
with $\ti{\cal G}$ (${\cal G}$) via this bilinear form.
From the space-time point of view, we have identified
the targets of the mutually
dual $\sigma$-models
with the cosets $D/G$ and $D/\tilde G$ \cite{KS6}.
Here $D$ denotes the Drinfeld double,
and $G$ and $\tilde G$ two its mutually dual isotropic subgroups. In the special
case when the decomposition $D=\tilde G G= G\tilde G$ holds globally, the
corresponding
cosets turn out to be the group manifolds $\tilde G$ and $G$, respectively
\cite{KS2}.
The actions of mutually dual $\sigma$-models
are encoded in a choice of an $n$-dimensional
linear subspace ${\cal R}$ of the $2n$-dimensional
Lie algebra ${\cal D}$ of the double $D$ which is transversal to both ${\cal G}$ and
$\ti{\cal G}$. The $\sigma$-model actions
on the targets $D/G$ and $D/\tilde G$
have a similar structure; indeed, on $D/G$ we have \cite{KS6}
\begin{equation} S={1\over 2}I(f)-{1\over 4\pi}\int d\xi^+ d\xi^- \langle
\partial_+ f~f^{-1},R_-^a\rangle (M_-^{-1})_{ab}\langle f^{-1}\partial_- f,T^b\rangle,\end{equation}
where $f\in D$ is some local section of the $D/G$ fibration which
parametrizes the points of the coset. Recall \cite{KS6} that
\begin{equation} M_{\pm}^{ab}\equiv \langle T^a ,f^{-1}R_{\pm}^b f\rangle\end{equation}
and $R_-^a$ ($R_+^a$) are vectors of an orthonormal basis of ${\cal R}$
(${\cal R}^{\perp}$):
\begin{equation} \langle R_{\pm}^a,R_{\pm}^b\rangle= \pm\delta^{ab},\qquad \langle R_+^a,R_-^b\rangle=0.\end{equation}
The action of the dual $\sigma$-model on the coset $D/\tilde G$ has
the same form; just the generators $T^a$ of ${\cal G}$ are replaced by the
generators
$\tilde T_a$ of $\ti{\cal G}$ and $f$ will parametrize $D/\tilde G$ instead of $D/G$.
We have referred to the $\sigma$-models of the form (16) as
those having a PL symmetry \cite{KS6}. There is an important feature
of such models, namely, their field equations
can be written as the zero curvature condition valued in the algebra
${\cal G}$. Indeed,
\begin{equation} d\lambda-\lambda^2=0,\end{equation}
where
\begin{equation} \lambda =\lambda_+ d\xi^+ +\lambda_- d\xi^-\end{equation}
and
\begin{equation} \lambda_{\pm}=-\langle \partial_{\pm} f~f^{-1},R_{\mp}^a\rangle
(M_{\mp}^{-1})_{ab}T^b.\end{equation}
So far we have been reviewing the results of \cite{KS6}; now a new
observation comes: If the subspace ${\cal R}$ is itself a Lie algebra
of a compact subgroup $R$ of the double $D$ then
the model (16) is essentially the WZNW model on the target $R$ for the both
choices $D/G$ and $D/\tilde G$!
The argument goes in two steps:
\noindent 1. ${\cal R}$
can be transported by the right action to the
tangent space of every point of the double.
Because ${\cal R}$ is the subalgebra, the distribution of the planes ${\cal R}$
in the tangent bundle of the double is integrable and it foliates
the double into fibration with fibres $R$ and basis $R\backslash D$.
Since ${\cal R}$ is transversal
to the both ${\cal G}$ and $\ti{\cal G}$ (which means that it intersects ${\cal G}$ and
$\ti{\cal G}$ only in $O$) , any fiber of the $R$ fibration
either intersects the fiber $G$ (or $\tilde G$) in some finite subgroup
$R\cap G$ of
$R$
or does not intersect it at all. The latter cannot be true, however,
if the group $R$ is compact. Indeed, $R$ acts on $D/G$ by the left action.
The $R$ orbit of the element of $D/G$ which has the unit
element of $D$ on its fiber is open. Since $R$ is compact this orbit
must be also closed which for connected doubles
imply that this orbit is the whole $D/G$. In other words,
there always exists an intersection of $R$ and $G$.
The argument for $D/\tilde G$ is the same.
If the finite subgroups $R\cap G$ and $R\cap \tilde G$ have only one element
for
both fibers $G$ and $\tilde G$, respectively, it si not dificult to see
that the both cosets $D/G$ and $D/\tilde G$ can be globally
identified with $R$. In general, the cosets $D/G$ and $D/\tilde G$ can be
identified with the discrete cosets $R/R\cap G$ and $R/R\cap \tilde G$,
respectively.
\noindent 2. For simplicity, consider only the case when $R$ can be directly
identified with $D/G$ and $D/\tilde G$. In this case, we can choose
the field $f(\sigma,\tau)$ in (16) to have values in $R$. Note that
we can choose the basis $R_-^a$ dependent on $f$ in such a way that
the combinations $f^{-1}R_-^a f$ are $f$ independent. Then we can choose
the basis $T^a$ in such a way that $M_-(f)$ is the identity matrix.
We have
\begin{equation} \langle \partial_+ f ~f^{-1}, R_-^a\rangle =\langle f^{-1} \partial_+ f, f^{-1}R_-^a
f\rangle\equiv (f^{-1} \partial_+ f)^a\end{equation}
and
\begin{equation} \langle f^{-1}\partial_- f , T^a\rangle=
\langle f^{-1}\partial_- f,f^{-1}R_-^c f\rangle M_-^{ca}=(f^{-1}\partial_- f)^a,\end{equation}
because $M_-$ is the identity matrix. Putting (16),(22) and (23) together,
we obtain
\begin{equation} S={1\over 2}I(f)-{1\over 4\pi}\int d\xi^+ d\xi^-
(\partial_+ f ~f^{-1})^a \delta_{ab}(\partial_- f ~f^{-1})^b=-{1\over 2}
I(f^{-1}
).\end{equation}
We conclude, that the mutually dual $\sigma$-models on the cosets $D/G$ and
$D/\tilde
G$ are the same, being equal to the WZNW model on $R$. In general,
$D/G$ ($D/\tilde G$) model is WZNW model on the target $R/R\cap G$
($R/R\cap \tilde G$).
\noindent {\it Notes}:
\noindent 1. The fact that the both models $D/G$ and $D/\tilde G$ may be
identical does not mean at all that the duality transformation is trivial.
In fact, the PL $T$-duality always implies an existence of
a non-trivial non-local
transformation on the phase space of the $WZNW$ model. We shall explicitly
describe this transformation in the next section.
\noindent 2. It often happens (cf. section 4) that a compact group
$R$ can be embedded in many inequivalent ways into various Drinfeld
doubles in such a way that the both cosets $D/G$ and $D/\tilde G$
can be identified with $R$. In this case we have the abundance
of the Poisson-Lie symmetries of the same WZNW model on the group manifold
$R$, each of them corresponding to the double into which $R$ is embedded.
\section{$D$-branes in WZNW models}
\subsection{General discussion}
For the further discussion of the $D$-branes, it is convenient to
recall \cite{KS6} the common `roof' of the both models described by (16).
They can be derived form the first order Hamiltonian action for
field configurations $l(\sigma,\tau)\in D$:
$$ S[l(\tau,\sigma)]= $$
\begin{equation} ={1\over 8\pi}\int \biggl\{\langle \partial_{\sigma} l~l^{-1},\partial_{\tau} l~l^{-1}\rangle+
{1\over 6}d^{-1}\langle dl~l^{-1},[dl~l^{-1},
dl~l^{-1}]\rangle -\langle \partial_{\sigma} l l^{-1},A\partial_{\sigma} l l^{-1}\rangle \biggl\}.\end{equation}
Here $A$ is a linear idempotent self-adjoint map from the Lie algebra
${\cal D}$ of the double into itself. It has two equally degenerated
eigenvalues $+1$ and $-1$, and the corresponding eigenspaces are just
${\cal R}^{\perp}$ and ${\cal R}$ respectively.
As it stands, the action (25)
is well defined only for the periodic functions of $\sigma$ because of the
WZNW term. This restriction corresponds to the case of closed strings
\cite{KS6} . The $\sigma$-model actions (16) are obtained from
the duality invariant first order action (25) as follows:
Consider the right coset
$D/G$ and parametrize it by the elements $f$ of $D$ \footnote{If there
exists no global section of this fibration, we can choose several
local sections covering the whole base space $D/G$.}.
With this parametrization of $D/G$ we may parametrize the surface
$l(\tau,\sigma)$ in the double as follows
\begin{equation} l(\tau,\sigma)= f(\tau,\sigma)g(\tau,\sigma),\quad g\in G.\end{equation}
The action $S$ then becomes
$$ S(f,\Lambda\equiv \partial_{\sigma} g g^{-1})={1\over 2}I(f) -{1\over 2\pi}
\int d\xi^+ d\xi^- \biggl\{\big\langle \Lambda -
{1\over 2} f^{-1}\cdot- f , \Lambda -{1\over 2}f^{-1}\cdot- f \big\rangle$$
\begin{equation} +\langle f\Lambda f^{-1} +\partial_{\sigma} f f^{-1}, R_-^a\rangle\langle R_-^a , f\Lambda f^{-1}
+\partial_{\sigma} f f^{-1}\rangle\biggl\}.\end{equation}
Now it is easy to eliminate $\Lambda$ from the action (27) and finish with
the $\sigma$-model action (16). In the case of the coset $D/\tilde G$, the procedure
is exactly analoguous.
Consider the case of open strings for a generic double $D$ with vanishing
second cohomology. In our
previous paper on the subject \cite{KS4}, we have studied only
the perfect doubles (cf. footnote 3) nevertheless we can easily generalize
the construction.
Let $F$ be a simply connected subgroup of the double $D$ whose Lie
algebra ${\cal F}$ is isotropic with respect to the bilinear form on ${\cal D}$.
This subgroup, as a manifold, can be shifted by the right action of some
element $d\in D$ (note that all non-equivalent shifts are parametrized by
the coset $F\backslash D$). We declare that the manifolds $F\hookrightarrow
D$ and $Fd\hookrightarrow D$ are $D$-branes in the double $D$.
Consider now oriented open strings in $D$ with the initial end-points
on $F$ and the final end-points on $Fd$. Their dynamics in the bulk is
governed by the action (25) which contains the WZNW term. As we have learnt
in the previous section such an action is well-defined provided
we choose some two-forms on the $D$-branes such that the exterior derivative
of them is equal to the restriction of the $WZNW$ three-form on the
$D$-branes. In our present case, this restriction of the WZNW form
vanishes in either of our $D$-branes because
$F$ and $Fd$ are the isotropic surfaces in $D$.
Thus we have to choose some closed two forms on $F$ and $Fd$; we choose them
to vanish identically. We summarize that our open string dynamics
is fully defined by the action (25), the $D$-branes boundary conditions
and the vanishing two-forms on the $D$-branes.
Much as in the closed string case, we can derive the open string $\sigma$-model
dynamics on the cosets $D/G$ and $D/\tilde G$ from (25) and the $D$-branes
data on the double; for concreteness let us consider the coset $D/G$:
As we have learnt in section 2, the WZNW model for open strings
is fully defined if we manage to compute the WZNW action
of the `diadem'. Recall that the diadem is composed of two evolving
open string world-sheets which are glued together at some initial
and final times. The edges of the diadem , swept by the end-points
of the open strings, lie in their corresponding $D$-branes.
Consider now the diadem in the double. We can choose some two-surface
$\Sigma$ ($\Sigma_d$) in the $D$-brane $F$ ($Fd$) whose boundary is just the
edge
of the diadem lying in $F$ ($Fd$). The diadem together with the surfaces
$\Sigma$ and $\Sigma_d$ form a boundary of some three-dimensional domain
$\gamma$.
We may write the action $S$ of the model (25) as
\begin{equation} S=S_0+S_{WZNW},\end{equation}
where $S_{WZNW}$ contains solely the term with the WZNW three-form $c$ on $D$.
Hence, the action of the diadem can be written as\footnote{Note that we
have included the factor $1/6$ from (25) in the definition of $c$.}
\begin{equation} S=S_0 + {1\over 8\pi}\int_{\gamma}c.\end{equation}
Again,
consider the parametrization of
$D/G$ by the elements $f$ of $D$.
A surface
$l(\tau,\sigma)$ in the double (respecting the $D$-branes boundary conditions),
can be written as
\begin{equation} l(\tau,\sigma)= f(\tau,\sigma)g(\tau,\sigma),\quad g\in G.\end{equation}
The
decomposition (30) induces two maps from $D$ into $D$: $f(l)=f$
and $g(l)=g$. Consider now the Polyakov-Wiegmann (PW) formula \cite{PW}
\begin{equation} (fg)^*c=f^*c +g^*c -d\langle f^*(l^{-1}dl) \stackrel {\wedge}{,}g^*(dl ~l^{-1})
\rangle,\end{equation}
where, as usual, $*$ denotes the pull-back of the forms
under the mappings to the group manifold $D$.
By using the PW formula, we can rewrite (29) as
\begin{equation} S=S_0(fg) +{1\over 8\pi}\int_{\gamma} f^*c -{1\over 8\pi}\int_{diad\cup
\Sigma
\cup \Sigma_d}\langle f^*(l^{-1}dl)\stackrel {\wedge}{,} g^*(dl~l^{-1})\rangle.\end{equation}
Note that $g^*c$ vanishes because of the isotropy of $G$. The action $S$ now
becomes
$$ S(f,\Lambda\equiv \partial_{\sigma} g g^{-1})={1\over 2\pi}\int_{diad}\biggl\{
{1\over 4}\langle \partial_+ f~f^{-1},\partial_- f~f^{-1}\rangle$$
$$ -\big\langle \Lambda -
{1\over 2} f^{-1}\cdot- f , \Lambda -{1\over 2}f^{-1}\cdot- f \big\rangle
+\langle f\Lambda f^{-1} +\partial_{\sigma} f f^{-1}, R_-^a\rangle\langle R_-^a , f\Lambda f^{-1}
+\partial_{\sigma} f f^{-1}\rangle\biggl\}$$
\begin{equation} +{1\over 8\pi}\int_{\gamma}f^*c
-{1\over 8\pi}\int_{\Sigma\cup \Sigma_d}\langle f^*(l^{-1} dl)\stackrel {\wedge}{,}
g^*(dl~l^{-1}
)\rangle.\end{equation}
Of course, this is a similar expression as before (cf. (27)). However,
the field $f$ respects different boundary conditions. A configuration
$f$ is an open string configuration; its end-points stick on $D$-branes
$D_i$ and $D_f$ in $D/G$ which are obviously obtained just by projecting the
$D$-branes
$F$ and $Fd$ from the double D into the basis $D/G$ parametrized by the
section $f$.
Now we have to realize that upon varying $\Lambda$
the last term in (33) vanishes! This follows from the
isotropy of $F$, $Fd$ and $G$. Indeed, if we have $fg\in F$ ($fg\in Fd$)
and vary $g\to g\delta g$
at fixed $f$ in such a way that $fg\delta g\in F$ ($fg\delta g \in Fd$),
we observe that the last term in (33) does not change\footnote{It is easy
to see that $\delta g\in F\cap G$ ($\delta g\in F\cap dGd^{-1}$).}.
Hence we can eliminate the field $\Lambda$ from (33)
in the same way as from (27). The result is
$$ S={1\over 8\pi}\int_{diad}
\biggl\{\langle \partial_+ f~f^{-1},\partial_- f~f^{-1}\rangle -2 \langle
\partial_+ f~f^{-1},R_-^a\rangle (M_-^{-1})_{ab}\langle f^{-1}\partial_- f,
T^b\rangle\biggl\}$$
$$ +{1\over 8\pi}\int_{\gamma}f^*c
-{1\over 8\pi}\int_{\Sigma\cup \Sigma_d}\langle f^*(l^{-1}dl)\stackrel {\wedge}{,}
g^*(dl~l^{-1})
\rangle.$$
Consider again the special situation in which the subspace ${\cal R}\equiv
{\rm Span} R_-$
is the Lie algebra of the compact group $R$, moreover, $R$ can be directly
identified with $D/G$ and $D/\tilde G$. Recall, that upon transporting
${\cal R}$ by the right action everywhere onto the double, we get the
fibration of $D$ with the fibers $R$ and the basis $R\backslash D$.
With some abuse of the notation, the fiber crossing the unit element
of the double we shall also denote as $R$.
We choose the parametrization of the double as follows
\begin{equation} l=rg,\quad r\in R,\quad g\in G.\end{equation}
This parametrization holds for every element $l$ of the double
and is unique by the assumption. Note that the restriction of the
WZNW three-form $c$ gives just the WZNW three-form $c_R$ on $R$.
It is easy to see that the $D$-branes $D_i$ and $D_f$ in $R$, being the
projections
of $F$ and $Fd$ to $R$, can be identified with the cosets
$F/F\cap G$ and $F/F\cap dGd^{-1}$ respectively. On the other hand we have
just seen (cf. footnote 9) that the variation $\delta g\in F\cap G$
($\delta g
\in F\cap dGd^{-1}$) leaves intact the two-form
$\omega \equiv (1/8\pi)\langle r^*(l^{-1}dl)\stackrel {\wedge}{,} g^*(dl~l^{-1}
)\rangle$ on $F$ ($Fd$). This means that this two-form is a pull-back of
some two-form $\alpha_i$ ($\alpha_f$) from the $D$-brane $D_i$ ($D_f$).
Of course, the notation is not accidental; the two-forms $\alpha_{i(f)}$ are
precisely those appearing in (5).
It is not difficult to find an explicit
expression for $\alpha_{i(f)}$. For this, consider a map $k_i$ ($k_f$)
from $D_i$ ($D_f$) into $G$ such that
\begin{equation} rk_{i(f)}(r) \in F(Fd), \quad r\in D_{i(f)}.\end{equation}
In general, the mapping $k_i$ ($k_f$) is not
defined unambiguously but it locally always exists since $D_i$ ($D_f$)
is just the projection of $F$ ($Fd$) on $R$.
Because two-form $\omega$ on $F$ ($Fd$) is invariant
under the variations from $F\cap G$ ($F\cap dGd^{-1}$) we can
locally\footnote{The two-form $\alpha_{i(f)}$ is defined {\it globally}
on $D_{i(f)}$ only the explicit expression for it in terms of $k_{i(f)}$
may, in general, be written only locally.} write
\begin{equation} \langle r^*(l^{-1}dl)\stackrel {\wedge}{,} g^*(dl~l^{-1})\rangle\vert_{F(Fd)}=
r^*\langle dr ~r^{-1}\stackrel {\wedge}{,} k_{i(f)}(r)^{-1}dk_{i(f)}(r)\rangle.\end{equation}
In other
words, (36) is true independently of the choice of the map $k_{i(f)}$.
Thus in our special situation, the action of the diadem can be written
as
$$ S=-{1\over 8\pi}\int_{diad}
\langle \partial_+ r~r^{-1},\partial_- r~r^{-1}\rangle +
{1\over 8\pi}\int_{r(\gamma)}c_R $$
\begin{equation} -{1\over 8\pi}\int_{D_i}\langle dr ~r^{-1}
\stackrel {\wedge}{,} k_i(r)^{-1}dk_i(r)\rangle
-{1\over 8\pi}\int_{D_f}\langle dr ~r^{-1}
\stackrel {\wedge}{,} k_f(r)^{-1}dk_i(r)\rangle.\end{equation}
We can read $\alpha_{i(f)}$ off directly from (37):
\begin{equation} \alpha_{i(f)}= {1\over 8\pi}\langle dr ~r^{-1}
\stackrel {\wedge}{,} k_{i(f)}(r)^{-1}dk_{i(f)}(r)\rangle.\end{equation}
It remains
to prove that
\begin{equation} d\alpha_{i(f)}={1\over 8\pi} c_R\vert_{D_{i(f)}}.\end{equation}
This is easy: take the PW formula (31) and restrict all forms
in it on the $D$-brane $F$ ($Fd$) in the double. Then the form $c$ vanishes
by the isotropy of $F$ ($Fd$). Hence
\begin{equation} r^*c\vert_{F(Fd)}=d\langle r^*(l^{-1}dl)\stackrel {\wedge}{,} g^*(dl~l^{-1})
\rangle\vert_{F(Fd)}=r^*d\langle dr ~r^{-1}\stackrel {\wedge}{,}
k_{i(f)}(r)^{-1}dk_{i(f)}(r)\rangle,\end{equation}
where the last equality follows from (36). Thus,
upon removing the pull-back map
$r^*$, we conclude that
\begin{equation} {1\over 8\pi}c_R\vert_{D_{i(f)}}={1\over 8\pi}d\langle r^{-1} dr
\stackrel {\wedge}{,}
dk_{i(f)}(r)k_{i(f)}(r)^{-1}\rangle=
d\alpha_{i(f)}.\end{equation}
\noindent {\it Remarks}:
\noindent 1. The model (37) has the `wrong' sign in front of its
first
term. Upon the change of variables $r\to r^{-1}$ it gives the
standard WZNW model on the group manifold $R$ (cf. (1)).
The $D$-branes $D_{i(f)}$
and the two
forms $\alpha_{i(f)}$ on them have to be transformed correspondingly.
\noindent 2. The geometry of the dual $D$-branes in $D/\tilde G$ is obtained
in the same way as in the case $D/G$; it is enough to replace
everywhere $G$ by $\tilde G$.
\noindent 3. We should mention that the Kiritsis-Obers duality \cite{KO} fits
in our formalism. The double is the direct product of a compact
group $R$ with itself and the invariant bilinear form
in the direct sum of the Lie algebras ${\cal R}+{\cal R}$ is the difference
between the Killing-Cartan forms on each algebra. Hence, the diagonal
embedding of $R$ in $R\times R$ is isotropic. So it is the
embedding in which second copy of ${\cal R}$ is twisted by some outer automorphism.
The resulting duality is a $D$-branes $D$-branes duality, i.e. the
$D$-branes have never the dimension of the group manifold.
\subsection{The classical solvability}
We wish to find the complete solution of the field equations
of the model (25) submitted to the $D$-branes boundary conditions.
It is not difficult to do that. The bulk equations following from (25) read
\begin{equation} \langle \partial_{\pm}l ~l^{-1}, {\cal R}_{\mp}\rangle=0.\end{equation}
We already know that after integrating away $g$ from the
decomposition (34) we get the WZNW model
on $R$, hence, the solution $l$ of (25) must look
like
\begin{equation} l(\sigma,\tau)=r_-(\xi^-)r_+(\xi^+)g(\xi^+).\end{equation}
The first two multiplicative terms on the right-hand-side follow from
the known bulk solution of the WZNW model on $R$ and the fact that
$g$ is only a function of $\xi^+$ follows from Eqs. (21).
Putting
\begin{equation} h(\xi^+)\equiv r_+(\xi^+)g(\xi^+)\end{equation}
and inserting $l=r_-(\xi^-)h(\xi^+)$ into (37),
we obtain
\begin{equation} \partial_+h ~h^{-1}\in {\cal R}_+\equiv {\cal R}^{\perp}.\end{equation}
Here we have used the fact that ${\cal R}_-(\equiv {\cal R})$ is
the Lie algebra of $R$. We conclude that every bulk solution of
(25) look like
\begin{equation} l=r_-(\xi^-)h(\xi^+), \qquad \partial_+h~h^{-1}\in {\cal R}_+.\end{equation}
It is important to note that ${\cal R}_+\equiv{\cal R}^{\perp}$ does not have to be a
Lie subalgebra
of ${\cal D}$; in general it is just a linear subspace of ${\cal D}$.
Now we can take into account the effect of the boundary conditions.
Recall that the initial point of the open string ($\sigma=0$) should
stick on the $D$-brane $F$ in the double and the final point ($\sigma=\pi$)
on the $D$-brane $Fd$; $d$ is a fixed element of the double $D$.
These two conditions can be rewritten as follows
\begin{equation} r_-(\tau)h(\tau)=f_i(\tau),\qquad r_-(\tau-\pi)h(\tau)=f_f(\tau)d,\end{equation}
where $f_i$ and $f_f$ are some functions with values in the group $F$.
It follows that
\begin{equation} h^{-1}(\tau-\pi)h(\tau)=f_i^{-1}(\tau-\pi)f_f(\tau)d.\end{equation}
By differentiating Eq. (48) with respect to $\tau$ we obtain
$$ -dh(\tau-\pi)h^{-1}(\tau-\pi)+dh(\tau)h^{-1}(\tau)=$$
\begin{equation} =h(\tau-\pi)[-f_i^{-1}(\tau-\pi)df_i(\tau-\pi) +f_i^{-1}(\tau-\pi)
df_f(\tau)f_f^{-1}(\tau)f_i(\tau-\pi)]h^{-1}(\tau-\pi).\end{equation}
Now we can bracket (49) with ${\cal R}$ which gives
\begin{equation} df_f(\tau)f_f^{-1}(\tau)-df_i(\tau-\pi)f_i^{-1}(\tau-\pi)=0.\end{equation}
For deriving (50), we have used Eq. (47) and the fact
that the Lie algebra ${\cal F}$ of
$F$ is transversal to ${\cal R}^{\perp}$.
By inserting (50) back in (49) we get a very important relation
\begin{equation} dh(\tau+\pi)h^{-1}(\tau+\pi)=dh(\tau)h^{-1}(\tau).\end{equation}
It expresses the periodicity of the ${\cal R}^{\perp}$-valued `connection'
$dh~h^{-1}$. The monodromy of this `connection is also constrained; indeed,
from (50) and (48) we conclude that
\begin{equation} h^{-1}(\tau-\pi)h(\tau)=fd,\end{equation}
where $f$ is some constant element of $F$. In words: the monodromy
$h^{-1}(\tau-\pi)h(\tau)$
is an element of the double $D$ which is equivalent to $d$ in the
sense of the coset $F\backslash D$.
\noindent {\it Summary}: The space of the solutions of the
field equations (42) submitted to the $D$-branes boundary conditions (47)
is given by an arbitrary element $p$ of the double $D$ and a
periodic field $\rho(\xi^+)(\equiv dh~h^{-1}(\xi^+))$
with values in the subspace ${\cal R}^{\perp}$ of ${\cal D}$ and with the monodromy
\begin{equation} P\exp{\int_{\tau -\pi}^{\tau}d\tau' \rho(\tau')}\equiv
h^{-1}(\tau-\pi)h(\tau)\end{equation}
equivalent to $d$ in the sense of the coset $F\backslash D$. Of course,
$P$ in (53) means the ordered exponent.
The full solution $l(\sigma,\tau)$ is then reconstructed as follows:
take $\rho(\xi^+)$ and $p\in D$ and construct
\begin{equation} h(\xi^+)=P\exp{\{\int_{\xi_0^+}^{\xi^+}d\xi^{+'} \rho(\xi^{+'})\}}\times
p.\end{equation}
Obviously, the choice of $\xi^+_0$ is irrelevant and can be compensated
by the corresponding change of $p$. Now we can reconstruct $r_-(\xi^-)$
by decomposing $h(\xi^-)$ as
\begin{equation} h(\xi^-)=r_-^{-1}(\xi^-)f(\xi^-), \qquad r\in R,\quad f\in F. \end{equation}
This decomposition is unique, because $R$ can be globally identified
with $D/F$. Finally
\begin{equation} l(\xi^+,\xi^-)=r_-(\xi^-)h(\xi^+).\end{equation}
It remains to recover from the solution (56) on the double the solutions
of the $\sigma$-models on the cosets $D/G$ and $D/\tilde G$. Recall that
the both $\sigma$-models are the WZNW models on the group manifold $R$.
In the $D/G$ case we have to decompose $h$ as
\begin{equation} h(\xi^+)=r_+(\xi^+)g(\xi^+), \qquad r\in R, \quad g\in G,\end{equation}
while in the $D/\tilde G$ case as
\begin{equation} h(\xi^+)=\tilde r_+(\xi^+)\tilde g(\xi^+), \qquad \tilde r\in R,
\quad \tilde g\in \tilde G.\end{equation}
Because $r_+\neq
\tilde r_+$ we indeed obtain a nontrivial map from
the phase space of the WZNW model with one set of the $D$-branes
boundary conditions into the phase space of the same WZNW model
but with the dual $D$-branes boundary conditions. The both phase
spaces can be identified with the set of all solutions $l(\xi^+,\xi^-)$
on the double. The system (25) is already written in the Hamiltonian
form, hence the mapping between the phase spaces is a canonical
transformation.
\noindent {\it Note}: It is interesting that the both left movers
$r_+(\xi^+)$
and right movers $r_-(\xi^-)$ are obtained from the master function
$h(\xi^+)$ in a very similar way. Recall that
\begin{equation} h(\xi^+)= r_+(\xi^+) g(\xi^+), \qquad h(\xi^-)= r_-^{-1}(\xi^-)
f(\xi^-), \quad g\in G,~f\in F.\end{equation}
In particular, if $G=F$ then the left and the right movers of the
$R$ WZNW model are given
by the same function , i.e.
\begin{equation} r(\sigma,\tau)=r_-(\xi^-)r_-^{-1}(\xi^+).\end{equation}
This means that the initial point $\sigma=0$ of the string sits at the origin
of the group $R$ for all times. Indeed, the corresponding $D$-brane
is just the group origin, being the projection of $F=G$ along $G$.
\subsection{Interacting $D$-brane diagrams}
Given a $D$-brane configuration on the target $R$ we can in principle
compute the WZNW path integral
over all topologically non-trivial world-sheets interpolating between
a set of fixed
open string
segments with end-points sitting on the $D$-branes and a set of fixed
loops in the target $R$.
We postpone the evaluation of
some of such diagrams (like the open string
propagator) to a forth-coming publication,
here we just discuss whether there are some topological obstructions
in doing that possibly
coming from the WZNW term $\Omega$ and the two-forms $\alpha_i$ and $\alpha_f$ defined
on the $D_i$ and $D_f$ by (38). We have learned in the section 2 that
the WZNW path integral is well-defined if the triplet $(\Omega,\alpha_i,\alpha_f)$
is an integer-valued cocycle in the relative singular cohomology
of the group manifold $R$ with respect to its submanifold $D_i\cup D_f$.
In general, we have found it to be a difficult topological problem
to identify for which $D$-brane configuration $D_i\cup D_f$ and which choice
of the
two-forms $\alpha_i$ and $\alpha_f$ the cocycle $(\Omega,\alpha_i,\alpha_f)$ is integer-valued.
Fortunately enough, if the maximal compact subgroup of $D$ is simple
and simply connected,
we have the key for solving our problem:
we draw the interacting $D$-brane diagrams directly in the double
and repeat the discussion of the section 2, using the duality invariant
first-order action (25). The action (25) also contains the WZNW term
but now the forms $\alpha$ vanish. This means that the pairing
of the cocycle $(c,\alpha_F=0,\alpha_{Fd}=0)$
with any relative cycle $\gamma$ is just
\begin{equation} \langle (c,\alpha_F,\alpha_{Fd}),\gamma\rangle=\int_{\gamma}c .\end{equation}
Recall that we assumed $\pi_1(F)=0$. By the Hurewicz isomorphism,
we obtain $H^2(F)=0$ , hence every cycle in the relative
singular homology $H_3(R,D_i\cup D_f)$ can be represented by a cycle
in $H_3(R)$. This means that what matters is only whether $c$ is the standard
integer-valued three-cocycle in the third de Rham cohomology $H^3(D)$
of the Drinfeld double. But it is, because $H^3(D)=H^3(K)$, where
$K$ is the simple simply connected maximal compact subgroup of $D$, and it
is known that
the WZNW three-form restricted to $K$ is the integer-valued cocycle.
We find quite appealing that the path integral for the $D$-branes
configurations seems to be topologically more easily tractable
by using the duality invariant formalism on the Drinfeld double.
On the other hand, an account of the local world-sheet phenomena,
like a short-distance behaviour, seems to be more difficult when
working with the non-manifestly Lorentzian
first order Hamiltonian action (25). We plan to study this issue
in detail in a near future.
\section{Example: $SU(N)$ WZNW model}
Now we shall study examples of this general construction
of the self-dual WZNW models. Consider the group $SL(N,{\bf C})$ viewed as
the real group and the following invariant non-degenerate bilinear form
on its algebra\footnote{The normalization of the bilinear form is always
such that the resulting action of the $SU(N)$ WZNW model will be properly
normalized in order to meet the requirement that the WZNW three-form is
the integer-valued cocycle.}
\begin{equation} \langle X,Y\rangle = {\rm Im}[(a^*)^2{\rm Tr} XY], \qquad {\rm Im} a^2=4.\end{equation}
The group $SL(N,{\bf C})$ equipped with the invariant bilinear form is the Drinfeld
double for every choice of the complex parameter $a$
satisfying the normalization constraint in (62). Two isotropic subalgebras
${\cal G}$ and $\ti{\cal G}$ of ${\cal D}$ are all upper and lower triangular matrices
respectively with diagonal elements being $\lambda_k a, ~\lambda_k\in {\bf R}$
for ${\cal G}$ and $\tilde\lambda_k ia, ~\tilde\lambda_k\in {\bf R}$ for $\ti{\cal G}$.
Obviously, the index $k$ denotes the position on the diagonal and
lambdas are constrained by the tracelessness condition.
An example of $SL(2,{\bf C})$:
\begin{equation} {\cal G}=\left(\matrix{\lambda a&z\cr 0& -\lambda a}\right),
\quad \ti{\cal G}=\left(\matrix{\tilde\lambda ia&0\cr \tilde z&-\tilde\lambda ia}\right),\end{equation}
where $z,\tilde z$ are arbitrary complex numbers.
The dual pair of the $\sigma$-models is encoded in the choice of the
half-dimensional subspace ${\cal R}$ of the Lie algebra ${\cal D}$ of the double.
We choose ${\cal R}$ to be the $su(N)$ subalgebra of the algebra $sl(N,{\bf C})$.
Following our discussion above, it is easy to find the principal
fibrations of $SL(N,{\bf C})\equiv D$, corresponding to the algebras ${\cal R}$,
${\cal G}$ and
$\ti{\cal G}$. The total space of the bundles is always the double $D$, the fibres
are $SU(N)$, $\exp{{\cal G}}\equiv G$ and $\exp{\ti{\cal G}}\equiv \tilde G$ and the bases are
$SU(N)\backslash D$, $D/G$ and $D/\tilde G$ respectively. Note that
every fiber of all three
fibrations can be obtained from the fiber crossing the unit element $e$ of $D$
by either the right (for $SU(N)$)
or the left (for $G$ and $\tilde G$) action of some element of the double $D$.
In the particular example of the double $SL(N,{\bf C})$, the intersection
of a fibre $SU(N)$ with fibres $G$ or $\tilde G$ occurs always
precisely at one point.
It is not difficult to prove this fact. We already know that the intersection
always exists because $SU(N)$ is compact and $SL(N,{\bf C})$ is connected
(cf. sec 3).
If both fibers $SU(N)$ and $G$ (or $\tilde G$) cross the unit element
of the double (which is the intersection point), it is obvious that a
non-unit element of $G$ (or $\tilde G$)
cannot be a unitary matrix. Thus the intersection is unique in this case.
Also an intersection $r$ of the $SU(N)$ fiber crossing
the unit element of $D$ with some fiber $G$ must
be unique. Indeed, the $G$ fiber can be then written as $rG$ where
$r\in SU(N)$. By the left action of $r^{-1}$ the $G$ fiber can be transported
to the origin of $D$ where there is just one intersection.
Hence we conclude: for our data $D=SL(N,{\bf C}),G,\tilde G$ and ${\cal R}=su(N)$,
the both models of the dual pair (16) are the standard $SU(N)$ WZNW models,
because the restriction of the bilinear form (62) to ${\cal R}$ is
nothing but the standard Killing-Cartan form on $su(N)$.
Now we may choose
the subgroup $F$ of $D$, which defines the $D$-branes in the
double, to be equal to $G$. Thus we have given a concrete meaning
to our so far abstract construction.
It may be of some interest to provide few
explicit formulas for the $SL(2,{\bf C})$ Drinfeld double.
The both cosets $D/G$ and $D/\tilde G$ can be identified with the group
$SU(2)$. Recall that the space of all $D$-branes corresponding to the choice
$F$ is parametrized by the {\it left} coset $F\backslash D$.
In our case $F\backslash D$ can also
be identified with $SU(2)$, hence a generic $D$-brane ($Fd$) in the
double is a set of $SL(2,{\bf C})$ matrices of the form
\begin{equation} Fd \equiv \left(\matrix{e^{\lambda a}&z\cr 0&e^{-\lambda a}}\right)
\left(\matrix{C&-E^*\cr E&C^*}\right),\end{equation}
where $C,E$ are fixed complex numbers satisfying
$CC^* +EE^*=1$, $\lambda$ is a real
and $z$ a complex number.
In order to get
the $D$-branes in the cosets $D/G=D/F$ and $D/\tilde G$, we have to project $Fd$
on $SU(2)$ along $G$ and $\tilde G$, respectively:
\begin{equation} Fd= \left(\matrix{A&-B^*\cr B&A^*}\right)
\left(\matrix{e^{\eta a}&w\cr 0&e^
{-\eta a}}\right), \quad \eta\in {\bf R}, \quad w\in {\bf C} ,\end{equation}
\begin{equation} Fd= \left(\matrix{\tilde A&-\tilde B^*\cr \tilde B&\tilde A^*}\right)
\left(\matrix{e^{\tilde\eta ia}
&0\cr \tilde w&e^{-\tilde\eta ia}}\right), \quad \tilde\eta\in {\bf R},
\quad \tilde w\in {\bf C} .\end{equation}
Here again
$AA^*+BB^*=1$ and the same constraint is of course true also for $\tilde A$ and
$\tilde B$. If $\lambda$ and $z$ vary then $A$ and $B$ sweep a submanifold
of $SU(2)$, which is just the $D$-brane in $R$ and $\tilde A$ and $\tilde B$ sweep
the dual $D$-brane in $R$.
There may occur three qualitatively different possibilities:
\noindent 1. Both $C$ and $E$ do not vanish (a generic case).
\noindent Then $B\neq 0$ and it is convenient to parametrize $D$ and $B$ as
\begin{equation} E=e^{E_1 a} e^{E_2 ia},\qquad B=e^{B_1 a} e^{B_2 ia},
\quad B_i,E_i\in{\bf R} .\end{equation}
The original $D$-brane is then a two-dimensional submanifold of $SU(2)$
characterized by the condition
\begin{equation} B_2=E_2.\end{equation}
The dual $D$-brane is a three dimensional submanifold of $SU(2)$ which
is complement of the circle $A=0$.
\noindent 2. $C=0$.
\noindent The original $D$-brane is the same as in 1. but the
dual $D$-brane is just the one-dimensional circle $A=0$.
\noindent 3. $E=0$.
\noindent The original $D$-brane is a point $A=C$ and the dual $D$-brane
is the same as in 1.
It is not difficult to compute also the two-form $\alpha$ on the $D$-brane
(cf. (38)).
For doing this, we have just to know the mapping $k(r)$ from
the original $D$-brane in $R$ to $G$
and the dual mapping $\tilde k(r)$ from the dual $D$-brane in
$R$ to $\tilde G$ (cf. (35)). We do that
for the case 3 choice $C=1$ and $E=0$. The original map $k$ is trivial,
since the $D$ brane is just the point $A=1$ but the dual map $\tilde k$
is nontrivial and it reads
\begin{equation} \tilde k(A,B)=\left(\matrix{e^{iA_2 a}&0\cr -e^{-A_1 a}B&e^{-iA_2 a}}\right).
\end{equation}
Here
\begin{equation} 0\neq A^*\equiv e^{A_1 a}e^{iA_2 a}, \quad A_1,A_2\in{\bf R}.\end{equation}
Now insert (69) in (38) and find
\begin{equation} 8\pi\tilde\alpha_f=
{\rm Im}[{a^*}^2\{(-BdB^*+B^*dB)a\wedge(A_1+idA_2)+
dB^*\wedge dB -2ia^2 dA_1\wedge dA_2\}].\end{equation}
It is easy to compute the exterior derivative of $\tilde\alpha_f$:
\begin{equation} 8\pi d\tilde\alpha_f=2{\rm Im}[{a^*}^2 a~ dB^*\wedge dB \wedge d(A_1+iA_2)]=
c_R\vert_{D_f}.\end{equation}
In words: the exterior derivative of $\alpha_f$ is equal to the restriction
of the WZNW three-form $c_R$ on the $D$-brane $D_f$.
It may be interesting to remark that in the case of the $SU(N)$ WZNW
models there are no topological obstructions in quantizing the
model on the topologically trivial open strip world-sheet. Thus, we do not
have to lift the $D$-brane configuration to the double in order to
make the argument but we can directly proceed at the level of the
$D/G\equiv SU(N)$ target.
Indeed, choose two different
surfaces lying in the same $D$-brane
and interpolating between the edges of the `diadem'. Their
oriented sum does not have a boundary and it is topologically the two-sphere.
If we happen to show that the second homotopy group $\pi_2(D_{i(f)})$ of
the $D$-brane $D_{i(f)}$ vanishes then the two interpolating surfaces are
homotopically equivalent and there is no ambiguity coming from the
WZNW term (cf. sec 2).
It is easy to prove that $\pi_2(D_f)$
vanishes if the $D$-brane $D_f$ was obtained by our method of projecting
the isotropic surface $Fd$ from the double. Our basic tool
is the long exact homotopy sequence \cite{Schw}:
\begin{equation} \pi_2(F)=0\to \pi_2(F/H)\to\pi_1(H)\to \pi_1(F)\to \pi_1(F/H)\to
\pi_0(H)\to 0=\pi_0(F),\end{equation}
which holds for a connected group $F$ and its arbitrary subgroup
$H$; note that $\pi_2$ of any Lie
group vanishes. Now the $D$-brane on $SU(N)$ is gotten by projection
of the surface $Fd$ from the double to $D/G$.
This means that topologically it can
be identified with the coset $F/dGd^{-1}\cap F\equiv F/H$.
We observe that in our $SL(N,{\bf C})$ context the group $F$ can be topologically
identified with its algebra ${\cal F}$ because the usual exponential
mapping $\exp{{\cal F}}=F$ is one-to-one. So it is one-to-one
for any its connected subgroup including the unity component of $H$.
Hence $\pi_1(H)=0$ and since $\pi_1(F)=0$,
from the sequence
(73) we conclude that $\pi_2$ of the $D$-brane in $SU(N)$ vanishes.
We should mention that from the exact sequence (73) it also follows
that $\pi_1(D_f)=\pi_0(H)$. In general, for
our $SU(N)$ case the group $H$ is not connected.
This means that, strictly speaking, the diadem in our argument must
be equivalent to the zero element of $H_2(R,D_f)$, or, in other, words
it must be a relative two-boundary.
If the diadem is a non-trivial relative
two-cycle we use the results of section 2
and evaluate its contribution
by choosing the extension $\tilde f:Z\to U$ of the homomorphism $f:B\to U$.
| proofpile-arXiv_065-462 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
\begin{figure}
\vspace*{13pt}
\vspace*{6.7truein}
\special{psfile=figa.ps
voffset= 240 hoffset= -40 hscale=50 vscale=50 angle = 0}
\special{psfile=figb.ps
voffset= 240 hoffset= 220 hscale=50 vscale=50 angle = 0}
\special{psfile=figc.ps
voffset= -40 hoffset= 90 hscale=50 vscale=50 angle = 0}
\caption{Histogram
of the number of models that yield a particular prediction
for $m_{\nu_{\mu}}^2- m_{\nu_{e}}^2$
assuming (a) small angle and (b) large angle solution to solar
neutrino problem. In (c) we
solve the solar neutrino problem via small angle $e$--$\tau$
oscillations and check whether this is compatible with
the LSND result.
}
\label{fig}
\end{figure}
In the Standard Model of elementary particles (SM)
both lepton number ($L$) and baryon number ($B$) are
conserved due to an accidental symmetry,
{\sl i.e.} there is no renormalizable, gauge-invariant
term that would break the symmetry.
In the minimal supersymmetric extension of the SM (MSSM)
the situation is different. Due to a the variety
of scalar partners the MSSM allows for a host of new
interactions many of which violate $B$ or $L$.
Since neither $B$ nor $L$ violation has been
observed in present collider experiments
these couplings are constrained from above.
More constraints arise from neutrino
physics
or cosmology.
Thus, all lepton and baryon number violating
interaction are often eliminated by imposing
a discrete, multiplicative symmetry called
$R$-parity,\cite{r-parity}
$R_p \equiv (-1)^{2S+3B+L}$, where $S$ is the spin.
One very attractive feature of
$R_p$ conserving models is that
the lightest supersymmetric particle (LSP)
is stable and a good cold dark matter candidate.\cite{cdm}
However, while the existence of a dark matter candidate
is a very desirable prediction, it does not prove
$R_p$ conservation and
one should consider more general models.
Here, we will investigate
the scenario where $R_p$ is broken explicitly via
the terms\cite{suzuki} $W = \mu_i L_i H$,
where $H$ is the Higgs coupling to up-type fermions
and $L_i$ ($i = 1,2,3$) are the left-handed lepton doublets.
Clearly, these Higgs-lepton mixing terms violate
lepton-number. As a result, majorana masses will be generated for
one neutrino at tree-level and for the remaining
two neutrinos at the one-loop level.
These masses were calculated in the frame-work of minimal supergravity
in ref.~\citenum{npb} and the numerical results will be
briefly summarized here.
There are three $R_P$ violating parameters which can be used to fix
1) the tree-level neutrino mass,
2) the $\mu$--$\tau$ mixing angle and
3) the $e$--$\mu$ mixing angle.
The question of whether e.g. the solar\cite{solarn}
and the atmospheric\cite{atmosphericn} neutrino puzzle
can be solved simultaneously depends on the prediction of
$m_{\nu_\mu}^2-m_{\nu_e}^2$.
In fig.~1 we have scanned the entire SUSY parameter space
consisting of the Higgsino (gaugino) mass parameter,
$\mu$ ($m_{1/2}$), the trilinear scalar interaction parameter $A_0$,
and the ratio of Higgs VEVs, $\tan\beta$. The universal
scalar mass parameter $m_0$ is fixed by minimizing the potential.
Plotted is the number of models yielding a particular prediction for
$m_{\nu_\mu}^2-m_{\nu_e}^2$ for
(a) sin$^2 2 \theta_{e \nu_\mu} = 0.008$ and
(b) sin$^2 2 \theta_{e \nu_\mu} = 1$.
We fix $m_{\nu_\tau}=0.1$~eV and
sin$^2 2 \theta_{\mu \nu_\tau} = 1$ in order to solve the
atmospheric neutrino problem.
We see that both
long wave-length oscillation (LWO)\cite{lwo}
($m_{\nu_\mu}^2-m_{\nu_e}^2=10^{-10}$~eV$^2$)
and MSW effect\cite{msw-effect}
($m_{\nu_\mu}^2-m_{\nu_e}^2=10^{-5}$~eV$^2$)
can be accommodated.
In fig.~1(c) we solve the solar neutrino problem via $e$--$\tau$
oscillations and we fix sin$^2 2 \theta_{e \nu_\mu} = 0.004$
in order to accommodate the LSND result.\cite{lsnd}
We see that most models are already ruled out
by collider constraints and even more by dark matter (DM) constraints.
However, a very small (but non-zero) number of models
yields a prediction compatible with the LSND
result (the dotted line is lower limit of LSND).
\noindent{\bf Acknowledgements}
This work was supported in parts by the DOE under
Grants No. DE-FG03-91-ER40674 and by the
Davis Institute for High Energy Physics.
\vskip0.3cm
\noindent{\bf References}
| proofpile-arXiv_065-463 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section*{Introduction}
To define quasilocal energy in general relativity, one can begin with
a suitable action functional for the time history ${\cal M}$ of a
spatially bounded system $\Sigma$. Here ``suitable'' means that in the
associated variational principle the induced metric on the
time history ${\cal T}$ of the system boundary $B =
\partial\Sigma$ is fixed. In particular, this means that the
lapse of proper time between the boundaries of the initial and final
states of the system $\Sigma$ must be fixed as boundary data.
The quasilocal energy (QLE) is then defined as minus
the rate of change of the classical action (or Hamilton-Jacobi
principal function) corresponding to a unit increase in proper
time.\cite{BY,Lau} So defined, the QLE is a functional on the gravitational
phase space of $\Sigma$, and is the value of the gravitational Hamiltonian
corresponding to unit lapse function and zero shift vector on the
system boundary $B$. Although other definitions of quasilocal energy
have been proposed (see, for example, the references listed in \cite{BY}),
the QLE considered here has the key property, which we consider crucial,
that it plays the role of internal energy in the thermodynamical
description of coupled gravitational and matter fields.\cite{thermo}
In this paper we define the energy of a perfectly isolated system at a
given retarded time as the suitable limit of the quasilocal energy $E$
for the partial system enclosed within a finite topologically spherical
boundary.\footnote{Hecht and Nester have also considered energy-momentum
(and ``spin'') at null infinity (for a class of generally covariant
theories including general relativity) {\em via} limits of quasilocal Hamiltonian
values.\cite{HechtNester} Their treatment of energy-momentum is based on a
differential-forms version of canonical gravity, often referred to as the
``covariant canonical formalism." For pure {\sc bms} translations our results
are in accord with those found by Hecht and Nester, although at the level of
{\em general} supertranslations they differ. We provide a careful analysis
of the zero-energy reference term (necessary for the QLE to have a finite
limit at null infinity), and this analysis is intimately connected with
our results concerning general supertranslations.} For our choice of
asymptotic reference frame the energy
that we compute equals what is usually called
the Bondi-Sachs mass.\cite{Goldberg,review} As we shall see, our asymptotic
reference frame defines precisely that infinitesimal generator of the
Bondi-Metzner-Sachs ({\sc bms}) group corresponding to a pure time translation.\cite{Sachs,review,PenroseRindler} We also show that in the
same null limit the lapse-arbitrary, shift-zero
Hamiltonian boundary value defines a physically meaningful
element in the space dual to
supertranslations. This dual space element, it turns out, coincides with
the ``supermomentum" discussed by Geroch.\cite{Geroch} Our results are
then specialized to an expression for the full Bondi-Sachs four-momentum
in terms of Hamiltonian values. It is already known that when
$B$ is the two-sphere at spacelike infinity, the quasilocal
and Arnowitt-Deser-Misner\cite{ADM} notions of energy-momentum
agree.\cite{BY,thermo} Our results therefore
indicate that the quasilocal formalism provides a unified
Hamiltonian framework
for describing the standard notions of gravitational
energy-momentum in asymptopia.
Before turning to the
technical details, let us first present a short overview
of our approach. Consider a spacetime ${\cal M}$ which is
asymptotically flat at future null infinity ${\cal I}^{+}$
and a system $(w,R,\theta,\phi)$ of Bondi coordinates
thereon.\cite{review} The retarded time $w$ labels a
one-parameter family of outgoing null hypersurfaces
${\cal N}(w)$. The coordinate $R$ is a luminosity
parameter (areal radius) along the outgoing null-geodesic
generators of the hypersurfaces ${\cal N}(w)$.
The Bondi coordinate system
also defines a two-parameter family of topologically spherical
two-surfaces $B(w,R)$. It suits our purposes to consider
only a single null hypersurface of the family ${\cal N}(w)$,
say ${\cal N}(w_{*})$, the one determined by setting $w$ equal
to an {\em arbitrary} constant $w_{*}$. The collection
$B(w_{*},R)$ of two-surfaces foliates ${\cal N}(w_{*})$, and
in the $R\to\infty$ limit these two-surfaces converge on an
infinite-radius {\em round} sphere $B(w_{*},\infty)$.
To streamline the presentation, we refer to our generic
null hypersurface simply as $\cal N$; and we use the
plain letter $B$ to denote both the $\cal N$-foliating
collection $B(w_{*},R)$ and a single generic two-surface of
this collection. Now, should we desire a more general
${\cal N}$-foliating collection of two-surfaces, we could, of course,
introduce a new radial coordinate $\bar{R}$. For a
fixed retarded time $w = w_{*}$ the new two-surfaces would
then arise as level surfaces of constant $\bar{R}$. However,
we shall not consider such a new radial coordinate, because
the new two-surfaces would not necessarily converge towards a round
sphere in the asymptotic limit. At any rate, we could handle such an
additional kinematical freedom, were it present, by assuming that
along each outgoing null ray $\bar{R}$ approached $R$ at a
sufficiently fast rate in the asymptotic limit.
Our first goal is to compute the QLE within a
two-surface $B$ in the limit as $B$ approaches a spherical
cut of ${\cal I}^{+}$ along the null surface $\cal N$, and to
show that this result coincides with the Bondi-Sachs mass:
\begin{equation}
M_{_{BS}}(w_{*}) = \lim_{R \rightarrow \infty}
\int_{B(w_{*},R)}
{\rm d}^{2}x \sqrt{\sigma} \varepsilon\, .
\label{limit}
\end{equation}
Here $\varepsilon
= (k - k |^{\scriptscriptstyle {\rm ref}})/\kappa$
is the quasilocal energy
surface density with $\kappa = 8\pi$ (in geometrical units)
and $\sigma$ is the determinant of the induced metric on $B$.
Recall that $k$ denotes the mean curvature of
$B$ as embedded in some {\em spacelike} spanning three-surface
$\Sigma$. Since both $B$ and $\Sigma$ are embedded in the
{\em physical} spacetime ${\cal M}$, we sometimes use the notation
$\varepsilon |^{\scriptscriptstyle {\rm phy}} = k/\kappa$. Also recall that
$k |^{\scriptscriptstyle {\rm ref}}$ denotes the mean curvature
of a surface which is isometric to $B$ but
which is embedded in a three-dimensional {\em reference} space
different than $\Sigma$. Here we choose the reference space to be
flat Euclidean space $E^{3}$, i.\ e.\
we assign a flat three-slice of Minkowski spacetime the zero value of
energy.\cite{BY} Although a definition of the zero-energy reference in
terms of flat space is neither always essential nor
appropriate\cite{Brown}, it is the correct choice for the analysis
of this paper.
In order to define $k$,
we must select such a three-surface spanning $B$ for each $R$
value. (For a single $B$ many different spanning
three-surfaces will determine
the same $k$. In fact, $k$ is determined solely by $B$ and a timelike
unit vector field $u^\mu$ on $B$, which can be considered as the unit
normal of a slice $\Sigma$. Thus, the continuation of $\Sigma$
away from $B$ is not needed; moreover, such a continuation of $\Sigma$ might
not be defined throughout the interior of ${\cal M}$. Therefore, though
we speak of choosing a $\Sigma$ three-surface to span $B$
for each $R$ value, we are really fixing only a timelike unit normal
vector field at $B$.) For generality, we leave the choice of
spanning three-surface $\Sigma$ essentially arbitrary at
each $R$ value, but we do enforce a definite choice
asymptotically. Heuristically, as $R \rightarrow \infty$
the $\Sigma$ three-surface spanning $B$ approaches an
asymptotic three-surface $\Sigma_{\infty}$ which spans a
round infinite-radius spherical cut of ${\cal I}^{+}$ (see the
figure). Our construction is, as expected, sensitive to the choice of
asymptotic three-surface $\Sigma_{\infty}$. Said another way, the
QLE depends on the fleet of observers at $B$ whose four-velocities
are orthogonal to the spanning three-surface at $B$. Therefore,
one expects {\em a priori} the expression on the right-hand side
of (\ref{limit}) to depend on the choice of asymptotic fleet
associated with the two-sphere at
${\cal I}^{+}$. The asymptotic fleet we choose
corresponds to a pure {\sc bms} time translation:
each member of the asymptotic fleet rides along
$\partial/\partial w$. Note that, although $\partial/\partial w$
is everywhere timelike in ${\cal M}$ (at least in the relevant exterior
regions), the {\em extension} of
$\partial/\partial w$ to ${\cal I}^{+}$ in a conformal completion
$\hat{\cal M}$ of the physical spacetime ${\cal M}$ is in fact a
null vector which lies in ${\cal I}^{+}$. (While we occasionally
find it clarifying to make reference to the concept of
a conformal completion, we
do not explicitly use conformal completions in this paper.)
Therefore, heuristically, one should envision $\Sigma_{\infty}$
as a spacelike slice which becomes null asymptotically
(see the figure).
This paper is organized as follows. In a preliminary
section we write down the familiar Bondi-Sachs
form\cite{Sachs,Chrusciel_et_al}
of the spacetime metric as well as asymptotic expansions
for the associated metric coefficients.
We also introduce on $\cal M$ two
future-pointing null vector fields $k^{\mu}$ and $l^{\mu}$
(do not confuse $k^{\mu}$ with the mean curvature
$k$). Both vector fields point everywhere normal to
our collection $B(w,R)$ of two-surfaces, and
$k_{\mu} l^{\mu} = - 1$.
Next, we construct on ${\cal N}$ a timelike vector field
$u^{\mu} := \frac{1}{2} k^{\mu} + l^{\mu}$ (equality restricted
to ${\cal N}$), which in
our analysis will {\em define} for each $B$ along $\cal N$
a spacelike spanning three-surface $\Sigma$. In $\S$II we use the $\Sigma$
three-surfaces determined by $u^{\mu}$ to define an unreferenced energy
surface density $\varepsilon |^{\scriptscriptstyle {\rm phy}} = k/\kappa$
for each $B$ slice of $\cal N$ and then examine the asymptotic
limit of $k/\kappa$.
In $\S$III we consider the asymptotic expression for the flat-space
reference density
$\varepsilon |^{\scriptscriptstyle {\rm ref}} =
k |^{\scriptscriptstyle {\rm ref}}/\kappa$, but give the
derivation of this expression in the Appendix. Next, we assemble
the results of the previous two sections and prove the main
claim (\ref{limit}). In $\S$IV we examine the ``smeared energy
surface density,'' which is the Hamiltonian value corresponding
to an arbitrary supertranslation. We then specialize our
result for the smeared energy surface density to express the
full Bondi-Sachs four-momentum in terms of Hamiltonian values. In $\S$V we
examine the smeared energy surface density
{\em via} the spin-coefficient formalism, and show that it equals the
``supermomentum'' of Geroch\cite{Geroch} as written by Dray and
Streubel.\cite{DrayStreubel} The Appendix is
devoted to a detailed analysis of
the reference term.
\section{Preliminaries}
In terms of a Bondi coordinate system the metric of our asymptotically
flat spacetime ${\cal M}$ takes the standard
form\cite{Sachs,Chrusciel_et_al}
\begin{equation}
g_{\mu\nu} {\rm d}x^{\mu} {\rm d}x^{\nu} =
- UV {\rm d}w^{2}
- 2 U {\rm d}w {\rm d}R
+ \sigma_{ab}
({\rm d}x^{a} + W^{a}{\rm d}w) ({\rm d}x^{b}
+ W^{b}{\rm d}w){\,} ,
\label{spacetimemetric}
\end{equation}
where $a,b$ are $B$ indices running over $\theta,\phi$. We assume the
following expansions for the various metric coefficients
above:\footnote{Up to the $\Delta$ remainder terms, our expansions in the
radial coordinate $R$ coincide with those given by Sachs;
however, we do not assume that the $\Delta$ remainder terms are
necessarily expandable in powers of inverse $R$, an assumption
which would be tantamount to what Sachs calls the
``outgoing radiation condition.''\cite{Sachs} Recently, Chru\'{s}ciel
{\em et al.}~have shown that ``polyhomogeneous''
expansions in terms of $R^{-i}\log^{j}\!R$ also provide a consistent
framework for solving the characteristic initial value problem of the
Bondi-Sachs type.\cite{Chrusciel_et_al} They argue that the
so-called outgoing radiation condition is overly restrictive.}
\begin{eqnarray}
V & = &
1
- 2{\rm m} R^{-1}
+ \Delta_{V}
\label{Sachs} \eqnum{\ref{Sachs}a} \\
U & = &
1
- {\textstyle \frac{1}{2}}(X^{2} + Y^{2})
R^{-2} + \Delta_{U}
\eqnum{\ref{Sachs}b}\\
W^{\theta} & = &
(2 X\cot \theta + X_{,\theta}
+ Y_{,\phi} \csc\theta)R^{-2}
+ \Delta_{W^{\theta}}
\nonumber \\
W^{\phi} & = &
\csc\theta\left(2 Y \cot\theta
+ Y_{,\theta} - X_{,\phi}\csc\theta\right)R^{-2}
+ \Delta_{W^{\phi}}
\eqnum{\ref{Sachs}c} \\
\sigma_{ab} & = &
R^{2}\delta_{ab}
+ \left[(2 X)\theta_{,a}\theta_{,b}
+ (4 Y \sin\theta)\theta_{,(a} \phi_{,b)}
- (2 X \sin^{2}\theta) \phi_{,a} \phi_{,b}\right]R
+ \Delta_{\sigma_{ab}}{\,} . \eqnum{\ref{Sachs}d}
\addtocounter{equation}{1}
\end{eqnarray}
Here $X(w,\theta,\phi)$ and $Y(w,\theta,\phi)$
are respectively the real and imaginary parts of the asymptotic
shear $c = X + {\rm i} Y$, ${\rm m}(w,\theta,\phi)$ is the
all-important {\em mass aspect}, $\delta_{ab}$ is the metric of a
unit-radius round sphere, and commas denote partial differentiation.
In the Appendix we examine the form of the two metric $\sigma_{ab}$ in
more detail. Remainder terms, denoted by the $\Delta$ symbol, always fall
off faster (or have slower growth, as the case may be) than the terms
which precede them. For instance, $\Delta_{V}$ denotes a term
which falls off {\em faster} than $O(R^{-1})$.
Introduce the future-directed null covector field
$k_{\mu} = - e^{\eta}
\nabla_{\mu} w$, where the scalar function
$\eta = \eta(w, R , \theta , \phi)$ is a point-dependent
boost parameter. The null covector $k_{\mu}$ is orthogonal
to the spheres $B(w,R)$, and the function $\eta$ gives us
complete freedom in choosing the extent of $k_{\mu}$ at each
point of any $B$ two-surface. We shall find it necessary later to assume that
$\eta$ falls off {\em faster} than $1/\sqrt{R}$ on every outgoing ray.
Also define another future-directed null
vector field $l^{\mu}$ which is
orthogonal to the $B(w,R)$ and normalized so that
$k_{\mu} l^{\mu} = - 1$. As one-forms these
null normals are
\begin{eqnarray}
k_{\mu} {\rm d}x^{\mu} & = &
- e^{\eta} {\rm d}w
\label{bassae} \eqnum{\ref{bassae}a} \\
l_{\mu} {\rm d}x^{\mu} & = &
- e^{-\eta} U {\rm d}R -
{\textstyle \frac{1}{2}} e^{-\eta} UV
{\rm d}w\, , \eqnum{\ref{bassae}b}
\addtocounter{equation}{1}
\end{eqnarray}
while as vector fields they are
\begin{eqnarray}
k^{\mu} \partial/\partial x^{\mu}
& = & e^{\eta} U^{-1}
\partial/\partial R \label{vectors}
\eqnum{\ref{vectors}a}\\
l^{\mu} \partial/\partial x^{\mu}
& = & e^{-\eta} \partial/\partial w -
{\textstyle \frac{1}{2}} e^{-\eta}
V \partial/\partial R - e^{-\eta} W^{a}
\partial/\partial x^{a} \eqnum{\ref{vectors}b}
\, .
\addtocounter{equation}{1}
\end{eqnarray}
Now define $u^\mu := \frac{1}{2} k^\mu + l^\mu$ and
$n^\mu := \frac{1}{2} k^\mu - l^\mu$ along ${\cal N}$ as the timelike
and spacelike unit normals of the $B$ two-surfaces. For each
slice $B$ of the null hypersurface ${\cal N}$, the normals $u^\mu$ and $n^\mu$
determine a spanning spacelike three-surface $\Sigma$. As mentioned
previously, the three-surface $\Sigma$ is not unique and, moreover,
need not be defined throughout ${\cal M}$. Indeed,
there is no guarantee that $u^\mu$ as defined is even surface-forming.
(That is, in general $u_\mu$ does not satisfy the Fr\"obinius condition
$u_{[\alpha}\nabla_{\mu} u_{\nu]}=0$.) Nevertheless, our construction
provides us with what we need: a unit timelike vector $u^{\mu}$ orthogonal
to $B$. We can therefore obtain an unreferenced
energy surface density $k/\kappa$ which is the same for any
slice or partial slice $\Sigma$ that
contains $B$ and has timelike unit normal which agrees
with $u^\mu$ at $B$.
Our construction implies
\begin{equation}
u^{\mu}\partial/\partial x^{\mu} \rightarrow
\partial/\partial w
\end{equation}
on each ray as $R \rightarrow \infty$.
Now, the standard realization of the {\sc bms}-group Lie algebra (as
vector fields on future null infinity) identifies the {\em extension} of
$\partial/\partial w$ to ${\cal I}^{+}$ (in a conformal completion
$\hat{\cal M}$ of ${\cal M}$) with a pure time translation.\cite{Sachs,review}
Therefore, asymptotically, our fiducial surface $\Sigma_{\infty}$ determines
precisely the pure time-translation generator of the {\sc bms} group.
We do {\em not} claim that $u^{\mu}$ generates an ``infinitesimal asymptotic
symmetry transformation'' in the sense of Sachs\cite{Sachs},
{\it i.~e.~}that the various coefficients associated with the transformed metric
$g_{\mu\nu} + 2\nabla_{(\mu} u_{\nu)}$ satisfy the fall-off conditions
(\ref{Sachs}); however, this is unimportant for our construction.
\section{Computation of the quasilocal energy surface density}
We now turn to the task of calculating an expression for
the unreferenced quasilocal energy surface density
$\varepsilon |^{\scriptscriptstyle {\rm phy}} = k/\kappa$.
Our starting point is the
definition $k := - \sigma^{\mu\nu} \nabla_{\mu} n_{\nu}$, where the
two-metric $\sigma^{\mu\nu} = g^{\mu\nu} + 2k^{(\mu} l^{\nu)}$
serves as the projection operator into $B$. We find it
convenient to write\footnote{Note that the definition of $k$ does not
depend on how $n^{\mu}$ is extended off $B$.} $k = 2\mu + \rho$,
where in the standard notation of the spin-coefficient
formalism\cite{PenroseRindler} $- \mu$ and $\rho$ are, respectively,
the expansions associated with the inward null normal and
outward null normal to $B$. These are given by the formulae
\begin{eqnarray}
\mu & = & {\textstyle \frac{1}{2}}
\sigma^{\mu\lambda} \nabla_{\mu} l_{\lambda}
= {\textstyle \frac{1}{2}}(\nabla_{\mu} l^{\mu} + k^{\nu} l^{\mu}
\nabla_{\mu} l_{\nu})\label{spin} \eqnum{\ref{spin}a} \\
\rho & = & - {\textstyle \frac{1}{2}}
\sigma^{\mu\lambda}\nabla_{\mu} k_{\lambda} =
- {\textstyle \frac{1}{2}}(\nabla_{\mu} k^{\mu}
+ l^{\nu} k^{\mu} \nabla_{\mu} k_{\nu})\, .
\eqnum{\ref{spin}b}
\addtocounter{equation}{1}
\end{eqnarray}
As a technical tool, it proves convenient to introduce
fiducial vector fields $\hat{k}^{\mu}$ and
$\hat{l}^{\mu}$ determined from (\ref{vectors})
by setting $\eta = 0$ on $\cal N$. {}From the
middle expressions above, it
is obvious that $\mu = e^{-\eta} \hat{\mu}$
and $\rho = e^{\eta} \hat{\rho}$, where the
easier-to-calculate expressions $\hat{\mu}$
and $\hat{\rho}$ are built exactly as in
(\ref{spin}) but with the fiducial
null normals $\hat{k}^{\mu}$ and
$\hat{l}^{\mu}$. Therefore, we may
assume that $\eta = 0$ while
calculating the spin coefficients in (\ref{spin}) and then simply
multiply the $\eta = 0$ results by
the appropriate factor to get the correct
general expressions. Let us sketch the calculation. First,
from (\ref{bassae}a) with
$\eta = 0$ note that $\hat{k}^{\mu} \nabla_{\mu} \hat{k}_{\nu} = 0$,
because $\hat{k}_{\nu}$ is a gradient. Next, using both expressions
(\ref{bassae}) with $\eta = 0$, one
can work the second term inside the parenthesis of
(\ref{spin}a) into the form $\hat{k}^{\nu}
\hat{l}^{\mu} \nabla_{\mu} \hat{l}_{\nu} = - U^{-1} \hat{l}^{\mu}
\nabla_{\mu} U + \frac{1}{2} U \hat{k}^{\mu} \nabla_{\mu} V$. Finally,
one writes the covariant-divergence terms as ordinary
divergences; for example, $\nabla_{\mu} \hat{k}^{\mu}
= (- g)^{-1/2} \partial_{\mu}(\sqrt{-g}
\hat{k}^{\mu})$, where the square root of
(minus) the determinant of the spacetime metric is
$\sqrt{- g} = U\sqrt{\sigma}$.
Following these steps and multiplying by
the appropriate boost factors at the end of the calculation, one finds
\begin{eqnarray}
\mu & = & {\textstyle \frac{1}{4}}
e^{-\eta} \sigma^{-1} \dot{\sigma}
- {\textstyle \frac{1}{8}}
e^{-\eta} V \sigma^{-1}\sigma'
- {\textstyle \frac{1}{2}} e^{-\eta}
\delta_{a} W^{a} \label{mu1} \eqnum{\ref{mu1}a} \\
\rho & = & - {\textstyle \frac{1}{4}} e^{\eta} U^{-1}
\sigma^{-1}\sigma'{\,} . \eqnum{\ref{mu1}b}
\addtocounter{equation}{1}
\end{eqnarray}
Here the over-dot denotes partial differentiation
by $\partial/\partial w$, the prime denotes partial
differentiation by $\partial/\partial R$, and $\delta_{a}$
denotes the $B$ covariant
derivative. Since $R$ is an areal radius, we may take
$\sigma = R^{4}\sin^{2}\theta$ [see the form of the $B$
metric given in (\ref{Bmetric})]. Therefore, we obtain the
compact expressions
\begin{eqnarray}
\mu & = & - {\textstyle \frac{1}{2}} e^{-\eta} V R^{-1}
- {\textstyle \frac{1}{2}} e^{-\eta} \delta_{a} W^{a}
\label{mu2} \eqnum{\ref{mu2}a} \\
\rho & = &
- e^{\eta} U^{-1} R^{-1}\, .
\eqnum{\ref{mu2}b}
\addtocounter{equation}{1}
\end{eqnarray}
Adding twice (\ref{mu2}a) to (\ref{mu2}b), we arrive
at our desired expression
\begin{equation}
k = - ( e^{-\eta} V + e^{\eta} U^{-1})
R^{-1} - e^{-\eta} \delta_{a} W^{a}\, ,
\end{equation}
which has the asymptotic form
\begin{equation}
k = - 2R^{-1} + 2{\rm m}(w_{*},\theta,\phi)
R^{-2} - \delta_{a}
W^{a} + \Delta_{k}\, .
\label{kexpansion}
\end{equation}
Note that we have chosen not to expand the
$O(R^{-2})$ pure divergence term $-\delta_{a}W^{a}$. Our assumption
about the fall-off of $\eta$ ensures that a term $-\eta^{2}/R$ which
appears in the asymptotic expression for $k$ can be
swept into $\Delta_{k}$.
\section{The Bondi-Sachs mass}
Write the total quasilocal energy as
$E = E |^{\scriptscriptstyle {\rm phy}} -
E |^{\scriptscriptstyle {\rm ref}}$,
with the total {\em unreferenced} quasilocal
energy $E |^{\scriptscriptstyle {\rm phy}}$ taken as
\begin{equation}
E |^{\scriptscriptstyle {\rm phy}}
= \frac{1}{\kappa} \int_{B(w_{*},R)}
{\rm d}^{2}x \sqrt{\sigma} k\, .
\label{unreferencedE1} \end{equation}
Plugging the expansion (\ref{kexpansion})
into the above expression, using
$\sqrt{\sigma} = R^{2}\sin\theta$ for our choice of coordinates,
and integrating term-by-term, one finds
\begin{equation}
E |^{\scriptscriptstyle {\rm phy}} = - R + M_{_{BS}}(w_{*})
+ \Delta_{E |^{\scriptscriptstyle {\rm phy}}}\, .
\label{unreferencedE2}
\end{equation}
Here the Bondi-Sachs mass associated with the $w = w_{*}$
cut of ${\cal I}^{+}$ is the two-surface
average of the mass aspect evaluated at $w = w_{*}$,\cite{Sachs,Goldberg}
\begin{equation}
M_{_{BS}}(w_{*})
= \frac{2}{\kappa} \int {\rm d}\Omega\,
{\rm m}(w_{*},\theta,\phi)\, .
\end{equation}
We use the notation $\int {\rm d}\Omega :=
\int^{\pi}_{0} {\rm d}\theta
\int^{2\pi}_{0}{\rm d}\phi\sin\theta$ to denote proper
integration over the unit sphere
(which is identified with a spherical cut of ${\cal I}^{+}$).
In passing from
(\ref{unreferencedE1}) to (\ref{unreferencedE2}),
we have made an appeal to Stokes' theorem to show that
the ``dangerous'' $O(R^{0})$ term that arises from proper
integration over the pure-divergence term
$\delta_{a} W^{a}$ in (\ref{kexpansion}) does indeed
vanish. Hence, this term does not contribute to the
Bondi-Sachs mass and does not spoil the result (\ref{unreferencedE2}).
The reference point contribution to the energy is
\begin{equation}
- E |^{\scriptscriptstyle {\rm ref}} =
- \frac{1}{\kappa} \int_{B(w_{*},R)}
{\rm d}^{2}x \sqrt{\sigma}
k |^{\scriptscriptstyle {\rm ref}}\, ,
\label{referencedE1}
\end{equation}
where the asymptotic expression for
$k |^{\scriptscriptstyle {\rm ref}}$ must
be determined from the specific asymptotic form (\ref{Bmetric}) of
the Sachs two-metric. We present this calculation in
the Appendix. The result is
\begin{equation}
- E |^{\scriptscriptstyle {\rm ref}} =
R + 0 \cdot R^{0} + \Delta_{E |^{\scriptscriptstyle {\rm ref}}}\, .
\label{referencedE2}
\end{equation}
Note the absence of an $O(R^{0})$ term in
$E |^{\scriptscriptstyle {\rm ref}}$. The result (3.5)
has just the right form, in that it removes the part
of $E |^{\scriptscriptstyle {\rm phy}}$ which becomes singular as
$R \rightarrow \infty$ but does not itself contribute
to the mass. Therefore the total quasilocal energy for large $R$ is
\begin{equation}
E = \int_{B(w_{*},R)}
{\rm d}^{2}x \sqrt{\sigma} \varepsilon
= M_{_{BS}}(w_{*}) + \Delta_{E}\, .
\end{equation}
This is the energy of the gravitational and matter fields associated with
the spacelike three-surface $\Sigma$ which spans a $B$ slice
of $\cal N$ and which tends toward $\Sigma_{\infty}$.
Our main claim (\ref{limit}) follows immediately from (3.6).
\section{Smeared Energy Surface Density}
Consider the expression $H_{B}$ for the on-shell value of the
gravitational Hamiltonian appropriate for a spatially bounded
three-manifold $\Sigma$, subject to the choice of a vanishing shift vector
at the boundary $\partial\Sigma = B$:
\begin{equation}
H_{B} = \int_{B}
{\rm d}^{2}x \sqrt{\sigma} N \varepsilon
{\,} .
\label{boundaryH}
\end{equation}
We refer to $H_B$ as the smeared energy surface density.
Addition of this boundary term to the smeared Hamiltonian
constraint ensures that as a whole the
sum is functionally differentiable.\cite{BY}
In this section we consider the $R\to\infty$ limit
of $H_B$ along the null hypersurface $\cal N$ in exactly the same
fashion as we considered the limit (1.1) of the quasilocal energy previously.
Before evaluating $\lim_{R \rightarrow \infty} H_{B(w_{*},R)}$,
let us discuss its physical significance. Consider a particular spherical
cut $B(w_{*},\infty)$ of ${\cal I}^{+}$. A general {\sc bms}
supertranslation pushes $B(w_{*},\infty)$ forward in retarded
time $w$ in a general angle-dependent fashion.
As is well-known, the infinitesimal generator corresponding
to such a supertranslation has the form
$\left. \alpha{\,} \partial/\partial w\right|_{{\cal I}^{+}}$,
where $\alpha(\theta,\phi)$ is any twice differentiable
function of the angular coordinates.\cite{Sachs}
As we have seen, $\partial/\partial w$ is heuristically the hypersurface
normal $u^{\mu}$ at $B(w_{*},\infty)$ of an asymptotic spanning
three-surface $\Sigma_{\infty}$. In other words, each member
of the fleet of observers at $B(w_{*},\infty)$ rides along
$\partial/\partial w$. Therefore, again heuristically, the
on-shell value of the Hamiltonian generator of a general {\sc bms}
supertranslation is
\begin{equation}
\int_{B(w_{*},\infty)}{\rm d}^{2}x
\sqrt{\sigma} \alpha \varepsilon\, .
\end{equation}
This symbolic expression coincides with the $R\to\infty$ limit
of the smeared energy surface density (4.1), where we
set $\alpha(\theta,\phi)
:= \lim_{R \rightarrow \infty} N(R,\theta,\phi)$ (suitable
fall-off behavior for $N$ is assumed). Thus,
$\lim_{R \rightarrow \infty} H_{B(w_{*},R)}$
defines a physically meaningful element in the dual space
of general supertranslations. In this respect it is like the
``supermomentum" of Geroch.\cite{Geroch} In the next section we show
explicitly that, in fact, $\lim_{R \rightarrow \infty}
H_{B(w_{*},R)}$ is precisely Geroch's ``supermomentum." Note,
however, that it might be
better to call such an expression the ``superenergy,'' as it arises
entirely from the ``energy sector'' of the Hamiltonian's boundary term
(that is, the sector with vanishing shift vector) but also incorporates
the ``many-fingered'' nature of time (that is, an arbitrary lapse function).
Let us now evaluate the $R\to\infty$ limit of the smeared energy surface
density $H_B$.
As we have stated, there is no $O(R^{0})$ contribution to
$E |^{\scriptscriptstyle {\rm ref}}$.
The absence of this contribution stems from
the fact that the two-sphere average of the coefficient
${}^{(2)}\!k |^{\scriptscriptstyle {\rm ref}}$ of the
$O(R^{-2})$ piece of the
reference term $k |^{\scriptscriptstyle {\rm ref}}$ vanishes.
As spelled out in the Appendix, this fact follows directly
from an equation governing the required isometric
embedding of $B$ into Euclidean three-space. Moreover, as seen in
Section 2, the coefficient ${}^{(2)}\! k$ of the $O(R^{-2})$ piece of the
physical $k$ is
not solely twice the mass aspect but also contains a unit-sphere
divergence term. Now, in the present case $\varepsilon
= (k - k |^{\scriptscriptstyle {\rm ref}})/\kappa$ is smeared
against a function
$N$, so one might worry that the limit is spoiled in some way by
the presence of the smearing function. However, as we now show,
{\em for solutions of the field equations}, the {\em unintegrated}
expression $4\pi R^{2}\varepsilon$ is precisely the mass aspect
of the system in the $R \rightarrow \infty$ limit. This striking
result rests on an exact cancellation between
${}^{(2)}\!k |^{\scriptscriptstyle {\rm ref}}$ and the aforementioned
unit-sphere divergence part of ${}^{(2)}\!k$.
With the machinery set up in the previous sections and the Appendix
[see in particular equations (\ref{kexpansion}) and (A2)], we find
the following limit:
\begin{equation}
\lim_{R \rightarrow \infty}
{\textstyle \frac{1}{2}}\kappa R^{2} \varepsilon =
{\rm m} (w_{*},\theta,\phi) -
{\textstyle \frac{1}{2}}
\left[\csc\theta\, \partial_{a}
(\sin\theta {}^{(2)}\! W^{a})
-{\textstyle \frac{1}{2}}
{}^{(3)}\!{\cal R} \right] \, .
\label{limk-k0}
\end{equation}
Here we set $\kappa = 8\pi$ (in geometrical units) and use
${}^{(3)}\!{\cal R}$ to denote the coefficient of the $O(R^{-3})$
piece of the $B$
Ricci scalar. Also, the coefficients ${}^{(2)}\! W^{a}$ of the leading
$O(R^{-2})$ pieces of $W^{a}$ are listed in (1.2c).
Inspection of (\ref{Sachs}c,d) shows that ${}^{(2)}\!W^{a}$
is expressed in terms of the same functions, $X$ and $Y$, that
appear in the $O(R^{-1})$ piece of $\sigma_{ab}/R^{2}$. Furthermore,
a short calculation with the $B$ metric shows that
for these solutions ${}^{(3)}\! {\cal R}$ may be
expressed in terms of ${}^{(2)}\! W^{a}$ as follows:
\begin{equation}
- {\textstyle \frac{1}{2}} {}^{(3)}\! {\cal R} =
- \csc\theta\, \partial_{a}(\sin\theta
{}^{(2)}\! W^{a}) \, .
\label{calR=dW}
\end{equation}
Therefore, the term in (\ref{limk-k0}) which is enclosed by square brackets
vanishes, and we obtain
\begin{equation}
\lim_{R \rightarrow \infty}
\int_{B(w_{*},R)} {\rm d}^{2}x \sqrt{\sigma} N
\varepsilon =
\frac{2}{\kappa}\int {\rm d}\Omega{\,} \alpha(\theta,\phi) {\rm m}
(w_{*},\theta,\phi)
{\,} , \label{limitresult}
\end{equation}
for the desired limit.
This result shows that the $R\to\infty$ limit of the smeared energy surface
density equals the smeared mass aspect. Coupled with the findings
of the next section, it
follows that Geroch's ``supermomentum" is just the smeared mass aspect.
This simple result does not appear to be widely known.
Finally, recall that the Bondi-Sachs
four-momentum components\footnote{Underlined Greek indices
refer to components of the total Bondi-Sachs four-momentum.}
$P_{_{BS}}^{\underline{\lambda}}$ correspond asymptotically to a pure
translation. In terms of the smeared energy surface density, one
obtains a pure translation for a judicious choice of lapse function
on $B(w_{*},\infty)$; namely, $\alpha(\theta,\phi)
= \epsilon_{\underline{\lambda}}
\alpha^{\underline{\lambda}}(\theta,\phi)$, where the
$\epsilon_{\underline{\lambda}}$ are constants and \cite{Goldberg}
\begin{eqnarray}
\alpha^{\underline{0}} & = & 1
\label{translations} \eqnum{\ref{translations}a} \\
\alpha^{\underline{1}} & = & \sin\theta\cos\phi
\eqnum{\ref{translations}b} \\
\alpha^{\underline{2}} & = & \sin\theta\sin\phi
\eqnum{\ref{translations}c} \\
\alpha^{\underline{3}} & = & \cos\theta
\eqnum{\ref{translations}d}\, .
\addtocounter{equation}{1}
\end{eqnarray}
Therefore, we write $\epsilon_{\underline{\lambda}}
P_{_{BS}}^{\underline{\lambda}}(w_{*}) =
\lim_{R \rightarrow \infty} H_{B(w_{*},R)}$ for the appropriate limiting
value of $N$, and thereby obtain the Bondi-Sachs four-momentum as a
Hamiltonian value.
\section{Supermomentum}
In this section we show that the null limit of the smeared Hamiltonian
boundary value, Eq.~(4.1), is the ``supermomentum" of Geroch.\cite{Geroch}
To be precise, we show that in the null limit $H_{B}$
equals Geroch's ``supermomentum" as written by Dray and
Streubel.\cite{DrayStreubel}
The spin-coefficient formalism is required for this
analysis.\footnote{Throughout $\S$V we deal
exclusively with smooth expansions in inverse powers of an
{\em affine} radius, as we know of no work examining the standard spin
coefficient approach to null infinity within a more general
framework such as the polyhomogeneous one.
The expansions we borrow from \cite{Dougan}
are valid for Einstein-Maxwell theory.}
Apart from a few minor notational changes we adopt the
conventions of Dougan.\cite{Dougan}
Geometrically, the scenario is nearly the same as the one described
in the previous sections. However, we now work with a
slightly different type of Bondi coordinates. Namely,
$(w,r,\zeta,\bar{\zeta})$, where $r$ is an {\em affine} parameter
along the null-geodesic generators of ${\cal N}$ and $\zeta =
e^{{\rm i}\phi}\cot(\theta/2)$ is the stereographic coordinate.
Dougan picks\footnote{Our $k_{\mu}$ and $l_{\mu}$
respectively correspond to
$l_{a} = \nabla_{a} u$ and $n_{a}$ in \cite{Dougan},
where $u$ is Dougan's
retarded time. The minus sign difference
between our definition for $k_{\mu}$ and Dougan's
definition for $l_{a}$ stems
from a difference in metric-signature conventions
[ours is $(-,+,+,+)$]. The convention for
metric signature does not affect the spin coefficients
(\ref{spinexpansions}).}
$k_{\mu} = - \nabla_{\mu} w$ as the first leg of a null tetrad,
which is the same normal as given in (\ref{bassae}a) if $\eta = 0$.
For convenience, in this section we ignore the kinematical freedom
associated with the $\eta$ parameter, setting it to zero throughout.
As before, the vector field $u^{\mu} := \frac{1}{2} k^{\mu} +
l^{\mu}$ (equality restricted to ${\cal N}$) defines a three-surface
$\Sigma$ spanning each $B$ slice of ${\cal N}$.
It follows that $u^{\mu}\partial/\partial x^{\mu}
\rightarrow \partial/\partial w$
as $r \rightarrow \infty$, and hence our asymptotic slice
$\Sigma_{\infty}$ again defines a pure {\sc bms} time translation.
Let us first collect the essential background results from
\cite{Dougan} which we will need. First, the required spin
coefficients have the following asymptotic expansions:\footnote{Note
the dual use of $\sigma$ as both the stem letter for
the $B$ two-metric and as the spin coefficient known as the shear.
We have used $\sigma$ twice in order to stick with the
conventions of our references as much as possible. In all but
equation (\ref{spinexpansions}), where $\sigma$ has the spin-coefficient
meaning, it carries a ``$0$'' superscript denoting the asymptotic piece.}
\begin{eqnarray}
\rho & = & - r^{-1} - \sigma^{0}
\bar{\sigma}{}^{0} r^{-3} + O(r^{-5})
\nonumber \\
\mu & = &
- {\textstyle \frac{1}{2}} r^{-1}
- [\Psi^{0}_{2} + \sigma^{0}
\dot{\bar{\sigma}}{}^{0}
+ \mbox{$\partial \hspace{-2mm} /$}^{2}_{{\scriptscriptstyle 0}}
\bar{\sigma}{}^{0}]r^{-2}
+ O(r^{-3})
\nonumber \\
\sigma & = &
\sigma^{0} r^{-2}
+ O(r^{-4})
\label{spinexpansions}
{\,} ,
\end{eqnarray}
where $\mu$ is Dougan's $- \rho'$, the term $\Psi^{0}_{2}$ is a certain
asymptotic component of the Weyl tensor, and $\sigma^{0}$
is the asymptotic piece of the shear. Like before, an over-dot denotes
differentiation by $\partial/\partial w$. As fully described in
\cite{Dougan}, $\ed{}_{\!\naught}$ is the standard differential operator from
the compacted spin-coefficient formalism, here defined on the {\em unit}
sphere. The expansion for the corresponding full operator in spacetime
is
\begin{equation}
\mbox{$\partial \hspace{-2mm} /$} = r^{-1} \ed{}_{\!\naught}
+ r^{-2} [s(\bar{\ed}{}_{\!\naught}\sigma^{0})
- \sigma^{0} \bar{\ed}{}_{\!\naught}] + O(r^{-3})
{\,} ,
\end{equation}
where $s =$ {\sc sw}$(\varphi)$, $\varphi$ being the spacetime
scalar on which $\mbox{$\partial \hspace{-2mm} /$}$ acts and {\sc sw} denoting {\em spin weight}.
The commutator of $\mbox{$\partial \hspace{-2mm} /$}$ and $\bar{\mbox{$\partial \hspace{-2mm} /$}}$ is
\begin{equation}
(\bar{\mbox{$\partial \hspace{-2mm} /$}}\mbox{$\partial \hspace{-2mm} /$} - \mbox{$\partial \hspace{-2mm} /$} \bar{\mbox{$\partial \hspace{-2mm} /$}})\varphi = {\textstyle \frac{1}{2}}
s {\cal R} \varphi {\,} .
\label{commutator}
\end{equation}
Now consider the following ansatz for the $B$ intrinsic
Ricci scalar:
\begin{equation}
{\cal R} = 2r^{-2} + {}^{(3)}\!{\cal R} r^{-3} + O(r^{-4})
{\,} .
\end{equation}
If we insert this expansion into (\ref{commutator}) and
expand both sides of the equation [assuming $\varphi =
{}^{(0)}\!\varphi + {}^{(1)}\!\varphi r^{-1} + O(r^{-2})$
with {\sc sw}$(\varphi)
= 1$], then to lowest order, namely $O(r^{-2})$, we get a trivial
equality. However, equality at the next order demands that
\begin{equation}
{\textstyle \frac{1}{2}} {}^{(3)}\!{\cal R}
= \bar{\ed}{}_{\!\naught}^{2} \sigma^{0}
+ \ed{}_{\!\naught}^{2} \bar{\sigma}{}^{0}
{\,} .
\label{result}
\end{equation}
This will prove to be a very important result for our purposes.
Finally, Dougan gives the following expansion for the $B$ volume element:
\begin{equation}
{\rm d}^{2}x\sqrt{\sigma} = {\rm d}\Omega {\,} r^{2}
(1 - \sigma^{0} \bar{\sigma}^{0} r^{-2})
+ O(r^{-2})
{\, } .
\end{equation}
(Here $\sigma$ is the determinant of the $B$ metric and $\sigma^0$
is the asymptotic piece of the shear.)
We now consider the spin-coefficient expression for the smeared energy
surface density introduced in $\S$IV. Again with $k = 2\mu + \rho$,
in the present notation we find
\begin{equation}
k = - 2r^{-1} -
2[\Psi^{0}_{2} + \sigma^{0} \dot{\bar{\sigma}}{}^{0}
+ \mbox{$\partial \hspace{-2mm} /$}^{2}_{{\scriptscriptstyle 0}}
\bar{\sigma}{}^{0}]r^{-2} + O(r^{-3})
{\,} .
\end{equation}
Moreover, by an argument identical to the one found in the last
paragraph of the Appendix (although here with the affine
radius $r$ rather than the areal radius $R$),
we know that the result (\ref{result}) determines
\begin{equation}
k |^{\scriptscriptstyle {\rm ref}} = - 2 r^{-1}
-(\bar{\ed}{}_{\!\naught}^{2} \sigma^{0}
+ \ed{}_{\!\naught}^{2} \bar{\sigma}{}^{0}) r^{-2} + O(r^{-3})
\end{equation}
as the appropriate asymptotic expansion for the reference term.
Therefore, ($\kappa$ times) the full quasilocal energy surface density is
\begin{equation}
\kappa\varepsilon =
- 2[\Psi^{0}_{2} + \sigma^{0} \dot{\bar{\sigma}}{}^{0}
+ {\textstyle \frac{1}{2}}\ed{}_{\!\naught}^{2}
\bar{\sigma}{}^{0} - {\textstyle \frac{1}{2}}\bar{\ed}{}_{\!\naught}^{2} \sigma^{0}
]r^{-2} + O(r^{-3})
{\,} .
\end{equation}
At this point we consider again
a smearing function $N$, with appropriate
fall-off behavior and limit $\alpha(\zeta,\bar{\zeta})
= \lim_{r \rightarrow \infty} N(r,\zeta,\bar{\zeta})$. Using the results
amassed up to now, one computes that the limit of the smeared energy
surface density is
\begin{equation}
\lim_{r \rightarrow \infty} \int_{B(w_{*}, r)}
{\rm d}^{2}x \sqrt{\sigma} N\varepsilon = - \frac{2}{\kappa} \int
{\rm d}\Omega{\,}\alpha\left. [\Psi^{0}_{2}
+ \sigma^{0} \dot{\bar{\sigma}}{}^{0}
+ {\textstyle \frac{1}{2}}\ed{}_{\!\naught}^{2}
\bar{\sigma}{}^{0} -
{\textstyle \frac{1}{2}}\bar{\ed}{}_{\!\naught}^{2} \sigma^{0}]\right|_{w = w_{*}}
{\,} .
\label{superE}
\end{equation}
The right-hand side of this equation is the ``supermomentum'' of Geroch as
written by Dray and Streubel [see equation (A1.12) of \cite{DrayStreubel}
and set their $b = 0$ for a Bondi frame as we have here]; and the ``supermomentum''
is known to be the ``charge integral" associated with the
Ashtekar-Streubel flux\cite{AshtekarStreubel}
of gravitational radiation at ${\cal I}^{+}$ (in the restricted case
when the flux is associated with a supertranslation).\cite{Shaw}
Dray and Streubel
have discussed the importance of the particular factors of $\frac{1}{2}$
which multiply the last two terms within the square brackets on the
right-hand side of equation (\ref{superE}). It is evident from our approach
that the origin of these $\frac{1}{2}$ factors stems from the flat-space
reference of the quasilocal energy (flat-space being the correct reference
in the present context).
When $\alpha$ determines
a pure {\sc bms} translation, the last two terms in the integrand integrate
to zero. For instance, setting $\alpha = 1$, one finds that the strict energy
\begin{equation}
E = \int_{B(w_{*},\infty)}
{\rm d}^{2}x\sqrt{\sigma}
\varepsilon = - \frac{2}{\kappa}
\int {\rm d}\Omega \left. [\Psi^{0}_{2} + \sigma^{0}
\dot{\bar{\sigma}}{}^{0}]\right|_{w = w_{*}}
{\,}
\end{equation}
is the standard spin-coefficient expression for the Bondi-Sachs mass
$M_{_{BS}}(w_{*})$.\cite{Dougan,ExtonNewmanPenrose}
\section*{Acknowledgments}
We thank H. Balasin, P. T. Chru\'{s}ciel, T. Dray, and N. \'{O} Murchadha
for helpful discussions and correspondence. We acknowledge support from National
Science Foundation grant 94-13207. S.\ R.\ Lau has been chiefly supported by
the ``Fonds zur F\"{o}r\-der\-ung der wis\-sen\-schaft\-lich\-en For\-schung''
in Austria (FWF project P 10.221-PHY and Lise Meitner Fellowship M-00182-PHY).
| proofpile-arXiv_065-464 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
proofpile-arXiv_065-465 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
|
\section{Introduction}
Blazars are
variable, polarized,
flat spectrum extragalactic radio sources with a non-thermal continuum
extending to $\gamma$-ray energies ({\it e.g.,\ } Urry \& Padovani 1995).
The radio emission from blazars is collimated into narrow beams composed of
many individual knots and an optically thick core (Kellerman \& Pauliny-T\"oth
1981).
In virtually all blazars, the radio knots appear to separate from the core
at speeds greater than the speed of light
(Urry \& Padovani 1995)
and this superluminal motion is strong evidence that blazars are relativistic
jets of magnetized plasma viewed along the jet axis
(Blandford \& K\"onigl 1979).
Although the jet model also accounts for such diverse blazar
properties as
the flat radio spectrum (Kellerman \& Pauliny-T\"oth 1981),
the short variability time scales
at all energies ({\it e.g.,\ } Quirrenbach, {\it et~al.\ } 1991, Maraschi, {\it et~al.\ } 1992),
the small synchrotron self-Compton X-ray fluxes (Marscher 1987,
Ghisellini, {\it et~al.\ } 1992) and the
large $\gamma$-ray fluxes (Maraschi, {\it et~al.\ } 1992),
the radiative processes which produce the blazar continuum have not been
identified. In this paper, we calculate the relative amplitudes of variations
in the radio and X-ray fluxes for
one of the most popular models of the blazar continuum,
synchrotron self-Compton scattering in a relativistic outflow
(Jones, O'Dell \& Stein 1974, Marscher 1977,
K\"onigl
1981, Ghisellini, {\it et~al.\ } 1985), and we compare the results to
observations of correlated
variations on the radio and X-ray flux from 3C279 (Grandi, etal 1995: G95).
Blazar spectra are nearly featureless and
a large number of models, which make very different assumptions for the
relevant radiation processes, agree qualitatively with
snapshots of the radio to $\gamma$-ray spectrum (Maraschi, {\it et~al.\ } 1992,
Maraschi, {\it et~al.\ } 1994,
Hartmann {\it et~al.\ } 1996). At low energies, the flat radio spectrum of compact
radio cores can be explained by inhomogeneous synchrotron emission
(Marscher 1977, K\"onigl
1981, Ghisellini, {\it et~al.\ } 1985) or by a homogeneous core that is optically thick
to induced
Compton scattering (Sincell \& Krolik 1994: SK94). The situation at high
energies is even more complicated. The X- and $\gamma$-ray emission may
be synchrotron emission by high energy electrons (K\"onigl 1981,
Ghisellini, {\it et~al.\ } 1985), radiation
from a pair cascade (Blandford \& Levinson 1995) or low frequency photons which are inverse
Compton scattered by relativistic electrons in the jet.
In the last case, the source
of the low energy photons could be synchrotron radiation from the jet (the
synchrotron self-Compton model, Maraschi {\it et~al.\ } 1992,
Maraschi, {\it et~al.\ } 1994, Bloom \& Marscher 1996), UV radiation from
a disk (Dermer, Schlickeiser \& Mastichiadis 1992), or some diffuse source of
radiation surrounding the jet
(Sikora, Begelman \& Rees 1994, Ghisellini \& Madau 1996).
We will consider only synchrotron self-Compton scattering in this paper and
ignore other sources for the high energy emission.
The synchrotron self-Compton model for the continuum emission
assumes that
a single population of relativistic electrons radiates synchrotron photons and
subsequently scatters a fraction of these to higher energies
(Jones, {\it et~al.\ } 1974).
Fluctuations in either the electron density, the magnetic field strength or
the Doppler factor of the emission region will affect the synchrotron and
self-Compton fluxes instantaneously. Therefore, variations in the low and
high energy fluxes which are uncorrelated, or have significant time delays,
cannot be the result of synchrotron self-Compton scattering. The relative
amplitudes of the variations in the synchrotron and self-Compton fluxes depends
upon which physical parameters change. For example, the fractional change in
the high energy flux is twice as large as the fractional change in the
optically thin synchrotron flux
when the electron density varies, whereas they are equal if the magnetic field
or the Doppler factor changes (SK94).
The amplitude of the variations in the synchrotron and self-Compton fluxes
also depends upon the relativistic electron distribution. The electron
spectrum will flatten with increasing radiation intensity because both the
synchrotron and inverse Compton cooling rates increase with increasing
radiation energy density.
However, these cooling processes are both too slow to have any
effect upon the electron distribution in the parsec scale jet (SK94).
Therefore,
we neglect the evolution of the electron spectrum in the calculations
presented in this paper.
SK94 demonstrated that induced Compton scattering can reduce the
amplitudes of variations in the
synchrotron radiation when the brightness temperature of the source
$T_B \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 2 \times 10^{11} \mbox{K}$.
The amplitudes of variations in the self-Compton X-ray flux are unaffected by
induced Compton scattering because the X-ray flux is dominated by
photons scattered from
the high-frequency, low $T_B$, end of the synchrotron spectrum, where
the stimulated scattering optical depth is small.
Thus, the relative amplitudes of the variations in the
synchrotron flux at the self-absorption turnover frequency and the self-Compton
flux at 1 keV are reduced from $\sim 0.4$ to $\sim 0.2$ when the induced
Compton scattering optical depth is large (SK94).
The continuum emission from 3C279, one of the most intensively monitored
blazars, varies coherently over its entire spectrum (Maraschi, {\it et~al.\ } 1992,
Maraschi {\it et~al.\ } 1994, Hartmann, {\it et~al.\ } 1996).
Recently, G95 used historical light curves to show that the
radio-mm and 1 keV X-ray fluxes from 3C279 are strongly correlated.
The maximum time resolution of the 3C279 light curves was $\sim 70 \mbox{d}$
and the absence of any detectable time delay
implies that the radio-mm and X-ray fluxes
are from physically related regions separated by $\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 0.06 \mbox{pc}$.
While this is consistent with the assumptions of the synchrotron self-Compton
theory, the relative amplitudes of the radio and X-ray variations are smaller
than predicted (Sincell 1996).
In this paper, we extend previous calculations (SK94, Sincell 1996)
and compute the relative amplitudes of
variations in the
synchrotron self-Compton flux as a function of the frequency of
the synchrotron emission. We first define the flux variability ratio (\S
\ref{sec: flux variability ratio}) and describe how it is calculated. This
ratio is computed for three simple types of variations in \S\ref{sec: results}
and the implications for 3C279 are discussed in \S \ref{sec: 3C279}. We
conclude in \S \ref{sec: conclusions}.
\section{\bf The Flux Variability Ratio}
\label{sec: flux variability ratio}
The time variability of the source spectrum is approximated as a sequence of
steady
state spectra.
We have used the code developed in SK94 to compute the steady-state
synchrotron spectrum and
self-Compton X-ray flux from a homogeneous sphere containing an isotropic
power-law distribution of relativistic electrons
\begin{equation}
\label{eq: electron distribution}
{\partial n_e \over \partial \gamma} = n_o \gamma^{-p}
\end{equation}
for $\gamma \geq \sqrt{2}$. We assume $p=2.5$ in all the simulations and the
normalization ($n_o$) is fixed by the assumed $\tau_T$.
This code incorporates synchrotron absorption and emission, inverse Compton
scattering and induced Compton scattering by the relativistic electrons.
Stimulated scattering becomes the dominant source of radio opacity when
$T_B \mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 2 \times 10^{11} \mbox{K}$ (SK94)
and must be included when calculating the spectra of compact radio sources.
Self-absorption reemerges as the dominant opacity source at low frequencies
and inverse Compton scattering of synchrotron photons by electrons with
$\gamma \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 10$ contributes to the radio flux above the self-absorption
turnover.
The flux variability ratio
\begin{equation}
R_m(\tau_T,B,\nu) = {\partial \log S_r(\tau_T,B,\nu)
\over \partial \log S_x(\tau_T,B)}|_m,
\end{equation}
is the ratio of the fractional change in the synchrotron flux at
$\nu$ ($S_r$) and
the self-Compton X-ray flux ($S_x$) at 1 keV caused by a fluctuation
in the physical
parameter $m$. In this paper we investigate three different variations:
the Thomson
depth ($\tau_T$) of the source varies at fixed magnetic field strength
($m=\tau$),
the magnetic field strength ($B$) varies at fixed $\tau_T$ ($m=B$) and equal
fractional changes in $\tau_T$ and $B$ $(m=S)$.
The third case approximates the
passage of a strong shock through the plasma, assuming that the field is
tangled ({\it e.g.,\ } Marscher \& Gear 1985).
We also assume that $\tau_T$ and $B$ are uniform throughout the source.
Simple analytic calculations of $R_m$ are possible for
synchrotron self-Compton scattering (SK94),
but $R_m$ must be calculated numerically when the induced Compton
scattering opacity is large (SK94).
We compute the variability ratio using the approximate formula
\begin{equation}
R_m(\nu) \simeq {S_r(m,\nu) - S_r(m+\Delta m,\nu) \over
S_r(m,\nu) + S_r(m+\Delta m,\nu)}
\cdot {S_x(m) + S_x(m+\Delta m) \over
S_x(m) - S_x(m+\Delta m)},
\end{equation}
and the radio spectra and X-ray fluxes from two models
with closely spaced values of
the parameter
$m$. In this paper we choose $\Delta m/m =0.1$ but the results are fairly
insensitive to $\Delta m$, even when $\Delta m/m \mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 1$ (see figures in
Sincell 1996).
\section{Results}
\label{sec: results}
We have calculated $R_{\tau}$, $R_{B}$ and $R_S$ for
$0.01 \leq \tau_T \leq 3.0$
and a range of $B$.
In figs. \ref{fig: tau variability}, \ref{fig: b variability}
and \ref{fig: shock variability} we plot $R_m$ as a function of $\nu / \nu_o$,
where $\nu_o$ is the synchrotron self-absorption turnover
frequency.
When the effects of induced and inverse Compton scattering on the radio
spectrum are neglected, $R_m(\nu/\nu_o)$ is independent of
the unperturbed values of $\tau_T$ and $B$. Including Compton scattering
in the computation of the radio spectrum
introduces a strong dependence upon $\tau_T$ but $R_m(\nu/\nu_o)$ remains
nearly independent of $B$ because the induced Compton scattering optical depth
at $\nu_o$ is $\propto T_B \tau_T \propto B^{-1/(p+4)} \tau_T^{(p+5)/(p+4)}$
(SK94).
The code was used to verify that $R_m$ changed by less than a few percent when
$B$ increased by two orders of magnitude.
Therefore, we present $R_m$ for $B=10^{-5} \mbox{G}$
and use the relation ({\it e.g.,\ } SK94)
\begin{equation}
\label{eq: nuo scaling}
\nu_o \propto
\left( {\delta \over 1+z} \right)
B^{(p+2)/(p+4)}
\end{equation}
to scale $R_m$ to any desired field strength,
Doppler boost ($\delta$) or redshift ($z$).
Although $R_m(\nu/\nu_o)$ is
independent of the unperturbed values of $\tau_T$ and $B$ when the effects
of Compton
scattering on the radio spectrum are neglected,
it does depend on which parameters vary.
At optically thin frequencies, $\nu \gg \nu_o$,
$R_{\tau} \simeq 0.5$, $R_{B} \simeq 1.0$ and $R_{S} \simeq 0.7$,
independent of frequency
(figs. \ref{fig: tau variability}, \ref{fig: b variability}
and \ref{fig: shock variability}).
These numerical values are in good agreement with the
analytic results in SK94.
Synchrotron self-absorption reduces $R_m$ at $\nu \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} \nu_o$.
This is because any
change in the physical parameters which increases the synchrotron
emissivity results
in a compensating increase in the opacity and decrease in the photospheric
depth.
Thus, the net flux at optically thick frequencies is less
sensitive to changes in the source parameters.
The ratio of the self-absorption opacity to the emissivity
increases rapidly as $\nu/\nu_o$ decreases ({\it e.g.,\ } SK94)
and
$R_{\tau}$ approaches zero at $\nu \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}}
\nu_o / 2$.
Increasing $B$ also increases $\nu_o$ (eq. \ref{eq: nuo scaling}) and
the increase in the self-absorption opacity at fixed $\nu$
overwhelms the increase in the
emissivity when $\nu \ll \nu_o$.
The resulting decrease in the synchrotron flux at $\nu$ appears as
negative values of $R_{B}$ and $R_{S}$
(figs. \ref{fig: b variability}
and \ref{fig: shock variability}).
Negative values of $R_m$ correspond to an anti-correlation
of the synchrotron and self-Compton fluxes.
Compton scattering changes the frequency dependence
of $R_m$ and introduces a dependence on $\tau_T$
(figs. \ref{fig: tau variability}, \ref{fig: b variability}
and \ref{fig: shock variability}).
Induced Compton scattering reduces $R_{\tau}$ over more than
a decade in frequency when $\tau_T \mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 0.1$
(fig. \ref{fig: tau variability}), relative to the
synchrotron self-Compton scattering model without stimulated scattering.
$R_{\tau}$ at $\nu \simeq \nu_o$ is reduced by almost
an order of magnitude when $\tau_T \mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 1$.
The stimulated scattering opacity at low frequencies and the
contribution of inverse Compton scattered photons at higher frequencies
results in a monotonic increase in $R_{\tau}$ from $0.01$ at $\nu \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}}
\nu_o$ to $0.5$ at $\nu \sim 100 \nu_o$ (fig. \ref{fig: tau variability}).
Even though synchrotron self-absorption is the dominant source of opacity,
stimulated scattering increases the photon
occupation numbers at
low frequencies. This increases the synchrotron flux at $\nu \ll \nu_o$
and the
anti-correlation of the synchrotron and self-Compton fluxes caused
by variations in $B$ disappears when $\tau_T \mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 1$.
Variations
in the magnetic field strength introduce local extrema into $R_{B,S}$ when
the stimulated scattering optical depth is large.
The largest contribution
to the induced Compton scattering opacity at $\nu \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} \nu_o$ is from
electrons
with
$\gamma_{*}
= {1 \over 2} \left( {\nu_m \over \nu} \right)^{1/2}$
where $\nu_m \mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} \nu_o$ is the peak of the spectrum (SK94).
The low energy cutoff $\gamma = \sqrt{2}$ reduces the stimulated scattering
opacity at frequencies
$ \nu_m/8 \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} \nu \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 8\nu_m$
because $\gamma_* < \sqrt{2}$ and there are no electrons which couple
$\nu$ to $\nu_m$.
When synchrotron self-absorption is the dominant source of opacity,
the optical depth of the plasma
decreases with frequency and $R$ increases with frequency.
A local maximum in $R$ appears at the frequency where the stimulated
scattering opacity is approximately equal to the synchrotron opacity. At
higher frequencies, induced Compton scattering
limits the variations in the synchrotron flux and reduces $R$.
The local minimum in $R$ occurs at $\nu \sim \nu_m/8$ where the stimulated
scattering opacity
reaches a maximum.
These local extrema are not as prominent in $R_{\tau}$ because variations
in $\tau_T$ increase the optical depth at all frequencies.
However, the kink in the $\tau_T = 0.1$ curve of $R_{\tau}$
(fig. \ref{fig: tau variability}) is also due to this effect.
\section{\bf 3C279}
\label{sec: 3C279}
G95 used historical light curves of 3C279 to show that variations in
the radio-mm and
X-ray fluxes are strongly correlated with a time delay of $\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 70$~days.
They also calculated the logarhithmic dispersion, or variability amplitude
($v(\nu)$), of
the available measurements and found that $v(\nu)$ increases systematically
with frequency.
However, these estimates of $v(\nu)$ are uncertain because many emission
components contribute to the observed flux at a given frequency ({\it e.g.,\ }
Unwin, {\it et~al.\ } 1989). These components may vary independently and the G95
data lacks the spatial resolution necessary to reliably subtract the
non-variable background.
Long-term monitoring at higher resolution (VLBA) is necessary to remove this
source of uncertainty. In the remainder of this paper we will assume that
the variable component dominates the observed flux, but it should be remembered
that a significant amount of non-variable flux at $\nu$ will reduce $v(\nu)$
below the value expected for the variable component alone.
Both the strong correlation of the radio and X-ray fluxes and the absence
of a detectable time delay between variations in the two bands
are consistent
with the assumptions of the synchrotron self-Compton model for the continuum
emission.
We used the variability amplitudes calculated by G95
to estimate
\begin{equation}
R \simeq {v(\nu) \over v(1 \mbox{keV})}
\end{equation}
at four frequencies in the range $14.5 \mbox{GHz} < \nu < 230 \mbox{GHz}$.
The estimated $R$ for the epochs 1988-1991.4 and 1991.4-1993.2
are plotted in fig. \ref{fig: observed variability}.
The model $R_{\tau}$ for $\tau_T = 1.0$, $B=10^{-3} \mbox{G}$ and
$\delta = 20$ is displayed on the same figure.
The relative amplitudes of the variations in the radio and X-ray fluxes from
3C279 are consistent with the synchrotron self-Compton model if $\tau_T$
varies in a fixed magnetic field and induced Compton scattering is the
dominant source of radio opacity.
It is immediately apparent from fig. \ref{fig: observed variability}
that the magnitudes of both $R_{B}$ and $R_{S}$ are
too large to fit the observed values of $R$ for 3C279.
In addition, neither the local extrema in $R$ or the anti-correlation
of the synchrotron and self-Compton fluxes are observed.
We also find that the increase in $R$ with
frequency is much slower than expected for the synchrotron self-Compton
model without induced Compton scattering
(fig. \ref{fig: observed variability}). However, both the magnitude
and the frequency dependence of $R$ are consistent with $R_{\tau}$ when
$\tau_T \mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 1$.
This implies that $\tau_T
\sim 1$ in the synchrotron self-Compton emission region.
The magnetic field strength and Doppler factor cannot be
calculated independently (eq. \ref{eq: nuo scaling}), but the values we have
assumed ($B = 10^{-3} \mbox{G}$ and $\delta = 20$) are consistent with
other estimates of the physical parameters for 3C279 (Ghisellini, etal 1985,
Maraschi, {\it et~al.\ } 1992, SK94). Larger magnetic fields require smaller Doppler
factors and vice versa.
We can set a lower limit on the electron density using the variability time
scale and the requirement $\tau_T \sim 1$. The linear dimension of the
emission region
$l \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} l_{max} = 1.8 \times 10^{17} \delta \mbox{cm}$
if the observed flux varies on a time scale of $\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 70$ days.
The stimulated scattering optical depth of the plasma will be large enough to
reduce the variability
amplitudes if the electron density
$n_e \mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 8 \times 10^6 \delta^{-1} \mbox{ ${\rm cm^{-3}}$}$.
The total particle energy density of the distribution in eq.
\ref{eq: electron distribution} is dominated by the rest mass energy so
$U_e \sim n_e m_p c^2 \sim 7 \delta^{-1} \left( {m_p \over m_e} \right)
\mbox{ergs ${\rm cm^{-3}}$}$,
where $m_{p,e}$ are the masses of the positively charged particle and an
electron, respectively. The magnetic energy
density $U_B \ll U_e$ and the plasma is far from equipartition.
A strong shock passing through the synchrotron emission region amplifies
both the electron density and the magnetic field strength if the magnetic
field is either tangled or aligned parallel to the shock front ({\it e.g.,\ }
Marscher \& Gear 1985). Our results
for 3C279 indicate that the variations in the flux are due to fluctuations
in the electron density alone. Thus, we conclude that if the variations
are caused by a shock the magnetic field must be aligned perpendicular to the
shock front. An alternative possibility is that the observed variations are
due to fluctuations in the local
electron density caused by a change in the particle injection rate.
\section{\bf Conclusions}
\label{sec: conclusions}
We have calculated $R_m(\nu)$, the relative amplitude of variations in the
synchrotron flux at
$\nu$ and the self-Compton
X-ray flux at 1 keV, for a homogeneous sphere
of relativistic electrons orbiting in a tangled magnetic field.
The index $m$ refers to the physical quantity which is assumed to vary and
in this paper we investigate three cases: variations in $\tau_T$ at fixed
$B$, variations in $B$ at fixed $\tau_T$ and equal fractional changes in both
quantities. The last case approximates the passage of a strong shock through
the plasma ({\it e.g.,\ } Marscher \& Gear 1985). We find $R_m$ to be useful for two
reasons. First, the frequency dependence of $R_m$
is determined by the optical depth of the plasma. Second, $R$ may be estimated
directly from observations of correlated radio and X-ray variability ({\it e.g.,\ }
G95).
If synchrotron self-absorption is the dominant source of
opacity,
the frequency dependence of $R_m$ is determined by $\nu_o$, the self-absorption
turnover frequency, and the physical parameter which is assumed to vary.
We find that $R_m$ is constant at all optically thin
frequencies ($\nu \gg \nu_o$) and, for our assumed electron distribution
(eq. \ref{eq: electron distribution}), $R_{\tau} \simeq 0.5$, $R_B \simeq 1.0$
and $R_S \simeq 0.7$. Self-absorption causes all the $R_m$ to drop sharply
at $\nu \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} \nu_o$ and both $R_{B,S}$ become negative at $\nu \ll \nu_o$.
Induced Compton scattering reduces $R_m$ over more than a decade in
frequency, relative to the synchrotron self-Compton model without
stimulated scattering, when $\tau_T \mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 0.1$.
Increasing $\tau_T$ reduces $R_{\tau}$ at frequencies near $\nu_o$
and $R_{\tau}$
can be an order of magnitude smaller than the self-absorbed value when $\tau_T
\mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 1$. We find that $R_{\tau}$ increases monotonically from low to high
frequencies, but a slight change in the stimulated scattering opacity at
$\nu \sim \nu_m / 8$ causes
local extrema in $R_{B,S}$.
Variations in the Thomson depth of a homogeneous source of synchrotron
self-Compton
radiation reproduces
the relative amplitudes of the correlated radio and X-ray flux variations in
3C279 (G95)
if $\tau_T \sim 1$ and the emission region is optically
thick to induced Compton scattering. Although $B$ and $\delta$ cannot be
independently constrained (eq. \ref{eq: nuo scaling}), the observed $R$ is
consistent with $B \sim 10^{-3} \mbox{G}$ and $\delta \sim 20$. If we
assume that the maximum linear dimension of the emission region is $l_{max}
\sim 0.06 \delta$ pc, as implied by the variability time scale, $\tau_T \sim 1$
requires that the electron
density be $n_e \mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 8 \times 10^6 \delta^{-1}$~cm$^{-3}$. In this case,
the particle energy density is much larger than the magnetic field energy
density.
Variations in the magnetic field strength result in values
of $R$ which are larger than observed. If the observed fluctuations are due
to the passage of a shock, the magnetic field must be oriented perpendicular
to the shock front. Variations in the local particle injection rate could
change the electron density without necessarily changing the field strength.
Finally, it has been argued that stimulated scattering cannot be important in
blazars with optically thick spectral indices of $\alpha = -5/2$
(Litchfield, {\it et~al.\ } 1995). This
argument is erroneous because synchrotron self-absorption {\it always}
becomes the
dominant source of opacity, and $\alpha = -5/2$, at low enough frequencies
(SK94). However, this points out the ambiguities inherent in attempting to
determine the
radio
opacity using spectral measurements alone. Additional multi-wavelength
variability
studies ({\it e.g.,\ } G95) are clearly necessary to determine the relevant
radiative processes in blazars.
\acknowledgements
We thank the referee, Laura Maraschi, for helpful comments and Stephen Hardy
for pointing out the error in the kinetic equation.
Support for this work was provided by
NASA
grants NAGW-1583 and NAG 5-2925, and NSF grant AST 93-15133. The simulations
were performed at the Pittsburgh Supercomputing Center (grant AST 960002P).
| proofpile-arXiv_065-466 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
We propose a new search for the time reversal violating
polarization of the muon normal to the plane of
the $K^+ \to \mu^+ \pi^0 \nu$ decay~\cite{proposal}.
The term $\vec{\sigma_\mu}
\cdot (\vec{P_\pi} \times \vec{P_\mu})$,
which is proportional to the projection of the muon polarization out
of the
decay plane,
changes sign upon time reversal; therefore a finite expectation value
for this quantity indicates a violation of time reversal invariance.
Moreover, since the Standard Model prediction for such polarization is zero,
and there is no final state interaction,
the observation of T-violation in the $K_{\mu 3}$ decay is a discovery of
T-violation beyond the Standard Model.
Through the CPT theorem we know that T-invariance is intimately
related to CP-invariance.
Although the only observed CP-violation in the
neutral kaon system can be attributed to the complex phase in the
CKM matrix within the Standard Model
the true nature of CP-violation is far from
being revealed by the current experimental data.
It is now accepted that the baryon asymmetry of the universe
requires a source of CP violation stronger than that embodied in the
CKM matrix.
Models of non-standard CP violation that produce the baryon asymmetry
could also produce effects observable in the transverse polarization.
Because of the very high sensitivity
of the experiment the possibility of discovering unexpected new
physics should not be underestimated.
The best previous experimental limits were obtained over 15 years
ago with 4 GeV charged kaons~\cite{campbell} at the
AGS, yielding a result of $P^T_\mu = 0.0031 \pm 0.0053$.
The high intensity kaon beams available now at the AGS
makes it possible to improve the limit on the polarization
by more than an order of magnitude.
\section{Detector Design}
\begin{figure}
\psfig{figure=pict1.eps,height=2.8in,width=4.5in,angle=90}
\caption{Schematic of the detector. A typical $K^+\rightarrow
\mu^+\pi^0\nu$ events is superimposed.}
\label{pict1}
\end{figure}
The experiment will be performed with 2 GeV/c electrostatically
seperated charged kaons
decaying in flight.
The beam intensity will be $2\times 10^7 K^+$'s/spill with
$3\times 10^{13}$ protons on target every 3.6 sec.
Figure \ref{pict1}
shows the plan view
of the experiment. The basic workings of the experiment
are the same as the experiment in Reference 2.
The detailed design is, however, optimized for a
high intensity 2 GeV beam.
The cylindrically symmetric detector
is centered on the kaon beam. The $K^+_{\mu 3}$ ~decays of interest
occur in the decay tank; the photons from the
$\pi^0$ decay are detected in the calorimeter; the muon stops
in the polarimeter.
The decay of the stopped muon is
detected in the polarimeter by wire chambers, which are
arranged radially with graphite wedges that serve as absorber medium.
The hit pattern in the polarimeter identifies the muon stop as well as
positron direction relative to the muon stop.
By selecting events with $\pi^0$ moving along the beam direction and
muon moving perpendicular to the beam direction in the $K^+$ center of
mass frame, the decay plane coincides with the radial wedges.
A non-zero transverse muon polarization causes an asymmetry between
the number of muons that decay
clockwise versus the number counter-clockwise.
To reduce systematic errors, a weak solenoidal magnetic field along
the beam direction
(70 gauss or an precessing period of $\sim 1 \mu s$) with
polarity reversal every spill is applied to the polarimeter.
The initial muon transverse polarization causes a small shift in the
phase of the sinusoidal oscillation in the measured asymmetry.
The difference in the asymmetry for the two polarities is proportional
to the muon polarization in the decay plane, while the sum is
proportional to the muon polarization normal to the decay plane.
Compared to the previous experiment, this experiment has
much better background rejection and event reconstruction.
The separated $K^+$ beam should greatly reduce the accidental rate.
The polarimeter is fine segmented and the analyzing power is higher.
The positron signature is defined by the coincidence of signals
in a pair of neighboring wedges.
The larger calorimeter makes it possible to reconstruct the $\pi^0$
momentum. Together with the muon trajectory, the event can be fully
reconstructed.
The detector
acceptance and background rejection is optimized using GEANT
simulation.
We expect to obtain 550 events/spill, with up to 20\%
background. With an analyzing power of over 30\%, we expect to
reach the statistical sensitivity of
$\delta P_t = 1.3\times 10^{-4}$.
At such a high statistical accuracy, much care has to be taken
in reducing systematic errors. We have studied various effects that
may give a false signal, such as misalignment within the polarimeter,
misalignment among and detector components and the beam, asymmetry in
or caused by the precessing field and inefficiencies.
We believe that
these errors can be made acceptably small by proper construction techniques
and by using symmetries of the apparatus to internally cancel the
systematic errors. In addition, we will use the T-conserving
component of the muon polarization to calibrate the detector analyzing
power, and samples of muon stops with no known transverse
polarization, such as muons from $K_{\mu 2}$ decays, to detect any
systematic bias.
The proposal has been submitted to the Laboratory in Aug 1996.
If approved and funded, we would like to have the first engineering
run in 1998. The physics data taking will take about 2000 hours
of running time.
| proofpile-arXiv_065-467 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
proofpile-arXiv_065-468 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
|
\section*{Table of Contents}
\noindent \S0 - INTRODUCTION
\medskip
\noindent \S1 - THEIR PROTOTYPE IS ${\frak K}^{tr}_{\langle\lambda_n:n< \omega
\rangle}$, NOT ${\frak K}^{tr}_\lambda$!
\medskip
\noindent \S2 - ON STRUCTURES LIKE $(\prod\limits_n
\lambda_n,E_m)_{m<\omega}$, $\eta E_m\nu =:\eta(m) = \nu(m)$
\medskip
\noindent \S3 - REDUCED TORSION FREE GROUPS; NON-EXISTENCE OF UNIVERSALS
\medskip
\noindent \S4 - BELOW THE CONTINUUM THERE MAY BE UNIVERSAL STRUCTURES
\medskip
\noindent \S5 - BACK TO ${\frak K}^{rs(p)}$, REAL NON-EXISTENCE RESULTS
\medskip
\noindent \S6 - IMPLICATIONS BETWEEN THE EXISTENCE OF UNIVERSALS
\medskip
\noindent \S7 - NON-EXISTENCE OF UNIVERSALS FOR TREES WITH SMALL DENSITY
\medskip
\noindent \S8 - UNIVERSALS IN SINGULAR CARDINALS
\medskip
\noindent \S9 - METRIC SPACES AND IMPLICATIONS
\medskip
\noindent \S10 - ON MODULES
\medskip
\noindent \S11 - OPEN PROBLEMS
\medskip
\noindent REFERENCES
\eject
\section{Introduction}
{\bf Context.}\hspace{0.15in} In this paper, model theoretic notions (like
superstable, elementary classes) appear in the introduction but not in the
paper itself (so the reader does not need to know them). Only naive set
theory and basic facts on Abelian groups (all in \cite{Fu}) are necessary
for understanding the paper. The basic definitions are reviewed at the end
of the introduction. On the history of the problem of the existence of
universal members, see Kojman, Shelah \cite{KjSh:409}; for more direct
predecessors see Kojman, Shelah \cite{KjSh:447}, \cite{KjSh:455} and
\cite{Sh:456}, but we do not rely on them. For other advances see
\cite{Sh:457}, \cite{Sh:500} and D\v{z}amonja, Shelah \cite{DjSh:614}.
Lately \cite{Sh:622} continue this paper.
\medskip
A class ${\frak K}$ is a class of structures with an embeddability notion.
If not said otherwise, an embedding, is a one to one function preserving
atomic relations and their negations. If ${\frak K}$ is a class and
$\lambda$ is a cardinal, then ${\frak K}_\lambda$ stands for the collection
of all members of ${\frak K}$ of cardinality $\lambda$.\\
We similarly define ${\frak K}_{\leq\lambda}$.
A member $M$ of ${\frak K}_\lambda$ is universal, if every $N\in {\frak
K}_{\le \lambda}$, embeds into $M$. An example is $M=:\bigoplus\limits_\lambda
{\Bbb Q}$, which is universal in ${\frak K}_\lambda$ if ${\frak K}$ is
the class of all torsion-free Abelian groups, under usual embeddings.
We give some motivation to the present paper by a short review of the above
references. The general thesis in these papers, as well as the present one
is:
\begin{Thesis}
\label{0.1}
General Abelian groups and trees with $\omega+1$ levels behave in universality
theorems like stable non-superstable theories.
\end{Thesis}
The simplest example of such a class is the class ${\frak K}^{tr} =:$ trees
$T$ with $(\omega+1)$-levels, i.e. $T\subseteq {}^{\omega\ge}\alpha$ for some
$\alpha$, with the relations $\eta E^0_n\nu =: \eta\restriction n=\nu
\restriction n\ \&\ \lg(\eta)\geq n$. For ${\frak K}^{tr}$ we know
that $\mu^+<\lambda={\rm cf}(\lambda)
<\mu^{\aleph_0}$ implies there is no universal for ${\frak K}^{tr}_\lambda$
(by \cite{KjSh:447}). Classes as ${\frak K}^{rtf}$ (defined in the title),
or ${\frak K}^{rs(p)}$ (reduced separable Abelian $p$-groups) are similar
(though they are not elementary classes) when we consider pure embeddings
(by \cite{KjSh:455}). But it is not less natural to consider usual embeddings
(remembering they, the (Abelian) groups under consideration, are reduced). The
problem is that the invariant has been defined using divisibility, and so
under non-pure embedding those seemed to be erased.
Then in \cite{Sh:456} the non-existence of universals is proved restricting
ourselves to $\lambda>2^{\aleph_0}$ and $(< \lambda)$-stable groups
(see there). These restrictions hurt the generality of the theorem; because
of the first requirement we lose some cardinals. The second requirement
changes the class to one which is not established among Abelian group
theorists (though to me it looks natural).
Our aim was to eliminate those requirements, or show that they are necessary.
Note that the present paper is mainly concerned essentially with results in
ZFC, but they have roots in ``difficulties" in extending independence results
thus providing a case for the
\begin{Thesis}
\label{0.2}
Even if you do not like independence results you better look at them, as you
will not even consider your desirable ZFC results when they are camouflaged
by the litany of many independence results you can prove things.
\end{Thesis}
Of course, independence has interest {\em per se}; still for a given problem in
general a solution in ZFC is for me preferable on an independence result. But
if it gives a method of forcing (so relevant to a series of problems) the
independence result is preferable (of course, I assume there are no other
major differences; the depth of the proof would be of first importance to me).
As occurs often in my papers lately, quotations of {\bf pcf} theory appear.
This paper is also a case of
\begin{Thesis}
\label{0.3}
Assumption of cases of not GCH at singular (more generally $pp\lambda>
\lambda^+$) are ``good", ``helpful" assumptions; i.e. traditionally uses of
GCH proliferate mainly not from conviction but as you can prove many theorems
assuming $2^{\aleph_0}=\aleph_1$ but very few from $2^{\aleph_0}>\aleph_1$,
but assuming $2^{\beth_\omega}>\beth^+_\omega$ is helpful in proving.
\end{Thesis}
Unfortunately, most results are only almost in ZFC as they use extremely weak
assumptions from {\bf pcf}, assumptions whose independence is not know. So
practically it is not tempting to try to remove them as they may be
true, and it is unreasonable to try to prove independence results before
independence results on {\bf pcf} will advance.
In \S1 we give an explanation of the earlier difficulties: the problem
(of the existence of universals for
${\frak K}^{rs(p)}$) is not like looking for ${\frak K}^{tr}$ (trees with
$\omega+1$ levels) but for ${\frak K}^{tr}_{\langle\lambda_\alpha:\alpha<
\omega\rangle}$ where
\begin{description}
\item[($\oplus$)] $\lambda^{\aleph_0}_n<\lambda_{n+1}<\mu$, $\lambda_n$
are regular and $\mu^+<\lambda=\lambda_\omega={\rm cf}(\lambda)<\mu^{\aleph_0}$
and ${\frak K}^{tr}_{\langle\lambda_n:n<\omega\rangle}$ is
\[\{T:T\mbox{ a tree with $\omega+1$ levels, in level $n< \omega$ there are
$\lambda_n$ elements}\}.\]
\end{description}
We also consider ${\frak K}^{tr}_{\langle\lambda_\alpha:\alpha\leq\omega
\rangle}$, which is defined similarly but the level $\omega$ of $T$ is
required to have $\lambda_\omega$ elements.
\noindent For ${\frak K}^{rs(p)}$ this is proved fully, for ${\frak K}^{rtf}$
this is proved for the natural examples.
\medskip
In \S2 we define two such basic examples: one is ${\frak K}^{tr}_{\langle
\lambda_\alpha:\alpha \le \omega \rangle}$, and the second is
${\frak K}^{fc}_{\langle \lambda_\alpha:\alpha\leq\omega\rangle}$.
The first is a tree with $\omega+1$ levels;
in the second we have slightly less restrictions. We have $\omega$ kinds of
elements and a function from the $\omega$-th-kind to the $n$th kind. We can
interpret a tree $T$ as a member of the second example: $P^T_\alpha = \{x:x
\mbox{ is of level }\alpha\}$ and
\[F_n(x) = y \quad\Leftrightarrow \quad x \in P^T_\omega\ \&\ y \in
P^T_n\ \&\ y <_T x.\]
For the second we recapture the non-existence theorems.
But this is not one of the classes we considered originally.
In \S3 we return to ${\frak K}^{rtf}$ (reduced torsion free Abelian groups)
and prove the non-existence of universal ones in $\lambda$ if $2^{\aleph_0}
< \mu^+<\lambda={\rm cf}(\lambda)<\mu^{\aleph_0}$ and an additional very weak set
theoretic assumption (the consistency of its failure is not known).
\medskip
\noindent Note that (it will be proved in \cite{Sh:622}):
\begin{description}
\item[($\otimes$)] if $\lambda<2^{\aleph_0}$ then ${\frak K}^{rtf}_\lambda$
has no universal members.
\end{description}
\noindent Note: if $\lambda=\lambda^{\aleph_0}$ then ${\frak K}^{tr}_\lambda$
has universal member also ${\frak K}^{rs(p)}_\lambda$ (see \cite{Fu}) but not
${\frak K}^{rtf}_\lambda$ (see \cite[Ch IV, VI]{Sh:e}).
\noindent We have noted above that for ${\frak K}^{rtf}_\lambda$ requiring
$\lambda\geq 2^{\aleph_0}$ is reasonable: we can prove (i.e. in ZFC) that
there is no universal member. What about ${\frak K}^{rs(p)}_\lambda$? By \S1
we should look at ${\frak K}^{tr}_{\langle\lambda_i:i\le\omega\rangle}$,
$\lambda_\omega=\lambda<2^{\aleph_0}$, $\lambda_n<\aleph_0$.
In \S4 we prove the consistency of the existence of universals for
${\frak K}^{tr}_{\langle\lambda_i:i \le\omega\rangle}$
when $\lambda_n\leq \omega$, $\lambda_\omega=\lambda< 2^{\aleph_0}$ but of
cardinality
$\lambda^+$; this is not the original problem but it seems to be a reasonable
variant, and more seriously, it shoots down the hope to use the present
methods of proving non-existence of universals. Anyhow this is
${\frak K}^{tr}_{\langle \lambda_i:i \le\omega\rangle}$ not
${\frak K}^{rs(p)}_{\lambda_\omega}$, so we proceed to reduce this problem to
the previous one under a mild variant of MA. The intentions are to deal with
``there is universal of cardinality $\lambda$" in D\v{z}amonja Shelah
\cite{DjSh:614}.
The reader should remember that the consistency of e.g.
\begin{quotation}
{\em
\noindent $2^{\aleph_0}>\lambda>\aleph_0$ and there is no $M$ such that $M\in
{\frak K}^{rs(p)}$ is of cardinality $<2^{\aleph_0}$ and universal for
${\frak K}^{rs(p)}_\lambda$
}
\end{quotation}
is much easier to obtain, even in a wider context (just add many Cohen reals).
\medskip
As in \S4 the problem for ${\frak K}^{rs(p)}_\lambda$ was reasonably resolved
for $\lambda<2^{\aleph_0}$ (and for $\lambda = \lambda^{\aleph_0}$, see
\cite{KjSh:455}), we now, in \S5 turn to $\lambda>2^{\aleph_0}$ (and
$\mu,\lambda_n$)
as in $(\oplus)$ above.
As in an earlier proof we use $\langle C_\delta: \delta\in S\rangle$ guessing
clubs for $\lambda$ (see references or later here), so $C_\delta$ is a subset
of $\delta$ (so the invariant depends on the representation of $G$ but this
disappears when we divide by suitable ideal on $\lambda$).
What we do is: rather than trying to code a subset of
$C_\delta$ (for $\bar G=\langle G_i:i<\lambda\rangle$ a representation or
filtration of the structure $G$ as the union of an increasing continuous
sequence of structures of smaller cardinality)
by an element of $G$, we do it, say, by some set $\bar x=\langle x_t:t\in
{\rm Dom}(I)\rangle$, $I$ an ideal on ${\rm Dom}(I)$ (really by $\bar x/I$). At first
glance if ${\rm Dom}(I)$ is infinite we cannot list {\em a priori} all possible
such sequences for a candidate $H$ for being a universal member, as their
number is $\ge\lambda^{\aleph_0}=\mu^{\aleph_0}$. But we can find a family
\[{\cal F}\subseteq\{\langle x_t:t\in A\rangle:\ A\subseteq{\rm Dom}(I),\ A\notin
I,\ x_t\in\lambda\}\]
of cardinality $<\mu^{\aleph_0}$ such that for any $\bar{x}=\langle x_t:t\in
{\rm Dom}(I)\rangle$, for some $\bar y\in {\cal F}$ we have $\bar y=\bar
x\restriction {\rm Dom}(\bar y)$.
\medskip
As in \S3 there is such ${\cal F}$ except when some set theoretic statement
related to {\bf pcf} holds. This statement is extremely strong, also in the
sense that we do not know to prove its consistency at present. But
again, it seems unreasonable to try to prove its consistency before the
{\bf pcf} problem was dealt with. Of course, we may try to improve the
combinatorics to avoid the use of this statement, but are naturally
discouraged by the possibility that the {\bf pcf} statement can be proved in
ZFC; thus we would retroactively get the non-existence of universals in ZFC.
\medskip
In \S6, under weak {\bf pcf} assumptions, we prove: if there is a universal
member in ${\frak K}^{fc}_\lambda$ then there is one in
${\frak K}^{rs(p)}_\lambda$; so making the connection between the
combinatorial structures and the algebraic ones closer.
\medskip
In \S7 we give other weak {\bf pcf} assumptions which suffice to prove
non-existence of universals in ${\frak K}^x_{\langle\lambda_\alpha:\alpha
\le\omega\rangle}$ (with $x$ one of the ``legal'' values):
$\max{\rm pcf}\{\lambda_n:n<\omega\}=\lambda$ and ${\cal P}(
\{\lambda_n:n<\omega\})/J_{<\lambda}\{\lambda_n:n<\omega\}$ is an infinite
Boolean Algebra (and $(\oplus)$ holds, of course).
\medskip
In \cite{KjSh:409}, for singular $\lambda$ results on non-existence of
universals (there on orders) can be gotten from these weak {\bf pcf}
assumptions.
\medskip
In \S8 we get parallel results from, in general, more complicated assumptions.
\medskip
In \S9 we turn to a closely related class: the class of metric spaces with
(one to one) continuous embeddings, similar results hold for it. We also
phrase a natural criterion for deducing the non-existence of universals
from one class to another.
\medskip
In \S10 we deal with modules and in \S11 we discuss the open problems of
various degrees of seriousness.
The sections are written in the order the research was done.
\begin{notation}
\label{0.4}
Note that we deal with trees with $\omega+1$ levels rather than, say, with
$\kappa+1$, and related situations, as those cases are quite popular. But
inherently the proofs of \S1-\S3, \S5-\S9 work for $\kappa+1$ as well (in
fact, {\bf pcf} theory is stronger).
\noindent For a structure $M$, $\|M\|$ is its cardinality.
\noindent For a model, i.e. a structure, $M$ of cardinality $\lambda$,
where $\lambda$ is regular uncountable, we say that $\bar M$ is a
representation (or filtration) of $M$ if $\bar M=\langle M_i:i<\lambda\rangle$
is an increasing continuous sequence of submodels of cardinality $<\lambda$
with union $M$.
\noindent For a set $A$, we let $[A]^\kappa = \{B:B \subseteq A \mbox{ and }
|B|=\kappa\}$.
\noindent For a set $C$ of ordinals,
$${\rm acc}(C)=\{\alpha\in C: \alpha=\sup(\alpha \cap C)\}, \mbox{(set of
accumulation points)}$$
$$
{\rm nacc}(C)=C\setminus {\rm acc}(C) \ (=\mbox{ the set of non-accumulation points}).
$$
We usually use $\eta$, $\nu$, $\rho$ for sequences of ordinals; let
$\eta\vartriangleleft\nu$ means $\eta$ is an initial segment of $\nu$.
Let ${\rm cov}(\lambda, \mu, \theta, \sigma)= \min\{|{\cal P}|: {\cal P}\subseteq
[\lambda]^{<\mu}$, and for every $A\in [\lambda]^{<\theta}$ for some $\alpha<
\sigma$ and $B_i\in {\cal P}$ for $i< \alpha$ we have $A\subseteq
\bigcup\limits_{i< \alpha} B_i\}$.
\noindent Remember that for an ordinal $\alpha$, e.g. a natural number,
$\alpha=\{\beta:\beta<\alpha\}$.
\end{notation}
\begin{notation}
\noindent ${\frak K}^{rs(p)}$ is the class of (Abelian) groups which are
$p$-groups (i.e. $(\forall x\in G)(\exists n)[p^nx = 0]$) reduced (i.e. have
no divisible non-zero subgroups) and separable (i.e. every cyclic pure
subgroup is a direct summand). See \cite{Fu}.
\noindent For $G\in{\frak K}^{rs(p)}$ define a norm $\|x\|=\inf\{2^{-n}:
p^n \mbox{ divides } x\}$. Now every $G\in {\frak K}^{rs(p)}$ has a
basic subgroup $B=\bigoplus\limits_{\scriptstyle n<\omega\atop\scriptstyle
i<\lambda_n} {\Bbb Z} x^n_i$, where $x^n_i$ has order $p^{n+1}$, and every
$x\in G$ can be represented as $\sum\limits_{\scriptstyle n<\omega\atop
\scriptstyle i<\lambda_n} a^n_ix^n_i$, where for each $n$, $w_n(x)=\{i<
\lambda_n:a^n_ix^n_i\ne 0\}$ is finite.
\noindent ${\frak K}^{rtf}$ is the class of Abelian groups which are reduced
and torsion free (i.e. $G \models nx = 0$, $n>0$\qquad
$\Rightarrow\qquad x = 0$).
\noindent For a group $G$ and $A\subseteq G$ let $\langle A\rangle_G$ be the
subgroup of $G$ generated by $A$, we may omit the subscript $G$ if
clear from the context.
\noindent Group will mean an Abelian group, even if not stated
explicitly.
\noindent Let $H\subseteq_{pr} G$ means $H$ is a pure subgroup of
$G$.
\noindent Let $nG=\{nx: x\in G\}$ and let $G[n]=\{x\in G: nx=0\}$.
\end{notation}
\begin{notation}
${\frak K}$ will denote a class of structures with the same
vocabulary, with a notion of embeddability, equivalently a notion
$\leq_{{\frak K}}$ of submodel.
\end{notation}
\section{Their prototype is ${\frak K}^{tr}_{\langle \lambda_n:n<\omega
\rangle}$ not ${\frak K}^{tr}$!}
If we look for universal member in ${\frak K}^{rs(p)}_\lambda$, thesis
\ref{0.1} suggests to us to think it is basically ${\frak K}^{tr}_\lambda$
(trees with $\omega+1$ levels, i.e. ${\frak K}^{tr}_{\lambda}$ is our
prototype), a way followed in \cite{KjSh:455}, \cite{Sh:456}. But, as
explained in the introduction, this does not give answer for the case
of usual embedding for the family of all such groups. Here we show
that for this case the thesis should be corrected. More concretely,
the choice of the prototype means the choice of what we expect is the
division of the possible classes. That is for a family of classes a
choice of a prototype assert that we believe that they all behave in
the same way.
We show that looking for a universal member $G$ in ${\frak K}^{rs(p)}_\lambda$
is like looking for it among the $G$'s with density $\le\mu$
($\lambda,\mu$, as usual, as in $(\oplus)$ from \S0). For
${\frak K}^{rtf}_\lambda$ we get weaker results which still cover the examples
usually constructed, so showing that the restrictions in \cite{KjSh:455} (to
pure embeddings) and \cite{Sh:456} (to $(<\lambda)$-stable groups) were
natural.
\begin{Proposition}
\label{1.1}
Assume that $\mu=\sum\limits_{n<\omega}\lambda_n=\lim\sup\limits_n\lambda_n$,
$\mu\le\lambda\le\mu^{\aleph_0}$, and $G$ is a reduced separable
$p$-group such that
\[|G|=\lambda\quad\mbox{ and }\quad\lambda_n(G)=:\dim((p^n G)[p]/
(p^{n+1}G)[p])\le\mu\]
(this is a vector space over ${\Bbb Z}/p {\Bbb Z}$, hence the dimension is
well defined). \\
{\em Then} there is a reduced separable $p$-group $H$ such that
$|H|=\lambda$, $H$ extends $G$ and $(p^nH)[p]/(p^{n+1}H)[p]$ is a
group of dimension $\lambda_n$ (so if $\lambda_n\geq \aleph_0$, this
means cardinality $\lambda_n$).
\end{Proposition}
\begin{Remark}
\label{1.1A}
So for $H$ the invariants from \cite{KjSh:455} are trivial.
\end{Remark}
\proof (See Fuchs \cite{Fu}). We can find $z^n_i$ (for
$n<\omega$, $i<\lambda_n(G)\le\mu$) such that:
\begin{description}
\item[(a)] $z^n_i$ has order $p^n$,
\item[(b)] $B=\sum\limits_{n,i}\langle z^n_i \rangle_G$ is a direct sum,
\item[(c)] $B$ is dense in $G$ in the topology induced by the norm
\[\|x\|=\min\{2^{-n}:p^n \mbox{ divides } x \mbox{ in } G\}.\]
\end{description}
For each $n<\omega$ and $i<\lambda_n(G)$ ($\le\mu$) choose $\eta^n_i\in
\prod\limits_{m<\omega}\lambda_m$, pairwise distinct such that for $(n^1,i^1)
\neq (n^2,i^2)$ for some $n(*)$ we have:
\[\lambda_n \ge \lambda_{n(*)}\qquad \Rightarrow\qquad \eta^{n^1}_{i^1}(n)
\neq \eta^{n^2}_{i^2}(n).\]
Let $H$ be generated by $G$, $x^m_i$ ($i<\lambda_m$, $m<\omega$),
$y^{n,k}_i$ ($i<\lambda_n$, $n<\omega$, $n\le k<\omega)$ freely except for:
\begin{description}
\item[($\alpha$)] the equations of $G$,
\item[($\beta$)] $y^{n,n}_i = z^n_i$,
\item[($\gamma$)] $py^{n,k+1}_i - y^{n,k}_i = x^k_{\eta^n_i(k)}$,
\item[($\delta$)] $p^{n+1}x^n_i = 0$,
\item[($\varepsilon$)] $p^{k+1}y^{n,k}_i = 0$.
\end{description}
Now check. \hfill$\square_{\ref{1.1}}$
\begin{Definition}
\label{1.2}
\begin{enumerate}
\item ${\bf t}$ denotes a sequence $\langle t_i:i<\omega\rangle$, $t_i$ a
natural number $>1$.
\item For a group $G$ we define
\[G^{[{\bf t}]}=\{x\in G:\bigwedge_{j<\omega}[x\in (\prod_{i<j} t_i)
G]\}.\]
\item We can define a semi-norm $\|-\|_{{\bf t}}$ on $G$
\[\|x\|_{{\bf t}}=\min\{2^{-i}:x\in (\prod_{j<i} t_j)G\}\]
and so the semi-metric
\[d_{{\bf t}}(x,y)=\|x-y\|_{{\bf t}}.\]
\end{enumerate}
\end{Definition}
\begin{Remark}
\label{1.2A}
So, if $\|-\|_{{\bf t}}$ is a norm, $G$ has a completion under $\|-\|_{{\bf t}}$,
which we call $\|-\|_{{\bf t}}$-completion; if ${\bf t}=\langle i!:i<\omega
\rangle$ we refer to $\|-\|_{{\bf t}}$ as $\Bbb Z$-adic norm, and this induces
$\Bbb Z$-adic topology, so we can speak of $\Bbb Z$-adic completion.
\end{Remark}
\begin{Proposition}
\label{1.3}
Suppose that
\begin{description}
\item[($\otimes_0$)] $\mu=\sum\limits_n\lambda_n$ and $\mu\le\lambda\le
\mu^{\aleph_0}$ for simplicity, $2<2\cdot\lambda_n\le\lambda_{n+1}$ (maybe
$\lambda_n$ is finite!),
\item[($\otimes_1$)] $G$ is a torsion free group, $|G|=\lambda$, and
$G^{[{\bf t}]}=\{0\}$,
\item[($\otimes_2$)] $G_0\subseteq G$, $G_0$ is free and $G_0$ is
${\bf t}$-dense in $G$ (i.e. in the topology induced by the metric $d_{{\bf t}}$),
where ${\bf t}$ is a sequence of primes.
\end{description}
{\em Then} there is a torsion free group $H$, $G\subseteq H$, $H^{[{\bf t}]}
=\{0\}$, $|H|=\lambda$ and, under $d_{{\bf t}}$, $H$ has density $\mu$.
\end{Proposition}
\proof Let $\{x_i:i<\lambda\}$ be a basis of $G_0$. Let $\eta_i\in
\prod\limits_{n<\omega} \lambda_n$ for $i<\mu$ be distinct such that
$\eta_i(n+1)\geq \lambda_n$ and
\[i\ne j\qquad \Rightarrow\qquad (\exists m)(\forall n)[m \le n \quad
\Rightarrow\quad \eta_i(n) \ne \eta_j(n)].\]
Let $H$ be generated by
\[G,\ \ x^m_i \mbox{ (for $i<\lambda_m$, $m<\omega$), }\ y^n_i \mbox{
(for $i<\mu$, $n<\omega$)}\]
freely except for
\begin{description}
\item[(a)] the equations of $G$,
\item[(b)] $y^0_i = x_i$,
\item[(c)] $t_n\, y^{n+1}_i + y^n_i = x^n_{\eta_i(n)}$.
\end{description}
\medskip
\noindent{\bf Fact A}\hspace{0.15in} $H$ extends $G$ and is torsion free.
\noindent [Why? As $H$ can be embedded into the divisible hull of $G$.]
\medskip
\noindent{\bf Fact B}\hspace{0.15in} $H^{[{\bf t}]}= \{0\}$.
\proof Let $K$ be a countable pure subgroup of $H$ such that $K^{[{\bf t}]}\ne
\{0\}$. Now without loss of generality $K$ is generated by
\begin{description}
\item[(i)] $K_1\subseteq G\cap\mbox{ [the $d_{{\bf t}}$--closure of $\langle x_i:
i\in I\rangle_G]$]}$, where $I$ is a countable infinite subset of $\lambda$
and $K_1\supseteq\langle x_i:i\in I\rangle_G$,
\item[(ii)] $y^m_i$, $x^n_j$ for $i\in I$, $m<\omega$ and $(n,j)\in
J$, where $J\subseteq \omega\times \lambda$ is countable and
\[i\in I,\ n<\omega\qquad\Rightarrow\qquad (n,\eta_i(n))\in J.\]
\end{description}
Moreover, the equations holding among those elements are deducible from the
equations of the form
\begin{description}
\item[(a)$^-$] equations of $K_1$,
\item[(b)$^-$] $y^0_i=x_i$ for $i \in I$,
\item[(c)$^-$] $t_n\,y^{n+1}_i+y^n_i=x^n_{\eta_i(n)}$ for $i\in I,n<\omega$.
\end{description}
\noindent We can find $\langle k_i:i<\omega\rangle$ such that
\[[n\ge k_i\ \&\ n\ge k_j\ \&\ i \ne j\qquad \Rightarrow\qquad \eta_i(n)\ne
\eta_j(n)].\]
Let $y \in K\setminus\{0\}$. Then for some $j$, $y\notin (\prod\limits_{i<j}
t_i)G$, so for some finite $I_0\subseteq I$ and finite $J_0\subseteq J$ and
\[y^* \in\langle\{x_i:i\in I_0\}\cup\{x^n_\alpha:(n,\alpha)\in J_0\}
\rangle_K\]
we have $y-y^*\in(\prod\limits_{i<j} t_i) G$. Without loss of generality $J_0
\cap\{(n,\eta_i(n)):i\in I,\ n\ge k_i\}=\emptyset$. Now there is a
homomorphism $\varphi$ from $K$ into the divisible hull $K^{**}$ of
\[K^* = \langle\{x_i:i\in I_0\}\cup\{x^n_j:(n,j)\in J_0\}\rangle_G\]
such that ${\rm Rang}(\varphi)/K^*$ is finite. This is enough.
\medskip
\noindent{\bf Fact C}\hspace{0.15in} $H_0=:\langle x^n_i:n<\omega,i<\lambda_n
\rangle_H$ is dense in $H$ by $d_{{\bf t}}$.
\proof Straight as each $x_i$ is in the $d_{{\bf t}}$-closure of $H_0$ inside $H$.
\medskip
Noting then that we can increase the dimension easily, we are done.
\hfill$\square_{\ref{1.3}}$
\section{On structures like $(\prod\limits_n \lambda_n,E_m)_{m<\omega}$,
$\eta E_m \nu =: \eta(m)=\nu(m)$}
\begin{Discussion}
\label{2.1}
We discuss the existence of universal members in cardinality $\lambda$,
$\mu^+<\lambda<\mu^{\aleph_0}$, for certain classes of groups. The claims
in \S1 indicate that the problem is similar not to the problem of the
existence of a universal member in ${\frak K}^{tr}_\lambda$ (the class of
trees with $\lambda$ nodes, $\omega+1$ levels) but to the one where the first
$\omega$ levels, are each with $<\mu$ elements. We look more carefully and
see that some variants are quite different.
The major concepts and Lemma (\ref{2.3}) are similar to those of \S3, but
easier. Since detailed proofs are given in \S3, here we give somewhat
shorter proofs.
\end{Discussion}
\begin{Definition}
\label{2.2}
For a sequence $\bar\lambda=\langle\lambda_i:i\le\delta\rangle$ of cardinals
we define:
\begin{description}
\item[(A)] ${\frak K}^{tr}_{\bar \lambda}=\{T:\,T$ is a tree with $\delta
+1$ levels (i.e. a partial order such that
\qquad\quad for $x\in T$, ${\rm lev}_T(x)=:{\rm otp}(\{y:y<x\})$ is an ordinal
$\le\delta$) such
\qquad\quad that:\quad ${\rm lev}_i(T)=:\{x\in T:{\rm lev}_T(x)=i\}$ has cardinality
$\le\lambda_i\}$,
\item[(B)] ${\frak K}^{fc}_{\bar\lambda}=\{M:\,M=(|M|,P_i,F_i)_{i\le\delta}$,
$|M|$ is the disjoint union of
\qquad\quad $\langle P_i:i\le\delta\rangle$, $F_i$ is a function from
$P_\delta$ to $P_i$, $\|P_i\|\le\lambda_i$,
\qquad\quad $F_\delta$ is the identity (so can be omitted)$\}$,
\item[(C)] If $[i\le\delta\quad \Rightarrow\quad \lambda_i=\lambda]$ then we
write $\lambda$, $\delta+1$ instead of
$\langle\lambda_i:i\le\delta\rangle$.
\end{description}
\end{Definition}
\begin{Definition}
\label{2.2A}
Embeddings for ${\frak K}^{tr}_{\bar\lambda}$, ${\frak K}^{fc}_{\bar\lambda}$
are defined naturally: for ${\frak K}^{tr}_{\bar\lambda}$ embeddings preserve
$x<y$, $\neg x<y$, ${\rm lev}_T(x)=\alpha$; for ${\frak K}^{fc}_{\bar\lambda}$
embeddings are defined just as for models.
If $\delta^1=\delta^2=\delta$ and $[i<\delta\quad\Rightarrow\quad\lambda^1_i
\le\lambda^2_i]$ and $M^\ell\in{\frak K}^{fc}_{\bar\lambda^\ell}$, (or $T^\ell
\in{\frak K}^{tr}_{\bar\lambda^\ell}$) for $\ell=1,2$, then an embedding of
$M^1$ into $M^2$ ($T^1$ into $T^2$) is defined naturally.
\end{Definition}
\begin{Lemma}
\label{2.3}
Assume $\bar\lambda=\langle\lambda_i:i\le\delta\rangle$ and $\theta$, $\chi$
satisfy (for some $\bar C$):
\begin{description}
\item[(a)] $\lambda_\delta$, $\theta$ are regular, $\bar C=\langle C_\alpha:
\alpha\in S\rangle$, $S\subseteq\lambda=:\lambda_\delta$, $C_\alpha\subseteq
\alpha$, for every club $E$ of $\lambda$ for some $\alpha$ we have $C_\alpha
\subseteq E$, $\lambda_\delta<\chi\le |C_\alpha|^\theta$ and ${\rm otp}(C_\alpha)
\ge\theta$,
\item[(b)] $\lambda_i\le\lambda_\delta$,
\item[(c)] there are $\theta$ pairwise disjoint sets $A\subseteq\delta$
such that $\prod\limits_{i\in A}\lambda_i\ge\lambda_\delta$.
\end{description}
{\em Then}
\begin{description}
\item[($\alpha$)] there is no universal member in ${\frak K}^{fc}_{\bar
\lambda}$;\quad moreover
\item[($\beta$)] if $M_\alpha\in {\frak K}^{fc}_{\bar\lambda}$ or even
$M_\alpha\in {\frak K}^{fc}_{\lambda_\delta}$ for $\alpha<\alpha^*<\chi$
{\em then} some $M\in {\frak K}^{fc}_{\bar\lambda}$ cannot be embedded into
any $M_\alpha$.
\end{description}
\end{Lemma}
\begin{Remark}
\label{2.3A}
Note that clause $(\beta)$ is relevant to our discussion in \S1: the
non-universality is preserved even if we increase the density and,
also, it is witnessed even by non-embeddability in many models.
\end{Remark}
\proof Let $\langle A_\varepsilon:\varepsilon<\theta\rangle$ be as in clause
(c) and let $\eta^\varepsilon_\alpha\in\prod\limits_{i\in A_\varepsilon}
\lambda_i$ for $\alpha<\lambda_\delta$ be pairwise distinct. We fix $M_\alpha
\in {\frak K}^{fc}_{\lambda_\delta}$ for $\alpha<\alpha^*<\chi$.
\noindent For $M\in {\frak K}^{fc}_{\bar\lambda}$, let $\bar M=(|M|,P^M_i,
F^M_i)_{i\le\delta}$ and let $\langle M_\alpha: \alpha<
\lambda_\delta\rangle$ be a representation (=filtration) of $M$; for
$\alpha\in S$, $x\in P^M_\delta$, let
\[\begin{ALIGN}
{\rm inv}(x,C_\alpha;\bar M)=\big\{\beta\in C_\alpha:&\mbox{for some }\varepsilon
<\theta\mbox{ and } y\in M_{\min(C_\alpha\setminus (\beta+1))}\\
&\mbox{we have }\ \bigwedge\limits_{i\in A_\varepsilon} F^M_i(x)=F^M_i(y)\\
&\mbox{\underbar{but} there is no such } y\in M_\beta\big\}.
\end{ALIGN}\]
\[{\rm Inv}(C_\alpha,\bar M)=\{{\rm inv}(x,C_\alpha,\bar M):x\in P^M_\delta\}.\]
\[{\rm INv}(\bar M,\bar C)=\langle{\rm Inv}(C_\alpha,\bar M):\alpha\in S\rangle.\]
\[{\rm INV}(\bar M,\bar C)={\rm INv}(\bar M,\bar C)/{\rm id}^a(\bar C).\]
Recall that
\[{\rm id}^a(\bar C)=\{T\subseteq\lambda:\mbox{ for some club $E$ of
$\lambda$ for no $\alpha\in T$ is $C_\alpha\subseteq E$}\}.\]
The rest should be clear (for more details see proofs in \S3), noticing
\begin{Fact}
\label{2.3B}
\begin{enumerate}
\item ${\rm INV}(\bar M,\bar C)$ is well defined, i.e. if $\bar M^1$, $\bar M^2$
are representations (=filtrations) of $M$ then ${\rm INV}(\bar M^1,\bar
C)={\rm INV}(\bar M^2,\bar C)$.
\item ${\rm Inv}(C_\alpha,\bar M)$ has cardinality $\le\lambda$.
\item ${\rm inv}(x,C_\alpha;\bar M)$ is a subset of $C_\alpha$ of cardinality
$\le \theta$.
\end{enumerate}
\end{Fact}
\hfill$\square_{\ref{2.3}}$
\begin{Conclusion}
\label{2.4}
If $\mu=\sum\limits_{n<\omega}\lambda_n$ and $\lambda^{\aleph_0}_n<
\lambda_{n+1}$ and $\mu^+<\lambda_\omega={\rm cf}(\lambda_\omega)<\mu^{\aleph_0}$,
{\em then} in ${\frak K}^{fc}_{\langle\lambda_\alpha:\alpha\le\omega\rangle}$
there is no universal member and even in ${\frak K}^{fc}_{\langle
\lambda_\omega:\alpha\le\omega\rangle}$ we cannot find a member universal
for it.
\end{Conclusion}
\proof Should be clear or see the proof in \S3.
\hfill$\square_{\ref{2.4}}$
\section{Reduced torsion free groups: Non-existence of universals}
We try to choose torsion free reduced groups and define invariants so that
in an extension to another such group $H$ something survives. To this end
it is natural to stretch ``reduced" near to its limit.
\begin{Definition}
\label{3.1}
\begin{enumerate}
\item ${\frak K}^{tf}$ is the class of torsion free (abelian) groups.
\item ${\frak K}^{rtf}=\{G\in {\frak K}^{tf}:{\Bbb Q}$ is not embeddable into
$G$ (i.e. $G$ is reduced)$\}$.
\item ${\bf P}^*$ denotes the set of primes.
\item For $x\in G$, ${\bf P}(x,G)=:\{p\in{\bf P}^*: \bigwedge\limits_n x\in
p^n G\}$.
\item ${\frak K}^x_\lambda=\{G\in{\frak K}^x:\|G\|=\lambda\}$.
\item If $H\in {\frak K}^{rtf}_\lambda$, we say $\bar H$ is a representation
or filtration of $H$ if $\bar H=\langle
H_\alpha:\alpha<\lambda\rangle$ is increasing continuous and
$H=\bigcup\limits_{\alpha<\lambda} H_\alpha$,
$H\in {\frak K}^{rtf}$ and each $H_\alpha$ has cardinality $<\lambda$.
\end{enumerate}
\end{Definition}
\begin{Proposition}
\label{3.2}
\begin{enumerate}
\item If $G\in {\frak K}^{rtf}$, $x\in G\setminus\{0\}$, $Q\cup{\bf P}(x,G)
\subsetneqq{\bf P}^*$, $G^+$ is the group generated by $G,y,y_{p,\ell}$ ($\ell
<\omega$, $p\in Q$) freely, except for the equations of $G$ and
\[y_{p,0}=y,\quad py_{p,\ell+1}=y_{p,\ell}\quad \mbox{ and }\quad
y_{p,\ell}=z\mbox{ when } z\in G,p^\ell z=x\]
{\em then} $G^+\in {\frak K}^{rtf}$, $G\subseteq_{pr}G^+$ (pure extension).
\item If $G_i\in {\frak K}^{rtf}$ ($i<\alpha$) is $\subseteq_{pr}$-increasing
{\em then} $G_i\subseteq_{pr}\bigcup\limits_{j<\alpha}G_j\in{\frak K}^{rtf}$
for every $i<\alpha$.
\end{enumerate}
\end{Proposition}
The proof of the following lemma introduces a method quite central to this
paper.
\begin{Lemma}
\label{3.3}
Assume that
\begin{description}
\item[$(*)^1_\lambda$] $2^{\aleph_0}+\mu^+<\lambda={\rm cf}(\lambda)<
\mu^{\aleph_0}$,
\item[$(*)^2_\lambda$] for every $\chi<\lambda$, there is $S\subseteq
[\chi]^{\le\aleph_0}$, such that:
\begin{description}
\item[(i)] $|S|<\lambda$,
\item[(ii)] if $D$ is a non-principal ultrafilter on $\omega$ and $f:D
\longrightarrow\chi$ {\em then} for some $a\in S$ we have
\[\bigcap \{X\in D:f(X)\in a\}\notin D.\]
\end{description}
\end{description}
{\em Then}
\begin{description}
\item[($\alpha$)] in ${\frak K}^{rtf}_\lambda$ there is no universal
member (under usual embeddings (i.e. not necessarily pure)),
\item[($\beta$)] moreover, \underbar{for any} $G_i\in {\frak
K}^{rtf}_\lambda$, for $i<i^*<\mu^{\aleph_0}$ \underbar{there is} $G\in
{\frak K}^{rtf}_\lambda$ not embeddable into any one of $G_i$.
\end{description}
\end{Lemma}
Before we prove \ref{3.3} we consider the assumptions of \ref{3.3} in
\ref{3.4}, \ref{3.5}.
\begin{Claim}
\label{3.4}
\begin{enumerate}
\item In \ref{3.3} we can replace $(*)^1_\lambda$ by
\begin{description}
\item[$(**)^1_\lambda$ (i)] $2^{\aleph_0}<\mu<\lambda={\rm cf}(\lambda)<
\mu^{\aleph_0}$,
\item[\qquad(ii)] there is $\bar C=\langle C_\delta:\delta\in S^*\rangle$
such that $S^*$ is a stationary subset of $\lambda$, each $C_\delta$ is
a subset of $\delta$ with ${\rm otp}(C_\delta)$ divisible by $\mu$,
$C_\delta$ closed in $\sup(C_\delta)$ (which normally $\delta$, but
not necessarily so) and
\[(\forall\alpha)[\alpha\in {\rm nacc}(C_\delta)\quad \Rightarrow\quad {\rm cf}(\alpha)
>2^{\aleph_0}]\]
(where ${\rm nacc}$ stands for ``non-accumulation points''),
and such that $\bar C$ guesses clubs of $\lambda$ (i.e. for every club $E$ of
$\lambda$, for some $\delta\in S^*$ we have $C_\delta\subseteq E$) and
$[\delta\in S^*\quad \Rightarrow\quad {\rm cf}(\delta)=\aleph_0]$.
\end{description}
\item In $(*)^1_\lambda$ and in $(*)^2_\lambda$, without loss of generality
$(\forall\theta<\mu)[\theta^{\aleph_0}<\mu]$ and ${\rm cf}(\mu)=\aleph_0$.
\end{enumerate}
\end{Claim}
\proof \ \ \ 1) This is what we actually use in the proof (see below).
\noindent 2) Replace $\mu$ by $\mu'=\min\{\mu_1:\mu^{\aleph_0}_1\ge\mu$
(equivalently $\mu^{\aleph_0}_1=\mu^{\aleph_0}$)$\}$.
\hfill$\square_{\ref{3.4}}$
Compare to, say, \cite{KjSh:447}, \cite{KjSh:455}; the new assumption is
$(*)^2_\lambda$, note that it is a very weak assumption, in fact it might be
that it is always true.
\begin{Claim}
\label{3.5}
Assume that $2^{\aleph_0}<\mu<\lambda<\mu^{\aleph_0}$ and $(\forall \theta<
\mu)[\theta^{\aleph_0}<\mu]$ (see \ref{3.4}(2)).
Then each of the following is a sufficient condition to $(*)^2_\lambda$:
\begin{description}
\item[($\alpha$)] $\lambda<\mu^{+\omega_1}$,
\item[($\beta$)] if ${\frak a}\subseteq{\rm Reg}\cap\lambda\setminus\mu$ and
$|{\frak a}|\le 2^{\aleph_0}$ then we can find $h:{\frak a}\longrightarrow
\omega$ such that:
\[\lambda>\sup\{\max{\rm pcf}({\frak b}):{\frak b}\subseteq {\frak a}\mbox{
countable, and $h\restriction {\frak b}$ constant}\}.\]
\end{description}
\end{Claim}
\proof Clause $(\alpha)$ implies Clause $(\beta)$: just use any
one-to-one function $h:{\rm Reg}\cap\lambda\setminus\mu\longrightarrow\omega$.
\smallskip
Clause $(\beta)$ implies (by \cite[\S6]{Sh:410} + \cite[\S2]{Sh:430}) that
for $\chi<\lambda$ there is $S\subseteq [\chi]^{\aleph_0}$, $|S|<\lambda$
such that for every $Y\subseteq\chi$, $|Y|=2^{\aleph_0}$, we can find
$Y_n$ such that $Y=\bigcup\limits_{n<\omega} Y_n$ and $[Y_n]^{\aleph_0}
\subseteq S$. (Remember: $\mu>2^{\aleph_0}$.) Without loss of generality
(as $2^{\aleph_0} < \mu < \lambda$):
\begin{description}
\item[$(*)$] $S$ is downward closed.
\end{description}
So if $D$ is a non-principal ultrafilter on $\omega$ and $f:D\longrightarrow
\chi$ then letting $Y={\rm Rang}(f)$ we can find $\langle Y_n:n<\omega\rangle$ as
above. Let $h:D\longrightarrow\omega$ be defined by $h(A)=\min\{n:f(A)\in
Y_n\}$. So
\[X\subseteq D\ \ \&\ \ |X|\le\aleph_0\ \ \&\ \ h\restriction X
\mbox{ constant }\Rightarrow\ f''(X)\in S\quad\mbox{(remember
$(*)$)}.\]
Now for each $n$, for some countable $X_n\subseteq D$ (possibly finite or
even empty) we have:
\[h \restriction X_n\ \mbox{ is constantly } n,\]
\[\ell <\omega \ \&\ (\exists A\in D)(h(A)=n\ \&\ \ell\notin A)
\Rightarrow (\exists B\in X_n)(\ell\notin B).\]
Let $A_n=:\bigcap\{A:A\in X_n\}=\bigcap\{A:A\in D\mbox{ and } h(X)=n\}$.
If the desired conclusion fails, then $\bigwedge\limits_{n<\omega}A_n\in D$.
So
\[(\forall A)[A\in D\quad \Leftrightarrow\quad\bigvee_{n<\omega} A\supseteq
A_n].\]
So $D$ is generated by $\{A_n:n<\omega\}$ but then $D$ cannot be a
non-principal ultrafilter.
\hfill$\square_{\ref{3.5}}$
\begin{Remark}
The case when $D$ is a principal ultrafilter is trivial.
\end{Remark}
\proof of Lemma \ref{3.3} Let $\bar C=\langle C_\delta:\delta\in S^*\rangle$
be as in $(**)^1_{\bar \lambda}$ (ii) from \ref{3.4} (for \ref{3.4}(1) its
existence is obvious, for \ref{3.3} - use \cite[VI, old III 7.8]{Sh:e}).
Let us suppose that $\bar A=\langle A_\delta:\delta\in S^*\rangle$, $A_\delta
\subseteq{\rm nacc}(C_\delta)$ has order type $\omega$ ($A_\delta$ like this will
be chosen later) and let $\eta_\delta$ enumerate $A_\delta$ increasingly. Let
$G_0$ be freely generated by $\{x_i:i<\lambda\}$.
Let $R$ be
\[\begin{ALIGN}
\big\{\bar a: &\bar a=\langle a_n:n<\omega\rangle\mbox{ is a sequence of
pairwise disjoint subsets of } {\bf P}^*,\\
&\mbox{with union }{\bf P}^* \mbox{ for simplicity, such that}\\
&\mbox{for infinitely many }n,\ a_n\ne\emptyset\big\}.
\end{ALIGN}\]
Let $G$ be a group generated by
\[G_0 \cup \{y^{\alpha,n}_{\bar a},z^{\alpha,n}_{\bar a,p}:\ \alpha<\lambda,
\ \bar a\in R,\ n<\omega,\ p \mbox{ prime}\}\]
freely except for:
\begin{description}
\item[(a)] the equations of $G_0$,
\item[(b)] $pz^{\alpha,n+1}_{\bar a,p}=z^{\alpha,n}_{\bar a,p}$ when
$p\in a_n$, $\alpha<\lambda$,
\item[(c)] $z^{\delta,0}_{\bar a,p}=y^{\delta,n}_{\bar a}-
x_{\eta_\delta(n)}$ when $p\in a_n$ and $\delta\in S^*$.
\end{description}
Now $G\in {\frak K}^{rtf}_\lambda$ by inspection.
\medskip
\noindent Before continuing the proof of \ref{3.3} we present a definition
and some facts.
\begin{Definition}
\label{3.7}
For a representation $\bar H$ of $H\in {\frak K}^{rtf}_\lambda$, and $x\in H$,
$\delta\in S^*$ let
\begin{enumerate}
\item ${\rm inv}(x,C_\delta;\bar H)=:\{\alpha\in C_\delta:$ for some $Q\subseteq
{\bf P}^*$, there is $y\in H_{\min[C_\delta\setminus(\alpha+1)]}$ such that
$Q\subseteq{\bf P}(x-y,H)$ but for no $y\in H_\alpha$ is $Q\subseteq{\bf P}(x-y,H)\}$
(so ${\rm inv}(x,C_\delta;\bar H)$ is a subset of $C_\delta$ of cardinality $\le
2^{\aleph_0}$).
\item ${\rm Inv}^0(C_\delta,\bar H)=:\{{\rm inv}(x,C_\delta;\bar H):x\in
\bigcup\limits_i H_i\}$.
\item ${\rm Inv}^1(C_\delta,\bar H)=:\{a:a\subseteq C_\delta$ countable and for
some $x\in H$, $a\subseteq{\rm inv}(x,C_\delta;\bar H)\}$.
\item ${\rm INv}^\ell(\bar H,\bar C)=:{\rm Inv}^\ell(H,\bar H,\bar C)=:\langle
{\rm Inv}^\ell(C_\delta;\bar H):\delta\in S^*\rangle$ for $\ell\in\{0,1\}$.
\item ${\rm INV}^\ell(H,\bar C)=:{\rm INv}^\ell(H,\bar H,\bar C)/{\rm id}^a(\bar C)$,
where
\[{\rm id}^a(\bar C)=:\{T\subseteq\lambda:\mbox{ for some club $E$ of
$\lambda$ for no $\delta\in T$ is }C_\delta\subseteq E\}.\]
\item If $\ell$ is omitted, $\ell = 0$ is understood.
\end{enumerate}
\end{Definition}
\begin{Fact}
\label{3.8}
\begin{enumerate}
\item ${\rm INV}^\ell(H,\bar C)$ is well defined.
\item The $\delta$-th component of ${\rm INv}^\ell(\bar H,\bar C)$ is a family
of $\le\lambda$ subsets of $C_\delta$ each of cardinality $\le 2^{\aleph_0}$
and if $\ell=1$ each member is countable and the family is closed under
subsets.
\item {\em If} $G_i\in{\frak K}^{rtf}_\lambda$ for $i<i^*$, $i^*<
\mu^{\aleph_0}$, $\bar G^i=\langle\bar G_{i,\alpha}:\alpha<\lambda\rangle$
is a representation of $G_i$,
{\em then} we can find $A_\delta\subseteq{\rm nacc}(C_\delta)$ of order type
$\omega$ such that: $i<i^*$, $\delta\in S^*\qquad \Rightarrow$\qquad for
no $a$ in the $\delta$-th component of ${\rm INv}^\ell(G_i,\bar G^i,\bar C)$ do
we have $|a \cap A_\delta|\ge\aleph_0$.
\end{enumerate}
\end{Fact}
\proof Straightforward. (For (3) note ${\rm otp}(C_\delta)\ge\mu$, so there
are $\mu^{\aleph_0}>\lambda$ pairwise almost disjoint subsets of
$C_\delta$ each
of cardinality $\aleph_0$ and every $A\in{\rm Inv}(C_\delta,\bar G^i)$
disqualifies at most $2^{\aleph_0}$ of them.)
\hfill$\square_{\ref{3.8}}$
\begin{Fact}
\label{3.9}
Let $G$ be as constructed above for $\langle A_\delta:\delta\in
S^*\rangle,A_\delta\subseteq{\rm nacc}(C_\delta)$, ${\rm otp}(A_\delta)=\omega$
(where $\langle A_\delta:\delta\in S^*\rangle$ are chosen as in \ref{3.8}(3)
for the sequence $\langle G_i:i<i^* \rangle$ given for proving
\ref{3.3}, see $(\beta)$ there).
\noindent Assume $G \subseteq H\in {\frak K}^{rtf}_\lambda$ and $\bar H$ is a
filtration of $H$. {\em Then}
\[\begin{array}{rr}
B=:\big\{\delta:A_\delta\mbox{ has infinite intersection with some}&\ \\
a\in{\rm Inv}(C_\delta,\bar H)\big\}&=\ \lambda\ \mod\ {\rm id}^a(\bar C).
\end{array}\]
\end{Fact}
\proof We assume otherwise and derive a contradiction. Let for $\alpha
<\lambda$, $S_\alpha\subseteq [\alpha]^{\le \aleph_0}$, $|S_\alpha|<\lambda$
be as guaranteed by $(*)^2_\lambda$.
Let $\chi>2^\lambda$, ${\frak A}_\alpha\prec (H(\chi),\in,<^*_\chi)$
for $\alpha<\lambda$ increasing continuous, $\|{\frak A}_\alpha\|<\lambda$,
$\langle {\frak A}_\beta:\beta\le\alpha\rangle\in {\frak A}_{\alpha+1}$,
${\frak A}_\alpha\cap\lambda$ an ordinal and:
\[\langle S_\alpha:\alpha<\lambda\rangle,\ G,\ H,\ \bar C,\ \langle
A_\delta:\delta\in S^* \rangle,\ \bar H,\ \langle x_i, y^\delta_{\bar a},
z^{\delta,n}_{\bar a,p}:\;i,\delta,\bar a,n,p \rangle\]
all belong to ${\frak A}_0$ and $2^{\aleph_0}+1\subseteq {\frak A}_0$.
Then $E=\{\delta<\lambda:{\frak A}_\delta \cap\lambda=\delta\}$ is a club
of $\lambda$. Choose $\delta\in S^* \cap E\setminus B$ such that $C_\delta
\subseteq E$. (Why can we? As to ${\rm id}^a(\bar C)$ belong all non
stationary subsets of $\lambda$, in particular $\lambda\setminus E$,
and $\lambda\setminus S^*$ and $B$, but $\lambda\notin {\rm id}^a(\bar
C)$.) Remember that $\eta_\delta$ enumerates $A_\delta$ (in the
increasing order). For each $\alpha\in A_\delta$ (so $\alpha\in E$
hence ${\frak A}_\alpha \cap \lambda=\alpha$ but $\bar H\in {\frak
A}_\alpha$ hence $H\cap {\frak A}_\alpha= H_\alpha$) and
$Q\subseteq{\bf P}^*$ choose, if possible, $y_{\alpha,Q}\in H_\alpha$ such that:
\[Q\subseteq{\bf P}(x_\alpha-y_{\alpha,Q},H).\]
Let $I_\alpha=:\{Q\subseteq{\bf P}^*:y_{\alpha,Q}$ well defined$\}$. Note (see
\ref{3.4} $(**)^1_\lambda$ and remember $\eta_\delta(n)\in A_\delta\subseteq
{\rm nacc}(C_\delta)$) that ${\rm cf}(\alpha)>2^{\aleph_0}$ (by (ii) of
\ref{3.4} $(**)^1_\lambda$) and hence for some $\beta_\alpha<\alpha$,
\[\{ y_{\alpha,Q}:Q\in I_\alpha\}\subseteq H_{\beta_\alpha}.\]
Now:
\begin{description}
\item[$\otimes_1$] $I_\alpha$ is downward closed family of subsets of
${\bf P}^*$, ${\bf P}^*\notin I_\alpha$ for $\alpha
\in A_\delta$.
\end{description}
[Why? See the definition for the first phrase and note also that $H$ is
reduced for the second phrase.]
\begin{description}
\item[$\otimes_2$] $I_\alpha$ is closed under unions of two members (hence
is an ideal on ${\bf P}^*$).
\end{description}
[Why? If $Q_1,Q_2\in I_\alpha$ then (as $x_\alpha\in G\subseteq H$ witnesses
this):
\[\begin{ALIGN}
({\cal H}(\chi),\in,<^*_\chi)\models &(\exists x)(x\in H\ \&\
Q_1\subseteq{\bf P}(x- y_{\alpha,Q_1},H)\ \&\\
&Q_2\subseteq{\bf P}(x-y_{\alpha,Q_2},H)).
\end{ALIGN}\]
All the parameters are in ${\frak A}_\alpha$ so there is $y\in
{\frak A}_\alpha\cap H$ such that
\[Q_1\subseteq{\bf P}(y-y_{\alpha,Q_1},H)\quad\mbox{ and }\quad Q_2\subseteq
{\bf P}(y-y_{\alpha,Q_2},H).\]
By algebraic manipulations,
\[Q_1\subseteq {\bf P}(x_\alpha-y_{\alpha,Q_1},H),\ Q_1\subseteq{\bf P}(y-y_{\alpha,
Q_1},H)\quad\Rightarrow\quad Q_1\subseteq{\bf P}(x_\alpha-y,H);\]
similarly for $Q_2$. So $Q_1\cup Q_2\subseteq{\bf P}(x_\alpha-y,H)$ and hence
$Q_1\cup Q_2\in I_\alpha$.]
\begin{description}
\item[$\otimes_3$] If $\bar Q=\langle Q_n:n\in\Gamma\rangle$ are pairwise
disjoint subsets of ${\bf P}^*$, for some infinite $\Gamma\subseteq\omega$,
then for some $n\in\Gamma$ we have $Q_n\in I_{\eta_\delta(n)}$.
\end{description}
[Why? Otherwise let $a_n$ be $Q_n$ if $n\in \Gamma$, and $\emptyset$
if $n\in \omega\setminus \Gamma$, and let $\bar a=\langle a_n: n<
\omega\rangle$. Now
$n\in\Gamma\quad\Rightarrow\quad\eta_\delta(n)\in
{\rm inv}(y^{\delta, 0}_{\bar a},C_\delta;\bar H)$ and hence
\[A_\delta\cap{\rm inv}(y^{\delta, 0}_{\bar a},C_\delta;\bar
H)\supseteq\{\eta_\delta(n):n\in \Gamma\},\]
which is infinite, contradicting the choice of $A_\delta$.]
\begin{description}
\item[$\otimes_4$] for all but finitely many $n$ the Boolean algebra
${\cal P}({\bf P}^*)/I_{\eta_\delta(n)}$ is finite.
\end{description}
[Why? If not, then by $\otimes_1$ second phrase, for each $n$ there are
infinitely many non-principal ultrafilters $D$ on ${\bf P}^*$ disjoint to
$I_{\eta_\delta(n)}$, so for $n<\omega$ we can find an ultrafilter $D_n$
on ${\bf P}^*$ disjoint to $I_{\eta_\delta(n)}$, distinct from $D_m$ for
$m<n$. Thus we can find $\Gamma\in [\omega]^{\aleph_0}$ and $Q_n\in D_n$
for $n\in\Gamma$ such that $\langle Q_n:n\in\Gamma\rangle$ are pairwise
disjoint (as $Q_n\in D_n$ clearly $|Q_n|=\aleph_0$). Why? Look: if $B_n
\in D_0\setminus D_1$ for $n\in\omega$ then
\[(\exists^\infty n)(B_n \in D_n)\quad\mbox{ or }\quad(\exists^\infty n)
({\bf P}^*\setminus B_n \in D_n),\]
etc. Let $Q_n=\emptyset$ for $n\in\omega\setminus\Gamma$, now $\bar Q=
\langle Q_n:n<\omega\rangle$ contradicts $\otimes_3$.]
\begin{description}
\item[$\otimes_5$] If the conclusion (of \ref{3.9}) fails, then for no
$\alpha\in A_\delta$ is ${\cal P}({\bf P}^*)/I_\alpha$ finite.
\end{description}
[Why? If not, choose such an $\alpha$ and $Q^*\subseteq{\bf P}^*$, $Q^*
\notin I_\alpha$ such that $I=I_\alpha\restriction Q^*$ is a maximal
ideal on $Q^*$. So $D=:{\cal P}(Q^*)\setminus I$ is a non-principal
ultrafilter. Remember $\beta=\beta_\alpha<\alpha$ is such that
$\{y_{\alpha,Q}:Q\in I_\alpha\}\subseteq H_\beta$. Now, $H_\beta\in
{\frak A}_{\beta+1}$, $|H_\beta|<\lambda$. Hence $(*)^2_\lambda$ from
\ref{3.3} (note that it does not matter whether we consider an ordinal
$\chi<\lambda$ or a cardinal $\chi<\lambda$, or any other set of
cardinality $< \lambda$) implies that there is $S_{H_\beta}\in
{\frak A}_{\beta+1}$, $S_{H_\beta}\subseteq [H_\beta]^{\le \aleph_0}$,
$|S_{H_\beta}|<\lambda$ as there. Now it does not matter if we deal with
functions from an ultrafilter on $\omega$ \underbar{or} an ultrafilter on
$Q^*$. We define $f:D\longrightarrow H_\beta$ as follows: for $U\in D$ we
let $f(U)=y_{\alpha,Q^* \setminus U}$. (Note: $Q^*\setminus U\in I_\alpha$,
hence $y_{\alpha,Q^* \setminus U}$ is well defined.) So, by the choice of
$S_{H_\beta}$ (see (ii) of $(*)^2_\lambda$), for some countable $f'
\subseteq f$, $f'\in {\frak A}_{\beta+1}$ and $\bigcap\{U:U\in{\rm Dom}(f')\}
\notin D$ (reflect for a minute). Let ${\rm Dom}(f')=\{U_0,U_1,\ldots\}$.
Then $\bigcup\limits_{n<\omega}(Q^*\setminus U_n)\notin I_\alpha$. But as
in the proof of $\otimes_2$, as
\[\langle y_\alpha,(Q^* \setminus U_n):n<\omega\rangle\in
{\frak A}_{\beta+1}\subseteq {\frak A}_\alpha,\]
we have $\bigcup\limits_{n<\omega}(Q^*\setminus U_n)\in I_\alpha$, an
easy contradiction.]
Now $\otimes_4$, $\otimes_5$ give a contradiction. \hfill$\square_{\ref{3.3}}$
\begin{Remark}
\label{3.10}
We can deal similarly with $R$-modules, $|R|<\mu$ \underbar{if} $R$ has
infinitely many prime ideals $I$. Also the treatment of
${\frak K}^{rs(p)}_\lambda$ is similar to the one for modules over
rings with one prime.
\noindent Note: if we replace ``reduced" by
\[x\in G \setminus\{0\}\quad \Rightarrow\quad (\exists p\in{\bf P}^*)(x\notin
pG)\]
then here we could have defined
\[{\bf P}(x,H)=:\{p\in {\bf P}^*:x\in pH\}\]
and the proof would go through with no difference (e.g. choose a fixed
partition $\langle {\bf P}^*_n: n< \omega\rangle$ of ${\bf P}^*$ to infinite
sets, and let ${\bf P}'(x, H)=\{n: x\in pH\mbox{ for every }p\in
{\bf P}^*_n\}$). Now the groups are less divisible.
\end{Remark}
\begin{Remark}
\label{3.11}
We can get that the groups are slender, in fact, the construction gives it.
\end{Remark}
\section{Below the continuum there may be universal structures}
Both in \cite{Sh:456} (where we deal with universality for $(<\lambda)$-stable
(Abelian) groups, like ${\frak K}^{rs(p)}_\lambda$) and in \S3, we restrict
ourselves to $\lambda>2^{\aleph_0}$, a restriction which does not appear
in \cite{KjSh:447}, \cite{KjSh:455}. Is this restriction necessary? In this
section we shall show that at least to some extent, it is.
We first show under MA that for $\lambda<2^{\aleph_0}$, any $G\in
{\frak K}^{rs(p)}_\lambda$ can be embedded into a ``nice" one; our aim
is to reduce the consistency of ``there is a universal in
${\frak K}^{rs(p)}_\lambda$" to ``there is a universal in
${\frak K}^{tr}_{\langle\aleph_0:n<\omega\rangle\char 94\langle \lambda
\rangle}$". Then we proceed to
prove the consistency of the latter. Actually a weak form of MA suffices.
\begin{Definition}
\label{4.2}
\begin{enumerate}
\item $G\in {\frak K}^{rs(p)}_\lambda$ is {\em tree-like} if:
\begin{description}
\item[(a)] we can find a basic subgroup $B=
\bigoplus\limits_{\scriptstyle i<\lambda_n\atop\scriptstyle n<\omega}
{\Bbb Z} x^n_i$, where
\[\lambda_n=\lambda_n(G)=:\dim\left((p^nG)[p]/p^{n+1}(G)[p]\right)\]
(see Fuchs \cite{Fu}) such that: ${\Bbb Z} x^n_i \cong {\Bbb Z}/p^{n+1}
{\Bbb Z}$ and
\begin{description}
\item[$\otimes_0$] every $x\in G$ has the form
\[\sum\limits_{n,i}\{a^n_i p^{n-k} x^n_i:n\in [k,\omega)\mbox{ and }
i<\lambda\}\]
where $a^n_i\in{\Bbb Z}$ and
\[n<\omega\quad \Rightarrow\quad w_n[x]=:\{i:a^n_i\, p^{n-k}x^n_i\neq 0\}
\mbox{ is finite}\]
\end{description}
(this applies to any $G\in {\frak K}^{rs(p)}_\lambda$ we considered so far;
we write $w_n[x]=w_n[x,\bar Y]$ when $\bar Y=\langle x^n_i:n,i\rangle$).
Moreover
\item[(b)] $\bar Y=\langle x^n_i:n,i\rangle$ is tree-like inside $G$,
which means
that we can find $F_n:\lambda_{n+1}\longrightarrow\lambda_n$ such that
letting $\bar F=\langle F_n:n<\omega\rangle$, $G$ is generated by some
subset of
$\Gamma(G,\bar Y,\bar F)$ where:
\[\hspace{-0.5cm}\begin{ALIGN}
\Gamma(G,\bar Y,\bar F)=\big\{x:&\mbox{for some }\eta\in\prod\limits_{n
<\omega}\lambda_n, \mbox{ for each } n<\omega \mbox{ we have}\\
&F_n(\eta(n+1))=\eta(n)\mbox{ and }x=\sum\limits_{n\ge k}
p^{n-k}x^n_{\eta(n)}\big\}.
\end{ALIGN}\]
\end{description}
\item $G\in {\frak K}^{rs(p)}_\lambda$ is {\em semi-tree-like} if above
we replace (b) by
\begin{description}
\item[(b)$'$] we can find a set $\Gamma\subseteq\{\eta:\eta$ is a partial
function from $\omega$ to $\sup\limits_{n<\omega} \lambda_n$ with
$\eta(n)< \lambda_n\}$ such that:
\begin{description}
\item[($\alpha$)] $\eta_1\in\Gamma,\ \eta_2\in\Gamma,\ \eta_1(n)=\eta_2(n)
\quad \Rightarrow\quad\eta_1\restriction n=\eta_2\restriction n$,
\item[($\beta$)] for $\eta\in\Gamma$ and $n\in{\rm Dom}(\eta)$, there is
\[y_{\eta,n}=\sum \{p^{m-n}x^m_{\eta(m)}:m \in {\rm Dom}(\eta)\mbox{ and }
m \ge n\}\in G,\]
\item[($\gamma$)] $G$ is generated by
\[\{x^n_i:n<\omega,i<\lambda_n\}\cup\{y_{\eta,n}:\eta\in\Gamma,n\in
{\rm Dom}(\eta)\}.\]
\end{description}
\end{description}
\item $G\in {\frak K}^{rs(p)}_\lambda$ is {\em almost tree-like} if
in (b)$'$ we add
\begin{description}
\item[($\delta$)] for some $A\subseteq\omega$ for every $\eta\in\Gamma$,
${\rm Dom}(\eta)=A$.
\end{description}
\end{enumerate}
\end{Definition}
\begin{Proposition}
\label{4.3}
\begin{enumerate}
\item Suppose $G\in {\frak K}^{rs(p)}_\lambda$ is almost tree-like, as
witnessed by $A\subseteq\omega$, $\lambda_n$ (for $n<\omega$), $x^n_i$ (for
$n\in A$, $i<\lambda_n$), and if $n_0<n_2$ are successive members of $A$,
$n_0<n<n_2$ then $\lambda_n\ge\lambda_{n_0}$ or just
\[\lambda_n\ge|\{\eta(n_0):\eta\in\Gamma\}|.\]
{\em Then} $G$ is tree-like (possibly with other witnesses).
\item If in \ref{4.2}(3) we just demand $\eta\in\Gamma\quad\Rightarrow
\quad\bigvee\limits_{n<\omega}{\rm Dom}(\eta)\setminus n=A\setminus n$;
then changing the $\eta$'s and the $y_{\eta,n}$'s we can regain the
``almost tree-like".
\end{enumerate}
\end{Proposition}
\proof 1) For every successive members $n_0<n_2$ of $A$ for
\[\alpha\in S_{n_0}=:\{\alpha:(\exists\eta)[\eta\in\Gamma\ \&\ \eta(n_0)
=\alpha]\},\]
choose ordinals $\gamma(n_0,\alpha,\ell)$ for $\ell\in (n_0,n_2)$ such that
\[\gamma(n_0,\alpha_1,\ell)=\gamma(n_0,\alpha_2,\ell)\quad\Rightarrow
\quad\alpha_1=\alpha_2.\]
We change the basis by replacing for $\alpha\in S_{n_0}$, $\{x^n_\alpha\}\cup
\{x^\ell_{\gamma(n_0,\alpha,\ell)}:\ell\in (n_0,n_2)\}$ (note: $n_0<n_2$
but possibly $n_0+1=n_2$), by:
\[\begin{ALIGN}
\biggl\{ x^{n_0}_\alpha + px^{n_0+1}_{\gamma(n_0,\alpha,n_0+1)},
&x^{n_0+1}_{\gamma(n_0,\alpha,n_0+1)} +
px^{n_0+2}_{\gamma(n_0,\alpha,n_0+2)},\ldots, \\
&x^{n_2-2}_{\gamma(n_0,\alpha,n_2-2)} +
px^{n_2-1}_{\gamma(n_0,\alpha,n_2-1)},
x^{n_2-1}_{\gamma(n_0,\gamma,n_2-1)} \biggr\}.
\end{ALIGN}\]
2) For $\eta\in \Gamma$ let $n(\eta)=\min\{ n: n\in A\cap{\rm Dom}(\eta)$
and ${\rm Dom}(\eta)\setminus n=A\setminus n\}$, and let
$\Gamma_n=\{\eta\in \Gamma: n(\eta)=n\}$ for $n\in A$. We choose by
induction on $n< \omega$ the objects $\nu_\eta$ for $\eta\in \Gamma_n$
and $\rho^n_\alpha$ for $\alpha< \lambda_n$ such that: $\nu_\eta$ is a
function with domain $A$, $\nu_\eta\restriction (A\setminus
n(\eta))=\eta\restriction (A\setminus n(\eta))$ and
$\nu_\eta\restriction (A\cap n(\eta))= \rho^n_{\eta(n)}$,
$\nu_\eta(n)< \lambda_n$ and $\rho^n_\alpha$ is a function with domain
$A\cap n$, $\rho^n_\alpha(\ell)< \lambda_\ell$ and $\rho^n_\alpha
\restriction (A\cap \ell) = \rho^\ell_{\rho^n_\alpha(\ell)}$ for
$\ell\in A\cap n$. There are no problems and $\{\nu_\eta: \eta\in
\Gamma_n\}$ is as required.
\hfill$\square_{\ref{4.3}}$
\begin{Theorem}[MA]
\label{4.1}
Let $\lambda<2^{\aleph_0}$. Any $G\in {\frak K}^{rs(p)}_\lambda$ can be
embedded into some $G'\in {\frak K}^{rs(p)}_\lambda$ with countable
density which is tree-like.
\end{Theorem}
\proof By \ref{4.3} it suffices to get $G'$ ``almost tree-like" and
$A\subseteq\omega$ which satisfies \ref{4.3}(1). The ability to make $A$
thin helps in proving Fact E below. By \ref{1.1} without loss of generality
$G$ has a base (i.e. a dense subgroup of the form)
$B=\bigoplus\limits_{\scriptstyle n<\omega\atop\scriptstyle i<\lambda_n}
{\Bbb Z} x^n_i$, where ${\Bbb Z} x^n_i\cong{\Bbb Z}/p^{n+1}{\Bbb Z}$ and
$\lambda_n=\aleph_0$ (in fact $\lambda_n$ can be $g(n)$ if $g\in
{}^\omega\omega$ is not bounded (by algebraic manipulations), this will be
useful if we consider the forcing from \cite[\S2]{Sh:326}).
Let $B^+$ be the extension of $B$ by $y^{n,k}_i$ ($k<\omega$, $n<\omega$,
$i<\lambda_n$) generated freely except for $py^{n,k+1}_i=y^{n,k}_i$ (for
$k<\omega$), $y^{n,\ell}_i=p^{n-\ell}x^n_i$ for $\ell\le n$, $n<\omega$,
$i<\lambda_n$. So $B^+$ is a divisible $p$-group, let $G^+ =:
B^+\bigoplus\limits_B
G$. Let $\{z^0_\alpha:\alpha<\lambda\}\subseteq G[p]$ be a basis of $G[p]$
over $\{p^n x^n_i:n,i<\omega\}$ (as a vector space over ${\Bbb Z}/p{\Bbb
Z}$ i.e. the two sets are disjoint, their union is a basis); remember
$G[p]=\{x\in G:px=0\}$. So we can find $z^k_\alpha\in G$ (for $\alpha<
\lambda$, $k<\omega$ and $k\ne 0$) such that
\[pz^{k+1}_\alpha-z^k_\alpha=\sum_{i\in w(\alpha,k)} a^{k,\alpha}_i
x^k_i,\]
where $w(\alpha,k)\subseteq\omega$ is finite (reflect on the Abelian group
theory).
We define a forcing notion $P$ as follows: a condition $p \in P$ consists
of (in brackets are explanations of intentions):
\begin{description}
\item[(a)] $m<\omega$, $M\subseteq m$,
\end{description}
[$M$ is intended as $A\cap\{0,\ldots,m-1\}$]
\begin{description}
\item[(b)] a finite $u\subseteq m\times\omega$ and $h:u\longrightarrow
\omega$ such that $h(n,i)\ge n$,
\end{description}
[our extensions will not be pure, but still we want that the group produced
will be reduced, now we add some $y^{n,k}_i$'s and $h$ tells us how many]
\begin{description}
\item[(c)] a subgroup $K$ of $B^+$:
\[K=\langle y^{n,k}_i:(n,i)\in u,k<h(n,i)\rangle_{B^+},\]
\item[(d)] a finite $w\subseteq\lambda$,
\end{description}
[$w$ is the set of $\alpha<\lambda$ on which we give information]
\begin{description}
\item[(e)] $g:w\rightarrow m + 1$,
\end{description}
[$g(\alpha)$ is in what level $m'\le m$ we ``start to think" about $\alpha$]
\begin{description}
\item[(f)] $\bar\eta=\langle\eta_\alpha:\alpha\in w\rangle$ (see (i)),
\end{description}
[of course, $\eta_\alpha$ is the intended $\eta_\alpha$ restricted to $m$ and
the set of all $\eta_\alpha$ forms the intended $\Gamma$]
\begin{description}
\item[(g)] a finite $v\subseteq m\times\omega$,
\end{description}
[this approximates the set of indices of the new basis]
\begin{description}
\item[(h)] $\bar t=\{t_{n,i}:(n,i)\in v\}$ (see (j)),
\end{description}
[approximates the new basis]
\begin{description}
\item[(i)] $\eta_\alpha\in {}^M\omega$, $\bigwedge\limits_{\alpha\in w}
\bigwedge\limits_{n\in M} (n,\eta_\alpha(n))\in v$,
\end{description}
[toward guaranteeing clause $(\delta)$ of \ref{4.2}(3) (see \ref{4.3}(2))]
\begin{description}
\item[(j)] $t_{n,i}\in K$ and ${\Bbb Z} t_{n,i} \cong {\Bbb Z}/p^n
{\Bbb Z}$,
\item[(k)] $K=\bigoplus\limits_{(n,i)\in v} ({\Bbb Z} t_{n,i})$,
\end{description}
[so $K$ is an approximation to the new basic subgroup]
\begin{description}
\item[(l)] if $\alpha\in w$, $g(\alpha)\le\ell\le m$ and $\ell\in M$ then
\[z^\ell_\alpha-\sum\{t^{n-\ell}_{n,\eta_\alpha(n)}:\ell\le n\in
{\rm Dom}(\eta_\alpha)\}\in p^{m-\ell}(K+G),\]
\end{description}
[this is a step toward guaranteeing that the full difference (when
${\rm Dom}(\eta_\alpha)$ is possibly infinite) will be in the closure of
$\bigoplus\limits_{\scriptstyle n\in [i,\omega)\atop\scriptstyle
i<\omega} {\Bbb Z} x^n_i$].
We define the order by:
\noindent $p \le q$ \qquad if and only if
\begin{description}
\item[$(\alpha)$] $m^p\le m^q$, $M^q \cap m^p = M^p$,
\item[$(\beta)$] $u^p\subseteq u^q$, $h^p\subseteq h^q$,
\item[$(\gamma)$] $K^p\subseteq_{pr} K^q$,
\item[$(\delta)$] $w^p\subseteq w^q$,
\item[$(\varepsilon)$] $g^p\subseteq g^q$,
\item[$(\zeta)$] $\eta^p_\alpha\trianglelefteq\eta^q_\alpha$,
(i.e. $\eta^p_\alpha$ is an initial segment of $\eta^q_\alpha$)
\item[$(\eta)$] $v^p\subseteq v^q$,
\item[$(\theta)$] $t^p_{n,i}=t^q_{n,i}$ for $(n,i)\in v^p$.
\end{description}
\medskip
\noindent{\bf A Fact}\hspace{0.15in} $(P,\le)$ is a partial order.
\medskip
\noindent{\em Proof of the Fact:}\ \ \ Trivial.
\medskip
\noindent{\bf B Fact}\hspace{0.15in} $P$ satisfies the c.c.c. (even is
$\sigma$-centered).
\medskip
\noindent{\em Proof of the Fact:}\ \ \ It suffices to observe the following.
Suppose that
\begin{description}
\item[$(*)$(i)] $p,q \in P$,
\item[\quad(ii)] $M^p=M^q$, $m^p=m^q$, $h^p=h^q$, $u^p=u^q$, $K^p=K^q$,
$v^p=v^q$, $t^p_{n,i}=t^q_{n,i}$,
\item[\quad(iii)] $\langle\eta^p_\alpha:\alpha\in w^p\cap w^q\rangle =
\langle\eta^q_\alpha:\alpha\in w^p\cap w^q\rangle$,
\item[\quad(iv)] $g^p\restriction (w^p \cap w^q)=g^q \restriction(w^p
\cap w^q)$.
\end{description}
Then the conditions $p,q$ are compatible (in fact have an upper bound with
the same common parts): take the common values (in (ii)) or the union (for
(iii)).
\medskip
\noindent{\bf C Fact}\hspace{0.15in} For each $\alpha<\lambda$ the set
${\cal I}_\alpha=:\{p\in P:\alpha\in w^p\}$ is dense (and open).
\medskip
\noindent{\em Proof of the Fact:}\ \ \ For $p\in P$ let $q$ be like $p$
except that:
\[w^q=w^p\cup\{\alpha\}\quad\mbox{ and }\quad g^q(\beta)=\left\{
\begin{array}{lll}
g^p(\beta) &\mbox{if}& \beta\in w^p \\
m^p &\mbox{if}& \beta=\alpha,\ \beta\notin w^p.
\end{array}\right.\]
\medskip
\noindent{\bf D Fact}\hspace{0.15in} For $n<\omega$, $i<\omega$ the
following set is a dense subset of $P$:
\[{\cal J}^*_{(n,i)}=\{p\in P:x^n_i\in K^p\ \&\ (\forall n<m^p)(\{n\}\times
\omega)\cap u^p \mbox{ has }>m^p\mbox{ elements}\}.\]
\medskip
\noindent{\em Proof of the Fact:}\ \ \ Should be clear.
\medskip
\noindent{\bf E Fact}\hspace{0.15in} For each $m<\omega$ the set ${\cal J}_m
=:\{p\in P:m^p\ge m\}$ is dense in $P$.
\medskip
\noindent{\em Proof of the Fact:}\ \ \ Let $p\in P$ be given such that $m^p
<m$. Let $w^p=\{\alpha_0,\ldots,\alpha_{r-1}\}$ be without repetitions;
we know that in $G$, $pz^0_{\alpha_\ell}=0$ and $\{z^0_{\alpha_\ell}:
\ell<r\}$ is independent $\mod\ B$, hence also in $K+G$ the set
$\{z^0_{\alpha_\ell}:\ell<r\}$ is independent $\mod\ K$. Clearly
\begin{description}
\item[(A)] $pz^{k+1}_{\alpha_\ell}=z^k_{\alpha_\ell}\mod\ K$
for $k\in [g(\alpha_\ell),m^p)$, hence
\item[(B)] $p^{m^p}z^{m^p}_{\alpha_\ell}=z^{g(\alpha_\ell)}_{\alpha_\ell}
\mod\ K$.
\end{description}
Remember
\begin{description}
\item[(C)] $z^{m^p}_{\alpha_\ell}=\sum\{a^{k,\alpha_\ell}_i p^{k-m^p} x^k_i:
k \ge m^p,i\in w(\alpha_\ell,k)\}$,
\end{description}
and so, in particular, (from the choice of $z^0_{\alpha_\ell}$)
\[p^{m^p+1}z^{m^p}_{\alpha_\ell}=0\quad\mbox{ and }\quad
p^{m^p}z^{m^p}_{\alpha_\ell}\ne 0.\]
For $\ell<r$ and $n\in [m^p,\omega)$ let
\[s^n_\ell=:\sum\big\{a^{k,\alpha_\ell}_i p^{k-m^p} x^k_i:k \ge m^p
\mbox{ but }k<n\mbox{ and } i\in w(\alpha_\ell,k)\big\}.\]
But $p^{k-m^p}x^k_i = y^{k,m^p}_i$, so
\[s^n_\ell=\sum\big\{a^{k,\alpha_\ell}_i y^{k,m^p}_i:k\in [m^p,n)
\mbox{ and }i\in (\alpha_\ell,k)\big\}.\]
Hence, for some $m^*>m,m^p$ we have: $\{p^m\,s^{m^*}_\ell:\ell<r\}$ is
independent in $G[p]$ over $K[p]$ and therefore in $\langle x^k_i:k\in
[m^p,m^*],i<\omega\rangle$. Let
\[s^*_\ell=\sum\big\{a^{k,\alpha_\ell}_i:k\in [m^p,m^*)\mbox{ and }i\in
w(\alpha_\ell,k)\}.\]
Then $\{ s^*_\ell:\ell<r\}$ is independent in
\[B^+_{[m,m^*)}=\langle y^{l,m^*-1}_i:k\in [m^p,m^*)\mbox{ and }i<\omega
\rangle.\]
Let $i^*<\omega$ be such that: $w(\alpha_\ell,k)\subseteq\{0,\ldots,i^*-1\}$
for $k\in [m^p,m^*)$, $\ell=1,\ldots,r$. Let us start to define $q$:
\[\begin{array}{c}
m^q=m^*,\quad M^q=M^p\cup\{m^*-1\},\quad w^q=w^p,\quad g^q=g^p,\\
u^q=u^p\cup ([m^p,m^*)\times\{0,\ldots,i^*-1\}),\\
h^q\mbox{ is } h^p\mbox{ on } u^p\mbox{ and }h^q(k,i)=m^*-1\mbox{
otherwise},\\
K^q\mbox{ is defined appropriately, let } K'=\langle x^n_i:n\in [m^p,m^*),
i<i^*\rangle.
\end{array}\]
Complete $\{s^*_\ell:\ell<r\}$ to $\{s^*_\ell:\ell<r^*\}$, a basis of $K'[p]$,
and choose $\{t_{n,i}:(n,i)\in v^*\}$ such that: $[p^mt_{n,i}=0\ \
\Leftrightarrow\ \ m>n]$, and for $\ell<r$
\[p^{m^*-1-\ell}t_{m^*-1,\ell} = s^*_\ell.\]
The rest should be clear.
\medskip
The generic gives a variant of the desired result: almost tree-like basis; the
restriction to $M$ and $g$ but by \ref{4.3} we can finish.
\hfill$\square_{\ref{4.2}}$
\begin{Conclusion}
[MA$_\lambda$($\sigma$-centered)]
\label{4.4}
For $(*)_0$ to hold it suffices that $(*)_1$ holds where
\begin{description}
\item[$(*)_0$] in ${\frak K}^{rs(p)}_\lambda$, there is a universal
member,
\item[$(*)_1$] in ${\frak K}^{tr}_{\bar\lambda}$ there is a universal
member, where:
\begin{description}
\item[(a)] $\lambda_n=\aleph_0$, $\lambda_\omega=\lambda$, $\ell g(\bar\lambda)
=\omega+1$\qquad \underbar{or}
\item[(b)] $\lambda_\omega=\lambda$, $\lambda_n\in [n,\omega)$, $\ell g
(\bar \lambda)=\omega+1$.
\end{description}
\end{description}
\end{Conclusion}
\begin{Remark}
\label{4.4A}
Any $\langle\lambda_n:n<\omega\rangle$, $\lambda_n<\omega$ which is not
bounded suffices.
\end{Remark}
\proof For case (a) - by \ref{4.1}.
\noindent For case (b) - the same proof. \hfill$\square_{\ref{4.4}}$
\begin{Theorem}
\label{4.5}
Assume $\lambda<2^{\aleph_0}$ and
\begin{description}
\item[(a)] there are $A_i\subseteq\lambda$, $|A_i|=\lambda$ for
$i<2^\lambda$ such that $i\ne j \Rightarrow |A_i \cap A_j| \le \aleph_0$.
\end{description}
Let $\bar\lambda=\langle \lambda_\alpha:\alpha\le\omega\rangle$, $\lambda_n
= \aleph_0$, $\lambda_\omega=\lambda$.
\noindent{\em Then} there is $P$ such that:
\medskip
\begin{description}
\item[$(\alpha)$] $P$ is a c.c.c. forcing notion,
\item[$(\beta)$] $|P|=2^\lambda$,
\item[$(\gamma)$] in $V^P$, there is $T\in {\frak K}^{tr}_{\bar\lambda}$
into which every $T' \in ({\frak K}^{tr}_{\bar \lambda})^V$ can be embedded.
\end{description}
\end{Theorem}
\proof Let $\bar T=\langle T_i:i<2^\lambda\rangle$ list the trees $T$ of
cardinality $\le\lambda$ satisfying
\[{}^{\omega >}\omega\subseteq T \subseteq {}^{\omega \ge} \omega\quad
\mbox{ and }\quad T\cap {}^\omega\omega\mbox{ has cardinality $\lambda$, for
simplicity.}\]
Let $T_i\cap {}^\omega\omega=\{\eta^i_\alpha:\alpha\in A_i \}$.
We shall force $\rho_{\alpha,\ell}\in {}^\omega\omega$ for $\alpha<
\lambda$, $\ell<\omega$, and for each $i<2^\lambda$ a function $g_i:A_i
\longrightarrow\omega$ such that: there is an automorphism $f_i$ of
$({}^{\omega>}\omega,\triangleleft)$ which induces an embedding of $T_i$
into $\left(({}^{\omega>}\omega)\cup \{\rho_{\alpha,g_i(\alpha)}:\alpha<
\lambda\},\triangleleft\right)$. We shall define $p\in P$ as an approximation.
\noindent A condition $p\in P$ consists of:
\begin{description}
\item[(a)] $m<\omega$ and a finite subset $u$ of ${}^{m \ge}\omega$, closed
under initial segments such that $\langle\rangle\in u$,
\item[(b)] a finite $w\subseteq 2^\lambda$,
\item[(c)] for each $i\in w$, a finite function $g_i$ from $A_i$ to
$\omega$,
\item[(d)] for each $i\in w$, an automorphism $f_i$ of
$(u,\triangleleft)$,
\item[(e)] a finite $v\subseteq\lambda\times\omega$,
\item[(f)] for $(\alpha,n)\in v$, $\rho_{\alpha,n}\in u\cap
({}^m\omega)$,
\end{description}
such that
\begin{description}
\item[(g)] if $i\in w$ and $\alpha\in{\rm Dom}(g_i)$ then:
\begin{description}
\item[$(\alpha)$] $(\alpha,g_i(\alpha))\in v$,
\item[$(\beta)$] $\eta^i_\alpha\restriction m\in u$,
\item[$(\gamma)$] $f_i(\eta^i_\alpha\restriction m)=\rho_{\alpha,
g_i(\alpha)}$,
\end{description}
\item[(h)] $\langle\rho_{\alpha,n}:(\alpha,n)\in v\rangle$ is with no
repetition (all of length $m$),
\item[(i)] for $i\in w$, $\langle\eta^i_\alpha\restriction m:\alpha\in
{\rm Dom}(g_i)\rangle$ is with no repetition.
\end{description}
The order on $P$ is: $p \le q$ if and only if:
\begin{description}
\item[$(\alpha)$] $u^p \subseteq u^q$, $m^p\le m^q$,
\item[$(\beta)$] $w^p \subseteq w^q$,
\item[$(\gamma)$] $f^p_i \subseteq f^q_i$ for $i\in w^p$,
\item[$(\delta)$] $g^p_i \subseteq g^q_i$ for $i\in w^p$,
\item[$(\varepsilon)$] $v^p \subseteq v^q$,
\item[$(\zeta)$] $\rho^p_{\alpha,n}\trianglelefteq\rho^q_{\alpha,n}$,
when $(\alpha,n) \in v^p$,
\item[$(\eta)$] if $i\ne j\in w^p$ then for every $\alpha\in A_i\cap
A_j\setminus ({\rm Dom}(g^p_i)\cap {\rm Dom}(g^p_j))$ we have $g^q_i(\alpha)\ne
g^q_j(\alpha)$.
\end{description}
\medskip
\noindent{\bf A Fact}\hspace{0.15in} $(P,\le)$ is a partial order.
\medskip
\noindent{\em Proof of the Fact:}\ \ \ Trivial.
\medskip
\noindent{\bf B Fact}\hspace{0.15in} For $i<2^\lambda$ the set $\{p:i\in
w^p\}$ is dense in $P$.
\medskip
\noindent{\em Proof of the Fact:}\ \ \ If $p\in P$, $i\in 2^\lambda
\setminus w^p$, define $q$ like $p$ except $w^q=w^p\cup\{i\}$,
${\rm Dom}(g^q_i)=\emptyset$.
\medskip
\noindent{\bf C Fact}\hspace{0.15in} If $p\in P$, $m_1\in
(m^p,\omega)$, $\eta^*\in u^p$, $m^*<\omega$, $i\in w^p$, $\alpha\in\lambda
\setminus{\rm Dom}(g^p_i)$ {\em then} we can find $q$ such that $p\le q\in
P$, $m^q>m_1$, $\eta^* \char 94\langle m^*\rangle\in u^q$ and $\alpha\in
{\rm Dom}(g_i)$ and $\langle\eta^j_\beta\restriction m^q:j\in w^q$ and $\beta\in
{\rm Dom}(g^q_j)\rangle$ is with no repetition, more exactly
$\eta^{j(1)}_{\beta_1}\setminus m^q= \eta^{j(2)}_{\beta_2}\restriction
m^q \Rightarrow \eta^{j(1)}_{\beta_1}=\eta^{j(2)}_{\beta_2}$.
\medskip
\noindent{\em Proof of the Fact:}\ \ \ Let $n_0\le m^p$ be maximal such that
$\eta^i_\alpha\restriction n_0 \in u^p$. Let $n_1<\omega$ be minimal such
that $\eta^i_\alpha\restriction n_1\notin\{\eta^i_\beta\restriction n_1:
\beta\in{\rm Dom}(g^p_i)\}$ and moreover the sequence
\[\langle\eta^j_\beta\restriction n_1:j\in w^p\ \&\ \beta\in{\rm Dom}(g^p_j)\
\ \mbox{ or }\ \ j=i\ \&\ \beta=\alpha\rangle\]
is with no repetition. Choose a natural number $m^q>m^p+1,n_0+1,n_1+2$ and
let $k^*=:3+\sum\limits_{i\in w^p}|{\rm Dom}(g^p_i)|$. Choose $u^q\subseteq
{}^{m^q\ge}\omega$ such that:
\begin{description}
\item[(i)] $u^p\subseteq u^q\subseteq {}^{m^q\ge}\omega$, $u^q$ is downward
closed,
\item[(ii)] for every $\eta\in u^q$ such that $\ell g(\eta)<m^q$, for
exactly $k^*$ numbers $k$, $\eta\char 94\langle k\rangle\in u^q\setminus
u^p$,
\item[(iii)] $\eta^j_\beta\restriction\ell\in u^q$ when $\ell\le m^q$
and $j\in w^p$, $\beta\in{\rm Dom}(g^p_j)$,
\item[(iv)] $\eta^i_\alpha\restriction\ell\in u^q$ for $\ell\le m^q$,
\item[(v)] $\eta^*\char 94\langle m^* \rangle\in u^q$.
\end{description}
Next choose $\rho^q_{\beta,n}$ (for pairs $(\beta,n)\in v^p)$ such that:
\[\rho^p_{\beta,n}\trianglelefteq\rho^q_{\beta,n}\in u^q\cap {}^{m^q}
\omega.\]
For each $j\in w^p$ separately extend $f^p_j$ to an automorphism $f^q_j$
of $(u^q,\triangleleft)$ such that for each $\beta\in{\rm Dom}(g^p_j)$ we have:
\[f^q_j(\eta^j_\beta\restriction m^q)=\rho^q_{\beta,g_j}(\beta).\]
This is possible, as for each $\nu\in u^p$, and $j\in w^p$, we can
separately define
\[f^q_j\restriction\{\nu':\nu\triangleleft\nu'\in u^q\ \mbox{ and }\ \nu'
\restriction (\ell g(\nu)+1)\notin u^p\}\]
--its range is
\[\{\nu':f^p_j(\nu)\triangleleft \nu'\in u^q\ \mbox{ and }\ \nu'
\restriction (\ell g(\nu)+1)\notin u^p\}.\]
The point is: by Clause (ii) above those two sets are isomorphic and for
each $\nu$ at most one $\rho^p_{\beta,n}$ is involved (see Clause (h) in the
definition of $p \in P$). Next let $w^q=w^p$, $g^q_j=g^p_j$ for $j\in w
\setminus\{i\}$, $g^q_i\restriction{\rm Dom}(g^p_i)=g^p_i$, $g^q_i(\alpha)=
\min(\{n:(\alpha,n)\notin v^p\})$, ${\rm Dom}(g^q_i)={\rm Dom}(g^p_i)\cup\{\alpha\}$,
and $\rho^q_{\alpha,g^q_i(\alpha)}=f^g_i(\eta^i_\alpha\restriction m^q)$ and
$v^q=v^p\cup\{(\alpha,g^q_i(\alpha))\}$.
\medskip
\noindent{\bf D Fact}\hspace{0.15in} $P$ satisfies the c.c.c.
\medskip
\noindent{\em Proof of the Fact:}\ \ \ Assume $p_\varepsilon\in P$ for
$\varepsilon<\omega_1$. By Fact C, without loss of generality each
\[\langle\eta^j_\beta\restriction m^{p_\varepsilon}:j\in w^{p_\varepsilon}
\mbox{ and }\beta\in{\rm Dom}(g^{p_\varepsilon}_j)\rangle\]
is with no repetition. Without loss of generality, for all $\varepsilon
<\omega_1$
\[U_\varepsilon=:\big\{\alpha<2^\lambda:\alpha\in w^{p_\varepsilon}\mbox{ or }
\bigvee_{i\in w^p}[\alpha\in{\rm Dom}(g_i)]\mbox{ or }\bigvee_k(k,\alpha)\in
v^{p_\varepsilon}\big\}\]
has the same number of elements and for $\varepsilon\ne\zeta<\omega_1$,
there is a unique one-to-one order preserving function from $U_\varepsilon$
onto $U_\zeta$ which we call ${\rm OP}_{\zeta,\varepsilon}$, which also maps
$p_\varepsilon$ to $p_\zeta$ (so $m^{p_\zeta}=m^{p_\varepsilon}$; $u^{p_\zeta}
=u^{p_\varepsilon}$; ${\rm OP}_{\zeta,\varepsilon}(w^{p_\varepsilon})=
w^{p_\zeta}$; if $i\in w^{p_\varepsilon}$, $j={\rm OP}_{\zeta,\varepsilon}(i)$,
then $f_i\circ{\rm OP}_{\varepsilon,\zeta}\equiv f_j$; and {\em if\/} $\beta=
{\rm OP}_{\zeta,\varepsilon}(\alpha)$ and $\ell<\omega$ {\em then}
\[(\alpha,\ell)\in v^{p_\varepsilon}\quad\Leftrightarrow\quad (\beta,\ell)
\in v^{p_\zeta}\quad\Rightarrow\quad\rho^{p_\varepsilon}_{\alpha,\ell}=
\rho^{p_\zeta}_{\beta,\ell}).\]
Also this mapping is the identity on $U_\zeta\cap U_\varepsilon$ and
$\langle U_\zeta:\zeta<\omega_1\rangle$ is a $\triangle$-system.
Let $w=:w^{p_0}\cap w^{p_1}$. As $i\ne j\ \Rightarrow\ |A_i\cap A_j|\le
\aleph_0$, without loss of generality
\begin{description}
\item[$(*)$] if $i\ne j\in w$ then
\[U_{\varepsilon}\cap (A_i\cap A_j)\subseteq w.\]
\end{description}
We now start to define $q\ge p_0,p_1$. Choose $m^q$ such that $m^q\in
(m^{p_\varepsilon},\omega)$ and
\[\begin{array}{ll}
m^q>\max\big\{\ell g(\eta^{i_0}_{\alpha_0}\cap\eta^{i_1}_{\alpha_1})+1:&
i_0\in w^{p_0},\ i_1 \in w^{p_1},\ {\rm OP}_{1,0}(i_0)=i_1,\\
\ &\alpha_0\in{\rm Dom}(g^{p_0}_{i_0}),\ \alpha_1\in{\rm Dom}(g^{p_1}_{i_1}),\\
\ &{\rm OP}_{1,0}(\alpha_0)=\alpha_1\big\}.
\end{array}\]
Let $u^q\subseteq {}^{m^q\ge}\omega$ be such that:
\begin{description}
\item[(A)] $u^q\cap\left({}^{m^{p_0}\ge}\omega\right)=u^q\cap\left(
{}^{m^{p_1}\ge}\omega\right)=u^{p_0}=u^{p_1}$,
\item[(B)] for each $\nu\in u^q$, $m^{p_0}\le\ell g(\nu)<m^q$, for exactly
two numbers $k<\omega$, $\nu\char 94 \langle k\rangle\in u^q$,
\item[(C)] $\eta^i_\alpha\restriction\ell\in u^q$ for $\ell\le m^q$
\underbar{when}: $i\in w^{p_0}$, $\alpha\in{\rm Dom}(g^{p_0}_i)$ \underbar{or}
$i\in w^{p_1}$, $\alpha\in{\rm Dom}(g^{p_1}_i)$.
\end{description}
[Possible as $\{\eta^i_\alpha\restriction m^{p_\varepsilon}:i\in
w^{p_\varepsilon},\alpha\in{\rm Dom}(g^{p_\varepsilon}_i)\}$ is with no
repetitions (the first line of the proof).]
Let $w^q=:w^{p_0}\cup w^{p_1}$ and $v^q=:v^{p_0}\cup v^{p_1}$ and for
$i \in w^q$
\[g^q_i=\left\{\begin{array}{lll}
g^{p_0}_i &\mbox{\underbar{if}}& i\in w^{p_0}\setminus w^{p_1},\\
g^{p_1}_i &\mbox{\underbar{if}}& i\in w^{p_1}\setminus w^{p_0},\\
g^{p_0}_i \cup g^{p_1}_i &\mbox{\underbar{if}}& i\in w^{p_0}\cap w^{p_1}.
\end{array}\right.\]
Next choose $\rho^q_{\alpha,\ell}$ for $(\alpha,\ell)\in v^q$ as follows.
Let $\nu_{\alpha,\ell}$ be $\rho^{p_0}_{\alpha,\ell}$ if defined,
$\rho^{p_1}_{\alpha,\ell}$ if defined (no contradiction). If $(\alpha,\ell)
\in v^q$ choose $\rho^q_{\alpha,\ell}$ as any $\rho$ such that:
\begin{description}
\item[$\otimes_0$] $\nu_{\alpha,\ell}\triangleleft\rho\in u^q\cap
{}^{(m^q)}\omega$.
\end{description}
But not all choices are O.K., as we need to be able to define $f^q_i$ for
$i\in w^q$. A possible problem will arise only when $i\in w^{p_0}\cap
w^{p_1}$. Specifically we need just (remember that $\langle
\rho^{p_\varepsilon}_{\alpha,\ell}:(\alpha,\ell)\in v^{p_\varepsilon}
\rangle$ are pairwise distinct by clause (b) of the Definition of $p\in P$):
\begin{description}
\item[$\otimes_1$] if $i_0\in w^{p_0},(\alpha_0,\ell)=(\alpha_0,
g_{i_0}(\alpha_0)),\alpha_0\in{\rm Dom}(g^{p_0}_{i_0})$, $i_1={\rm OP}_{1,0}(i_0)$
and $\alpha_1={\rm OP}_{1,0}(\alpha_0)$ and $i_0=i_1$
{\em then} $\ell g(\eta^{i_0}_{\alpha_0}\cap\eta^{i_1}_{\alpha_1})=\ell g
(\rho^q_{\alpha_0,\ell}\cap\rho^q_{\alpha_1,\ell})$.
\end{description}
We can, of course, demand $\alpha_0\neq \alpha_1$ (otherwise the
conclusion of $\otimes_1$ is trivial).
Our problem is expressible for each pair $(\alpha_0,\ell),(\alpha_1,\ell)$
separately as: first the problem is in defining the
$\rho^q_{(\alpha,\ell)}$'s and second, if $(\alpha'_1,\ell')$,
$(\alpha'_2,\ell)$ is another such pair then $\{(\alpha_1,\ell),
(\alpha_2,\ell)\}$, $\{(\alpha'_1,\ell'),(\alpha'_2,\ell')\}$ are either
disjoint or equal. Now for a given pair $(\alpha_0,\ell),(\alpha_1,\ell)$ how
many $i_0=i_1$ do we have? Necessarily $i_0\in w^{p_0}\cap w^{p_1}=w$. But
if $i'_0\ne i''_0$ are like that then $\alpha_0\in A_{i'_0}\cap
A_{i''_0}$, contradicting $(*)$ above because $\alpha_0\neq
\alpha_1={\rm OP}_{1,0}(\alpha_0)$. So there is at most one candidate
$i_0=i_1$, so there is no problem to satisfy $\otimes_1$. Now we can define
$f^q_i$ (i$\in w^q$) as in the proof of Fact C.
\medskip
The rest should be clear. \hfill$\square_{\ref{4.4}}$
\begin{Conclusion}
\label{4.6}
Suppose $V\models GCH$, $\aleph_0<\lambda<\chi$ and $\chi^\lambda=\chi$.
Then for some c.c.c. forcing notion $P$ of cardinality $\chi$, not collapsing
cardinals nor changing cofinalities, in $V^P$:
\begin{description}
\item[(i)] $2^{\aleph_0}=2^\lambda=\chi$,
\item[(ii)] ${\frak K}^{tr}_\lambda$ has a universal family of cardinality
$\lambda^+$,
\item[(iii)] ${\frak K}^{rs(p)}_\lambda$ has a universal family of cardinality
$\lambda^+$.
\end{description}
\end{Conclusion}
\proof First use a preliminary forcing $Q^0$ of Baumgartner \cite{B},
adding $\langle A_\alpha:\alpha<\chi\rangle$, $A_\alpha\in
[\lambda]^\lambda$, $\alpha\neq\beta\quad \Rightarrow\quad |A_\alpha\cap
A_\beta|\le\aleph_0$ (we can have $2^{\aleph_0}=\aleph_1$ here, or
$[\alpha\ne \beta\quad \Rightarrow\quad A_\alpha\cap A_\beta$ finite], but
not both). Next use an FS iteration $\langle P_i,\dot{Q}_i:i<\chi\times
\lambda^+\rangle$ such that each forcing from \ref{4.4} appears and each
forcing as in \ref{4.5} appears. \hfill$\square_{\ref{4.6}}$
\begin{Remark}
\label{4.7}
We would like to have that there is a universal member in
${\frak K}^{rs(p)}_\lambda$; this sounds very reasonable but we did not try.
In our framework, the present result shows limitations to ZFC results which
the methods applied in the previous sections can give.
\end{Remark}
\section{Back to ${\frak K}^{rs(p)}$, real non-existence results}
By \S1 we know that if $G$ is an Abelian group with set of elements
$\lambda$, $C\subseteq\lambda$, then for an element $x\in G$ the distance
from $\{y:y<\alpha\}$ for $\alpha\in C$ does not code an appropriate invariant.
If we have infinitely many such distance functions, e.g. have infinitely many
primes, we can use more complicated invariants related to $x$ as in \S3.
But if we have one prime, this approach does not help.
If one element fails, can we use infinitely many? A countable subset $X$ of
$G$ can code a countable subset of $C$:
\[\{\alpha\in C:\mbox{ closure}(\langle X\rangle_G)\cap\alpha\nsubseteq
\sup(C\cap\alpha)\},\]
but this seems silly - we use heavily the fact that $C$ has many countable
subsets (in particular $>\lambda$) and $\lambda$ has at least as many.
However, what if $C$ has a small family (say of cardinality $\le\lambda$ or
$<\mu^{\aleph_0}$) of countable subsets such that every subset of cardinality,
say continuum, contains one? Well, we need more: we catch a countable subset
for which the invariant defined above is infinite (necessarily it is at most
of cardinality $2^{\aleph_0}$, and because of \S4 we are not trying any more
to deal with $\lambda\le 2^{\aleph_0}$). The set theory needed is expressed
by $T_J$ below, and various ideals also defined below, and the result itself
is \ref{5.7}.
Of course, we can deal with other classes like torsion free reduced groups,
as they have the characteristic non-structure property of unsuperstable
first order theories; but the relevant ideals will vary: the parallel to
$I^0_{\bar\mu}$ for $\bigwedge\limits_n \mu_n=\mu$, $J^2_{\bar\mu}$ seems
to be always O.K.
\begin{Definition}
\label{5.1}
\begin{enumerate}
\item For $\bar\mu=\langle\mu_n:n<\omega\rangle$ let $B_{\bar\mu}$ be
\[\bigoplus\{K^n_\alpha:n<\omega,\alpha<\mu_n\},\qquad K^n_\alpha=
\langle{}^* t^n_\alpha\rangle_{K^n_\alpha}\cong {\Bbb Z}/p^{n+1} {\Bbb Z}.\]
Let $B_{\bar\mu\restriction n}=\bigoplus\{K^m_\alpha:\alpha<\mu_m,m<n\}
\subseteq B_{\bar\mu}$ (they are in ${\frak K}^{rs(p)}_{\le\sum\limits_{n}
\mu_n}$). Let $\hat B$ be the $p$-torsion completion of $B$ (i.e. completion
under the norm $\|x\|=\min\{2^{-n}:p^n\mbox{ divides }x\}$).
\item Let $I^1_{\bar \mu}$ be the ideal on $\hat B_{\bar \mu}$
generated by $I^0_{\bar \mu}$, where
\[\begin{array}{ll}
I^0_{\bar\mu}=\big\{A\subseteq\hat B_{\bar\mu}:&\mbox{for every large
enough }n,\\
&\mbox{for no }y\in\bigoplus\{K^m_\alpha:m\le n\mbox{ and }\alpha<\mu_m\}\\
&\mbox{but }y\notin\bigoplus\{K^m_\alpha:m<n\mbox{ and }\alpha<\mu_m\}
\mbox{ we have}:\\
&\mbox{for every }m\mbox{ for some }z\in\langle A\rangle\mbox{ we have:}\\
&p^m\mbox{ divides }z-y\big\}.
\end{array}\]
(We may write $I^0_{\hat B_{\bar\mu}}$, but the ideal depends also on
$\langle\bigoplus\limits_{\alpha<\mu_n} K^n_\alpha:n<\omega\rangle$ not
just on $\hat B_{\bar\mu}$ itself).
\item For $X,A\subseteq\hat B_{\bar\mu}$,
\[\mbox{ recall }\ \ \langle A\rangle_{\bar
B_{\bar\mu}}=\big\{\sum\limits_{n<n^*} a_ny_n:y_n
\in A,\ a_n\in{\Bbb Z}\mbox{ and } n^*\in{\Bbb N}\big\},\]
\[\mbox{ and let }\ \ c\ell_{\hat B_{\bar\mu}}(X)=\{x:(\forall
n)(\exists y\in X)(x-y\in p^n \hat B_{\bar \mu})\}.\]
\item Let $J^1_{\bar \mu}$ be the ideal which $J^{0.5}_{\bar \mu}$
generates, where
\[\begin{array}{ll}
J^{0.5}_{\bar\mu}=\big\{A\subseteq\prod\limits_{n<\omega}\mu_n:
&\mbox{for some }n<\omega\mbox{ for no }m\in [n,\omega)\\
&\mbox{and }\beta<\gamma<\mu_m\mbox{ do we have}:\\
&\mbox{for every }k\in [m,\omega)\mbox{ there are }\eta,\nu\in A\mbox{
such}\\
&\mbox{that:}\ \ \eta(m)=\beta,\,\nu(m)=\gamma,\,\eta\restriction m=
\nu\restriction m\\
&\mbox{and }\eta\restriction (m,k)=\nu\restriction (m,k)\big\}.
\end{array}\]
\item
\[\begin{array}{rr}
J^0_{\bar\mu}=\{A\subseteq\prod\limits_{n<\omega}\mu_n:
&\mbox{for some }n<\omega\mbox{ and } k,\mbox{ the mapping }\eta\mapsto
\eta \restriction n\\
&\mbox{is }(\le k)\mbox{-to-one }\}.
\end{array}\]
\item $J^2_{\bar\mu}$ is the ideal of nowhere dense subsets of
$\prod\limits_n\mu_n$ (under the following natural topology: a
neighbourhood of $\eta$ is $U_{\eta,n}=\{\nu:\nu\restriction n=\eta
\restriction n\}$ for some $n$).
\item $J^3_{\bar \mu}$ is the ideal of meagre subsets of $\prod\limits_n
\mu_n$, i.e. subsets which are included in countable union of members of
$J^2_{\bar \mu}$.
\end{enumerate}
\end{Definition}
\begin{Observation}
\label{5.2}
\begin{enumerate}
\item $I^0_{\bar\mu}$, $J^0_{\bar\mu}$, $J^{0.5}_{\bar\mu}$ are
$(<\aleph_1)$-based, i.e. for $I^0_{\bar \mu}$: if $A\subseteq\hat
B_{\bar\mu}$, $A\notin I^0_{\bar\mu}$ then there is a countable $A_0
\subseteq A$ such that $A_0\notin I^0_{\bar\mu}$.
\item $I^1_{\bar\mu}$, $J^0_{\bar\mu}$, $J^1_{\bar\mu}$,
$J^2_{\bar\mu}$, $J^3_{\bar\mu}$ are ideals, $J^3_{\bar\mu}$ is
$\aleph_1$-complete.
\item $J^0_{\bar\mu}\subseteq J^1_{\bar\mu}\subseteq J^2_{\bar\mu}
\subseteq J^3_{\bar\mu}$.
\item There is a function $g$ from $\prod\limits_{n<\omega}\mu_n$ into
$\hat B_{\bar\mu}$ such that for every $X\subseteq\prod\limits_{n<\omega}
\mu_n$:
\[X\notin J^1_{\bar\mu}\quad \Rightarrow\quad g''(X)\notin
I^1_{\bar\mu}.\]
\end{enumerate}
\end{Observation}
\proof E.g. 4)\ \ Let $g(\eta)=\sum\limits_{n<\omega}p^n({}^*t^n_{\eta(n)})$.
Let $X \subseteq\prod\limits_{n<\omega}\mu_n$, $X\notin J^1_{\bar\mu}$.
Assume $g''(X)\in\bar I^1_{\bar\mu}$, so for some $\ell^*$ and $A_\ell
\subseteq\hat B_{\bar\mu}$, ($\ell<\ell^*$) we have $A_\ell\in I^0_{\bar\mu}$,
and $g''(X)\subseteq\bigcup\limits_{\ell<\ell^*} A_\ell$, so $X=
\bigcup\limits_{\ell<\ell^*} X_\ell$, where
\[X_\ell=:\{\eta\in X:g(\eta)\in A_\ell\}.\]
As $J^1_{\bar \mu}$ is an ideal, for some $\ell<\ell^*$, $X_\ell\notin
J^1_{\bar\mu}$. So by the definition of $J^1_{\bar\mu}$, for some infinite
$\Gamma\subseteq\omega$ for each $m\in\Gamma$ we have $\beta_m<\gamma_m<
\mu_m$ and for every $k\in [m,\omega)$ we have $\eta_{m,k},\nu_{m,k}$, as
required in the definition of $J^1_{\bar \mu}$. So $g(\eta_{m,k}),
g(\nu_{m,k}) \in A_\ell$ (for $m\in \Gamma$, $k\in (m,\omega)$). Now
\[{}^* t^m_{\gamma_m} - {}^* t^m_{\beta_m}=g(\eta_{m,k})-g(\nu_{m,k})
\mod\ p^k \hat B_{\bar \mu},\]
but $g(\eta_{m,k})-g(\nu_{m,k})\in\langle A_\ell\rangle_{\hat B_{\bar\mu}}$.
Hence
\[(\exists z\in\langle A_\ell\rangle_{\hat B_{\bar\mu}})[{}^* t^m_{\gamma_m}
-{}^* t^m_{\beta_m}=z\mod\ p^k \hat B_{\bar\mu}],\]
as this holds for each $k$, ${}^* t^m_{\gamma_m}-{}^* t^m_{\beta_m}\in
c \ell(\langle A_\ell\rangle_{\hat B_{\bar\mu}})$.
This contradicts $A_\ell
\in I^0_{\bar \mu}$. \hfill$\square_{\ref{5.2}}$
\begin{Definition}
\label{5.3}
Let $I\subseteq{\cal P}(X)$ be downward closed (and for simplicity $\{\{x\}:
x\in X\}\subseteq I$). Let $I^+={\cal P}(X)\setminus I$. Let
\[\begin{array}{ll}
{\bf U}^{<\kappa}_I(\mu)=:\min\big\{|{\cal P}|:&{\cal P}\subseteq [\mu]^{<\kappa},
\mbox{ and for every } f:X\longrightarrow\mu\mbox{ for some}\\
&Y\in {\cal P},\mbox{ we have }\{x\in X:f(x)\in Y\}\in
I^+\big\}.
\end{array}\]
Instead of $<\kappa^+$ in the superscript of ${\bf U}$ we write $\kappa$. If
$\kappa>|{\rm Dom}(I)|^+$, we omit it (since then its value does not matter).
\end{Definition}
\begin{Remark}
\label{5.4}
\begin{enumerate}
\item If $2^{<\kappa}+|{\rm Dom}(I)|^{<\kappa}\le\mu$ we can find $F\subseteq$
partial functions from ${\rm Dom}(I)$ to $\mu$ such that:
\begin{description}
\item[(a)] $|F|={\bf U}^{<\kappa}_I(\mu)$,
\item[(b)] $(\forall f:X\longrightarrow\mu)(\exists Y\in I^+)[f\restriction
Y \in F]$.
\end{description}
\item Such functions (as ${\bf U}^{<\kappa}_I(\mu)$) are investigated in {\bf
pcf} theory (\cite{Sh:g}, \cite[\S6]{Sh:410}, \cite[\S2]{Sh:430},
\cite{Sh:513}).
\item If $I\subseteq J\subseteq {\cal P}(X)$, then ${\bf U}^{<\kappa}_I(\mu)\le
{\bf U}^{<\kappa}_J(\mu)$, hence by \ref{5.2}(3), and the above
\[{\bf U}^{<\kappa}_{J^0_{\bar\mu}}(\mu)\le {\bf U}^{<\kappa}_{J^1_{\bar\mu}}(\mu)
\le {\bf U}^{<\kappa}_{J^2_{\bar\mu}}(\mu)\le
{\bf U}^{<\kappa}_{J^3_{\bar\mu}}(\mu)\]
and by \ref{5.2}(4) we have ${\bf U}^{<\kappa}_{I^1_{\bar \mu}}\leq
{\bf U}^{<\kappa}_{J^1_{\bar \mu}}(\mu).$
\item On ${\rm IND}_\theta(\bar\kappa)$ (see \ref{5.5A} below) see \cite{Sh:513}.
\end{enumerate}
\end{Remark}
\begin{Definition}
\label{5.5A}
${\rm IND}'_\theta(\langle\kappa_n:n<\omega\rangle)$ means that for every model
$M$ with universe $\bigcup\limits_{n<\omega}\kappa_n$ and $\le\theta$
functions, for some $\Gamma\in [\omega]^{\aleph_0}$ and $\eta\in
\prod\limits_{n<\omega}\kappa_n$ we have:
\[n\in\Gamma\quad\Rightarrow\quad\eta(n)\notin c\ell_M\{\eta(\ell):\ell
\ne n\}.\]
\end{Definition}
\begin{Remark}
Actually if $\theta\geq \aleph_0$, this implies that we can fix
$\Gamma$, hence replacing $\langle \kappa_n: n< \omega\rangle$ by an
infinite subsequence we can have $\Gamma=\omega$.
\end{Remark}
\begin{Theorem}
\label{5.5}
\begin{enumerate}
\item If $\mu_n\rightarrow (\kappa_n)^2_{2^\theta}$ and ${\rm IND}'_\theta(\langle
\kappa_n:n<\omega\rangle)$ {\em then} $\prod\limits_{n<\omega}\mu_n$ is not
the union of $\le\theta$ sets from $J^1_{\bar \mu}$.
\item If $\theta=\theta^{\aleph_0}$ and $\neg{\rm IND}'_\theta(\langle\mu_n:
n<\omega\rangle$) then $\prod\limits_{n<\omega}\mu_n$ is the union of
$\le\theta$ members of $J^1_{\bar\mu}$.
\item If $\lim\sup\limits_n \mu_n$ is $\ge 2$, then $\prod\limits_{n<\omega}
\mu_n\notin J^3_{\bar\mu}$ (so also the other ideals defined above are not
trivial by \ref{5.2}(3), (4)).
\end{enumerate}
\end{Theorem}
\proof 1)\ \ Suppose $\prod\limits_{n<\omega}\mu_n$ is
$\bigcup\limits_{i<\theta} X_i$, and each $X_i\in J^1_{\bar\mu}$. We define
for each $i<\theta$ and $n<k<\omega$ a two-place relation $R^{n,k}_i$ on
$\mu_n$:
\qquad $\beta R^{n,k}_i \gamma$ if and only if
\qquad there are $\eta,\nu\in X_i\subseteq\prod\limits_{\ell<k}\mu_\ell$
such that
\[\eta\restriction [0,n)=\nu\restriction [0,n)\quad\mbox{and }\
\eta\restriction (n,k)=\nu\restriction (n,k)\quad\mbox{and }\ \eta(n)
=\beta,\ \nu(n)=\gamma.\]
Note that $R^{n,k}_i$ is symmetric and
\[n<k_1<k_2\ \&\ \beta R^{n,k_2}_i \gamma\quad \Rightarrow\quad \beta
R^{n,k_1}_i \gamma.\]
As $\mu_n\rightarrow (\kappa_n)^2_{2^\theta}$, we can find $A_n\in
[\mu_n]^{\kappa_n}$ and a truth value ${\bf t}^{n,k}_i$ such that for all
$\beta<\gamma$ from $A_n$, the truth value of $\beta R^{n,k}_i\gamma$ is
${\bf t}^{n,k}_i$. If for some $i$ the set
\[\Gamma_i=:\{n<\omega:\mbox{ for every }k\in (n,\omega)\mbox{ we have }
{\bf t}^{n,k}_i=\mbox{ true}\}\]
is infinite, we get a contradiction to ``$X_i\in J^1_{\bar \mu}$", so for
some $n(i)<\omega$ we have $n(i)=\sup(\Gamma_i)$.
For each $n<k<\omega$ and $i<\theta$ we define a partial function
$F^{n,k}_i$ from $\prod\limits_{\scriptstyle \ell<k,\atop\scriptstyle\ell
\ne n} A_\ell$ into $A_n$:
\begin{quotation}
\noindent $F(\alpha_0\ldots\alpha_{n-1},\alpha_{n+1},\ldots,\alpha_k)$ is
the first $\beta\in A_n$ such that for some $\eta\in X_i$ we have
\[\begin{array}{c}
\eta\restriction [0,n)=\langle\alpha_0,\ldots,\alpha_{n-1}\rangle,\quad
\eta(n)=\beta,\\
\eta\restriction (n,k)=\langle\alpha_{n+1},\ldots,\alpha_{k-1}\rangle.
\end{array}\]
\end{quotation}
So as ${\rm IND}'_\theta(\langle\kappa_n:n<\omega\rangle)$ there is $\eta=\langle
\beta_n:n<\omega\rangle\in\prod\limits_{n<\omega} A_n$ such that for
infinitely many $n$, $\beta_n$ is not in the closure of $\{\beta_\ell:\ell
<\omega,\,\ell\ne n\}$ by the $F^{n,k}_i$'s. As $\eta\in\prod\limits_{n<
\omega} A_n\subseteq\prod\limits_{n<\omega}\mu_n=\bigcup\limits_{i<\theta}
X_i$, necessarily for some $i<\theta$, $\eta\in X_i$. Let $n\in(n(i),\omega)$
be such that $\beta_n$ is not in the closure of
$\{\beta_\ell:\ell<\omega\mbox{ and }\ell\neq n\}$
and let $k>n$ be such that ${\bf t}^{n,k}_i=\mbox{ false}$. Now $\gamma=:
F^{n,k}_i(\beta_0,\ldots,\beta_{n-1},\beta_{n+1},\ldots,\beta_{k-1})$ is well
defined $\le\beta_n$ (as $\beta_n$ exemplifies that there is such $\beta$) and
is $\ne \beta_n$ (by the choice of $\langle\beta_\ell:\ell<\omega\rangle$), so
by the choice of $n(i)$ (so of $n$, $k$ and earlier of ${\bf t}^{n, k}_i$
and of $A_n$) we get
contradiction to ``$\gamma<\beta_n$ are from $A_n$".
\noindent 2)\ \ Let $M$ be an algebra with universe $\sum\limits_{n<\omega}
\mu_n$ and $\le\theta$ functions (say $F^n_i$ for $i<\theta$, $n<\omega$,
$F^n_i$ is $n$-place) exemplifying $\neg{\rm IND}'_\theta(\langle\mu_n:n<\omega
\rangle)$. Let
\[\Gamma=:\{\langle(k_n,i_n):n^*\le n<\omega\rangle:n^*<\omega\mbox{ and }
\bigwedge_n n<k_n<\omega\mbox{ and }i_n<\theta\}.\]
For $\rho=\langle(k_n,i_n):n^*\le n<\omega\rangle\in\Gamma$ let
\[\begin{ALIGN}
A_\rho=:&\big\{\eta\in\prod\limits_{n<\omega}\mu_n:\mbox{for every }n
\in [n^*,\omega)\mbox{ we have}\\
&\qquad\eta(n)=F^{k_n-1}_{i_n}\left(\eta(0),\ldots,\eta(n-1),\eta(n+1),
\ldots,\eta(k_n)\right)\big\}.
\end{ALIGN}\]
So, by the choice of $M$, $\prod\limits_{n<\omega}\mu_n=\bigcup\limits_{\rho
\in\Gamma} A_\rho$. On the other hand, it is easy to check that $A_\rho\in
J^1_{\bar \mu}$. \hfill$\square_{\ref{5.5}}$
\begin{Theorem}
\label{5.6}
If $\mu=\sum\limits_{n<\omega}\lambda_n$, $\lambda^{\aleph_0}_n<\lambda_{n+1}$
and $\mu<\lambda={\rm cf}(\lambda)<\mu^{+\omega}$\\
then ${\bf U}^{\aleph_0}_{I^0_{\langle\lambda_n:n<\omega\rangle}}(\lambda)=
\lambda$ and even ${\bf U}^{\aleph_0}_{J^3_{\langle \lambda_n:n<\omega\rangle}}
(\lambda)=\lambda$.
\end{Theorem}
\proof See \cite[\S6]{Sh:410}, \cite[\S2]{Sh:430}, and \cite{Sh:513}
for considerably more.
\begin{Lemma}
\label{5.7}
Assume $\lambda>2^{\aleph_0}$ and
\begin{description}
\item[$(*)$(a)] $\prod\limits_{n<\omega}\mu_n<\mu$ and $\mu^+<\lambda=
{\rm cf}(\lambda)<\mu^{\aleph_0}$,
\item[\ \ (b)] $\hat B_{\bar\mu}\notin I^0_{\bar\mu}$ and $\lim_n\sup\mu_n$
is infinite,
\item[\ \ (c)] ${\bf U}^{\aleph_0}_{I^0_{\bar\mu}}(\lambda)=\lambda$
(note $I^0_{\bar \mu}$ is not required to be an ideal).
\end{description}
{\em Then} there is no universal member in ${\frak K}^{rs(p)}_\lambda$.
\end{Lemma}
\proof Let $S\subseteq\lambda$, $\bar C=\langle C_\delta:\delta\in S\rangle$
guesses clubs of $\lambda$, chosen as in the proof of \ref{3.3} (so $\alpha
\in{\rm nacc}(C_\delta)\ \Rightarrow\ {\rm cf}(\alpha)>2^{\aleph_0}$). Instead of
defining the relevant invariant we prove the theorem directly, but we could
define it, somewhat cumbersomely (like \cite[III,\S3]{Sh:e}).
Assume $H\in {\frak K}^{rs(p)}_\lambda$ is a pretender to universality;
without loss of generality with the set of elements of $H$ equal to
$\lambda$.
Let $\chi=\beth_7(\lambda)^+$, ${\bar{\frak A}}=\langle {\frak A}_\alpha:
\alpha<\lambda\rangle$ be an increasing continuous sequence of elementary
submodels of $({\cal H}(\chi),\in,<^*_\chi)$, ${\bar {\frak A}}\restriction
(\alpha+1)\in {\frak A}_{\alpha+1}$, $\|{\frak A}_\alpha\|<\lambda$, ${\frak
A}_\alpha\cap\lambda$ an ordinal, ${\frak A}=\bigcup\limits_{\alpha<\lambda}
{\frak A}_\alpha$ and $\{H,\langle\mu_n:n<\omega\rangle,\mu,\lambda\}\in
{\frak A}_0$, so $B_{\bar \mu},\hat B_{\bar \mu} \in {\frak A}_0$ (where $\bar
\mu=\langle\mu_n:n<\omega\rangle$, of course).
For each $\delta\in S$, let ${\cal P}_\delta=:[C_\delta]^{\aleph_0}\cap
{\frak A}$. Choose $A_\delta\subseteq C_\delta$ of order type $\omega$
almost disjoint from each $a\in {\cal P}_\delta$, and from $A_{\delta_1}$ for
$\delta_1\in\delta\cap S$; its existence should be clear as $\lambda< \mu^{\aleph_0}$. So
\begin{description}
\item[$(*)_0$] every countable $A\in {\frak A}$ is almost disjoint to
$A_\delta$.
\end{description}
By \ref{5.2}(2), $I^0_{\bar\mu}$ is $(<\aleph_1)$-based so by \ref{5.4}(1) and
the assumption (c) we have
\begin{description}
\item[$(*)_1$] for every $f:\hat B_{\bar\mu}\longrightarrow\lambda$ for some
countable $Y\subseteq \hat B_{\bar\mu}$, $Y\notin I^0_{\bar\mu}$, we have
$f\restriction Y\in {\frak A}$
\end{description}
(remember $(\prod\limits_{n<\omega}\mu_n)^{\aleph_0}=\prod\limits_{n<\omega}
\mu_n$).
\noindent Let $B$ be $\bigoplus\{G^n_{\alpha,i}:n<\omega,\alpha<\lambda,\,i<
\sum\limits_{k<\omega}\mu_k\}$, where
\[G^n_{\alpha,i}=\langle x^n_{\alpha,i}\rangle_{G^n_{\alpha,i}}\cong{\Bbb
Z}/p^{n+1}{\Bbb Z}.\]
\noindent
So $B$, $\hat B$, $\langle(n,\alpha,i,x^n_{\alpha,i}):n<\omega,\alpha<\lambda,
i<\sum\limits_{k<\omega}\mu_k\rangle$ are well defined. Let $G$ be the
subgroup of $\hat B$ generated by:
\[\begin{ALIGN}
B\cup\big\{x\in\hat B: &\mbox{for some }\delta\in S,\, x\mbox{ is in the
closure of }\\
&\bigoplus\{G^n_{\alpha,i}:n<\omega,i<\mu_n,\alpha\mbox{ is the }n\mbox{th
element of } A_\delta\}\big\}.
\end{ALIGN}\]
As $\prod\limits_{n<\omega}\mu_n<\mu<\lambda$, clearly $G\in {\frak
K}^{rs(p)}_\lambda$, without loss of generality the set of elements of $G$ is
$\lambda$ and let $h:G\longrightarrow H$ be an embedding. Let
\[E_0=:\{\delta<\lambda:({\frak A}_\delta,h \restriction \delta,\;G
\restriction\delta)\prec({\frak A},h,G)\},\]
\[E=:\{\delta<\lambda:{\rm otp}(E_0\cap\delta)=\delta\}.\]
They are clubs of $\lambda$, so for some $\delta\in S$, $C_\delta\subseteq E$
(and $\delta\in E$ for simplicity). Let $\eta_\delta$ enumerate $A_\delta$
increasingly.
There is a natural embedding $g = g_\delta$ of $B_{\bar \mu}$ into $G$:
\[g({}^* t^n_i) = x^n_{\eta_\delta(n),i}.\]
Let $\hat g_\delta$ be the unique extension of $g_\delta$ to an embedding of
$\hat B_{\bar\mu}$ into $G$; those embeddings are pure, (in fact $g''_\delta
(\hat B_{\bar\mu})\setminus g''_\delta(B_\mu)\subseteq G\setminus G\cap
{\frak A}_\delta$). So $h\circ\hat g_\delta$ is an embedding of $\hat B_{\bar
\mu}$ into $H$, not necessarily pure but still an embedding, so the distance
function can become smaller but not zero and
\[h\circ\hat g_\delta(\hat B_{\bar\mu})\setminus h\circ g_\delta(B_\mu)
\subseteq H\setminus {\frak A}_\delta.\]
Remember $\hat B_{\bar\mu}\subseteq {\frak A}_0$ (as it belongs to ${\frak
A}_0$ and has cardinality $\prod\limits_{n<\omega}\mu_n<\lambda$ and
$\lambda\cap {\frak A}_0$ is an ordinal). By $(*)_1$ applied to
$f=h\circ\hat g$ there is a countable $Y \subseteq \hat B_{\bar \mu}$
such that $Y \notin I^0_{\bar\mu}$ and $f \restriction Y \in {\frak
A}$. But, from $f \restriction Y$ we shall below reconstruct
some countable sets not almost disjoint to $A_\delta$, reconstruct meaning in
${\frak A}$, in contradiction to $(*)_0$ above.
As $Y\notin I^0_{\bar \mu}$ we can find an infinite
$S^*\subseteq\omega\setminus m^*$ and for $n\in
S^*$, $z_n\in\bigoplus\limits_{\alpha<\mu_n} K^n_\alpha\setminus\{0\}$ and
$y^\ell_n\in\hat B_{\bar\mu}$ (for $\ell<\omega$) such that:
\begin{description}
\item[$(*)_2$] $z_n+y_{n,\ell}\in\langle Y\rangle_{\hat B_{\bar\mu}}$,\qquad
and
\item[$(*)_3$] $y_{n,\ell}\in p^\ell\,\hat B_{\bar\mu}$.
\end{description}
Without loss of generality $pz_n=0\ne z_n$ hence $p\,y^\ell_n=0$. Let
\[\nu_\delta(n)=:\min(C_\delta\setminus (\eta_\delta(n)+1)),\quad z^*_n=
(h\circ\hat g_\delta)(z_n)\quad\mbox{ and }\quad y^*_{n,\ell}=(h\circ\hat
g_\delta)(y_{n,\ell}).\]
Now clearly $\hat g_\delta(z_n)=g_\delta(z_n)=x^n_{\eta_\delta(n),i}\in
G\restriction\nu_\delta(n)$, hence $(h\circ\hat g_\delta)(z_n)\notin H
\restriction\eta_\delta(n)$, that is $z^*_n\notin
H\restriction\eta_\delta(n)$.
So $z^*_n\in H_{\nu_\delta(n)}\setminus H_{\eta_\delta(n)}$ belongs to
the $p$-adic closure of ${\rm Rang}(f\restriction Y)$. As $H$, $G$, $h$ and
$f\restriction Y$ belongs to ${\frak A}$, also $K$, the closure of
${\rm Rang}(f\restriction Y)$ in $H$ by the $p$-adic topology belongs to
${\frak A}$, and clearly $|K|\leq 2^{\aleph_0}$, hence
\[A^*=\{\alpha\in C_\delta: K\cap H_{\min(C_\delta\setminus
(\alpha+1))}\setminus H_\alpha \mbox{ is not empty}\}\]
is a subset of $C_\delta$ of cardinality $\leq 2^{\aleph_0}$ which
belongs to ${\frak A}$, hence $[A^*]^{\aleph_0}\subseteq {\frak A}$
but $A_\delta\subseteq A^*$ so $A_\delta\in {\frak A}$, a contradiction.
\hfill$\square_{\ref{5.7}}$
\section{Implications between the existence of universals}
\begin{Theorem}
\label{6.1}
Let $\bar n=\langle n_i:i<\omega\rangle$, $n_i\in [1,\omega)$. Remember
\[J^2_{\bar n}=\{A\subseteq\prod_{i<\omega} n_i:A \mbox{ is nowhere
dense}\}.\]
Assume $\lambda\ge 2^{\aleph_0}$, $T^{\aleph_0}_{J^3_{\bar n}}(\lambda)=
\lambda$ or just $T^{\aleph_0}_{J^2_{\bar n}}(\lambda)=\lambda$ for every
such $\bar n$, and
\[n<\omega\quad\Rightarrow\quad\lambda_n\le\lambda_{n+1}\le\lambda_\omega
=\lambda\quad\mbox{ and}\]
\[\lambda\le\prod_{n<\omega}\lambda_n\quad\mbox{ and }\quad\bar\lambda=
\langle\lambda_i:i\le\omega\rangle.\]
\begin{enumerate}
\item If in ${\frak K}^{fc}_{\bar \lambda}$ there is a universal member
{\em then} in ${\frak K}^{rs(p)}_\lambda$ there is a universal member.
\item If in ${\frak K}^{fc}_\lambda$ there is a universal member for
${\frak K}^{fc}_{\bar \lambda}$
{\em then} in
\[{\frak K}^{rs(p)}_{\bar\lambda}=:\{G\in {\frak K}^{rs(p)}_\lambda:\lambda_n
(G)\le\lambda\}\]
there is a universal member (even for ${\frak K}^{rs(p)}_\lambda$).
\end{enumerate}
($\lambda_n(G)$ were defined in \ref{1.1}).
\end{Theorem}
\begin{Remark}
\begin{enumerate}
\item Similarly for ``there are $M_i\in {\frak K}_{\lambda_1}$ ($i<\theta$)
with $\langle M_i:i<\theta\rangle$ being universal for ${\frak K}_\lambda$''.
\item The parallel of \ref{1.1} holds for ${\frak K}^{fc}_\lambda$.
\item By \S5 only the case $\lambda$ singular or $\lambda=\mu^+\ \&\
{\rm cf}(\mu)=\aleph_0\ \& \ (\forall \alpha<
\mu)(|\alpha|^{\aleph_0}<\mu)$
is of interest for \ref{6.1}.
\end{enumerate}
\end{Remark}
\proof 1)\ \ By \ref{1.1}, (2) $\Rightarrow$ (1).
More elaborately, by part (2) of \ref{6.1} below there is $H\in {\frak
K}^{rs(p)}_{\bar \lambda}$ which is universal in ${\frak
K}^{rs(p)}_{\bar \lambda}$. Clearly $|G|=\lambda$ so $H\in {\frak
K}^{rs(p)}_\lambda$, hence for proving part (1) of \ref{6.1} it
suffices to prove that $H$ is a universal member of ${\frak
K}^{rs(p)}_\lambda$. So let $G\in {\frak K}^{rs(p)}_\lambda$, and we
shall prove that it is embeddable into $H$. By \ref{1.1} there is $G'$
such that $G\subseteq G'\in {\frak K}^{rs(p)}_{\bar \lambda}$. By the
choice of $H$ there is an embedding $h$ of $G'$ into $H$. So
$h\restriction G$ is an embedding of $G$ into $H$, as required.
\noindent 2)\ \ Let $T^*$ be a universal member of ${\frak K}^{fc}_{\bar
\lambda}$ (see \S2) and let $P_\alpha = P^{T^*}_\alpha$.
Let $\chi>2^\lambda$. Without loss of generality $P_n=\{n\}\times
\lambda_n$, $P_\omega=\lambda$. Let
\[B_0=\bigoplus\{G^n_t:n<\omega,t\in P_n \},\]
\[B_1=\bigoplus \{G^n_t: n< \omega\mbox{ and }t\in P_n\},\]
where $G^n_t\cong {\Bbb Z}/p^{n+1}{\Bbb Z}$, $G^n_t$ is generated by
$x^n_t$. Let ${\frak B}\prec ({\cal H}(\chi),\in,<^*_\chi)$, $\|{\frak B}\|=
\lambda$, $\lambda+1\subseteq {\frak B}$, $T^*\in {\frak B}$, hence
$B_0$, $B_1\in {\frak B}$ and $\hat B_0, \hat B_1\in {\frak B}$ (the
torsion completion of $B$). Let $G^* =\hat B_1\cap {\frak B}$.
Let us prove that $G^*$ is universal for ${\frak K}^{rs(p)}_{\bar
\lambda}$ (by \ref{1.1} this suffices).
Let $G \in {\frak K}^{rs(p)}_{\lambda}$, so by \ref{1.1} without loss of
generality $B_0 \subseteq G\subseteq\hat B_0$. We define $R$:
\[\begin{ALIGN}
R=\big\{\eta:&\eta\in\prod\limits_{n<\omega}\lambda_n\mbox{ and for
some }
x\in G\mbox{ letting }\\
&x=\sum\{a^n_i\,p^{n-k}\,x^n_i:n<\omega,i\in w_n(x)\}\mbox{ where }\\
&w_n(x)\in [\lambda_n]^{<\aleph_0},a^n_i\,p^{n-k}\,x^n_i\ne 0\mbox{
we have }\\
&\bigwedge\limits_n\eta(n)\in w_n(x)\cup \{\ell:
\ell+|w_n(x)|\leq n\}\big\}.
\end{ALIGN}\]
Lastly let $M =:(R\cup\bigcup\limits_{n<\omega}\{n\}\times\lambda_n,\,P_n,\,
F_n)_{n<\omega}$ where $P_n=\{n\}\times\lambda_n$ and $F_n(\eta)=(n,\eta(n))$,
so clearly $M\in {\frak K}^{fc}_{\bar\lambda}$. Consequently, there is an
embedding $g:M\longrightarrow T^*$, so $g$ maps $\{n\}\times\lambda_n$ into
$P^{T^*}_n$ and $g$ maps $R$ into $P^{T^*}_\omega$. Let $g(n,\alpha)=
(n,g_n(\alpha))$ (i.e. this defines $g_n$). Clearly $g\restriction (\cup
P^M_n)=g\restriction (\bigcup\limits_n\{n\}\times\lambda_n)$ induces an
embedding $g^*$ of $B_0$ to $B_1$ (by mapping the generators into the
generators).
\noindent The problem is why:
\begin{description}
\item[$(*)$] if $x=\sum\{a^n_i\,p^{n-k}\,x^n_i:n<\omega,i\in w_n(x)\}\in G$
{\em then} $g^*(x)=\sum\{a^n_i\,p^{n-k}\,g^*(x^n_i):n<\omega,i\in w_n(x)\}\in
G^*$.
\end{description}
As $G^*=\hat B_1\cap {\frak B}$, and $2^{\aleph_0}+1\subseteq {\frak B}$, it is
enough to prove $\langle g^{\prime\prime}(w_n(x)):n<\omega\rangle\in
{\frak B}$. Now for
notational simplicity $\bigwedge\limits_n [|w_n(x)|\ge n+1]$ (we can add an
element of $G^*\cap {\frak B}$ or just repeat the arguments). For each $\eta
\in\prod\limits_{n<\omega} w_n(x)$ we know that
\[g(\eta)=\langle g(\eta(n)):n<\omega\rangle\in T^*\quad\mbox{ hence is in }
{\frak B}\]
(as $T^*\in {\frak B}$, $|T^*|\le\lambda$). Now by assumption there is
$A\subseteq\prod\limits_{n<\omega} w_n(x)$ which is not nowhere dense
such that $g
\restriction A\in {\frak B}$, hence for some $n^*$ and $\eta^* \in
\prod\limits_{\ell<n^*}w_\ell(x)$, $A$ is dense above $\eta^*$ (in
$\prod\limits_{n<\omega} w_n(x)$). Hence
\[\langle\{\eta(n):\eta\in A\}:n^* \le n<\omega\rangle=\langle w_n[x]:n^*\le
n<\omega\rangle,\]
but the former is in ${\frak B}$ as $A\in {\frak B}$, and from the latter the
desired conclusion follows. \hfill$\square_{\ref{6.1}}$
\section{Non-existence of universals for trees with small density}
For simplicity we deal below with the case $\delta=\omega$, but the proof
works in general (as for ${\frak K}^{fr}_{\bar\lambda}$ in \S2). Section 1
hinted we should look at ${\frak K}^{tr}_{\bar\lambda}$ not only for the case
$\bar\lambda=\langle\lambda:\alpha\le\omega\rangle$ (i.e. ${\frak
K}^{tr}_\lambda$), but in particular for
\[\bar\lambda=\langle\lambda_n:n<\omega\rangle\char 94\langle\lambda\rangle,
\qquad \lambda^{\aleph_0}_n<\lambda_{n+1}<\mu<\lambda={\rm cf}(\lambda)<
\mu^{\aleph_0}.\]
Here we get for this class (embeddings are required to preserve levels),
results stronger than the ones we got for the classes of Abelian groups we
have considered.
\begin{Theorem}
\label{7.1}
Assume that
\begin{description}
\item[(a)] $\bar\lambda=\langle\lambda_\alpha:\alpha\le\omega\rangle$,
$\lambda_n<\lambda_{n+1}<\lambda_\omega$, $\lambda=\lambda_\omega$,
all are regulars,
\item[(b)] $D$ is a filter on $\omega$ containing cobounded sets,
\item[(c)] ${\rm tcf}(\prod \lambda_n /D)=\lambda$ (indeed, we mean $=$, we could
just use $\lambda\in{\rm pcf}_D(\{\lambda_n:n<\omega\})$),
\item[(d)] $(\sum\limits_{n<\omega}\lambda_n)^+<\lambda<\prod\limits_{n<
\omega}\lambda_n$.
\end{description}
{\em Then} there is no universal member in ${\frak K}^{tr}_{\bar\lambda}$.
\end{Theorem}
\proof We first notice that there is a sequence $\bar P=\langle P_\alpha:
\sum\limits_{n<\omega}\lambda_n<\alpha<\lambda\rangle$ such that:
\begin{enumerate}
\item $|P_\alpha|<\lambda$,
\item $a\in P_\alpha\quad\Rightarrow\quad a$ is a closed subset of $\alpha$
of order type $\leq\sum\limits_{n<\omega}\lambda_n$,
\item $a\in\bigcup\limits_{\alpha<\lambda} P_\alpha\ \&\ \beta\in{\rm nacc}(a)
\quad \Rightarrow\quad a\cap\beta\in P_\beta$,
\item For all club subsets $E$ of $\lambda$, there are stationarily many
$\delta$ for which there is an $a\in\bigcup\limits_{\alpha<\lambda} P_\alpha$
such that
\[{\rm cf}(\delta)=\aleph_0\ \&\ a\in P_\delta\ \&\ {\rm otp}(a)=\sum_{n<\omega}
\lambda_n\ \&\ a\subseteq E.\]
\end{enumerate}
[Why? If $\lambda=(\sum\limits_{n<\omega}\lambda_n)^{++}$, then it is the
successor of a regular, so we use \cite[\S4]{Sh:351}, i.e.
\[\{\alpha<\lambda:{\rm cf}(\alpha)\le(\sum_{n<\omega}\lambda_n)\}\]
is the union of $(\sum\limits_{n<\omega}\lambda_n)^+$ sets with squares.\\
If $\lambda>(\sum\limits_{n<\omega}\lambda_n)^{++}$, then we can use
\cite[\S1]{Sh:420}, which guarantees that there is a stationary $S\in
I[\lambda]$.]
We can now find a sequence
\[\langle f_\alpha,g_{\alpha,a}:\alpha<\lambda,a\in P_\alpha\rangle\]
such that:
\begin{description}
\item[(a)] $\bar f=\langle f_\alpha:\alpha<\lambda\rangle$ is a
$<_D$-increasing cofinal sequence in $\prod\limits_{n<\omega}\lambda_n$,
\item[(b)] $g_{\alpha,a}\in\prod\limits_{n<\omega}\lambda_n$,
\item[(c)] $\bigwedge\limits_{\beta<\alpha} f_\beta<_D g_{\alpha,a}<_D
f_{\alpha+1}$,
\item[(d)] $\lambda_n>|a|\ \&\ \beta\in{\rm nacc}(a)\quad \Rightarrow\quad
g_{\beta,a\cap\beta}(n)<g_{\alpha,a}(n)$.
\end{description}
[How? Choose $\bar f$ by ${\rm tcf}(\prod\limits_{n<\omega}\lambda_n/D)=\lambda$.
Then choose $g$'s by induction, possibly throwing out some of the
$f$'s; this is from \cite[II, \S1]{Sh:g}.]
Let $T\in {\frak K}^{tr}_{\bar \lambda}$.
We introduce for $x\in{\rm lev}_\omega(T)$ and $\ell<\omega$ the notation
$F^T_\ell(x) = F_\ell(x)$ to denote the unique member of ${\rm lev}_\ell(T)$ which
is below $x$ in the tree order of $T$.
\noindent For $a\in\bigcup\limits_{\alpha<\lambda} P_\alpha$, let $a=\{
\alpha_{a,\xi}:\xi<{\rm otp}(a)\}$ be an increasing enumeration. We shall consider
two cases. In the first one, we assume that the following statement $(*)$
holds. In this case, the proof is easier, and maybe $(*)$ always holds for
some $D$, but we do not know this at present.
\begin{description}
\item[{{$(*)$}}] There is a partition $\langle A_n:n < \omega \rangle$ of
$\omega$ into sets not disjoint to any member of $D$.
\end{description}
In this case, let for $n\in\omega$, $D_n$ be the filter generated by $D$ and
$A_n$. Let for $a\in\bigcup\limits_{\alpha<\lambda} P_\alpha$ with ${\rm otp}(a)=
\sum\limits_{n<\omega}\lambda_n$, and for $x\in{\rm lev}_\omega(T)$,
\[{\rm inv}(x,a,T)=:\langle\xi_n(x,a,T):n<\omega\rangle,\]
where
\[\begin{ALIGN}
\xi_n(x,a,T)=:\min\big\{\xi<{\rm otp}(a)\!:&\mbox{ for some }m<\omega
\mbox{ we have }\\
& \langle F^T_\ell(x):
\ell<\omega\rangle<_{D_n} g_{\alpha', a'}\mbox{ where }\\
&\alpha'=\alpha_{a,\omega\xi+m}\mbox{ and } a'=
a\cap\alpha'\big\}.
\end{ALIGN}\]
Let
\[{\rm INv}(a,T)=:\{{\rm inv}(x,a,T):x\in T\ \&\ {\rm lev}_T(x)=\omega\},\]
\[\begin{ALIGN}
{\rm INV}(T)=:\big\{c:&\mbox{for every club } E\subseteq\lambda,\mbox{ for some }
\delta\mbox{ and } a\in P\\
&\mbox{we have }{\rm otp}(a)=\sum\lambda_n\ \&\ a\subseteq E\ \&\ a\in P_\delta\\
&\mbox{and for some } x\in T\mbox{ of } {\rm lev}_T(x)=\omega,\ c={\rm inv}(x,a,T)
\big\}.
\end{ALIGN}\]
(Alternatively, we could have looked at the function giving each $a$ the value
${\rm INv}(a,T)$, and then divide by a suitable club guessing ideal as in
the proof in \S3, see Definition \ref{3.7}.)
\noindent Clearly
\medskip
\noindent{\bf Fact}:\hspace{0.15in} ${\rm INV}(T)$ has cardinality $\le\lambda$.
\medskip
The main point is the following
\medskip
\noindent{\bf Main Fact}:\hspace{0.15in} If ${\bf h}:T^1\longrightarrow T^2$ is
an embedding, {\em then\/}
\[{\rm INV}(T^1)\subseteq{\rm INV}(T^2).\]
\medskip
\noindent{\em Proof of the {\bf Main Fact} under $(*)$}\ \ \ We define for $n
\in\omega$
\[E_n=:\big\{\delta<\lambda_n:\,\delta>\bigcup_{\ell<n}\lambda_\ell\mbox{ and
}\left(\forall x\in{\rm lev}_n(T^1)\right)({\bf h}(x)<\delta \Leftrightarrow x<\delta)
\big\}.\]
We similarly define $E_\omega$, so $E_n$ ($n\in\omega$) and $E_\omega$ are
clubs (of $\lambda_n$ and $\lambda$ respectively). Now suppose $c\in{\rm INV}(T_1)
\setminus{\rm INV}(T_2)$. Without loss of generality $E_\omega$ is (also) a club of
$\lambda$ which exemplifies that $c\notin{\rm INV}(T_2)$. For $h\in
\prod\limits_{n<\omega}\lambda_n$, let
\[h^+(n)=:\min(E_n\setminus h(n)),\quad\mbox{ and }\quad\beta[h]=\min\{\beta
<\lambda:h<f_\beta\}.\]
(Note that $h<f_{\beta[h]}$, not just $h<_D f_{\beta[h]}$.) For a sequence
$\langle h_i:i<i^*\rangle$ of functions from $\prod\limits_{n<\omega}
\lambda_n$, we use $\langle h_i:i<i^* \rangle^+$ for $\langle h^+_i:i<i^*
\rangle$. Now let
\[E^*=:\big\{\delta<\lambda:\mbox{if }\alpha<\delta\mbox{ then }
\beta[f^+_\alpha]<\delta\mbox{ and }\delta\in{\rm acc}(E_\omega)\big\}.\]
Thus $E^*$ is a club of $\lambda$. Since $c\in{\rm INV}(T_1)$, there is $\delta<
\lambda$ and $a\in P_\delta$ such that for some $x\in{\rm lev}_\omega(T_1)$ we have
\[a\subseteq E^*\ \&\ {\rm otp}(a)=\sum_{n<\omega}\lambda_n\ \&\ c={\rm inv}(x,a,T_1).\]
Let for $n\in\omega$, $\xi_n=:\xi_n(x,a,T_1)$, so $c=\langle\xi_n:n<\omega
\rangle$. Also let for $\xi<\sum\limits_{n<\omega}\lambda_n$, $\alpha_\xi=:
\alpha_{a,\xi}$, so $a=\langle\alpha_\xi:\xi<\sum\limits_{n<\omega}\lambda_n
\rangle$ is an increasing enumeration. Now fix an $n<\omega$ and consider
${\bf h}(x)$. Then we know that for some $m$
\begin{description}
\item[($\alpha$)] $\langle F^{T_1}_\ell(x):\ell<\omega\rangle<_{D_n}
g_{\alpha'}$ where $\alpha'=\alpha_{\omega\xi_n+m}$\qquad and
\item[($\beta$)] for no $\xi<\xi_n$ is there such an $m$.
\end{description}
Now let us look at $F^{T_1}_\ell(x)$ and $F^{T_2}_\ell({\bold h}(x))$. They are
not necessarily equal, but
\begin{description}
\item[($\gamma$)] $\min(E_\ell\setminus F^{T_1}_\ell(x))=\min(E_\ell
\setminus F^{T_2}_\ell({\bf h}(x))$
\end{description}
(by the definition of $E_\ell$). Hence
\begin{description}
\item[($\delta$)] $\langle F^{T_1}_\ell(x):\ell<\omega\rangle^+=\langle
F^{T_2}_\ell({\bf h}(x)):\ell<\omega\rangle^+$.
\end{description}
Now note that by the choice of $g$'s
\begin{description}
\item[($\varepsilon$)] $(g_{\alpha_\varepsilon,a\cap\alpha_\varepsilon})^+
<_{D_n} g_{\alpha_{\varepsilon+1},a\cap\alpha_{\varepsilon+1}}$.
\end{description}
\relax From $(\delta)$ and $(\varepsilon)$ it follows that $\xi_n({\bf h}(x),a,
T^2)=\xi_n(x,a,T^1)$. Hence $c\in{\rm INV}(T^2)$.
\hfill$\square_{\mbox{Main Fact}}$
\medskip
Now it clearly suffices to prove:
\medskip
\noindent{\bf Fact A:}\hspace{0.15in} For each $c=\langle\xi_n:n<\omega
\rangle\in {}^\omega(\sum\limits_{n<\omega}\lambda_n)$ we can find a $T\in
{\frak K}^{tr}_{\bar\lambda}$ such that $c\in{\rm INV}(T)$.
\medskip
\noindent{\em Proof of the Fact A in case $(*)$ holds}\ \ \ For each $a\in
\bigcup\limits_{\delta<\lambda} P_\delta$ with ${\rm otp}(a)=\sum\limits_{n\in
\omega}\lambda_n$ we define $x_{c,a}=:\langle x_{c,a}(\ell):\ell<\omega
\rangle$ by:
\begin{quotation}
if $\ell\in A_n$, then $x_{c,a}(\ell)=\alpha_{a,\omega\xi_n+\delta}$.
\end{quotation}
Let
\[T=\bigcup_{n<\omega}\prod_{\ell<n}\lambda_\ell\cup\big\{x_{c,a}:a\in
\bigcup_{\delta<\lambda} P_\delta\ \&\ {\rm otp}(a)=\sum_{n<\omega}\lambda_n
\big\}.\]
We order $T$ by $\triangleleft$.
It is easy to check that $T$ is as required. \hfill$\square_A$
\medskip
Now we are left to deal with the case that $(*)$ does not hold. Let
\[{\rm pcf}(\{\lambda_n:n<\omega\})=\{\kappa_\alpha:\alpha\le\alpha^*\}\]
be an enumeration in increasing order so in particular
\[\kappa_{\alpha^*}=\max{\rm pcf}(\{\lambda_n:n<\omega\}).\]
Without loss of generality $\kappa_{\alpha^*}=\lambda$ (by throwing out some
elements if necessary) and $\lambda\cap{\rm pcf}(\{\lambda_n:n<\omega\})$ has no
last element (this appears explicitly in \cite{Sh:g}, but is also
straightforward from the pcf theorem). In particular, $\alpha^*$ is a limit
ordinal. Hence, without loss of generality
\[D=\big\{A\subseteq\omega:\lambda>\max{\rm pcf}\{\lambda_n:n\in\omega\setminus A\}
\big\}.\]
Let $\langle {\frak a}_{\kappa_\alpha}:\alpha\le\alpha^*\rangle$ be a
generating sequence for ${\rm pcf}(\{\lambda_n:n<\omega\})$, i.e.
\[\max{\rm pcf}({\frak a}_{\kappa_\alpha})=\kappa_\alpha\quad\mbox{ and }\quad
\kappa_\alpha\notin{\rm pcf}(\{\lambda_n:n<\omega\}\setminus {\frak
a}_{\kappa_\alpha}).\]
(The existence of such a sequence follows from the pcf theorem). Without
loss of generality,
\[{\frak a}_{\alpha^*} = \{ \lambda_n:n < \omega\}.\]
Now note
\begin{Remark}
If ${\rm cf}(\alpha^*)=\aleph_0$, then $(*)$ holds.
\end{Remark}
Why? Let $\langle\alpha(n):n<\omega\rangle$ be a strictly increasing cofinal
sequence in $\alpha^*$. Let $\langle B_n:n<\omega\rangle$ partition $\omega$
into infinite pairwise disjoint sets and let
\[A_\ell=:\big\{k<\omega:\bigvee_{n\in B_\ell}[\lambda_k\in {\frak
a}_{\kappa_{\alpha(n)}}\setminus\bigcup_{m<n} {\frak a}_{\kappa_{\alpha(m)}}]
\big\}.\]
To check that this choice of $\langle A_\ell:\ell<\omega\rangle$ works, recall
that for all $\alpha$ we know that $\alpha_{\kappa_\alpha}$ does not belong to
the ideal generated by $\{{\frak a}_{\kappa_\beta}: \beta<\alpha\}$ and use
the pcf calculus. \hfill$\square$
Now let us go back to the general case, assuming ${\rm cf}(\alpha^*)>\aleph_0$.
Our problem is the possibility that
\[{\cal P}(\{\lambda_n:n<\omega\})/J_{<\lambda}[\{\lambda_n:n<\omega\}].\]
is finite. Let now $A_\alpha=:\{n:\lambda_n\in {\frak a}_\alpha\}$, and
\[\begin{array}{lll}
J_\alpha &=: &\big\{A\subseteq\omega:\max{\rm pcf}\{\lambda_\ell:\ell\in A\}<
\kappa_\alpha\big\}\\
J'_\alpha &=: &\big\{A\subseteq\omega:\max{\rm pcf}\{\lambda_\ell:\ell\in A\}\cap
{\frak a}_{\kappa_\alpha}<\kappa_\alpha\big\}.
\end{array}\]
We define for $T\in {\frak K}^{tr}_{\bar\lambda}$, $x\in{\rm lev}_\omega(T)$,
$\alpha<\alpha^*$ and $a\in\bigcup\limits_{\delta<\lambda} P_\delta$:
\[\begin{ALIGN}
\xi^*_\alpha(x,a,T)=:\min\big\{\xi:&\bigvee_m [\langle F^T_\ell(x):\ell<
\omega\rangle <_{J'_\alpha} g_{\alpha', a'}\mbox{ where }\\
&\qquad \alpha'=\alpha_{a,\omega \xi+m}\mbox{ and }a'=a\cap
\alpha'\big\}.
\end{ALIGN}
\]
Let
\[{\rm inv}_\alpha(x,a,T)=:\langle\xi^*_{\alpha+n}(x,a,T):n<\omega\rangle,\]
\[{\rm INv}(a,T)=:\big\{{\rm inv}_\alpha(x,a,T):x\in T\ \&\ \alpha<\alpha^* \ \&\
{\rm lev}_T(x)=\omega\big\},\]
and
\[\begin{ALIGN}
{\rm INV}(T)=\big\{c:&\mbox{for every club } E^*\mbox{ of }\lambda\mbox{ for some
} a\in\bigcup\limits_{\delta<\lambda} P_\delta\\
&\mbox{with }{\rm otp}(a)=\sum\limits_{n<\omega}\lambda_n\mbox{ for arbitrarily
large }\alpha<\alpha^*,\\
&\mbox{there is }x\in{\rm lev}_\omega(T)\mbox{ such that }{\rm inv}_\alpha(x,a,T)=
c\big\}.
\end{ALIGN}\]
As before, the point is to prove the Main Fact.
\medskip
\noindent{\em Proof of the {\bf Main Fact} in general}\ \ \ Suppose ${\bf h}:
T^1\longrightarrow T^2$ and $c\in{\rm INV}(T^1)\setminus{\rm INV}(T^2)$. Let $E'$ be
a club of $\lambda$ which witnesses that $c\notin{\rm INV}(T^2)$. We define
$E_n,E_\omega$ as before, as well as $E^*$ ($\subseteq E_\omega\cap
E'$). Now let us choose $a\in
\bigcup\limits_{\delta<\lambda} P_\delta$ with $a\subseteq E^*$ and ${\rm otp}(a)=
\sum\limits_{n<\omega}\lambda_n$. So $a=\{\alpha_{a,\xi}:\xi<\sum\limits_{n<
\omega}\lambda_n\}$, which we shorten as $a=\{\alpha_\xi:\xi<\sum\limits_{n<
\omega}\lambda_n\}$. For each $\xi<\sum\limits_{n<\omega}\lambda_n$, as
before, we know that
\[(g_{\alpha_\xi,a\cap\alpha_\xi})^+<_{J^*_\alpha}g_{\alpha_{\xi+1},a\cap
\alpha_{\xi+1}}.\]
Therefore, there are $\beta_{\xi,\ell}<\alpha^*$ ($\ell<\ell_\xi$) such that
\[\{\ell:g^+_{\alpha_\xi,a\cap\alpha_\xi}(\ell)\ge g_{\alpha_{\xi+1},a\cap
\alpha_{\xi+1}}(\ell)\}\supseteq\bigcup_{\ell<\ell_\xi} A_{\beta_{\xi,\ell}}.\]
Let $c=\langle\xi_n:n<\omega\rangle$ and let
\[\Upsilon=\big\{\beta_{\xi,\ell}:\mbox{for some } n\mbox{ and } m\mbox{ we
have }\xi=\omega\xi_n+m\ \&\ \ell<\omega\big\}.\]
Thus $\Upsilon\subseteq\alpha^*$ is countable. Since ${\rm cf}(\alpha^*)>\aleph_0$,
the set $\Upsilon$ is bounded in $\alpha^*$. Now we know that $c$ appears as
an invariant for $a$ and arbitrarily large $\delta<\alpha^*$, for some
$x_{a,\delta}\in{\rm lev}_\omega(T_1)$. If $\delta>\sup(\Upsilon)$, $c\in{\rm INV}(T^2)$
is exemplified by $a,\delta,{\bf h}(x_{\alpha,\delta})$, just as before.
\hfill$\square$
\medskip
We still have to prove that every $c=\langle\xi_n:n<\omega\rangle$ appears as
an invariant; i.e. the parallel of Fact A.
\medskip
\noindent{\em Proof of Fact A in the general case:}\ \ \ Define for each $a\in
\bigcup\limits_{\delta<\lambda} P_\delta$ with ${\rm otp}(a)=\sum\limits_{n<\omega}
\lambda_n$ and $\beta<\alpha^*$
\[x_{c,a,\beta}=\langle x_{c,a,\beta}(\ell):\ell<\omega\rangle,\]
where
\[x_{c,a,\beta}(\ell)=\left\{
\begin{array}{lll}
\alpha_{a,\omega\xi_n+\delta} &\mbox{\underbar{if}} &\lambda_\ell\in {\frak
a}_{\beta+k}\setminus\bigcup\limits_{k'<k} {\frak a}_{\beta+k'}\\
0 &\mbox{\underbar{if}} &\lambda_\ell\notin {\frak a}_{\beta+k}\mbox{ for
any } k<\omega.
\end{array}\right.\]
Form the tree as before. Now for any club $E$ of $\lambda$, we can find $a\in
\bigcup\limits_{\delta<\lambda} P_\delta$ with ${\rm otp}(a)=\sum\limits_{n<\omega}
\lambda_n$, $a\subseteq E$ such that $\langle x_{c,a,\beta}:\beta<\alpha^*
\rangle$ shows that $c\in{\rm INV}(T)$. \hfill$\square_{\ref{7.1}}$
\begin{Remark}
\begin{enumerate}
\item Clearly, this proof shows not only that there is no one $T$ which is
universal for ${\frak K}^{tr}_{\bar\lambda}$, but that any sequence of
$<\prod\limits_{n<\omega}\lambda_n$ trees will fail. This occurs generally in
this paper, as we have tried to mention in each particular case.
\item The case ``$\lambda<2^{\aleph_0}$" is included in the theorem, though
for the Abelian group application the $\bigwedge\limits_{n<\omega}
\lambda^{\aleph_0}_n<\lambda_{n+1}$ is necessary.
\end{enumerate}
\end{Remark}
\begin{Remark}
\label{7.1A}
\begin{enumerate}
\item If $\mu^+< \lambda={\rm cf}(\lambda)<\chi<\mu^{\aleph_0}$ and
$\chi^{[\lambda]}<\mu^{\aleph_0}$
(or at least $T_{{\rm id}^a(\bar C)}(\chi)<\mu^{\aleph_0}$) we can get the results
for ``no $M \in {\frak K}^x_\chi$ is universal for ${\frak K}^x_\lambda$", see
\S8 (and \cite{Sh:456}).
\end{enumerate}
\end{Remark}
\noindent We can below (and subsequently in \S8) use $J^3_{\bar m}$ as in \S6.
\begin{Theorem}
\label{7.2}
Assume that $2^{\aleph_0}<\lambda_0$, $\bar\lambda=\langle\lambda_n:n<\omega
\rangle\char 94\langle\lambda\rangle$, $\mu=\sum\limits_{n<\omega}\lambda_n$,
$\lambda_n<\lambda_{n+1}$, $\mu^+<\lambda={\rm cf}(\lambda)<\mu^{\aleph_0}$.\\
\underbar{If}, for simplicity, $\bar m=\langle m_i:i<\omega\rangle=\langle
\omega:i<\omega\rangle$ (actually $m_i\in [2,\omega]$ or even $m_i\in
[2,\lambda_0)$, $\lambda_0<\lambda$ are O.K.) and ${\bf U}^{<\mu}_{J^2_{\bar m}}
(\lambda)=\lambda$ (remember
\[J^2_{\bar m}=\{A\subseteq\prod_{i<\omega} m_i:A \mbox{ is nowhere dense}\}\]
and definition \ref{5.3}),\\
\underbar{then} in ${\frak K}^{tr}_{\bar\lambda}$ there is no universal
member.
\end{Theorem}
\proof 1)\ \ Let $S\subseteq\lambda$, $\bar C=\langle C_\delta:\delta\in S
\rangle$ be a club guessing sequence on $\lambda$ with ${\rm otp}(C_\delta)\ge\sup
\lambda_n$. We assume that we have ${\bar{\frak A}}=\langle {\frak A}_\alpha:
\alpha<\lambda\rangle$, $J^2_{\bar m}$, $T^*\in {\frak A}_0$ ($T^*$ is a
candidate for the universal), $\bar C=\langle C_\delta:\delta\in S\rangle\in
{\frak A}_\alpha$, ${\frak A}_\alpha\prec({\cal H}(\chi),\in,<^*_\chi)$, $\chi=
\beth_7(\lambda)^+$, $\|{\frak A}_\alpha\|<\lambda$, ${\frak A}_\alpha$
increasingly continuous, $\langle {\frak A}_\beta:\beta\le\alpha\rangle\in
{\frak A}_{\alpha+1}$, ${\frak A}_\alpha\cap\lambda$ is an ordinal, ${\frak A}
=\bigcup\limits_{\alpha<\lambda}{\frak A}_\alpha$ and
\[E=:\{\alpha:{\frak A}_\alpha\cap\lambda=\alpha\}.\]
Note: $\prod\bar m\subseteq {\frak A}$ (as $\prod\bar m\in {\frak A}$ and
$|\prod\bar m|=2^{\aleph_0}$).
\noindent NOTE: By ${\bf U}^{<\mu}_{J^2_{\bar m}}(\lambda)=\lambda$,
\begin{description}
\item[$(*)$] \underbar{if} $x_\eta\in{\rm lev}_\omega(T^*)$ for $\eta\in\prod\bar
m$
\underbar{then} for some $A\in (J^2_{\bar m})^+$ the set $\langle(\eta,
x_\eta):\eta\in A\rangle$ belongs to ${\frak A}$. But then for some $\nu\in
\bigcup\limits_k\prod\limits_{i<k} m_i$, the set $A$ is dense above $\nu$ (by
the definition of $J^2_{\bar m}$) and hence: if the mapping $\eta\mapsto
x_\eta$ is continuous then $\langle x_\rho:\nu\triangleleft\rho\in\prod\bar m
\rangle\in {\frak A}$.
\end{description}
For $\delta\in S$ such that $C_\delta \subseteq E$ we let
\[\begin{ALIGN}
P^0_\delta=P^0_\delta({\frak A})=\biggl\{\bar x:&\bar x=\langle x_\rho:\rho
\in t\rangle\in {\frak A}\mbox{ and } x_\rho\in{\rm lev}_{\ell g(\rho)}T^*,\\
&\mbox{the mapping }\rho\mapsto x_\rho\mbox{ preserves all of the
relations:}\\
&\ell g(\rho)=n,\rho_1\triangleleft\rho_2, \neg(\rho_1\triangleleft\rho_2),
\neg(\rho_1=\rho_2),\\
&\rho_1\cap\rho_2=\rho_3\mbox{ (and so }\ell g(\rho_1\cap\rho_2)=n\mbox{ is
preserved});\\
&\mbox{and } t\subseteq\bigcup\limits_{\alpha\le\omega}\prod\limits_{i<
\alpha} m_i\biggr\}.
\end{ALIGN}\]
Assume $\bar x=\langle x_\rho:\rho\in t\rangle\in P^0_\delta$. Let
\[{\rm inv}(\bar x,C_\delta,T^*,{\bar {\frak A}})=:\big\{\alpha\in C_\delta:
(\exists\rho\in{\rm Dom}(\bar x))(x_\rho\in {\frak A}_{\min(C_\delta
\setminus (\alpha+1))}\setminus {\frak A}_\alpha\big)\}.\]
Let
\[\begin{array}{l}
{\rm Inv}(C_\delta,T^*,\bar{\frak A})=:\\
\big\{a:\mbox{for some }\bar x\in P^0_\delta,\ \ a\mbox{ is a
countable subset of }{\rm inv}(\bar x,C_0,T^*,\bar{\frak A})\big\}.
\end{array}\]
Note: ${\rm inv}(\bar x,C_0,T^*,\bar{\frak A})$ has cardinality at most continuum,
so ${\rm Inv}(C_0,T^*,\bar{\frak A})$ is a family of $\le 2^{\aleph_0}\times
|{\frak A}|=\lambda$ countable subsets of $C_\delta$.
We continue as before. Let $\alpha_{\delta,\varepsilon}$ be the
$\varepsilon$-th member of $C_\delta$ for $\varepsilon<\sum\limits_{n<\omega}
\lambda_n$. So as $\lambda< \mu^{\aleph_0}$,$\mu> 2^{\aleph_0}$
clearly $\lambda< {\rm cf}([\lambda]^{\aleph_0}, \subseteq)$ (equivalently
$\lambda<{\rm cov}(\mu,\mu,\aleph_1,2)$) hence we can find $\gamma_n\in
(\bigcup\limits_{\ell<n}\lambda_\ell,\lambda_n)$ limit so such that
for each $\delta\in S$, $a\in{\rm Inv}(C_\delta,T^*,\bar{\frak A})$ we have
$\{\gamma_n+\ell:n<\omega\mbox{ and }\ell<m_i\}\cap a$ is bounded in $\mu$.
Now we can find $T$ such that ${\rm lev}_n(T)=\prod\limits_{\ell<n}\lambda_\ell$
and
\[\begin{ALIGN}
{\rm lev}_\omega(T)=\big\{\bar\beta:
&\bar\beta=\langle\beta_\ell:\ell<\omega
\rangle,\mbox{ and for some }\delta \in S,\mbox{ for every }\ell<\omega\\
&\mbox{we have }\gamma'_\ell\in\{\alpha_{\delta,\gamma_\ell+m}:m<m_i\}
\big\}.
\end{ALIGN} \]
So, if $T^*$ is universal there is an embedding $f:T\longrightarrow T^*$,
and hence
\[E'=\{\alpha\in E:{\frak A}_\alpha\mbox{ is closed under } f\mbox{ and }
f^{-1}\}\]
is a club of $\lambda$. By the choice of $\bar C$ for some $\delta\in S$ we
have $C_\delta\subseteq E'$. Now use $(*)$ with $x_\eta=f(\bar\beta^{\delta,
\eta})$, where $\beta^{\delta,\eta}_\ell=\alpha_{\delta,\gamma_\ell+
\eta(\ell)}\in{\rm lev}_\omega(T)$. Thus we get $A\in (J^2_{\bar m})^+$ such that
$\{(\eta,x_\eta):\eta\in A\}\in{\frak A}$, there is $\nu\in\bigcup\limits_k
\prod\limits_{i<k} m_i$ such that $A$ is dense above $\nu$, hence as $f$ is
continuous, $\langle(\eta,x_\eta):\nu\triangleleft\eta\in\prod\bar m\rangle
\in {\frak A}$. So $\langle x_\eta:\eta\in\prod\bar m,\nu\trianglelefteq\eta
\rangle\in P^0_\delta({\frak A})$, and hence the set
\[\{\alpha_{\delta,\gamma_\ell+m}:\ell\in [\ell g(\nu),\omega)\mbox{ and } m<
m_\ell\}\cup\{\alpha_{\delta,\gamma_i+\nu(i)}:\ell<\ell g(\nu)\}\]
is ${\rm inv}(\bar x,C_\delta,T^*,{\frak A})$. Hence
\[a=\{\alpha_{\delta,\gamma_\ell}\!:\ell\in [\ell g(\nu),\omega)\}\in{\rm Inv}(
C_\delta,T^*,{\frak A}),\]
contradicting
``$\{\alpha_{\delta,\gamma_\ell}:\ell<\omega\}$ has finite intersection with
any $a\in{\rm Inv}(C_\delta,T^*,{\frak A})$''.
\begin{Remark}
\label{7.3}
We can a priori fix a set of $\aleph_0$ candidates and say more on their order
of appearance, so that ${\rm Inv}(\bar x,C_\delta,T^*,{\bar{\frak A}})$ has order
type $\omega$. This makes it easier to phrase a true invariant, i.e. $\langle
(\eta_n,t_n):n<\omega\rangle$ is as above, $\langle\eta_n:n<\omega\rangle$
lists ${}^{\omega >}\omega$ with no repetition, $\langle t_n\cap {}^\omega
\omega:n<\omega\rangle$ are pairwise disjoint. If $x_\rho\in{\rm lev}_\omega(T^*)$
for $\rho\in {}^\omega\omega$, $\bar T^*=\langle\bar T^*_\zeta:\zeta<\lambda
\rangle$ representation
\[\begin{array}{l}
{\rm inv}(\langle x_\rho:\rho\in {}^\omega\omega\rangle,C_\delta,\bar T^*)=\\
\big\{\alpha\in C_\delta:\mbox{for some } n,\ (\forall\rho)[\rho\in t_n\cap
{}^\omega\omega\quad\Rightarrow\quad x_\rho\in T^*_{\min(C_\delta\setminus
(\alpha+1))}\setminus T^*_\alpha]\big\}.
\end{array}\]
\end{Remark}
\begin{Remark}
\label{7.4}
If we have $\Gamma\in (J^2_{\bar m})^+$, $\Gamma$ non-meagre, $J=J^2_m
\restriction\Gamma$ and ${\bf U}^2_J(\lambda)<\lambda^{\aleph_0}$ then we
can weaken the cardinal assumptions to:
\[\bar\lambda=\langle\lambda_n:n<\omega\rangle\char 94\langle\lambda\rangle,
\qquad \mu=\sum_n\lambda_n,\qquad\lambda_n<\lambda_{n+1},\]
\[\mu^+<\lambda={\rm cf}(\lambda)\qquad\mbox{ and }\qquad {\bf U}^2_J(\lambda)<{\rm cov}(\mu,
\mu,\aleph_1,2) (\mbox{see }\ref{0.4}).\]
The proof is similar.
\end{Remark}
\section{Universals in singular cardinals}
In \S3, \S5, \ref{7.2}, we can in fact deal with ``many'' singular cardinals
$\lambda$. This is done by proving a stronger assertion on some regular
$\lambda$. Here ${\frak K}$ is a class of models.
\begin{Lemma}
\label{8.1}
\begin{enumerate}
\item There is no universal member in ${\frak K}_{\mu^*}$ if for some $\lambda
<\mu^*$, $\theta\ge 1$ we have:
\begin{description}
\item[\mbox{$\otimes_{\lambda,\mu^*,\theta}[{\frak K}]$}] not only there is no
universal member in ${\frak K}_\lambda$ but if we assume:
\[\langle M_i:i<\theta\rangle\ \mbox{ is given, }\ \|M_i\|\le\mu^*<\prod_n
\lambda_n,\ M_i\in {\frak K},\]
then there is a structure $M$ from ${\frak K}_\lambda$ (in some cases
of a simple form) not embeddable in any $M_i$.
\end{description}
\item Assume
\begin{description}
\item[$\otimes^\sigma_1$] $\langle\lambda_n:n<\omega\rangle$ is given,
$\lambda^{\aleph_0}_n<\lambda_{n+1}$,
\[\mu=\sum_{n<\omega}\lambda_n<\lambda={\rm cf}(\lambda)\leq
\mu^*<\prod_{n<\omega}\lambda_n\]
and $\mu^+<\lambda$ or at least there is a club guessing $\bar C$ as
in $(**)^1_\lambda$ (ii) of \ref{3.4} for
$(\lambda,\mu)$.
\end{description}
\underbar{Then} there is no universal member in ${\frak K}_{\mu^*}$ (and
moreover $\otimes_{\lambda,\mu^*,\theta}[{\frak K}]$ holds) in the following
cases
\begin{description}
\item[$\otimes_2$(a)] for torsion free groups, i.e. ${\frak K}={\frak
K}^{rtf}_{\bar\lambda}$ if ${\rm cov}(\mu^*,\lambda^+,\lambda^+,\lambda)<
\prod\limits_{n<\omega}\lambda_n$, see notation \ref{0.4} on ${\rm cov}$)
\item[\quad(b)] for ${\frak K}={\frak K}^{tcf}_{\bar\lambda}$,
\item[\quad(c)] for ${\frak K}={\frak K}^{tr}_{\bar\lambda}$ as in \ref{7.2} -
${\rm cov}({\bf U}_{J^3_{\bar m}}(\mu^*),\lambda^+,\lambda^+,\lambda)<\prod\limits_{n<
\omega}\lambda_n$,
\item[(d)] for ${\frak K}^{rs(p)}_{\bar\lambda}$: like case (c) (for
appropriate ideals), replacing $tr$ by $rs(p)$.
\end{description}
\end{enumerate}
\end{Lemma}
\begin{Remark}
\label{8.1A}
\begin{enumerate}
\item For \ref{7.2} as $\bar m=\langle\omega:i<\omega\rangle$ it is clear that
the subtrees $t_n$ are isomorphic. We can use $m_i\in [2,\omega)$, and use
coding; anyhow it is immaterial since ${}^\omega \omega,{}^\omega 2$ are
similar.
\item We can also vary $\bar\lambda$ in \ref{8.1} $\otimes_2$, case (c).
\item We can replace ${\rm cov}$ in $\otimes_2$(a),(c) by
\[\sup{\rm pp}_{\Gamma(\lambda)}(\chi):{\rm cf}(\chi)=\lambda,\lambda<\chi\le
{\bf U}_{J^3_{\bar m}}(\mu^*)\}\]
(see \cite[5.4]{Sh:355}, \ref{2.3}).
\end{enumerate}
\end{Remark}
\proof Should be clear, e.g.\\
{\em Proof of Part 2), Case (c)}\ \ \ Let $\langle T_i:i<i^* \rangle$ be
given, $i^*<\prod\limits_{n<\omega}\lambda_n$ such that
\[\|T_i\|\le\mu^*\quad\mbox{ and }\quad\mu^\otimes=:{\rm cov}({\bf U}_{J^3_{\bar
m}}(\mu^*),\lambda^+,\lambda^+,\lambda)<\prod_{n<\omega}\lambda_n.\]
By \cite[5.4]{Sh:355} and ${\rm pp}$ calculus (\cite[2.3]{Sh:355}), $\mu^\otimes=
{\rm cov}(\mu^\otimes,\lambda^+,\lambda^+,\lambda)$. Let
$\chi=\beth_7(\lambda)^+$. For $i<i^*$ choose ${\frak B}_i\prec (H(\chi)\in
<^*_\chi)$, $\|{\frak B}_i\|=\mu^\otimes$, $T_i\in {\frak B}_i$, $\mu^\otimes
+1\subseteq {\frak B}_i$. Let $\langle Y_\alpha: \alpha<\mu^\otimes\rangle$ be
a family of subsets of $T_i$ exemplifying the Definition of $\mu^\otimes=
{\rm cov}(\mu^\otimes,\lambda^+,\lambda^+,\lambda)$.\\
Given $\bar x=\langle x_\eta:\eta\in {}^\omega\omega\rangle$, $x_\eta\in
{\rm lev}_\omega(T_i)$, $\eta\mapsto x_\eta$ continuous (in our case this means
$\ell g(\eta_1\cap\eta_2)=\ell g(x_{\eta_1}\cap x_{\eta_2})=:\ell g(\max
\{\rho:\rho\triangleleft\eta_1\ \&\ \rho\triangleleft\eta_2\})$. Then for some
$\eta\in {}^{\omega>}\omega$,
\[\langle x_\rho:\eta\triangleleft\rho\in {}^\omega\omega\rangle\in{\frak
B}.\]
So given $\left<\langle x^\zeta_\eta:\eta\in {}^\omega\omega\rangle:\zeta<
\lambda\right>$, $x^\zeta_\eta\in{\rm lev}_\omega(T_i)$ we can find $\langle (\alpha_j,\eta_j):j<j^*<\lambda\rangle$ such that:
\[\bigwedge_{\zeta<\lambda}\bigvee_j\langle x^\zeta_\eta:\eta_j\triangleleft
\eta\in {}^\omega\omega\rangle\in Y_\alpha.\]
Closing $Y_\alpha$ enough we can continue as usual. \hfill$\square_{\ref{8.1}}$
\section{Metric spaces and implications}
\begin{Definition}
\label{9.1}
\begin{enumerate}
\item ${\frak K}^{mt}$ is the class of metric spaces $M$ (i.e. $M=(|M|,d)$,
$|M|$ is the set of elements, $d$ is the metric, i.e. a two-place function
from $|M|$ to ${\Bbb R}^{\geq 0}$ such that
$d(x,y)=0\quad\Leftrightarrow\quad x=0$ and
$d(x,z)\le d(x,y)+d(y,z)$ and $d(x,y) = d(y,x)$).
An embedding $f$ of $M$ into $N$ is a one-to-one function from $|M|$ into
$|N|$ which is continuous, i.e. such that:
\begin{quotation}
\noindent if in $M$, $\langle x_n:n<\omega\rangle$ converges to $x$
\noindent then in $N$, $\langle f(x_n):n<\omega\rangle$ converges to $f(x)$.
\end{quotation}
\item ${\frak K}^{ms}$ is defined similarly but ${\rm Rang}(d)\subseteq\{2^{-n}:n
<\omega\}\cup\{0\}$ and instead of the triangular inequality we require
\[d(x,y)=2^{-i},\qquad d(y,z)=2^{-j}\quad \Rightarrow\quad d(x,z) \le
2^{-\min\{i-1,j-1\}}.\]
\item ${\frak K}^{tr[\omega]}$ is like ${\frak K}^{tr}$ but $P^M_\omega=|M|$
and embeddings preserve $x\;E_n\;y$ (not necessarily its negation) are
one-to-one, and remember $\bigwedge\limits_n
x\;E_n\;y\quad\Rightarrow\quad x \restriction n = y \restriction n$).
\item ${\frak K}^{mt(c)}$ is the class of semi-metric spaces $M=(|M|,d)$,
which means that for the constant $c\in\Bbb R^+$ the triangular inequality is
weakened to $d(x,z)\le cd(x,y)+cd(y,z)$ with embedding as in \ref{9.1}(1)
(so for $c=1$ we get ${\frak K}^{mt}$).
\item ${\frak K}^{mt[c]}$ is the class of pairs $(A,d)$ such that $A$ is a
non-empty set, $d$ a two-place symmetric function from $A$ to ${\Bbb
R}^{\ge 0}$ such that $[d(x,y)=0\ \Leftrightarrow\ x=y]$ and
\[d(x_0,x_n)\le c\sum\limits_{\ell<n} d(x_\ell,x_{\ell+1})\ \ \mbox{ for any
$n<\omega$ and $x_0,\ldots,x_n\in A$.}\]
\item ${\frak K}^{ms(c)}$, ${\frak K}^{ms[c]}$ are defined parallely.
\item ${\frak K}^{rs(p),\mbox{pure}}$ is defined like ${\frak K}^{rs(p)}$ but
the embeddings are pure.
\end{enumerate}
\end{Definition}
\begin{Remark}
There are, of course, other notions of embeddings; isometric embeddings if $d$
is preserved, co-embeddings if the image of an open set is open, bi-continuous
means an embedding which is a co-embedding. The isometric embedding is the
weakest, its case is essentially equivalent to the ${\frak
K}^{tr}_{\lambda}$ case (as in \ref{9.3}(3)); for the open case there
is a universal:
discrete space. The universal for ${\frak K}^{mt}_\lambda$ under bicontinuous
case exist in cardinality $\lambda^{\aleph_0}$, see \cite{Ko57}.
\end{Remark}
\begin{Definition}
\label{9.1A}
\begin{enumerate}
\item ${\rm Univ}^0({\frak K}^1,{\frak K}^2)=\{(\lambda,\kappa,\theta):$ there are
$M_i\in {\frak K}^2_\kappa$ for $i<\theta$ such that any $M\in {\frak
K}^1_\lambda$ can be embedded into some $M_i\}$. We may omit $\theta$
if it is 1. We may omit the superscript 0.
\item ${\rm Univ}^1({\frak K}^1,{\frak K}^2)=\{(\lambda,\kappa,\theta):$ there are
$M_i\in {\frak K}^2_\kappa$ for $i<\theta$ such that any $M\in {\frak
K}^1_\lambda$ can be represented as the union of $<\lambda$ sets $A_\zeta$
($\zeta<\zeta^*<\lambda)$ such that each $M\restriction A_\zeta$ can be
embedded into some $M_i\}$ and is a $\leq_{{\frak K}^1}$-submodel of $M$.
\item If above ${\frak K}^1={\frak K}^2$ we write it just once; (naturally we
usually assume ${\frak K}^1 \subseteq {\frak K}^2$).
\end{enumerate}
\end{Definition}
\begin{Remark}
\begin{enumerate}
\item We prove our theorems for $Univ^0$, we can get parallel things for
${\rm Univ}^1$.
\item Many previous results of this paper can be rephrased using a pair of
classes.
\item We can make \ref{9.2} below deal with pairs and/or function $H$ changing
cardinality.
\item ${\rm Univ}^\ell$ has the obvious monotonicity properties.
\end{enumerate}
\end{Remark}
\begin{Proposition}
\label{9.2}
\begin{enumerate}
\item Assume ${\frak K}^1,{\frak K}^2$ has the same models as their members
and every embedding for ${\frak K}^2$ is an embedding for ${\frak K}^1$.\\
Then ${\rm Univ}({\frak K}^2)\subseteq{\rm Univ}({\frak K}^1)$.
\item Assume there is for $\ell=1,2$ a function $H_\ell$ from ${\frak K}^\ell$
into ${\frak K}^{3-\ell}$ such that:
\begin{description}
\item[(a)] $\|H_1(M_1)\|=\|M_1\|$ for $M_1\in {\frak K}^1$,
\item[(b)] $\|H_2(M_2)\|=\|M_2\|$ for $M_2\in {\frak K}^2$,
\item[(c)] if $M_1\in {\frak K}^1$, $M_2\in {\frak K}^2$, $H_1(M_1)\in {\frak
K}^2$ is embeddable into $M_2$ \underbar{then} $M_1$ is embeddable into
$H_2(M_2)\in {\frak K}^1$.
\end{description}
\underbar{Then}\quad ${\rm Univ}({\frak K}^2)\subseteq{\rm Univ}({\frak K}^1)$.
\end{enumerate}
\end{Proposition}
\begin{Definition}
\label{9.2A}
We say ${\frak K}^1\le {\frak K}^2$ if the assumptions of \ref{9.2}(2)
hold. We say ${\frak K}^1\equiv {\frak K}^2$ if ${\frak K}^1\le {\frak K}^2
\le {\frak K}^1$ (so larger means with fewer cases of universality).
\end{Definition}
\begin{Theorem}
\label{9.3}
\begin{enumerate}
\item The relation ``${\frak K}^1\le {\frak K}^2$'' is a quasi-order
(i.e. transitive and reflexive).
\item If $({\frak K}^1,{\frak K}^2)$ are as in \ref{9.2}(1) then ${\frak K}^1
\le {\frak K}^2$ (use $H_1 = H_2 =$ the identity).
\item For $c_1>1$ we have ${\frak K}^{mt(c_1)}\equiv {\frak K}^{mt[c_1]}\equiv
{\frak K}^{ms[c_1]}\equiv {\frak K}^{ms(c_1)]}$.
\item ${\frak K}^{tr[\omega]} \le {\frak K}^{rs(p)}$.
\item ${\frak K}^{tr[\omega]} \le {\frak K}^{tr(\omega)}$.
\item ${\frak K}^{tr(\omega)} \le {\frak K}^{rs(p),\mbox{pure}}$.
\end{enumerate}
\end{Theorem}
\proof 1)\ \ Check.\\
2)\ \ Check. \\
3)\ \ Choose $n(*)<\omega$ large enough and ${\frak K}^1,{\frak K}^2$ any two
of the four. We define $H_1,H_2$ as follows. $H_1$ is the identity. For $(A,d)
\in{\frak K}^\ell$ let $H_\ell((A,d))=(A,d^{[\ell]})$ where $d^{[\ell]}(x,y)=
\inf\{1/(n+n(*)):2^{-n}\ge d(x,y)\}$ (the result is not necessarily a metric
space, $n(*)$ is chosen so that the semi-metric inequality holds). The point
is to check clause (c) of \ref{9.2}(2); so assume $f$ is a function which
${\frak K}^2$-embeds $H_1((A_1,d_1))$ into $(A_2,d_2)$; but
\[H_1((A_1,d_1))=(A_1,d_1),\quad H_2((A_2,d_2))=(A_2,d^{[2]}_2),\]
so it is enough to check that $f$ is a function which ${\frak K}^1$-embeds
$(A_1,d^{[1]}_1)$ into $(A_2,d^{[2]}_2)$ i.e. it is one-to-one (obvious) and
preserves limit (check).\\
4)\ \ For $M=(A,E_n)_{n<\omega}\in {\frak K}^{tr[\omega]}$, without loss of
generality $A\subseteq {}^\omega\lambda$ and
\[\eta E_n\nu\qquad\Leftrightarrow\qquad\eta\in A\ \&\ \nu\in A\ \&\
\eta\restriction n=\nu\restriction n.\]
Let $B^+=\{\eta\restriction n:\eta\in A\mbox{ and } n<\omega\}$. We define
$H_1(M)$ as the (Abelian) group generated by
\[\{x_\eta:\eta\in A\cup B\}\cup\{y_{\eta,n}:\eta\in A,n<\omega\}\]
freely except
\[\begin{array}{rcl}
p^{n+1}x_\eta=0 &\quad\mbox{\underbar{if}}\quad &\eta\in B, \ell g(\eta)=n\\
y_{\eta,0}=x_\eta &\quad\mbox{\underbar{if}}\quad &\eta\in A\\
py_{\eta,n+1}-y_\eta=x_{\eta\restriction n} &\quad\mbox{\underbar{if}}\quad
&\eta\in A, n<\omega\\
p^{n+1}y_{\eta, n}=0 &\quad\mbox{\underbar{if}}\quad &\eta\in B, n<\omega.
\end{array}\]
For $G\in {\frak K}^{rs(p)}$ let $H_2(G)$ be $(A,E_n)_{n<\omega}$ with:
\[A = G,\quad xE_ny\quad\underbar{iff}\quad G\models\mbox{``}p^n\mbox{ divides
}(x-y)\mbox{''.}\]
$H_2(G)\in {\frak K}^{tr[\omega]}$ as ``$G$ is separable" implies $(\forall
x)(x \ne 0\ \Rightarrow\ (\exists n)[x\notin p^nG])$. Clearly clauses
(a), (b) of
Definition \ref{9.1}(2) hold. As for clause (c), assume $(A,E_n)_{n<\omega}
\in {\frak K}^{tr[\omega]}$. As only the isomorphism type counts without loss
of generality $A\subseteq {}^\omega\lambda$. Let $B=\{\eta\restriction n:n<
\omega:\eta\in A\}$ and $G=H_1((A,E_n)_{n<\omega})$ be as above. Suppose that
$f$ embeds $G$ into some $G^*\in {\frak K}^{rs(p)}$, and let $(A^*,E^*_n)_{n<
\omega}$ be $H_2(G^*)$. We should prove that $(A,E_n)_{n<\omega}$ is
embeddable into $(A^*,E^*_n)$.\\
Let $f^*:A\longrightarrow A^*$ be $f^*(\eta)=x_\eta\in A^*$. Clearly $f^*$ is
one to one from $A$ to $A^*$; if $\eta E_n \nu$ then $\eta\restriction n=\nu
\restriction n$ hence $G \models p^n \restriction (x_\eta-x_\nu)$ hence
$(A^*,A^*_n)_{n<\omega}\models\eta E^*_n\nu$. \hfill$\square_{\ref{9.3}}$
\begin{Remark}
\label{9.3A}
In \ref{9.3}(4) we can prove ${\frak K}^{tr[\omega]}_{\bar\lambda}\le{\frak
K}^{rs(p)}_{\bar\lambda}$.
\end{Remark}
\begin{Theorem}
\label{9.4}
\begin{enumerate}
\item ${\frak K}^{mt} \equiv {\frak K}^{mt(c)}$ for $c \ge 1$.
\item ${\frak K}^{mt} \equiv {\frak K}^{ms[c]}$ for $c > 1$.
\end{enumerate}
\end{Theorem}
\proof 1)\ \ Let $H_1:{\frak K}^{mt}\longrightarrow {\frak K}^{mt(c)}$ be the
identity. Let $H_2:{\frak K}^{mt(c)}\longrightarrow {\frak K}^{mt}$ be defined
as follows:\\
$H_2((A,d))=(A,d^{mt})$, where
\[\begin{array}{l}
d^{mt}(y,z)=\\
\inf\big\{\sum\limits^n_{\ell=0} d(x_\ell,x_{\ell,n}):n<\omega\ \&\
x_\ell\in A\mbox{ (for $\ell\le n$)}\ \&\ x_0=y\ \&\ x_n=z\big\}.
\end{array}\]
Now
\begin{description}
\item[$(*)_1$] $d^{mt}$ is a two-place function from $A$ to ${\Bbb
R}^{\ge 0}$,
is symmetric, $d^{mt}(x,x)=0$ and it satisfies the triangular inequality.
\end{description}
This is true even on ${\frak K}^{mt(c)}$, but here also
\begin{description}
\item[$(*)_2$] $d^{mt}(x,y) = 0 \Leftrightarrow x=0$.
\end{description}
[Why? As by the Definition of ${\frak K}^{mt[c]},d^{mt}(x,y)\ge{\frac 1c}
d(x,y)$. Clearly clauses (a), (b) of \ref{9.2}(2) hold.]\\
Next,
\begin{description}
\item[$(*)_3$] If $M_1,N\in {\frak K}^{mt}$, $f$ is an embedding (for ${\frak
K}^{mt}$) of $M_1$ into $N$ then $f$ is an embedding (for ${\frak K}^{mt[c]}$)
of $H_1(M)$ into $H_1(N)$
\end{description}
[why? as $H_1(M)=M$ and $H_2(N)=N$],
\begin{description}
\item[$(*)_4$] If $M,N\in {\frak K}^{mt[c]}$, $f$ is an embedding (for
${\frak K}^{mt[c]}$) of $M$ into $N$ then $f$ is an embedding (for ${\frak
K}^{mt}$) of $H_2(M)$ into $H_1(M)$
\end{description}
[why? as $H^*_\ell$ preserves $\lim\limits_{n\to\infty} x_n=x$ and
$\lim\limits_{n\to\infty} x_n\ne x$].
So two applications of \ref{9.2} give the equivalence. \\
2)\ \ We combine $H_2$ from the proof of (1) and the proof of \ref{9.3}(3).
\hfill$\square_{\ref{9.4}}$
\begin{Definition}
\label{9.6}
\begin{enumerate}
\item If $\bigwedge\limits_n\mu_n=\aleph_0$ let
\[\hspace{-0.5cm}\begin{array}{ll}
J^{mt}=J^{mt}_{\bar\mu}=\big\{A\subseteq\prod\limits_{n<\omega}\mu_n:&
\mbox{for every } n \mbox{ large enough, } \\
\ & \mbox{ for every }\eta\in\prod\limits_{\ell
<n}\mu_\ell\\
\ &\mbox{the set }\{\eta'(n):\eta\triangleleft\eta'\in A\}\mbox{ is finite}
\big\}.
\end{array}\]
\item Let $T=\bigcup\limits_{\alpha\le\omega}\prod\limits_{n<\alpha}\mu_n$,
$(T,d^*)$ be a metric space such that
\[\prod_{\ell<n}\mu_\ell\cap\mbox{closure}\left(\bigcup_{m<n}\prod_{\ell<m}
\mu_\ell\right)=\emptyset;\]
now
\[\begin{ALIGN}
I^{mt}_{(T,d^*)}=:\big\{A\subseteq\prod\limits_{n<\omega}\mu_n:&\mbox{ for
some }n,\mbox{ the closure of } A\mbox{ (in $(T,d^*)$)}\\
&\mbox{ is disjoint to }\bigcup\limits_{m\in [n,\omega)}\prod\limits_{\ell
<m} \mu_\ell\big\}.
\end{ALIGN}\]
\item Let $H\in {\frak K}^{rs(p)}$, $\bar H=\langle H_n:n<\omega\rangle$, $H_n
\subseteq H$ pure and closed, $n<m\ \Rightarrow\ H_n\subseteq H_m$ and $\bigcup\limits_{n<\omega} H_n$ is dense in $H$. Let
\[\begin{array}{ll}
I^{rs(p)}_{H,\bar H}=:\big\{A\subseteq H:&\mbox{for some } n\mbox{ the closure
of }\langle A\rangle_H\mbox{ intersected with}\\
&\bigcup\limits_{\ell<\omega}H_\ell\mbox{ is included in }H_n\big\}.
\end{array}\]
\end{enumerate}
\end{Definition}
\begin{Proposition}
\label{9.5}
Suppose that $2^{\aleph_0}<\mu$ and $\mu^+<\lambda={\rm cf}(\lambda)<
\mu^{\aleph_0}$ and
\begin{description}
\item[$(*)_\lambda$] ${\bf U}_{J^{mt}_{\bar\mu}}(\lambda)=\lambda$ or at least
${\bf U}_{J^{mt}_{\bar\mu}}(\lambda)<\lambda^{\aleph_0}$ for some $\bar\mu=\langle
\mu_n:n<\omega\rangle$ such that $\prod\limits_{n<\omega}\mu_n<\lambda$.
\end{description}
Then ${\frak K}^{mt}_\lambda$ has no universal member.
\end{Proposition}
\begin{Proposition}
\label{9.7}
\begin{enumerate}
\item $J^{mt}$ is $\aleph_1$-based.
\item The minimal cardinality of a set which is not in the
$\sigma$-ideal generated by $J^{mt}$ is ${\frak b}$.
\item $I^{mt}_{(T,d^*)},I^{rs(p)}_{H,\bar H}$ are $\aleph_1$-based.
\item $J^{mt}$ is a particular case of $I^{mt}_{(T,d^*)}$ (i.e. for some
choice of $(T,d^*)$).
\item $I^0_{\bar \mu}$ is a particular case of $I^{rs(p)}_{H,\bar H}$.
\end{enumerate}
\end{Proposition}
\proof of \ref{9.5}. Let
$$
\begin{array}{ll}
T_\alpha=\{(\eta, \nu)\in{}^\alpha\lambda\times {}^\alpha(\omega+
1):& \mbox{ for every }n\mbox{ such that }n+1< \alpha\\
\ & \mbox{ we have
}\nu(n)< \omega\}
\end{array}
$$
and for $\alpha\le\omega$ let $T=\bigcup\limits_{\alpha\le
\omega}T_\alpha$. We define on $T$ the relation $<_T$:
\[(\eta_1,\nu_1)\le(\eta_1,\nu_2)\quad\mbox{ iff }\quad\eta_1\trianglelefteq
\eta_2\ \&\ \nu_1\triangleleft\nu_2.\]
We define a metric:\\
if $(\eta_1,\nu_1)\ne(\eta_2,\nu_2)\in T$ and $(\eta,\nu)$ is their maximal
common initial segment and $(\eta,\nu)\in T$ then necessarily $\alpha=
\ell g((\eta,\nu))<\omega$ and we let:
\begin{quotation}
\noindent if $\eta_1(\alpha)\ne\eta_2(\alpha)$ then
\[d\left((\eta_1,\nu_1),(\eta_2,\nu_2)\right)=2^{-\sum\{\nu(\ell):\ell<
\alpha\}},\]
if $\eta_1(\alpha)=\eta_2(\alpha)$ (so $\nu_1(\alpha)\ne\nu_2(\alpha)$ then
\[d\left((\eta_1,\nu_1),(\eta_2,\nu_2)\right)=2^{-\sum\{\nu(\ell):\ell<\alpha
\}}\times 2^{-\min\{\nu_1(\alpha),\nu_2(\alpha)\}}.\]
\end{quotation}
Now, for every $S\subseteq\{\delta<\lambda:{\rm cf}(\delta)=\aleph_0\}$, and $\bar
\eta=\langle\eta_\delta:\delta\in S\rangle$, $\eta_\delta\in {}^\omega
\delta$, $\eta_\delta$ increasing let $M_\eta$ be $(T,d)\restriction A_{\bar
\eta}$, where
\[A_{\bar\eta}=\bigcup_{n<\omega} T_n\cup\{(\eta_\delta,\nu):\delta\in S,\;\nu
\in {}^\omega\omega\}.\]
The rest is as in previous cases (note that $\langle(\eta\char 94\langle
\alpha \rangle,\nu\char 94\langle n\rangle):n<\omega\rangle$ converges to
$(\eta\char 94\langle\alpha\rangle,\nu\char 94\langle\omega\rangle)$
and even if $(\eta\char 94\langle \alpha\rangle, \nu\char 94\langle
n\rangle)\leq (\eta_n, \nu_n)\in T_\omega$ then $\langle(\eta_n,
\nu_n): n<\omega\rangle$ converge to $(\eta\char 94 \langle
\alpha\rangle, \nu\char 94\langle \omega\rangle)$).
\hfill$\square_{\ref{9.7}}$
\begin{Proposition}
\label{9.8}
If ${\rm IND}_{\chi'}(\langle\mu_n:n<\omega\rangle)$, then $\prod\limits_{n<\omega}
\mu_n$ is not the union of $\le\chi$ members of $I^0_{\bar\mu}$ (see
Definition \ref{5.5A} and Theorem \ref{5.5}).
\end{Proposition}
\proof Suppose that $A_\zeta=\{\sum\limits_{n<\omega}
p^nx^n_{\alpha_n}:\langle \alpha_n:n<\omega\rangle\in X_\zeta\}$ and
$\alpha_n<\mu_n$ are such that if $\sum
p^nx^n_{\alpha_n}\in A_\zeta$ then for infinitely many $n$ for every
$k<\omega$ there is $\langle \beta_n:n<\omega\rangle$,
\[(\forall\ell<k)[\alpha_\ell=\beta_\ell\ \ \Leftrightarrow\ \ \ell=n]\qquad
\mbox{ and }\qquad\sum_{n<\omega}p^nx^n_{\beta_n}\in A_\zeta\ \ \mbox{ (see
\S5).}\]
This clearly follows. \hfill$\square_{\ref{9.8}}$
\section{On Modules}
Here we present the straight generalization of the one prime case like Abelian
reduced separable $p$-groups. This will be expanded in \cite{Sh:622}
(including the proof of \ref{10.4new}).
\begin{Hypothesis}
\label{10.1}
\begin{description}
\item[(A)] $R$ is a ring, $\bar{\frak e}=\langle{\frak e}_n:n<\omega\rangle$,
${\frak e}_n$ is a definition of an additive subgroup of $R$-modules by an
existential positive formula (finitary or infinitary) decreasing with $n$, we
write ${\frak e}_n(M)$ for this additive subgroup, ${\frak e}_\omega(M)=
\bigcap\limits_n {\frak e}_n(M)$.
\item[(B)] ${\frak K}$ is the class of $R$-modules.
\item[(C)] ${\frak K}^*\subseteq {\frak K}$ is a class of $R$-modules, which
is closed under direct summand, direct limit and for which there is $M^*$, $x^*
\in M^*$, $M^*=\bigoplus\limits_{\ell\le n} M^*_\ell\oplus M^{**}_n$, $M^*_n
\in {\frak K}$, $x^*_n\in {\frak e}_n(M^*_n)\setminus {\frak e}_{n+1}(M^*)$,
$x^*-\sum\limits_{\ell<n} x^*_\ell\in {\frak e}_n(M^*)$.
\end{description}
\end{Hypothesis}
\begin{Definition}
\label{10.2}
For $M_1,M_2\in {\frak K}$, we say $h$ is a $({\frak K},\bar {\frak
e})$-homomorphism from $M_1$ to $M_2$ if it is a homomorphism and it maps $M_1
\setminus {\frak e}_\omega(M_1)$ into $M_2\setminus {\frak
e}_\omega(M_2)$;
we say $h$ is an $\bar {\frak e}$-pure homomorphism if for each $n$ it
maps $M_1\setminus {\frak e}_n(M_1)$ into $M_2\setminus {\frak
e}_n(M_2)$.
\end{Definition}
\begin{Definition}
\label{10.3}
\begin{enumerate}
\item Let $H_n \subseteq H_{n+1} \subseteq H$, $\bar H=\langle H_n:n<\omega
\rangle$, $c\ell$ is a closure operation on $H$, $c\ell$ is a function from
${\cal P}(H)$ to itself and
\[X \subseteq c \ell(X) = c \ell(c \ell(X)).\]
Define
\[I_{H,\bar H,c\ell}=\big\{A\subseteq H:\mbox{for some }k<\omega\mbox{ we have
} c\ell(A)\cap\bigcup_{n<\omega} H_n\subseteq H_k\big\}.\]
\item We can replace $\omega$ by any regular $\kappa$ (so $H=\langle H_i:i<
\kappa\rangle$).
\end{enumerate}
\end{Definition}
\begin{Claim}
\label{10.4new}
Assume $|R|+\mu^+< \lambda = {\rm cf}(\lambda)< \mu^{\aleph_0}$, then for
every $M\in {\frak K}_\lambda$ there is $N\in {\frak K}_\lambda$ with
no $\bar {\frak e}$-pure homomorphism from $N$ into $M$.
\end{Claim}
\begin{Remark}
In the interesting cases $c\ell$ has infinitary character.\\
The applications here are for $\kappa=\omega$. For the theory, $pcf$
is nicer for higher $\kappa$.
\end{Remark}
\section{Open problems}
\begin{Problem}
\begin{enumerate}
\item If $\mu^{\aleph_0}\ge\lambda$ then any $(A,d)\in {\frak K}^{mt}_\lambda$
can be embedded into some $M'\in {\frak K}^{mt}_\lambda$ with density
$\le\mu$.
\item If $\mu^{\aleph_0}\ge\lambda$ then any $(A,d)\in {\frak K}^{ms}_\lambda$
can be embedded into some $M'\in {\frak K}^{ms}_\lambda$ with density
$\le\mu$.
\end{enumerate}
\end{Problem}
\begin{Problem}
\begin{enumerate}
\item Other inclusions on ${\rm Univ}({\frak K}^x)$ or show consistency of non
inclusions (see \S9).
\item Is ${\frak K}^1\le {\frak K}^2$ the right partial order? (see \S9).
\item By forcing reduce consistency of ${\bf U}_{J_1}(\lambda)>\lambda+
2^{\aleph_0}$ to that of ${\bf U}_{J_2}(\lambda)>\lambda+2^{\aleph_0}$.
\end{enumerate}
\end{Problem}
\begin{Problem}
\begin{enumerate}
\item The cases with the weak ${\rm pcf}$ assumptions, can they be resolved in
ZFC? (the $pcf$ problems are another matter).
\item Use \cite{Sh:460}, \cite{Sh:513} to get ZFC results for large enough
cardinals.
\end{enumerate}
\end{Problem}
\begin{Problem}
If $\lambda^{\aleph_0}_n<\lambda_{n+1}$, $\mu=\sum\limits_{n<\omega}
\lambda_n$, $\lambda=\mu^+<\mu^{\aleph_0}$ can $(\lambda,\lambda,1)$ belong to
${\rm Univ}({\frak K})$? For ${\frak K}={\frak K}^{tr},{\frak K}^{rs(p)},{\frak
K}^{trf}$?
\end{Problem}
\begin{Problem}
\begin{enumerate}
\item If $\lambda=\mu^+$, $2^{<\mu}=\lambda<2^\mu$ can $(\lambda,\lambda,1)
\in {\rm Univ}({\frak K}^{\mbox{or}}=$ class of linear orders)?
\item Similarly for $\lambda=\mu^+$, $\mu$ singular, strong limit, ${\rm cf}(\mu)=
\aleph_0$, $\lambda<\mu^{\aleph_0}$.
\item Similarly for $\lambda=\mu^+$, $\mu=2^{<\mu}=\lambda^+ <2^\mu$.
\end{enumerate}
\end{Problem}
\begin{Problem}
\begin{enumerate}
\item Analyze the existence of universal member from ${\frak
K}^{rs(p)}_\lambda$, $\lambda<2^{\aleph_0}$.
\item \S4 for many cardinals, i.e. is it consistent that:
$2^{\aleph_0}> \aleph_\omega$ and for every $\lambda< 2^{\aleph_0}$
there is a universal member of ${\frak K}^{rs(p)}_\lambda$?
\end{enumerate}
\end{Problem}
\begin{Problem}
\begin{enumerate}
\item If there are $A_i\subseteq\mu$ for $i<2^{\aleph_0}$, $|A_i\cap A_j|<
\aleph_0$, $2^\mu=2^{\aleph_0}$ find forcing adding $S\subseteq [{}^\omega
\omega]^\mu$ universal for $\{(B, \vartriangleleft):
{}^{\omega>}\omega \subseteq B\subseteq {}^{\omega\geq}\omega, |B|\leq
\lambda\}$ under (level preserving) natural embedding.
\end{enumerate}
\end{Problem}
\begin{Problem}
For simple countable $T$, $\kappa=\kappa^{<\kappa}<\lambda\subseteq \kappa$
force existence of universal for $T$ in $\lambda$ still $\kappa=\kappa^{<
\kappa}$ but $2^\kappa=\chi$.
\end{Problem}
\begin{Problem}
Make \cite[\S4]{Sh:457}, \cite[\S1]{Sh:500} work for a larger class of
theories more than simple.
\end{Problem}
See on some of these problems \cite{DjSh:614}, \cite{Sh:622}.
\bibliographystyle{lit-plain}
| proofpile-arXiv_065-469 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{INTRODUCTION}
In Quantum Chromodynamics (QCD), quarks ($q$) and gluons ($g$) are coloured
objects that carry different colour charges. Quarks have a single colour
index while gluons are tensor objects carrying two colour indices. Due to
this fact, quarks and gluons differ in their relative coupling strength to
emit additional gluons, and, in consequence, jets originating from the
fragmentation of energetic quarks and gluons are expected to show differences
in their final particle multiplicities, energies and angular distributions.
The masses of quarks are also fundamental parameters of the QCD
lagrangian not predicted by the theory. The definition of the quark
masses is however not unique because quarks are not free particles and
various scenarios are possible. The perturbative pole mass, M$_q$, and the
running mass, m$_q$, of the $\overline{MS}$ scheme are among the most
currently used. At first order in $\alpha_s$
the predicted expression for an observable is not able to resolve
the mass ambiguity and, only when second or higher order terms are
included, the mass definition becomes known. At orders higher than one
the renormalization scheme used as the baseline of the calculation has
to be chosen and this contains the information about the mass
definition. Earlier calculations of the three-jet cross section in
$e^+e^-$ including mass terms already exist at O($\alpha_s$) \cite{tor,val}
and have been used to evaluate
mass effects for the $b$-quark when testing the universality of
the strong coupling constant, $\alpha_s$. They could not however be
used to evaluate the mass of the $b$-quark, m$_b$, because these
calculations are ambiguous in this parameter. Recently, expressions
at O($\alpha_s^2$), for the multi-jet production rate in $e^+e^-$
are available \cite{german} and, thus, they enable measuring
m$_b$ in case the flavour independence of $\alpha_s$ is assumed and
enough experimental precision is achieved.
There are well known existing difficulties to measure
all the above parameters in quantitative agreement with the
predictions from perturbative QCD, since partons, quarks and gluons, are
not directly observed in nature and only the stable particles, produced
after the fragmentation process, are experimentally detected. However, the
massive statistics and improved jet tagging techniques available
at LEP presently allow overcomig these difficulties by applying restrictive
selection criteria which lead to quark and gluon jet samples with
high purities. The selected data samples are almost background
free and small corrections to account for impurities are needed. A
smaller model dependence than ever is now achieved, bringing the
possibility to perform quantitative studies of quark and gluon
fragmentation according to perturbative QCD.
The analyses reported in here include more than 3 million Z$^0$ decays as
collected by DELPHI at center-of-mass energies of $\sqrt{s}\approx$M$_Z$.
In the first analysis, the ratio between the gluon jet multiplicity and
the quark jet multiplicity, $r=\langle N_g \rangle /\langle N_q \rangle$, is
presented and discussed in comparison with other detector results. In the
second study, preliminary values of errors associated to the determination
of m$_b$ at the M$_Z$ scale are given.
\section{EVENT SELECTION}
Gluon and quark jets were selected using hadronic three-jet events. Jets
were mainly reconstructed using the {\sc Durham} algorithm although the
{\sc Jade} algorithm was also used \cite{delphi_qg}, in particular,
to observe the effects due to different angular particle acceptance of
the various algorithms.
In the gluon splitting process ($g\rightarrow q\bar{q}$), the heavy quark
production is strongly suppressed \cite{gluo_qq}. Gluon jets can thus be extracted
from $q\bar{q}g$ events by applying $b$ tagging techniques. The two jets
which satisfy the experimental signatures of being initiated by $b$ quarks
are associated to the quark jets and the remaining one is, $by$ $definition$,
assigned to be the gluon jet without any further requirement. Algorithms
for tagging $b$ jets exploit the fact that the decay products of long lived
B hadrons have large impact parameters and/or contain inclusive high momentum
leptons coming from the semileptonic decays of the B hadrons. Gluon
purities of 94\% and 85\% are achieved when using these techniques,
respectively. Obviously, the quark jets belonging to these events cannot be
used to represent an unbiased quark sample. Thus the quark jets whose
properties are to be compared with the gluon jets must be selected
from other sources which in any case should preserve the same
kinematics. Two possibilities have been proposed in the
current literature. One consists in selecting symmetric three-jet event
configurations \cite{delphi_qg,opal_qg,aleph_qg} in which one (Y) or the two
(Mercedes) quark jets have similar energy to that of the gluon jet. The quark
jet purities reached are $\sim$52\% and $\sim$66\%, for Y and for Mercedes
events, respectively. In a second
solution \cite{delphi_qg,opal_qg,l3_qqgamma} radiative $q\bar{q}\gamma$ events
are selected, allowing a sample of quark jets with variable energy to be
collected. In this latter case, misidentifications of $\gamma$'s due to the
$\pi^\circ$ background and radiative $\tau^+\tau^-\gamma$ contamination give
rise to quark jet purities of $\sim$92\%. This method gives a higher purity
but unfortunately suffers from the lack of statistics.
The $b$-quark purity in the $b\bar{b}g$ sample reached in the DELPHI analyses
is $\sim$93\% and for the light $uds$-quarks is $\sim$80\%. Table
\ref{tab:events} summarizes the number of events selected and their
corresponding energy intervals.
{\small
\begin{table}[t]
\begin{center}
\caption{The three-jet event samples and their corresponding energy intervals as
used in the analysis.}
\begin{tabular}{lrc}
\hline
Event type & \# events & Jet energy range \\
\hline
$q\bar{q}\gamma$ & $\begin{array}{r} 2,237 \\ \end{array}$
& 7.5~GeV - 42.5~GeV \\
$(uds)\overline{(uds)}g$ & $\begin{array}{r} 552,645 \\ \end{array}$
& 7.5~GeV - 42.5~GeV \\
$b\bar{b}g$ & $\begin{array}{r} 104,081 \\ \end{array}$
& 7.5~GeV - 42.5~GeV \\
Y & $\begin{array}{r} 74,164 \\ \end{array}$
& 19.6~GeV - 28.8~GeV \\
Mercedes & $\begin{array}{r} 9,264 \\ \end{array}$ &
27.4~GeV - 33.4~GeV \\
\hline
\end{tabular}
\label{tab:events}
\end{center}
\end{table}
}
\section{MULTIPLICITIES OF QUARK AND GLUON JETS}
Results on the charged multiplicity of quark and gluon
jets \cite{opal_qg,aleph_qg} using symmetric Y configurations and
reconstructed with {\sc Durham} at 24 GeV gluon jet energy, give a ratio of
$r \approx 1.23 \pm 0.04 \mbox{(stat.+syst.)}$ which does not depend on the
cut-off parameter ($y_{cut}$) selected to reconstruct jets \cite{opal_qg}. It
is significantly higher than one, which indicates that quark and gluons in
fact fragment differently, but it remains far from the asymptotic lowest
order expectation of $C_F/C_A = 9/4$, suggesting that higher order
corrections and non-perturbative effects are very important to understand the
measured value. A next-to-leading order correction \cite{mueller} in MLLA
(Modified Leading Log Approximation) at O$(\sqrt{\alpha_s})$ already
lowers the prediction towards $r$ values slightly below two and exhibits a
small energy dependence due to the running of $\alpha_s$. However this is
still insufficient to explain the value of $r$ determined by the experiments.
Solutions based on the Monte Carlo method give a better
approximation \cite{delphi_qg}. The parton shower option of the {\sc Jetset}
generator \cite{jetset} which uses the Altarelli-Parisi splitting functions
for the evolution of the parton shower reduces the theoretical
prediction \cite{delphi_qg} for $r$. At parton level, at 24 GeV jet energy,
the expected value is $\sim$1.4 and it is further reduced to $\sim$1.3 if the
value of $r$ is computed after the fragmentation process. In both cases there
is a clear dependence of $r$ with the jet energy \cite{delphi_qg} which can be
parametrized using straight lines with slopes of ${\Delta r}/{\Delta E} =
(+90 \pm 3\mbox{(stat.)})\cdot 10^{-4} \ \mbox{GeV}^{-1}$ at parton level and
${\Delta r}/{\Delta E} = (+76 \pm 2\mbox{(stat.)})\cdot 10^{-4}\
\mbox{GeV}^{-1}$ after fragmentation. The absolute value of $r$ predicted at
parton level is however largely affected by the choice on the $Q_0$ parameter
(cut-off at which the parton evolution stops) but has negligible influence on
its relative variation with the energy, i.e., the slope. The DELPHI analysis
uses symmetric and non-symmetric three-jet event configurations with quark
and gluon jets of variable energy, allowing thus all these properties and
predictions to be tested. A value of $r=1.23\pm0.03\ \mbox{(stat.+syst.)}$ is
measured corresponding to an average jet energy of $\sim$27 GeV. The energy
dependence of $r$ is also suggested at 4$\sigma$ significance level, with a
fitted slope of ${\Delta r}/{\Delta E} = (+104\pm 25\mbox{(stat.+syst.)})\cdot
10^{-4}\ \mbox{GeV}^{-1}$.
{\small
\begin{figure}[hbt]
\begin{center}
\mbox{\epsfig{file=mont_mult.eps,height=11.cm,width=7.5cm}}
\end{center}
\caption{ (a) Mean charged multiplicity of quark and gluon jets and
(b) multiplicity ratio $r$ as a function of the jet energy}
\label{fig:proc_mult}
\end{figure}
}
In a recent review \cite{bruseles} all published data from various
experiments \cite{delphi_qg,opal_qg,aleph_qg,hrs_qg,sld_qg} were used to
perform a general study of $r$ as a function of the jet energy. At
present, more data can be
added to this comparison. These are the new analysed DELPHI data sample
presented above and
the most recent measurements of $r$ performed by CLEO \cite{cleo_qg} and
OPAL \cite{opal_hem} at 4-7 GeV and 39 GeV average jet energies,
respectively. The updated new DELPHI analysis incorporates two times
more statistics than the previous analysis \cite{delphi_qg}, therefore, significantly
reduces the statistical errors. The analysis from CLEO compares the charged particle
multiplicity in $\Upsilon(1S) \rightarrow gg\gamma$ decays
to that observed in $e^+e^- \rightarrow q\overline{q}\gamma$ just in the
continuum. This study does not rely on the Monte Carlo simulation to associate
the final hadrons to the initial partons and can consequently be fairly considered as
being model independent. The obtained value is
$r=1.04\pm0.05\ \mbox{(stat.+syst.)}$. The OPAL analysis uses a new
technique \cite{gary} which selects gluon jets at $\sim$39 GeV by
dividing the events into two hemispheres. While one of these hemispheres
is required to contain two tagged quark jets, the other is left untouched
being regarded as the gluon jet. The result from OPAL, expressed for only light
$uds$-quarks, is $r_{uds}=1.55\pm0.07\ \mbox{(stat.+syst.)}$. As
it can be observed in figure \ref{fig:proc_mult}.b all these data agree
with the predicted energy behaviour of \cite{delphi_qg,fodor,bruseles} when
the correction to the quark multiplicity to account for the same flavour
composition is applied. In our case it is: 56\% $uds$'s, 33\% $c$'s
and 11\% $b$'s. The OPAL number considering this quark mixture becomes
$r=1.48\pm0.07\ \mbox{(stat.+syst.)}$.
All these results thus give evidence for an energy dependence
of $r$. The measured increase is
\[
\frac{\Delta r}{\Delta E} =
(+110 \pm 13\ \mbox{(stat.+syst.)})\cdot10^{-4}\ \mbox{GeV}^{-1},
\]
representing a $\sim$8$\sigma$ effect.
The measured value of $r$ remains systematically lower than the {\sc Jetset}
prediction over the whole energy range, having an average value of
\[
r= 1.23\pm0.01\ (stat.) \pm0.03\ (syst.),
\]
which corresponds to an average energy of $\sim$23 GeV. This ratio can be further
expressed as
\[
r_{uds}= 1.30 \pm0.01\ (stat.) \pm0.04\ (syst.),
\]
if $r$ is computed only for the light $uds$-quarks, extracting the
$b$ and $c$ quark contribution to the quark jet multiplicity.
The absolute value of $r$ depends on the reconstruction jet algorithm. For
both the {\sc Jade} and {\sc Cone} schemes different results are obtained
w.r.t. the {\sc Durham} scheme \cite{delphi_qg,opal_qg}. This is due to the
combined effect of the different sensitivity of the various jet
reconstruction algorithms to soft particles at large angles and of the
expected different angular and energy spectra of the emitted soft gluons in
the quark and gluon jets. A precise deconvolution of both effects is, at
present, impossible \cite{ruso}. This jet algorithm dependence of $r$
becomes however less apparent as the jet energy increases. The results from
OPAL \cite{opal_hem}, $r=1.48\pm0.07\ \mbox{(stat.+syst.)}$ and those from
DELPHI \cite{delphi_qg} at $\sim$40 GeV presented in this conference,
$r=1.43\pm0.07\ \mbox{(stat.+syst.)}$ for {\sc Durham} and
$r=1.52\pm0.11\ \mbox{(stat.+syst.)}$ for {\sc Jade}, agree within errors
for the various methods and algorithms used. For the low energy
interval, the {\sc Jade} and {\sc Durham} jet algorithms give a different
description of the gluon jet properties \cite{delphi_qg}, although the
{\sc Durham} algorithm is in better agreement to
those, $model$ $independent$, results obtained by CLEO. Hence, the
{\sc Durham} jet algorithm seems to be better suited to decribe the
intermediate energy region than the {\sc Jade} algorithm is.
The interpretation of these results in combination with those obtained by
OPAL \cite{opal_bg} and ALEPH \cite{aleph_bg} restrict the validity of the
statement that gluon and $b$-quark jets have similar properties to the jet
energy interval around 24 GeV and cannot be applied to the whole
jet energy spectrum.
\section{GLUON RADIATION IN $b$-QUARKS}
For many observable quantities at LEP energies, $\sqrt{s} \raisebox{0.5ex}{$>$}\hspace{-1.7ex}\raisebox{-0.5ex}{$\sim$}$M$_Z$,
quark mass effects usually appear in terms proportional to
m$_q^2/$M$_Z^2$. This represents a $\sim$3\raisebox{0.5ex}{\tiny $0$}$\!/$\raisebox{-0.3ex}{\tiny $\! 00$}{\normalsize \hspace{1ex}} correction
for m$_q$=m$_b$ which in most of the cases can be savely
neglected. This argument, for instance is true for the total
hadronic cross section \cite{val} but cannot be applied for the
differential multi-jet cross sections that depend on
the jet-resolution parameter, $y_c$. The reason being the
new scale, $E_c = M_Z \sqrt{y_c}$, introduced in the analysis
by the new variable which enhances the mass effects in the form
$m_b^2/E_c^2 = (m_b^2/M_Z^2)/y_c$. At $\sqrt{s} \approx$M$_Z$
the three-jet production rate for $b$-quarks is in fact suppressed
by a factor $\sim5-10$\% w.r.t that of light
quarks \cite{tor,val,delphi_ab}. This difference can then be
expressed as a function of m$_b$ \cite{val} and, therefore, used to
measure its value.
\begin{figure}[ht]
\begin{center}
\mbox{\epsfig{file=systematics.eps,width=7.5cm} }
\end{center}
\caption{Relative systematics uncertainties in $R_3^{bd}$ due to fragmentation}
\label{fig:systematics}
\end{figure}
The experimental observation of such effects is however difficult
and delicate because the effect is after all small and furthermore the
correct theoretical framework to resolve the mass definition
ambiguities is needed. This means that the observable has to be calculated
including mass effects at O($\alpha_s^2$). For this purpose a recent
calculation \cite{german} of the ratio of the normalized
three-jet cross sections between $b$-quarks and light $uds$-quarks
\[
R_3^{bd} \equiv \frac{\Gamma_{3j}^{Z^0\rightarrow b\bar{b}g}(y_c)/
\Gamma_{tot}^{Z^0\rightarrow b\bar{b}}}
{\Gamma_{3j}^{Z^0\rightarrow d\bar{d}g}(y_c)/
\Gamma_{tot}^{Z^0\rightarrow d\bar{d}}}
\label{eq:r3bd}
\label{rtheta}
\]
has been performed
\begin{figure}[ht]
\begin{center}
\mbox{\epsfig{file=restes.eps,width=7.5cm} }
\end{center}
\caption{$R_3^{bd}$ distribution}
\label{fig:r3bd}
\end{figure}
The normalization in $R_3^{bd}$ to the total decay rates is introduced to
cancel possible weak corrections depending on the top quark mass \cite{top}
and the ratio of the three-jet cross sections between $b$ and light
$uds$-quarks minimizes uncertainties due to the hadronization process. In
figure \ref{fig:systematics} the dependence of these uncertainties w.r.t the
$y_{cut}$ is shown and seen not to exceed 3\raisebox{0.5ex}{\tiny $0$}$\!/$\raisebox{-0.3ex}{\tiny $\! 00$}{\normalsize \hspace{1ex}} for
large enough values of $y_{cut}$. The $R_3^{bd}$ distribution corrected for detector and
fragmentation effects is also displayed in figure \ref{fig:r3bd}. The solid curves
drawn in the figure are the theoretical O($\alpha_s$) prediction in steps of 1
GeV. The values of m$_b$ used to produce these curves are meaningless since
they correspond to a calculation at O($\alpha_s$). They can nevertheless be used to
evaluate the experimental precision assuming the difference between
the theoretical curves remains similar to that at O($\alpha_s^2$). As can be
observed the experimental error corresponds then to approximately 300 MeV for
reasonably high values of $y_{cut}$.
| proofpile-arXiv_065-470 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
Topological field theories offer an intriguing possibility to combine
ideas from physics and mathematics. They are quantum field theories
with no physical degrees of freedom and their properties are fully
determined by the global structure of the manifold they are defined
on. A remarkable feature is that for many topological theories, like
the Donaldson theory \cite{Wit-TQFT} and Chern-Simons (CS) theory,
the expectation values of the
observables are topological invariants.
The Chern-Simons theory provides a
three-dimensional interpretation of the theory of knots: the
correlators of its observables, Wilson loops, are related to the
Jones polynomials of knot theory \cite{Wit-Jones}. Another important
application of CS theory is $2+1$ dimensional gravity.
CS action with Poincar\'e group as the gauge
group is the Einstein-Hilbert action \cite{2+1}, giving a gauge
theory interpretation of gravity in $2+1$ dimensions.
However, the
Chern-Simons theory is defined only in three dimensions. Its
generalization to arbitrary dimensions \cite{KaRo-arb,Horo-ex} are
called BF-models or antisymmetric tensor models. They, like the
CS theory, describe the moduli space of flat connections
and their observables are related to the linking and intersection
numbers of manifolds. The supersymmetric BF-theories (SBF) were
introduced in \cite{Wit-topgra} as a supersymmetric version of $2+1$
dimensional topological gravity. There it was also shown that the
partition function of three dimensional SBF computes a topological
invariant, the Casson invariant. Generalizations of SBF to other
dimensions were considered in \cite{BBT} and \cite{Wal-alg,Horo-ex}.
In this article we study supersymmetric BF models. We are particularly
interested in finding new observables and possible topological
invariants for $3d$ SBF-theories, besides the partition function. By
formulating the theory in superspace a large set
of observables, including previously unknown ones, can be derived form
the superspace curvature. In \cite{BiRa-JGeom} a vector-like
supersymmetry
similar to that found in ordinary BF-models and Chern-Simons theory
\cite{DGS-3d,BRT-ren}
was constructed for the SBF-models. In particular, the hierarchy of
observables constructed from the supercurvature can be derived from
one initial observable with the help of the vector
supersymmetry. Using the superspace formulation we extend this
construction to include also the anti-BRST and anti-vector
supersymmetries, in addition to the usual BRST and
vector supersymmetries.
This article is organized
as follows: in section 2 we will introduce the model and write
it in the superspace. It turns out that in the superspace formalism
many features of
the CS theory can be generalized directly to SBF-theory. In
section 3 we derive the set of observables and discuss their relation
to topological invariants. In section 4 we generalize the vector
supersymmetry to SBF and show how it can be used to construct
of new observables.
\section{Supersymmetric BF-theories}
\la{SBF}
The classical action or non-supersymmetric BF-models in $d$
dimensions is \begin{equation} S_0 = \int \d^dx\, B_n^0 F_A, \nonumber \end{equation} where $B_n^0$
is a $n=d-2$ form (with ghost number zero) and $F_A$ is the curvature
two form $F_A = \d A + \frac{1}{2} [A,A]$. In addition to the normal
Yang-Mills gauge symmetry $ A \to A + \d_A \omega_0$ the action is
invariant under the transformation $ B_n \to B_n+ \d_A\omega_{n-1}$
caused by the Bianchi identity. In dimensions higher than three this
symmetry is reducible:
$$
\omega_{n-1} \to \d_A\omega_{n-2} \quad\hbox{\rm etc. }
$$
and additional ghost fields are needed in order to fix the gauge
according to the Batalin-Vilkovisky procedure.
In three dimensions the BF theory is closely related to the
Chern-Simons theory: CS-theory for the tangent group $TG \simeq (G,
\underline{g})$ is equivalent to the BF-theory for $G$
\cite{BBRT-TQFT}. In $TG$ the Chern-Simons connection one-form
splits into two parts $A$ and $B$, the basic fields of the BF-theory.
This makes it possible to construct the classical action of BF-theories, find
the BRST-transformations and fix the gauge easily by studying the CS
theory for the tangent group.
For the supersymmetric extension of three-dimensional BF-model the
situation is quite similar --- the action and many properties of the
theory can be expressed in terms of super CS-theory. This is done
elegantly by formulating the theory in superspace with two
anticommuting Grassmannian coordinates $\theta$, $\bar\theta$ in
addition to the normal space time coordinates $x_\mu$. Here we will
mainly concentrate in the three-dimensional case but with slight
modifications the method is suited for SBF-models in other
dimensions.
The integration over the Grassmannian variables is normalized as
\begin{equation}
\int \d\bar\theta\d\theta
\left\{\begin{array}{c}
1 \\
\theta \\
\bar\theta \\
\theta\bar\theta
\end{array}\right\} =
\left\{\begin{array}{c}
0\\
0\\
0\\
1
\end{array}\right\}.
\end{equation}
If the coordinates $\theta$ and $\bar \theta$are associated with
ghost numbers $-1$ and 1 the superspace connection one-form $\hat{\cal A}$
in $3+2$ dimensions is written\footnote{
Note that we will use graded differential forms $X^q_p$ with ordinary
form degree $p$ and ghost number $q$. Two graded forms satisfy $X^q_p
Y^r_s = (-1)^{(q+p)(r+s)}Y^r_sX^q_p $. All the commutators should also
be considered graded.}
\begin{equation}
\hat{{\cal A}} = \hat{ A}^0_\mu\d x^\mu + \hat{A}^1_\theta \d\theta +
\hat{A}^{-1}_{\bar\theta}\d\bar\theta
\la{suconn}
\end{equation}
where
the superfields $\hat A^0_\mu\d x^\mu ,\ \hat A^1_\theta $ and
$\hat A^{-1}_{\bar\theta}$ can be further expanded as:
\begin{eqnarray}
\hat A^0_\mu &=& A_\mu -\theta\psi_\mu + \bar\theta\chi_\mu +
\theta\bar\theta B_\mu\nonumber\\
\hat A^1_\theta &=& c -\theta\phi + \bar\theta\rho +
\theta\bar\theta\eta\nonumber\\
\hat A^{-1}_{\bar\theta} &=& \bar\eta -\theta\bar\rho - \bar\theta\bar\phi
+ \theta\bar\theta \bar c.\nonumber
\end{eqnarray}
The components
can be identified with the fields of three dimensional super
BF-theory: $\psi^1_\mu$ and $\chi^{-1}_\mu$ are the superpartners of
the connection $A^0_\mu$ and field $B^0_\mu$, while $ \rho^0_0,
\bar\rho^0_0$ and $\phi^2_0, \bar\phi^{-2}_0$ are their corresponding
ghosts and antighosts.
With these definitions the classical action of the SBF-model can be
written as
the action of the super-CS theory:
\begin{equation} S_{cl} = \int \d^3x (BF_A -
\chi\d_A\psi)= \frac{1}{2}\int\d^3x \d^2\theta (\hat{\cal A} \hat\d\hat{\cal A} +
\frac{2}{3} \hat{\cal A}[\hat{\cal A},\hat{\cal A}] ) .
\la{SBFcl}
\end{equation}
To obtain the quantum action one has to fix the gauge symmetry $\hat
A^0 \to \hat A^0 + \d_{\hat A^0} \omega$ by adding to the action a
BRST-exact gauge fixing term.
The BRST transformations of the fields can be derived from the
superspace curvature two-form
using a method similar to that of
\cite{BiRa-JGeom,Wal-alg,MaiNie,HiNiTi,unicon,BiRaTh-red} for
Donaldson theory and
Witten type topological theories. But because of the $N=2$ superspace
with two anticommuting coordinates of opposite ghost numbers we can
extend this method to include also the anti-BRST symmetry. We
define the superspace curvature as
\begin{equation}
\hat{\cal F} = (\d x^\mu \partial_\mu + \d\theta \delta +\d\bar\theta
\bar\delta )
\hat {\cal A} + \frac{1}{2} [\hat{\cal A}, \hat {\cal A} ]
\la{scurv}
\end{equation}
and impose the `horizontality condition''
\begin{equation}
\hat{\cal F} \equiv \hat F_{\mu\nu}\d x^\mu \d x^\nu -
(\d\theta\partial_\theta + \d\bar\theta\partial_{\bar\theta})\hat {\cal A} ,
\la{restrict}
\end{equation}
which truncates the curvature to the physical part independent of
$\d\theta,\ \d\bar\theta$ (and consequently of the ghost fields),
and identifies the BRST-operator $\delta$
with $\partial_\theta$ and $\bar\delta$ with
$\partial_{\bar\theta}$.
This gives the BRST-transformations for the component fields:
\begin{equation}
\begin{array}{llll}
\delta A &=-\d_A c +\psi \quad
&\delta B &= -\d_A \eta - [c,B] + [\phi,\chi]+
[\psi,\rho]\cr
\delta c &= -\frac{1}{2} [c,c] +\phi \quad
&\delta \eta &=-[c,\eta] +[\phi,\rho] \cr
\delta \psi &= -\d_A \phi - [c,\psi] \quad
&\delta \chi &= -\d_A\rho - [c,\chi] +B\cr
\delta \phi &= -[c,\phi] \quad
&\delta \rho &= -[c,\rho] +\eta
\end{array}
\la{BRST1}
\end{equation}
which have to be supplemented with the transformations of the
anti-ghosts and Lagrange multipliers $\lambda_0^0, b_0^0,
\beta_0^{-1}, \sigma_0^1$ for the gauge fixing conditions of the
fields $A,B,\psi$ and $\chi$. The Lagrange multipliers
can be combined into a superfield
\begin{equation}
\hat\Lambda_0^0 = \lambda -\theta \sigma
-\bar\theta\beta +\theta\bar\theta b .
\la{lambda}
\end{equation}
The simplest choice for the BRST transformations would be
$$
\delta \hat{\cal A}^{-1}_0 = -\hat\Lambda, \qquad\delta \hat\Lambda=0
$$
but with suitable field redefinitions
these can be put into a form which will be more convenient later on:
\begin{equation}
\begin{array}{llll}
\delta \bar c &= -b \quad
&\delta \bar\eta&= -\lambda -[c,\bar\eta] + \bar\rho\\
\delta b &= 0\quad
&\delta\lambda &= -[c,\lambda] -[\phi,\bar\eta]
-\sigma \\
\delta \bar\phi &= \beta - \bar c \quad
&\delta \bar\rho&= \sigma -[c,\bar\rho] \\
\delta \beta &= -b \quad
&\delta \sigma &= -[c,\sigma] +[\phi,\bar\rho].
\end{array}
\la{BRST2}
\end{equation}
The gauge fixing part of the supersymmetric
action is chosen to be
\begin{eqnarray}
S_{gf} &=& \int \d^2\theta\,
\delta(\hat A^{-1}_{\bar\theta} \d * \hat A^0 )\cr
&=&\int \d^3x \,(-b \d *A -\lambda \d *B
+ \beta\d *\psi + \sigma\d *\chi
+ \bar c \d *\d_A c +\bar \eta \d *\d_A \eta \la{SBFgf}\\
&+& \bar \phi\d *\d_A \phi + \bar \rho \d *\d_A \rho
- \bar\eta [\d c, *B] + \bar\eta \d [\phi, *\chi]
- \bar\eta \d [\rho, *\psi]+ \bar\rho [\d c, *\chi]
+\bar\phi \d [c, *\psi])
\nonumber
\end{eqnarray}
Note also that unlike in the ordinary BF-model the classical action
\nr{SBFcl}
is now BRST-exact
\[
S_{cl} = \int \d^3x\, \delta(\chi F_A ).
\]
This shows that the supersymmetric BF model is a Witten type
topological theory with a $\delta$-exact action, whereas the ordinary
non-abelian BF models are Schwartz type theories \cite{BBRT-TQFT}.
The gauge fixing term $S_{gf}= \int \d^2\theta\,
\delta(\hat A^{-1}_{\bar\theta} \d * \hat A^0)$ is formally
similar to that of Chern-Simons theory quantized in Landau gauge $\d
*{\cal A} =0$:
\[
S^{CS}_{gf}= \int \d^3 x (\delta \bar {\cal C} \d * {\cal A})
\]
In CS theory the BRST and anti-BRST operators are related by
transformation obtained by integrating the quantum action by parts
\cite{BiRa-vecSUSY}
\[
S^{CS}_q = \int \d^3 x ({\cal A} \d{\cal A} +
\frac{2}{3} {\cal A}[{\cal A},{\cal A}] - \Lambda \d * {\cal A} - \bar {\cal C} \d *\d_{\cal A} {\cal C}).
\]
The
integrated action is equivalent to the original action after a change of
fields which leaves the connection ${\cal A}$ unchanged but takes the
ghosts ${\cal C}$ to the antighosts $\bar{\cal C}$ and $\bar{\cal C}$ to $-{\cal C}$. The
Lagrange multiplier field $\Lambda$ transforms as $\Lambda \to \Lambda
- [{\cal C}, \bar{\cal C}]$. This transformation of the fields maps $\delta$ to
$\bar\delta$ .
For the super-CS and consequently for the three dimensional
SBF-theory the situation is again analogous. Integrating the gauge
fixed quantum action $S_{q} = S_{cl}+ S_{gf}$
(\ref{SBFcl},\ref{SBFgf}) by parts we find the superspace version of
the transformation which relates BRST- and anti-BRST operators. In
superspace language the transformation rules can be expressed
compactly by demanding that under the ``conjugation'' of the Grassmann
variables
\begin{equation}
\theta \to \bar \theta,\qquad \bar\theta \to - \theta
\la{conjugation}
\end{equation}
the total superspace connection $\hat{\cal A}$ stays the same while the
operators change as $\delta\to\bar\delta,\ \bar\delta\to
-\delta$. The transformations for the Lagrange multipliers are somewhat
more complicated
\begin{equation}
\begin{array}{llll}
\lambda &\to \lambda + [c, \bar\eta]\qquad
& b &\to b -[c, \bar c]-[\eta , \bar \eta ]
-[\rho , \bar\rho ]-[\phi ,\bar\phi] \cr
\sigma &\to \beta + [c, \bar\phi] \qquad &\beta &\to -\sigma
+[\phi,\bar\eta].
\la{trans}
\end{array}
\nonumber
\end{equation}
For BF-theories in dimensions other than three the situation is
slightly more complicated because the $A$ and $B$
fields cannot be combined
into one connection. In $d$ dimensions $B$ is a $d-2$ form
and additional fields will be needed to take care of the reducibility.
It is however possible to use truncated fields and write the
components of $\hat{\cal A}$ as
\begin{eqnarray}
\hat A_\mu^0 &=& A_\mu - \theta \psi_\mu \cr
\hat A_\theta^1 &=& c - \theta \phi \cr
\hat A_{\bar\theta}^{-1} &=& -\bar\theta\bar\phi + \theta\bar\theta \bar
c
\la{trunc}
\end{eqnarray}
and similarly for the $d-2$ superform $\hat{\cal B}$, which now contains in
addition to $B$, $\chi$ and their ghosts also the whole tower of
ghosts for ghosts from the Batalin-Vilkovisky gauge fixing. The
curvature of the $B$ sector is defined as
\begin{equation} \hat{\cal R} =(\d x^\mu
\partial_\mu + \d\theta\delta )\hat{\cal B} + [\hat{\cal A} , \hat{\cal B} ].
\end{equation}
It
satisfies a Bianchi identity, and again after imposing the
horizontality condition similar to \nr{restrict} it reproduces the
correct nilpotent BRST-transformations. Since the $A$ and $B$ sectors
do not appear symmetrically in the action there exists no partial
integration symmetry and thus no anti-BRST operator $\bar\delta$.
\section{Observables}
\la{Obs}
In order to establish that the observables of the theory are indeed
topological invariants it must be checked that they are BRST-closed,
their expectation values do not depend on variations of the metric
and, if they are integrals of some local functionals, that their
BRST-cohomology depends only on homology class of the integration
contour. The partition function of the three dimensional SBF-model
$$
Z_{3d} = \int e^{iS_q}.
$$
obviously satisfies all the requirements and can be shown to equal
the Casson invariant of the manifold \cite{Wit-topgra,BlaTho-Casson}.
We will now derive a set of other observables for $3d$ SBF form the
superspace curvature \nr{scurv} and see if they too could correspond
to topological invariants.
The Bianchi identity
\begin{equation}
(\d x^\mu \partial_\mu +\d\theta\delta + \d\bar\theta\bar\delta)\hat{\cal F}
+ [\hat{\cal A},\hat{\cal F}] =0
\nonumber
\end{equation}
guarantees that the powers of $\hat{\cal F}$ obey
\begin{equation}
(\d x^\mu \partial_\mu +\d\theta\delta + \d\bar\theta\bar\delta )
\hbox{\rm Tr\ } {\hat {\cal F}}^n =0.
\la{trcurv}
\end{equation}
The simplest one is the superspace 4-form ${\hat{\cal F}}^2$. It can be
expanded in powers of $\d\theta$ and $\d\bar\theta$:
\begin{equation}
\frac{1}{2} \hbox{\rm Tr\ } {\hat{\cal F}}^2 = \sum_{i,j; i+j\le 4}
W^{i,j} {\d\theta}^i{\d\bar\theta}^j .
\nonumber
\end{equation}
Equation \nr{trcurv} gives
\begin{equation}
\d W^{i,j} + \delta W^{i-1,j} + \bar\delta W^{i,j-1} = 0.
\la{cohom}
\end{equation}
When $j=0$ the integrals of the $(4-i)$-form $W^{i,0}$ over a
$(4-i)$ cocycle $\gamma$ are BRST-closed:
\begin{equation}
\int_\gamma \d W^{i,0} + \delta \int_\gamma W^{i-1,0} =
\int_{\partial\gamma} W^{i,0} + \delta \int_\gamma W^{i-1,0} =
\delta\int_\gamma W^{i-1,0} =0. \nonumber
\end{equation}
Because of \nr{cohom} the BRST-cohomology of $\int W$ depends only on the
homology class of $\gamma$, making the vacuum expectation values and
correlation functions of $\int W$ good candidates for topological
invariants.
Note that because of the symmetry of the three-dimensional action
(\ref{SBFcl},\ref{SBFgf}) under the
partial integration transformation the expectation values of any
observable $\O $ and its ``conjugate''
$\overline{\O}$
are the same. This can be seen by making a change of variables (with a unit
Jacobian) in the path integral taking all the fields to their
conjugates and using the invariance of the action. The condition
$\delta \O =0$ changes under this transformation to $\bar\delta
\overline{ \O} =0$. Therefore objects that are either $\delta$- or
$\bar\delta$-closed qualify as observables of $3d$ SBF. In particular,
we can thus identify $\overline{W}^{i,j} = (-1)^j W^{j,i}$.
The expansion of ${\hat{\cal F}}^2$ gives using \nr{restrict}
\begin{eqnarray}
W^{00} &=& \frac{1}{2} F^2 \cr
W^{10} &=& \psi F - \bar\theta (BF - \chi \d_A \psi)\cr
W^{20} &=& \frac{1}{2} \psi^2 +\phi F + \theta (\phi\d_A \psi)
- \bar\theta (\psi B + \phi\d_A\chi + F\eta)\cr
&+& \theta\bar\theta (
\phi \d_A B - \phi [\psi,\chi] + \d_A \psi \eta) \cr
W^{30} &=& \psi\phi -\bar\theta (\phi B + \psi\eta) \cr
W^{40} &=& \frac{1}{2} \phi^2 -\bar\theta (\phi\eta)
\la{W}
\end{eqnarray}
{}from which we can extract 11 observables. The previously unknown ones
are the $\theta$- and
$\theta\bar\theta$-components of $W^{20}$. They are a particular to
three-dimensional theories and unlike the others
which, or rather their generalizations involving
all the Batalin-Vilkovisky ghosts, can be obtained from the truncated
supercurvatures $\hat{\cal F}$ and $ \hat{\cal R}$ of $A$ and $B$-sectors in
all dimensions. Nevertheless, the $\theta$ component of $W^{20}$
seems to be BRST-closed also in higher dimensions:
the ghosts for ghosts and other fields appear only in the
transformations for the $B$-sector.
Interestingly, some of the
observables above are formally the same as for Donaldson theory. This is no
surprise since the BRST-structure of Donaldson theory is similar to
that of the $A$-sector of the SBF. In fact, the SBF can be thought as
reduction of the Donaldson theory to three dimensions
\cite{BiRaTh-red,MaiNie}.
As characteristic for the Witten type topological theories the
expectation values of observables
\begin{equation}
<\O> = \int [dX] \O e^{i/g^2 S_q}
\nonumber
\end{equation}
are independent of the coupling $g^2$. The integral can be calculated
in the $g^2 \to 0$ limit where it localizes to
the classical equations of motion
\begin{equation}
F_A=0, \qquad \d_A \psi =0, \qquad \d_A B - [\psi,\chi]=0,\qquad
\d_A\chi =0
\la{EQM}
\end{equation}
{\it i.e.\, }it is now calculated over the moduli space of flat
connections ${\cal M}$. In the limit $g^2 \to 0$ the fields in \nr{W} are
replaced by their classical values \nr{EQM}. Then the non-vanishing
observables are
\begin{equation}
\begin{array}{llllll}
\omega_0^4 &= \frac{1}{2}\phi^2\quad &\omega_1^3 &= \int\psi\phi \quad
&\omega_2^2 &= \frac{1}{2}\int \psi^2\\
\omega_0^3 &= \phi\eta \quad &\omega_1^2 &=\int \psi\eta +
B\phi\quad
&\omega_2^1 &= \int\psi B.
\end{array}
\nonumber
\end{equation}
To evaluate the expectation values one has to take care of the zero
modes of the fermions. Especially in dimensions higher than two the
zero-modes of the other fields complicate matters considerably. We
will not perform the calculations here but refer to
\cite{BBRT-TQFT} and references therein for
discussions on topological invariants of the Donaldson theory. The
considerations there are quite similar to those for the observables
$\omega_0^4,\ \omega_1^3,\ \omega_2^2$ of the $A$-sector of SBF. The
invariant corresponding to $\omega^2_2$ has been evaluated in
\cite{BBRT-TQFT} for $2d$ BF and found to be the symplectic volume of
the moduli space. Its products with $\omega^4_0$ produce linking and
intersection numbers of moduli spaces.
\section{Vector supersymmetry and the tower of observables}
\la{VecSUSY}
A peculiar feature of Chern-Simons and BF-theories is a vector-like
supersymmetry of the action \cite{BRT-ren}. It gives rise to new Ward
identities which have been utilized in proving the theories to be
finite, renormalizable and free of anomalies \cite{vecSUSY-1,
vecSUSY-2}. However, this supersymmetry is valid only in flat
space. One might argue though that because the theories are
topological their physical quantities are not dependent on the metric
of the manifold. In any case, the vector supersymmetry has been
established as a common feature of many topological theories
\cite{BiRa-JGeom} and a
useful tool, not only in the study of renormalization and related
topics but also in finding new observables.
The vector supersymmetry for non-supersymmetric CS theory quantized
in Landau gauge is generated by operator $s$ with ghost number and
form degree 1:
\begin{equation}
\begin{array}{llll}
s {\cal A} &= *\d{\cal C} \qquad &s {\cal C} &= 0 \nonumber\\
s \bar {\cal C} &= {\cal A} \qquad &s \Lambda &= -\delta {\cal A},
\end{array}
\la{vectorSUSY}
\end{equation}
or written in component form
\[
s = s_\alpha\d x^\alpha \qquad
*\d {\cal C} =-\epsilon_{\mu\alpha\beta} \partial^\beta{\cal C} \d x^\mu\d
x^\alpha.
\]
Using the partial integration for the CS theory one can obtain
the anti-supersymmetry $\bar s$:
\begin{equation}
\begin{array}{llll}
\bar s {\cal A} &= *\d\bar{\cal C} \qquad & \bar s \bar{\cal C} &= 0 \nonumber\\
\bar s {\cal C} &= -{\cal A} \qquad & \bar s \Lambda &= - \bar\delta {\cal A} -
[{\cal A},\bar {\cal C}]
\end{array}
\la{avectorSUSY}
\end{equation}
The anti\-commutation relations of the operators $\delta,\bar\delta$
and $s, \bar s$ are
\begin{eqnarray}
[s_\alpha , s_\beta ]&=&[\bar s_\alpha , \bar s_\beta ]=
[\delta, \bar\delta ] = [s_\alpha ,\bar s_\beta ]
= [\delta , s_\alpha ] = [\bar\delta, \bar s_\alpha ] =0
\la{commu}
\\
{} [\delta, \bar s_\alpha ] &=& -[\bar\delta , s_\alpha ]
= \partial_\alpha +
\hbox{\rm \, terms vanishing modulo the equations of motion}.
\nonumber
\end{eqnarray}
Together with the BRST-operators $\delta$ and $\bar\delta$ the
operators $s$ and $\bar s$ can be combined to form a generator of
$N=2$ supersymmetry algebra \cite{DGS-3d,BiRa-vecSUSY,BiRa-JGeom}. The
vector
supersymmetries can be formulated also for the non-supersymmetric
BF-theories. In dimensions other than three there exists no vector
supersymmetry $s$ but the $\bar s$ operator can still be constructed
\cite{vecSUSY-1}.
The anti-vector supersymmetry can be generalized to the
supersymmetric BF-theory in arbitrary dimensions. In $3d$ it can be
derived easily using \nr{avectorSUSY} for the superfields $\hat A^0,
\hat A^1_\theta, \hat A^{-1}_{\bar\theta}, \hat\Lambda$:
\begin{equation}
\begin{array}{llllll}
\bar s_\alpha A_\mu
&=&-\epsilon_{\mu\alpha\beta}\partial^\beta\bar\eta\quad
&\bar s_\alpha B
&=& -\epsilon_{\mu\alpha\beta}\partial^\beta\bar c \cr
\bar s_\alpha c &=& A_\alpha\quad
&\bar s_\alpha \eta &=& B_\alpha\cr
\bar s_\alpha \bar c &=&0 \quad
&\bar s_\alpha \bar\eta &=&0 \cr
\bar s_\alpha b &=& -\partial_\alpha \bar c\quad
&\bar s_\alpha \lambda &=& D_\alpha \bar\eta\cr
\bar s_\alpha \psi_\mu
&=& \epsilon_{\mu\alpha\beta}\partial^\beta\bar\rho \quad
&\bar s_\alpha \chi
&=&\epsilon_{\mu\alpha\beta}\partial^\beta\bar\phi \cr
\bar s_\alpha \phi &=& -\psi_\alpha\quad
&\bar s_\alpha \rho &=& -\chi_\alpha \cr
\bar s_\alpha \bar \phi &=&0 \quad
&\bar s_\alpha \bar\rho &=&0 \cr
\bar s_\alpha \beta &=& \partial_\alpha \bar \phi\quad
&\bar s_\alpha \sigma &=& D_\alpha \bar \rho
\end{array}
\la{aSUSY}
\end{equation}
This is a symmetry of the quantum action (\ref{SBFcl},\ref{SBFgf}) and
satisfies the anticommutation relations \nr{commu} with the
BRST-operator (\ref{BRST1},\ref{BRST2}). The analysis done on the
renormalization, finiteness and anomalies of ordinary BF-theories
using vector supersymmetry can thus be applied directly to the
supersymmetric BF-theories.
It is interesting to note that the vector supersymmetry of the SBF can
be useful also in constructing new observables (see also
\cite{BiRa-JGeom} for a slightly different approach). Whenever there exists
a BRST-closed object $\omega$ also $\bar s\omega$ is BRST-closed as
a result of the anticommutation relations \nr{commu}. So in principle
it is possible to find an observable, like $\omega_0^4$ and
$\omega_0^3$, and apply $\bar s_\alpha$ successively to get new
ones. Also, since
\begin{equation}
\omega_0^4 = \frac{1}{2}\delta (c\phi -\frac{1}{6}c[c,c]),\qquad
\omega_0^3 = \frac{1}{2}\delta ( \phi \rho +c\eta -\frac{1}{2} \rho [c,c] )=
\delta (\phi\rho)
\nonumber
\end{equation}
all observables obtained by acting with $\bar s$ are in fact
BRST-exact --- modulo equations of motion and surface terms. This is
valid only locally and does not mean that the observables should be
trivial.
{}From \nr{W} we see that modifying slightly the anti-supersymmetry
transformations for $\psi$ and $B$ in \nr{aSUSY} as
\[
\bar s B = *\d \bar c -2\d_A \chi, \qquad \bar s\psi = -*\d\bar\rho
-2 F
\]
and leaving the others intact the anti-supersymmetry still remains a
symmetry of the action. By denoting the metric independent part of
the modified $\bar s$ operator by $\bar v$:
\begin{equation}
\begin{array}{llllll}
\bar v B &= -2\d_A\chi \quad &\bar v\eta &= -B\quad &\bar v\rho &= -\chi \cr
\bar v\psi&= -2F \quad &\bar v c &= -A\quad &\bar v\phi &= -\psi
\end{array}
\la{mods}
\end{equation}
and applying successively $\frac{1}{k!}(-\bar v)^k$ to $\frac{1}{2}\phi^2$ and
$\phi\eta$
it is possible to derive all the observables in \nr{W},
except the $\theta$ and $\theta\bar\theta$ components of $W^{02}$.
The symmetry $\bar v$
acts as a vertical (in the direction of the form
degree) transformation along the components $W^{ij}$ of the
supersurvature \nr{scurv}.
It is easily seen that also a horizontal (ghost number direction)
transformation $\bar h$ can be defined:
\begin{equation}
\begin{array}{llll}
\bar h A &=\chi \quad &\bar h c &= -\rho\cr
\bar h \psi &= -B \quad &\bar h\phi &= -\eta\cr
\bar h\bar\eta &=-\bar\phi\quad &\bar h\bar\rho &= -\bar c .
\end{array}
\la{horizontal}
\end{equation}
This is a symmetry of the action and commutes with the
BRST-operator. It thus allows us to construct all possible
observables starting from the element $\frac{1}{2} \phi^2$ of highest ghost
number and lowest form degree --- again excluding the $\theta$ and
$\theta\bar\theta$ components of $W^{20}$.
The vertical transformation has geometrical interpretation as the
equivariant derivative of BRST-model acting on the curvature of the
universal bundle over the space of gauge connections
\cite{BiRa-uni}, which can be identified with the
supercurvature $\hat F$. Therefore the vertical transformation can be
defined for all Witten type topological theories. The horizontal
transformation $\bar h$ which can be constructed only in three
dimensions is in fact part of the anti-BRST operator
$-\delta$: only those terms that are not composites of
fields and do not contain Lagrange multipliers are included.
Acting on
$\phi^2$ with the total
vector supersymmetry transformation $\bar s$ instead of the vertical
transformation we get even a larger set
of observables. In addition to theose present in \nr{W} these include
observables that depend explicitely on the metric.
Since we already know that the observables in \nr{W}
are BRST-closed also the metric dependent ones should be closed
separately. Moreover, it can be shown that the metric
variations of these observables can be written as BRST-exact terms,
a necessary requirement for the observables to be topological
invariants \cite{BBRT-TQFT}. So we can conclude that the expectation
values of these observables are of topological nature.
\section{Conclusions}
We have studied three dimensional supersymmetric BF-theories using
superspace formalism. This proved out to be a powerful method for
studying the properties of the theory and especially for finding new
symmetries and observables. The superspace curvature gives rise to a
hierarchy of observables, which could be derived starting from one initial
observable using the two transformations we constructed.
The transformations have a geometrical interpretation as
vertical and horizontal transformations acting on the components of
the supercurvature, and can be identified as parts of
more general symmetries of the action, the vector supersymmetry
and anti-BRST symmetry.
\subsection*{Acknowledgements}
We thank prof. A. J. Niemi for useful comments on the manuscript and
the referee for pointing out \cite{BiRa-JGeom}.
\baselineskip 0.4cm
| proofpile-arXiv_065-471 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
Some charge-transfer salts are understood as strongly correlated
one-dimensional electron systems for half-filled bands
(Mott-Hubbard insulators)
(Farges 1994); (Alc\'{a}cer, Brau, and Farges 1994).
The extended Hubbard model for interacting electrons
on a Peierls-distorted
chain at half-filling is considered appropriate for these
materials (Mazumdar and Dixit 1986);
(Fritsch and Ducasse 1991); (Mila 1995).
There are only few studies of the optical, i.~e., finite
frequency properties
of correlated electron systems
since their calculation
is a formidable task
(Kohn 1964); (Maldague 1977);
(Lyo and Galinar 1977); (Lyo 1978); (Galinar 1979);
(Campbell, Gammel, and Loh 1988); (Mahan 1990);
(Shastry and Sutherland 1990);
(Stafford, Millis, and Shastry 1991);
(Fye, Martins, Scalapino, Wagner, and Hanke 1992);
(Stafford and Millis 1993).
In this second article on the optical absorption
of electrons in half-filled Peierls-distorted chains
we present a detailed analysis of the optical
absorption in the limit of strong correlations and for a half-filled band
where the charge and spin dynamics decouple.
We find that the physics of the half-filled Hubbard model at strong
correlations is determined by the upper and lower Hubbard band
for the charges which are {\em parallel\/} bands.
This essential feature and its significant consequences
have been missed in earlier analytical and numerical investigations.
In this work we include a finite lattice
dimerization and nearest-neighbor interaction between the electrons,
i.~e., we analyze the extended dimerized Hubbard model at
strong correlations.
The paper is organized as follows.
In section~\ref{Hamilts} we address the
Hubbard model at strong coupling from which we derive the Harris-Lange model
which determines the motion of the charge degrees of freedom.
In the ground state there are no free charges but only singly occupied
lattice sites.
The exact spectrum and eigenstates of the Harris-Lange model
are presented in section~\ref{exactsolution} for the translational
invariant and the dimerized case.
The results can be interpreted in terms of two {\em parallel\/}
Hubbard bands for the charges which are eventually split
into Peierls subbands. Optical absorption can now formally
be treated as if we had {\em independent\/} (spinless) Fermions.
Unfortunately, this simple band structure interpretation is blurred by
the spin degrees of freedom which enter the expressions
for the optical absorption in terms
of a very complicated ground state expectation value.
In section~\ref{optabsHL} we treat the case of the Harris-Lange
model where all
spin configurations are equally possible ground states.
This corresponds to the Hubbard model at temperatures
large compared to the spin exchange energy.
It allows the calculation of
the optical absorption even in
the presence of a Peierls distortion and a nearest-neighbor
interaction between the charges.
A summary and outlook closes our presentation.
Some details of the calculations are left to the appendices.
\section{Strongly correlated Mott-Hubbard insulators}
\label{Hamilts}
\subsection{Tight-binding electrons on a Peierls-distorted chain}
\label{Peidistrot}
For narrow-band materials
the electron transfer is limited to nearest neighbors only.
In standard notation of second quantization
the Hamiltonian for electrons in the tight-binding approximation reads
\begin{equation}
\hat{T}(\delta)= - t \sum_{l=1,\sigma}^{L}
\left(1+ (-1)^l\delta\right)
\left(
\hat{c}_{l,\sigma}^+ \hat{c}_{l+1,\sigma}^{\phantom{+}}
+ \hat{c}_{l+1,\sigma}^+ \hat{c}_{l,\sigma}^{\phantom{+}}
\right)
\end{equation}
where $\delta$ describes the effect of
bond-length alternation on the electron transfer amplitudes.
As usual the Hamiltonian can be diagonalized in momentum space.
We apply periodic boundary conditions, and introduce the
Fourier transformed electron operators
as $\hat{c}_{k,\sigma}^+=\sqrt{1/L} \sum_{l=1}^L
\exp(ikla) \hat{c}_{l,\sigma}^+$ for the~$L$ momenta~$k=2\pi m/(La)$,
$m=-(L/2),\ldots (L/2)-1$.
We may thus write
\begin{equation}
\hat{T}(\delta) = \sum_{|k|\leq \pi/(2a),\sigma}
\epsilon(k)\left(
\hat{c}_{k,\sigma}^+\hat{c}_{k,\sigma}^{\phantom{+}}
-
\hat{c}_{k+\pi/a,\sigma}^+\hat{c}_{k+\pi/a,\sigma}^{\phantom{+}}
\right)
-i \Delta(k)
\left( \hat{c}_{k+\pi/a,\sigma}^+\hat{c}_{k,\sigma}^{\phantom{+}}
-
\hat{c}_{k,\sigma}^+\hat{c}_{k+\pi/a,\sigma}^{\phantom{+}}\right)
\label{Tink}
\end{equation}
with the dispersion relation~$\epsilon(k)$ and hybridization
function~$\Delta(k)$ defined as
\begin{mathletters}
\begin{eqnarray}
\epsilon(k) &=& -2t\cos(ka)\label{gl3a} \\[3pt]
\Delta(k) &=& 2t\delta\sin(ka)\; .
\end{eqnarray}
\end{mathletters}%
The Hamiltonian can easily be diagonalized in $k$-space.
The result is (Gebhard, Bott, Scheidler, Thomas, and Koch I 1996)
\begin{equation}
\hat{T}(\delta)
= \sum_{|k|\leq \pi/(2a),\sigma} E(k) (\hat{a}_{k,\sigma,+}^+
\hat{a}_{k,\sigma,+}^{\phantom{+}}
- \hat{a}_{k,\sigma,-}^+ \hat{a}_{k,\sigma,-}^{\phantom{+}})
\; .
\label{Tdiapeierls}
\end{equation}
Here, $\pm E(k)$ is the dispersion relation for the upper~($+$)
and lower~($-$) Peierls band,
\begin{equation}
E(k) = \sqrt{\epsilon(k)^2 + \Delta(k)^2} \; .
\label{peierlsen}
\end{equation}
The new Fermion quasi-particle operators $\hat{a}_{k,\sigma,\pm}^{+}$
for these two bands are related to the original electron operators by
\begin{mathletters}
\label{mixamp}
\begin{eqnarray}
\hat{a}_{k,\sigma,-}^{\phantom{+}} &=&
\alpha_k \hat{c}_{k,\sigma}^{\phantom{+}} + i \beta_k
\hat{c}_{k+\pi,\sigma}^{\phantom{+}} \\[6pt]
\hat{a}_{k,\sigma,+}^{\phantom{+}} &=&
\beta_k \hat{c}_{k,\sigma}^{\phantom{+}} -i \alpha_k
\hat{c}_{k+\pi,\sigma}^{\phantom{+}}
\end{eqnarray}
\end{mathletters}%
with
\begin{mathletters}
\label{alphabeta}
\begin{eqnarray}
\alpha_k &=& \sqrt{ \frac{1}{2} \left( 1 - \frac{\epsilon(k)}{E(k)}
\right) }\\[6pt]
\beta_k &=& - \sqrt{ \frac{1}{2} \left( 1 + \frac{\epsilon(k)}{E(k)}
\right)} {\rm sgn}\left(\Delta(k)\right)
\end{eqnarray}
\end{mathletters}%
Details of the upper transformation and the optical absorption
of this model are presented in (Gebhard {\em et al.} I 1996).
\subsection{Hubbard model}
\label{Hubbard-Model}
The only spinful interacting electron model that can be solved exactly
for all values of the interaction strength is the
Hubbard model in one dimension
(Hubbard 1963);
(Gebhard and Ruckenstein 1992);
(E\ss ler and Korepin 1994);
(Gebhard, Girndt, and Ruckenstein 1994);
(Bares and Gebhard 1995).
For narrow-band materials
the electron transfer is limited to nearest neighbors only,
and the interaction is supposed to be described by the
purely local (Hub\-bard-)interaction of strength~$U$,
\begin{eqnarray}
\hat{H}_{\rm Hubbard}&=& \hat{T} +U \hat{D}\nonumber\\[6pt]
\label{Hubb-Model}
\hat{D}&=&\sum_{l} \hat{D}_{l}
= \sum_l \hat{n}_{l,\uparrow}\hat{n}_{l,\downarrow} \; ,
\end{eqnarray}
where
$\hat{n}_{l,\sigma}=\hat{c}_{l,\sigma}^+\hat{c}_{l,\sigma}^{\phantom{+}}$
is the local density of $\sigma$-electrons, and
$\hat{T}=\hat{T}(\delta=0)$.
The model~(\ref{Hubb-Model}) poses a very difficult many-body problem.
Its spectrum and, in particular,
its elementary excitations
can be obtained from the Bethe Ansatz solution
(Lieb and Wu 1968); (Shastry, Jha, and Singh 1985); (Andrei 1995).
Its low-energy properties including the DC-conductivity,
$\sigma_{\rm DC}={\rm Re}\{\sigma(\omega=0)\}$,
can explicitly be obtained from the corresponding $g$-ology
Hamiltonian (Schulz 1990); (Schulz 1991) or from conformal field theory
(Frahm and Korepin 1990); (Frahm and Korepin 1991);
(Kawakami and Yang 1990); (Kawakami and Yang 1991).
The Hubbard model
describes a (correlated) metal for all~$U>0$ for less than half-filling.
Unfortunately, the Bethe Ansatz solution
does not allow the direct calculation of transport properties at
finite frequencies.
At half-filling the one-dimensional Hubbard model
describes a Mott-insulator which implies
that~$\sigma_{\rm DC}=0$ for all~$U>0$.
The density of states for charge excitations
displays two bands,
the upper and lower Hubbard band, separated by the Mott-Hubbard gap.
This gap is defined as the jump in the chemical
potential at half filling,
\begin{mathletters}
\begin{eqnarray}
\Delta_{\rm MH}&=&\mu^+(N=L)-\mu^-(N=L) \nonumber \\[6pt]
&=& \left[ E(N=L+1)-E(N=L)\right] - \left[ E(N=L)-E(N=L-1)\right] \quad.
\end{eqnarray}
As shown by Ovchinnicov (Ovchinnicov 1969) the Mott-Hubbard gap can be
obtained from the Lieb-Wu solution (Lieb {\em et al.} 1968) in the form
\begin{eqnarray}
\Delta_{\rm MH} &=& \frac{16t}{U} \int_{1}^{\infty}
\frac{dy \sqrt{y^2-1}}{\sinh(2\pi t y/U)} \\[9pt]
&=& \left\{ \begin{array}{ccr}
(2W/\pi) \sqrt{4U/W} \exp(-\pi W/(2U)) & \hbox{for} & U \ll W=4t \\[6pt]
U - W + \ln(2)W^2/(2U) + {\cal O}(W^3/U^2) & \hbox{for} & U \gg W=4t
\end{array}
\right. \quad .
\end{eqnarray}
\end{mathletters}%
It is obvious that optical absorption is only
possible if $\omega \geq \Delta^{\rm MH}$.
It is further seen that the upper and lower Hubbard band
are well separated for $U\gg W$.
One might expect that the optical absorption for large interactions, $U \gg W$,
and high temperatures, $k_{\rm B}T\gg J={\cal O}(W^2/U)$,
shows the signature of a broad band-to-band
transition for $U-W\leq \omega\leq U+W$
(units $\hbar\equiv 1$),
similar to the Peierls insulator (Gebhard {\em et al.} I 1996).
Such considerations seemed to be supported by
analytical (Lyo {\em et al.} 1977); (Lyo 1978); (Galinar 1979)
and numerical calculations (Campbell, Gammel, and Loh 1989).
Below we will calculate~$\sigma(\omega>0)$ in the limit $U\gg W$,
and show that the linear absorption is actually
dominated by a singular contribution at~$\omega=U$ because the
upper and lower Hubbard band are in fact {\em parallel\/} bands.
The situation changes for $k_{\rm B}T\ll J$ which we will consider in
(Gebhard, Bott, Scheidler, Thomas, and Koch III 1996).
\subsection{Harris-Lange model}
In the following we will address
the limit $W/U\to 0$
where matters considerably simplify since the
charge and spin degrees of freedom completely decouple
(Ogata and Shiba 1990); (Parola and Sorella 1990).
For example, for less than half-filling, $N \leq L$, and $U=\infty$
the eigenenergies become those
of a Fermi gas of $N_h=L-N$ holes with dispersion~$\epsilon(k)$,
and each energy level is $2^N$-fold degenerate in the thermodynamical
limit (Beni, Pincus, and Holstein 1973); (Klein 1973);
(Ogata {\em et al.} 1990); (Parola {\em et al.} 1990).
To facilitate the discussion of the strong coupling
limit we map the Hubbard model onto a problem for which the number
of double occupancies is {\em conserved}.
For a large on-site Coulomb repulsion $W/U \to 0$
it is natural to start with a spectral decomposition
of operators into those which solely act in the upper or lower Hubbard band,
and to perturbatively eliminate those parts in $\hat{H}_{\rm Hubbard}$
which couple the two bands.
For the Hubbard model this has first been achieved by Harris and
Lange (Harris and Lange 1967); (van Dongen 1994),
and the resulting effective Hamiltonian
to lowest order in~$W/U$ will thus be called the ``Harris-Lange'' model.
It offers several advantages,
both for analytical and numerical calculations.
To carry out the spectral decomposition
we start from the case $t=0$.
The Fermi annihilation operator can be split into a part
which destroys an electron on
a single occupied site and does not
change the energy of the state, and another part
which destroys an electron on a double occupied site
and thus decreases the energy by $U$,
\begin{equation}
\hat{c}_{l,\sigma} = \hat{n}_{l,-\sigma} \hat{c}_{l,\sigma} +
(1 - \hat{n}_{l,-\sigma}) \hat{c}_{l,\sigma} \quad .
\end{equation}
The corresponding creation operator can be treated accordingly.
If we now turn on the hopping of electrons ($t\neq 0$)
we may split the kinetic energy operator into
\begin{mathletters}
\label{HLM}
\begin{eqnarray}
\hat{T} &=& \hat{T}_{\rm LHB} + \hat{T}_{\rm UHB}
+ \hat{T}^+ + \hat{T}^- \\[6pt]
\hat{T}_{\rm LHB} &=& (-t) \sum_{l,\sigma} \left(1-\hat{n}_{l,-\sigma}\right)
\left(
\hat{c}_{l,\sigma}^+ \hat{c}_{l+1,\sigma}^{\phantom{+}} +
\hat{c}_{l+1,\sigma}^+ \hat{c}_{l,\sigma}^{\phantom{+}}
\right)
\left( 1-\hat{n}_{l+1,-\sigma}\right) \\[6pt]
\hat{T}_{\rm UHB} &=& (-t) \sum_{l,\sigma} \hat{n}_{l,-\sigma}
\left(
\hat{c}_{l,\sigma}^+ \hat{c}_{l+1,\sigma}^{\phantom{+}} +
\hat{c}_{l+1,\sigma}^+ \hat{c}_{l,\sigma}^{\phantom{+}}
\right)
\hat{n}_{l+1,-\sigma}\\[6pt]
\hat{T}^+ &=& (-t) \sum_{l,\sigma} \left[
\hat{n}_{l,-\sigma}
\hat{c}_{l,\sigma}^+ \hat{c}_{l+1,\sigma}^{\phantom{+}}
\left( 1-\hat{n}_{l+1,-\sigma} \right)
+
\hat{n}_{l+1,-\sigma}
\hat{c}_{l+1,\sigma}^+ \hat{c}_{l,\sigma}^{\phantom{+}}
\left( 1- \hat{n}_{l,-\sigma}\right) \right]
\\[6pt]
\hat{T}^- &=& \left(\hat{T}^+\right)^+ \; .
\end{eqnarray}
\end{mathletters}%
The operator~$\hat{T}_{\rm LHB}$ for the lower
Hubbard band describes the hopping of holes while
doubly occupied sites can move in the
upper Hubbard band via $\hat{T}_{\rm UHB}$.
Their number is conserved by both hopping processes.
These two bands will constitute the basis for our approach.
The operator~$\hat{T}^+$ ($\hat{T}^-$) increases (decreases)
the number of double occupancies by one.
Similar to the Foldy-Wouthuysen transformation for the Dirac
equation (Bjorken and Drell 1964)
we apply a canonical transformation that eliminates
the operators~$\hat{T}^{\pm}$ to a given order in~$t/U$,
\begin{equation}
\hat{c}_{l,\sigma} = e^{i\hat{S}(\bar{c})} \bar{c}_{l,\sigma}
e^{-i\hat{S}(\bar{c})}
\end{equation}
with $\left(\hat{S}(\bar{c})\right)^+=\hat{S}(\bar{c})$.
As shown by Harris and Lange
(Harris {\em et al.} 1967); (van Dongen 1994) the operator
to lowest order in~$t/U$ reads
\begin{equation}
\hat{S}(\bar{c}) = \frac{it}{U}
\left( \hat{T}^{\bar{c},+} -\hat{T}^{\bar{c},-} \right)
\end{equation}
which can easily be verified since $\left[\hat{D},\hat{T}^{\pm}\right]_{-}=
\pm \hat{T}^{\pm}$.
The transformed Hamilton operator in the new Fermions becomes
the Harris-Lange model
\begin{equation}
\hat{H}_{\rm HL}^{\bar{c}} =
\hat{T}_{\rm LHB}^{\bar{c}} + \hat{T}_{\rm UHB}^{\bar{c}}
+ U \hat{D}^{\bar{c}} \quad ,
\end{equation}
if we neglect all correction terms to order~$t/U$ and higher.
The energies obtained from the Harris-Lange model thus agree with
those of the Hubbard model to order~$t (t/U)^{-1}$ and~$t (t/U)^0$.
For all other physical operators which do not contain a factor of~$U/t$
we may replace
\begin{equation}
\hat{c}_{l,\sigma} = \bar{c}_{l,\sigma}
\quad .
\end{equation}
because the error is only of order $\left(t/U\right)$.
In the following we will thus make no distinction between the operators
$\hat{c}_{l,\sigma}$ and $\bar{c}_{l,\sigma}$ to lowest order in~$t/U$.
The Hamiltonian has the following symmetry. The particle-hole
transformation
\begin{mathletters}
\label{parthole}
\begin{eqnarray}
{\cal T}_{\rm ph} \hat{c}_{l,\sigma}^+ {\cal T}_{\rm ph}^{-1}& =
& i \lambda_{\sigma}
e^{i\pi l} \hat{c}_{l,-\sigma}^{\phantom{+}}
\\[6pt]
{\cal T}_{\rm ph} \hat{c}_{k,\sigma}^+ {\cal T}_{\rm ph}^{-1}& =
& i\lambda_{\sigma}
\hat{c}_{\pi/a-k,-\sigma}^{\phantom{+}}
\label{watchout}
\end{eqnarray}
with $\lambda_{\uparrow}=-\lambda_{\downarrow}=1$
is generated with the help of
\begin{eqnarray}
{\cal T}_{\rm ph} &=& e^{i\pi/2 (\hat{C}^+ +\hat{C}^-)}=
\prod_l \left[ 1-(\hat{D}_l+\hat{H}_l)
+ i(\hat{C}_l^+ +\hat{C}_l^-)\right]
\\[6pt]
\hat{C}^+&=&\left(\hat{C}^-\right)^+=\sum_l \hat{C}_l^+=
\sum_l (-1)^l \hat{c}_{l,\uparrow}^+ \hat{c}_{l,\downarrow}^+ \quad
\quad;\quad
\hat{H}_l=(1-\hat{n}_{l,\uparrow})(1-\hat{n}_{l,\downarrow})
\; .
\label{defofthecs}
\end{eqnarray}
\end{mathletters}%
The additional phase factors $i\lambda_\sigma$ are irrelevant global phases,
and can be ignored since there is always an equal number
of Fermion creation and annihilation operators of each spin species.
The operators for the motion of holes and double occupancies
are mapped into each other,
\begin{equation}
\hat{T}_{\rm UHB} \mapsto \hat{T}_{\rm LHB} \qquad
\hat{T}_{\rm LHB} \mapsto \hat{T}_{\rm UHB} \; .
\end{equation}
Furthermore,
$\left[ \hat{T}_{\rm UHB}+\hat{T}_{\rm LHB}, \hat{C}^{\pm}\right]_-=0$.
This symmetry allows for an exact solution of the model
since there is essentially
no difference in the motion of double occupancies
in the upper Hubbard band and holes in the lower Hubbard band.
The discussion above is readily generalized to the case of
dimerization in the Harris-Lange model.
The model Hamiltonian reads
\begin{equation}
\hat{H}_{\rm HL}^{\rm dim} = \hat{T}_{\rm LHB}(\delta) +
\hat{T}_{\rm UHB}(\delta) + U\hat{D}
\end{equation}
in an obvious generalization of the kinetic operators for the
upper and lower Hubbard bands.
\subsection{Optical absorption and optical conductivity}
The dielectric function~$\widetilde{\epsilon}(\omega)$
and the coefficient for the linear optical
absorption~$\widetilde{\alpha}(\omega)$ are
given by (Haug and Koch 1990)
\begin{mathletters}
\begin{eqnarray}
\widetilde{\epsilon}(\omega) &=& 1 +\frac{4\pi i \sigma(\omega)}{\omega}
\label{epssigma}\\[6pt]
\widetilde{\alpha}(\omega) &=&
\frac{4\pi {\rm Re}\{\sigma(\omega)\}}{n_b c}
\end{eqnarray}
\end{mathletters}%
where ${\rm Re}\{\ldots\}$ denotes the real part and
$n_b$ is the background refractive index.
It is supposed to be frequency independent near a resonance.
Hence, the real part of the optical conductivity
directly gives the absorption spectrum of the system.
The standard result (Maldague 1977); (Mahan 1990)
for the real part of the optical
conductivity in terms of the current-current correlation
function~$\chi(\omega)$ is
\begin{eqnarray}
{\rm Re}\{ \sigma(\omega) \} &=&\frac{{\rm Im}\{\chi(\omega)\}}{\omega}
\\[6pt]
\chi(\omega) & =& \frac{{\cal N}_{\perp}}{La}
i \int_0^{\infty} dt e^{i\omega t} \langle
\left[\hat{\jmath}(t),\hat{\jmath}\right]_- \rangle
\end{eqnarray}
where~${\cal N}_{\perp}$ is the number of chains per unit area
perpendicular to the chain direction.
The current-current correlation function can be spectrally
decomposed in terms of exact eigenstates of the system as
\begin{equation}
\chi(\omega) = \frac{{\cal N}_{\perp}}{La}
\sum_n |\langle 0 | \hat{\jmath} | n\rangle|^2
\left[ \frac{1}{\omega +(E_n-E_0) +i\gamma} -
\frac{1}{\omega -(E_n-E_0) +i\gamma} \right] \; .
\label{decomp}
\end{equation}
Here, $|0\rangle$ is the exact ground state (energy $E_0$),
$|n\rangle$ are exact excited states (energy $E_n$),
and $\left|\langle 0 | \hat{\jmath} | n\rangle\right|^2$
are the oscillator strengths for optical transitions between them.
Although $\gamma =0^+$ is infinitesimal we may introduce $\gamma>0$
as a phenomenological broadening
of the resonances at $\omega = \pm(E_n-E_0)$.
The spectral decomposition of the real part of the optical conductivity
reads
\begin{equation}
{\rm Re}\{ \sigma(\omega) \} = \frac{{\cal N}_{\perp} \pi}{La \omega}
\sum_n\left| \langle 0 | \hat{\jmath} | n\rangle\right|^2
\left[ \delta\left( \omega
-(E_n-E_0)\right) -\delta\left( \omega +(E_n-E_0)\right) \right]
\label{speccomp}
\end{equation}
which is positive for all~$\omega$.
In the following we will always plot the dimensionless
reduced optical conductivity
\begin{equation}
\sigma_{\rm red}(\omega >0) =
\frac{\omega {\rm Re}\{\sigma(\omega>0)\} }%
{{\cal N}_{\perp}a e^2 W } \; .
\label{sigmared}
\end{equation}
Furthermore we replace the energy conservation~$\delta(x)$
by the smeared function
\begin{equation}
\widetilde{\delta}(x) = \frac{\gamma}{\pi(x^2+\gamma^2)}
\end{equation}
to include effects of phonons, and experimental
resolution. For all figures we graphically checked that the sum rules of
appendix~\ref{appsumrule} were fulfilled.
\subsection{Current operator}
As derived in (Gebhard {\em et al.} I 1996)
the current operator is given by
\begin{equation}
\hat{\jmath} = -e \sum_{l,\sigma}
ita \left(1+ (-1)^l\delta\right)\left(1+ (-1)^l\eta\right)
\left(
\hat{c}_{l+1,\sigma}^+ \hat{c}_{l,\sigma}^{\phantom{+}}
- \hat{c}_{l,\sigma}^+ \hat{c}_{l+1,\sigma}^{\phantom{+}}
\right)
\label{currenta}
\end{equation}
where $\eta=-|R_{l+1}-R_l-a|/a<0$ is the relative change of lattice distances
due to the Peierls distortion.
Note that $\delta$ and $\eta$ always have opposite sign.
The current operator can be split into two parts,
$\hat{\jmath}= \hat{\jmath}_{\rm intra}^{\rm H}
+\hat{\jmath}_{\rm inter}^{\rm H}$,
where $\hat{\jmath}_{\rm intra}^{\rm H}$ moves electrons between
neighboring sites without changing the
number of double occupancies or holes.
This (Hubbard-)intraband current does not change the
number of double occupancies. Hence it can be ignored
for the optical absorption in the Harris-Lange model
at half-filling.
The current operator between the two Hubbard bands
$\hat{\jmath}_{\rm inter}^{\rm H}$ can be written as
\begin{mathletters}
\begin{eqnarray}
\hat{\jmath}_{\rm inter}^{\rm H} &=& \hat{\jmath}_{{\rm inter},+}^{\rm H}
+ \hat{\jmath}_{{\rm inter},-}^{\rm H} \\[6pt]
\hat{\jmath}_{{\rm inter},+}^{\rm H} &=&
-(itea) \sum_{l,\sigma}\left(1+(-1)^l \delta\right) \left(1+(-1)^l \eta\right)
\nonumber \\[3pt]
&& \phantom{-(itea)\sum}
\left[
\hat{n}_{l+1,-\sigma} \hat{c}_{l+1,\sigma}^+ \hat{c}_{l,\sigma}^{\phantom{+}}
\left( 1- \hat{n}_{l,-\sigma} \right)
-
\hat{n}_{l,-\sigma} \hat{c}_{l,\sigma}^+ \hat{c}_{l+1,\sigma}^{\phantom{+}}
\left( 1- \hat{n}_{l+1,-\sigma} \right)
\right]
\\[6pt]
\hat{\jmath}_{{\rm inter},-}^{\rm H} &=&
-(itea) \sum_{l,\sigma}\left(1+(-1)^l \delta\right) \left(1+(-1)^l \eta\right)
\nonumber \\[3pt]
&& \phantom{-(itea)\sum}
\left[
\left(1- \hat{n}_{l+1,-\sigma}\right)
\hat{c}_{l+1,\sigma}^+ \hat{c}_{l,\sigma}^{\phantom{+}}
\hat{n}_{l,-\sigma}
-
\left(1 -\hat{n}_{l,-\sigma} \right)
\hat{c}_{l,\sigma}^+ \hat{c}_{l+1,\sigma}^{\phantom{+}}
\hat{n}_{l+1,-\sigma}
\right]
\end{eqnarray}
\end{mathletters}%
where~$\hat{\jmath}_{\rm inter,\pm}^{\rm H}$ create and destroy
a neighboring pair of double occupancy and hole, respectively.
Next we study the action of $\hat{\jmath}_{{\rm inter},+}^{\rm H}$
on a pair of neighboring spins in a state~$|\Psi\rangle$ in
position space. It is a sequence of
singly occupied sites ($\sigma$), holes ($\circ$), and double occupancies
($\bullet$) from site $1$ to~$L$, e.g.,
\begin{equation}
|\Psi\rangle = |\uparrow_1, \bullet_2, \circ_3,\circ_4,\downarrow_5,
\ldots\uparrow_{L-3},\bullet_{L-2},\downarrow_{L-1},\bullet_{L}\rangle
\quad .
\label{defofstate}
\end{equation}
We introduce the notations
\begin{mathletters}
\begin{eqnarray}
|\ldots,(\uparrow_l,\downarrow_{l+1}\pm \downarrow_{l},\uparrow_{l+1}),
\ldots\rangle
&=& |\ldots, \uparrow_l, \downarrow_{l+1},\ldots \rangle
\pm |\ldots,\downarrow_{l},\uparrow_{l+1},\ldots\rangle\\[3pt]
|{\rm S}_{l,l+1}=1,S^z_{l,l+1}=1\rangle&=& |\ldots,\uparrow_l,\uparrow_{l+1},
\ldots\rangle\\[3pt]
|{\rm S}_{l,l+1}=1,S^z_{l,l+1} =0\rangle &=& |\ldots,(\uparrow_l,\downarrow_{l+1}
+\downarrow_{l},\uparrow_{l+1}),\ldots\rangle\\[3pt]
|{\rm S}_{l,l+1}=1,S^z_{l,l+1}=-1\rangle &=& |
\ldots,\downarrow_l,\downarrow_{l+1},
\ldots\rangle\\[3pt]
|{\rm S}_{l,l+1}=0,S^z_{l,l+1} =0\rangle &=&
|\ldots,(\uparrow_l,\downarrow_{l+1}
-\downarrow_{l},\uparrow_{l+1}),\ldots\rangle
\end{eqnarray}
as the local spin triplet and spin singlet states.
Furthermore,
\begin{equation}
|C_{l,l+1}=1,C^z_{l,l+1}=0\rangle
= |\ldots,(\bullet_{l},\circ_{l+1} - \circ_l,\bullet_{l+1}),\ldots\rangle
\end{equation}
\end{mathletters}%
denotes the local charge
triplet state since
$\hat{C}^+|C_{l,l+1}=1,C^z_{l,l+1}=0\rangle \neq 0$.
With these definitions one finds
\begin{mathletters}
\begin{eqnarray}
\hat{\jmath}_{\rm inter,+}^{\rm H} |{\rm S}_{l,l+1}=1\rangle &=&0 \\[6pt]
\hat{\jmath}_{\rm inter,+}^{\rm H} |{\rm S}_{l,l+1}=0\rangle &=&
-itea (1+(-1)^l\delta)(1+(-1)^l\eta)(-2)
|C_{l,l+1}=1,C^z_{l,l+1}=0\rangle \; .
\end{eqnarray}
\end{mathletters}%
It is thus seen that
$\hat{\jmath}_{\rm inter}^{\rm H}$ preserves the spin of a neighboring pair
such that $\Delta S=\Delta S^z=0$ is the selection rule for the spin sector.
The selection rule for the charge sector is found as
$\Delta C=1$, $\Delta C^z=0$.
Note that the current operator does {\em not\/}
commute with~$\hat{C}^{\rm \pm}$ as defined in eq.~(\ref{defofthecs}).
Finally, the current operator is invariant against translations
by one unit cell and thus preserves the total momentum
modulo a reciprocal lattice vector ($Q=2\pi/a$ for~$\delta=0$,
$Q=\pi/a$ for~$\delta\neq 0$).
However, the current operator can
create or destroy a charge excitation with momentum~$q$,
and create or destroy a spin excitation with momentum~$-q$.
Although there is charge-spin separation in the Hubbard model
for strong coupling the current operator mixes both degrees of freedom.
This renders the calculation of the optical absorption
of the Hubbard model a very difficult problem even in the limit
of strong correlations.
\section{Exact solution of the Harris-Lange model}
\label{exactsolution}
\subsection{Translational invariant case}
The Harris-Lange model
can exactly be solved by an explicit construction of all eigenstates.
This has recently been shown by
(de Boer, Korepin, and Schadschneider 1995) and
(Schadschneider 1995) for periodic, and by (Aligia and Arrachea 1994)
for open boundary conditions.
Since optical excitations conserve total momentum we work with periodic
boundary conditions where the total momentum is a good quantum number.
The number~$N_S$ of sites with spin (singly occupied sites),
and the number~$N_C=N_d+N_h$ of sites with charge (double occupancies
and holes) are separately conserved in the Harris-Lange model.
We have $N_C+N_S=L$ lattice sites, $N=N_S+2N_d$ electrons,
and choose~$L$ to be even such that our lattice is bipartite for all~$L$.
In the sequence of singly occupied sites,
double occupancies and holes of the state~$|\Psi\rangle$
in eq.~(\ref{defofstate})
we may identify subsequences for the spins and the
charges only (independent of the position on a special site).
Additionally, the indices $l_j$ indicate the actual position
of the charges $C_j$.
The positions occupied by the
spins are then the ones left over by the charges:
\begin{mathletters}
\begin{equation}
|\Psi\rangle=
| ({l_1},{l_2}, \ldots{l_{N_{C}-1}},{l_{N_C}});
(C_{1},C_{2},\ldots C_{N_{C}-1},C_{N_C});
(S_{1},S_{2},\ldots S_{N_{S}-1},S_{N_S})\rangle
\end{equation}
where in our example
\begin{equation}
(S_{1},S_{2},\ldots S_{N_{S}-1},S_{N_S})=
(\uparrow,\downarrow,\ldots\uparrow,\downarrow)
\end{equation}
is the subsequence for the spins and
\begin{equation}
(C_{1},C_{2},C_{3},\ldots C_{N_{C}-1},C_{N_C})=
(\bullet,\circ,\circ,\ldots\bullet,\bullet)
\end{equation}
for the charges. The sequence for the positions occupied by charges is
\begin{equation}
({l_1},{l_2},{l_3},\ldots
{l_{N_C}-1},{l_{N_C}})=(2,3,4,\ldots L-2,L) \qquad .
\end{equation}
\end{mathletters}%
Since there is nearest-neighbor hopping only
both the spin and charge
subsequences are separately {\em conserved\/}
up to cyclic permutations due to the periodic boundary conditions.
To include the boundary effect we follow
(de Boer {\em et al.} 1995)
and (Schadschneider 1995)
and introduce the properly symmetrized spin and charge sequences.
To this end we define the operator for a cyclic permutation
of the spin sequence,
\begin{equation}
\hat{\cal T}_S (S_1,S_2,\ldots S_{N_S})
= (S_{N_S},S_1,\ldots S_{N_S-1}) \quad ,
\end{equation}
and, equivalently, $\hat{\cal T}_C$ for the charge sequence.
Let $K_S$ and $K_C$ be the smallest positive integers such that
$\left(\hat{\cal T}_S\right)^{K_S}$ and $\left(\hat{\cal T}_C\right)^{K_C}$
act as identity operators on a given spin and charge sequence,
respectively.
Then we define the~$K_S K_C$ states
{\arraycolsep=0pt\begin{eqnarray}
&&\sqrt{K_SK_C} |({l_1}, \ldots {l_{N_C}});
(C_1,\ldots C_{N_C})_{k_C};
(S_1,\ldots S_{N_S})_{k_S}\rangle
= \label{realspace}\\[9pt]
&& e^{\sum_l i\pi l \hat{D}_{l}}
\sum_{\nu_S=0}^{K_S-1} \sum_{\nu_C=0}^{K_C-1}
e^{ia(\nu_S k_S+\nu_C k_C)}
\hat{\cal T}_S^{\nu_S} \hat{\cal T}_C^{\nu_C}
|({l_1}, \ldots {l_{N_C}});
(C_1,\ldots C_{N_C});(S_1,\ldots S_{N_S})\rangle
\nonumber
\end{eqnarray}}%
with the momentum shifts $k_S=2\pi m_S/(K_S a)$, $m_S=0,1,\ldots (K_S-1)$,
$k_C=2\pi m_C/(K_C a)$, $m_C=0,1,\ldots (K_C-1)$.
An extra phase factor~$(-1)^{l_j}$
for each double occupancy at site~$l_j$ has been included here
through the operator
$\exp(\sum_l i\pi l \hat{D}_{l})$.
This allows to make direct contact
to the model considered by (de Boer {\em et al.} 1995);
(Schadschneider 1995), and (Aligia {\em et al.} 1994).
There the hopping amplitudes for lower and upper Hubbard band have
opposite signs.
We transform the states of eq.~(\ref{realspace})
into momentum space with respect to the charge coordinates.
The exact eigenstates can now be classified
according to their
spin and charge sequence, their momentum shifts $k_C$, $k_S$,
and~$N_C$ momenta from the set
of~$k_{j}=2\pi m_j/(L a)$, $m_j=-(L/2),\ldots (L/2)-1$.
The normalized eigenstates read
{\arraycolsep=0pt\begin{eqnarray}
&&L^{N_C/2}|k_1,\ldots k_{N_C};
(C_1,\ldots C_{N_C})_{k_C};
(S_1,\ldots S_{N_S})_{k_S}\rangle
= \label{eigenstates} \\[9pt]
&&\sum_{l_1< \ldots <l_{N_C}} \!\! \sum_{{\cal P}}(-1)^{\cal P}
\exp \! \left(ia\sum_{j=1}^{N_C} l_{{\cal P}(j)}\left(k_j+\Phi_{CS}\right) \!
\right) \!
|({l_1}, \ldots {l_{N_C}});
(C_1,\ldots C_{N_C})_{k_C};
(S_1,\ldots S_{N_S})_{k_S}\rangle
\nonumber
\end{eqnarray}}%
where the permutations ${\cal P}$ generate a simple Slater determinant
for the momenta and the positions of the
$N_C$ charges, and $\Phi_{CS}=(k_C-k_S)/L$ is an additional momentum shift
which vanishes in the thermodynamical limit.
It is straightforward
(de Boer {\em et al.} 1995); (Schadschneider 1995)
but lengthy to explicitly show that
the states in eq.~(\ref{eigenstates}) have energy and momentum
\begin{mathletters}
\begin{eqnarray}
E&=&\sum_{j=1}^{N_C} \epsilon(k_j+\Phi_{CS}) +U N_d
\label{eigenenHL}\\[3pt]
P&=& k_S +(\pi/a) (N_d-1)+ \sum_{j=1}^{N_C} (k_j+\Phi_{CS})
+ (N {\rm \ mod\ } 2) \pi/a
\qquad {\rm mod\ }2\pi/a
\label{eigenmomHL}
\end{eqnarray}
\end{mathletters}%
with $\epsilon(k)$ given by equation~(\ref{gl3a}).
The essential arguments are repeated in the appendices~\ref{appmomentum}
and~\ref{appenerg}.
The above set of eigenstates is complete.
After summing over the subspaces with different~$K_S$,
$K_C$ the number of states which become degenerate in energy
in the thermodynamical limit is given by
$2^{N_S} N_C!/(N_d!N_h!)$. The number of possible choices for the
momenta is~$L!/(N_C!(L-N_C)!)$ since the momenta are those of a
gas of spinless Fermions. Altogether one finds for an even number of
electrons~$N$
\begin{equation}
\sum_{N_d=0}^{N/2} {N_C \choose N_d} 2^{N-2N_d} {L \choose {L-N+2N_d}}
= {2L \choose N}
\end{equation}
where eq.~(10.33.5) of (Hansen 1975) and eq.~(22.2.3) of
(Abramovitz and Stegun 1970) have been used.
This exhausts the Hilbert space for fixed number of electrons~$N$.
\subsection{Dimerized Harris-Lange model}
The Harris-Lange model can also be solved in the
presence of a finite lattice distortion as long as
there is hopping between nearest neighbors only.
For the $U=\infty$ Hubbard model at less
than half filling this has already been realized
some time ago (Bernasconi, Rice, Schneider, and Str\"{a}\ss ler 1975).
The exact eigenenergies are those of spinless Fermions
on a dimerized chain.
For the Harris-Lange model
with~$N_C$ charge excitations
we choose momenta~$|k_j|\leq \pi/(2a)$ of the reduced Brillouin zone,
and one of the~$2^{N_C}$ sequences $(\tau_1,\ldots \tau_{N_C})$
with~$\tau_j=\pm 1$.
Let us introduce the operator~$\hat{\Pi}_r$ by
{\arraycolsep=0pt\begin{eqnarray}
\hat{\Pi}_r |k_1,\ldots k_r,\ldots k_{N_C}; &&
(C_1,\ldots C_{N_C})_{k_C};
(S_1,\ldots S_{N_S})_{k_S}\rangle
\nonumber \\[6pt]
&& = |k_1,\ldots k_r+\pi/a,\ldots k_{N_C};
(C_1,\ldots C_{N_C})_{k_C};
(S_1,\ldots S_{N_S})_{k_S}\rangle
\end{eqnarray}}%
and the four functions~$\xi_1^r(1)=\beta_{k_r}$,
$\xi_2^r(1)=i\alpha_{k_r}$,
$\xi_1^r(-1)=\alpha_{k_r}$, and $\xi_2^r(-1)=-i\beta_{k_r}$,
compare eq.~(\ref{alphabeta}).
The eigenstates for fixed number of charge excitations~$N_C$
can then be written as
\begin{eqnarray}
|k_1,\tau_1;\ldots &&k_{N_C},\tau_{N_C};
(C_1,\ldots C_{N_C})_{k_C};
(S_1,\ldots S_{N_S})_{k_S}\rangle
= \nonumber \\[6pt]
&&\left[
\prod_{r=1}^{N_C} \left( \xi_1^r(\tau_r) + \xi_2^r(\tau_r)\hat{\Pi}_r\right)
\right]
|k_1,\ldots k_{N_C};
(C_1,\ldots C_{N_C})_{k_C};
(S_1,\ldots S_{N_S})_{k_S}\rangle \; .
\end{eqnarray}
This state corresponds to $(N_C+\sum_r\tau_r)$ ($(N_C-\sum_r\tau_r)/2)$
spinless Fermions in the upper (lower) Peierls subband.
The corresponding energies and momenta of these states
are obviously given by
\begin{mathletters}
\begin{eqnarray}
E&=&\sum_{j=1}^{N_C} E(k_j+\Phi_{CS})\tau_j +U N_d\\[3pt]
P&=& k_S + \sum_{j=1}^{N_C} (k_j+\Phi_{CS})
\quad {\rm mod\ }\pi/a
\end{eqnarray}
\end{mathletters}%
where~$E(k)$ has been given in eq.~(\ref{peierlsen}).
\subsection{Band picture interpretation of the spectrum}
\label{bandpictureinterpretation}
\subsubsection{Translational invariant case}
The exact solution for the Harris-Lange model
can be interpreted in terms of upper and lower Hubbard bands.
To simplify the discussion we will ignore the momentum shift~$\Phi_{CS}$
in this subsection.
For $U>W$ the ground state of the half-filled band~$N=L$ has energy zero
and is $2^L$-fold degenerate, and we may choose the fully polarized
ferromagnetic state as our reference state,
$|{\rm FM}\rangle=|\uparrow,\ldots \uparrow\rangle$.
Note that this state has momentum~$\pi/a$
on a chain with an even number of sites~$L$, see eq.~(\ref{eigenmomHL}).
We may now add an electron.
We obtain all exact eigenstates for $N=L+1$ electrons, $N_d=1$, and
all spins up as
\begin{mathletters}
\begin{equation}
|k;(\bullet)_{k_C=0},(\uparrow,\ldots \uparrow)_{k_S=0}\rangle
= \hat{c}_{k,\downarrow}^+ |{\rm FM}\rangle
\label{plusone}
\end{equation}
with momentum~$P=k+\pi/a$ and energy~$E=\epsilon(k)+U$.
The state in eq.~(\ref{plusone}) is interpreted
as a {\em particle\/} at momentum~$k$ in the upper Hubbard
band which has the dispersion relation~$\epsilon(k)+U$.
The momentum~$\pi/a$ is attributed to the ferromagnetic reference state.
We may equally well take out an electron from the fully polarized state.
We obtain all exact eigenstates for $N=L-1$ electrons, $N_h=1$,
and all spins up as
\begin{equation}
|k;(\circ)_{k_C=0},(\uparrow,\ldots \uparrow)_{k_S=0}\rangle
= - \hat{c}_{\pi/a-k,\uparrow} |{\rm FM}\rangle
\label{minusone}
\end{equation}
\end{mathletters}%
with momentum~$P=-(\pi/a-k)+\pi/a$ and energy~$E=\epsilon(k)$.
Note that the states of eq.~(\ref{minusone})
and those of eq.~(\ref{plusone})
can be generated from each other by the
particle-hole transformation of eq.~(\ref{parthole}).
Their momenta differ by~$\pi/a$
since their respective numbers of double occupancies differ by one.
Since the ground state corresponds to
a completely filled lower Hubbard band
we {\em interpret\/} the state in eq.~(\ref{minusone})
as a hole in the lower Hubbard band at~$k_h=\pi/a-k$,
and the momentum~$\pi/a$ is again attributed to the ferromagnetic reference
state.
The lower Hubbard band must have the dispersion relation~$\epsilon(k)$
for {\em particles\/} because a {\em hole\/} at $k_h=\pi/a-k$
has momentum~$P=-(\pi/a-k)$ and energy~$E=-\epsilon(k_h)=
-\epsilon(-k+\pi/a)=\epsilon(k)$.
The band structure for the Harris-Lange model is
depicted in figure~\ref{HarrisLangedis}.
It displays the parallel upper and lower Hubbard bands
with band width~$W$ separated by a distance~$U$.
It is amusing that the celebrated Hubbard-I
approximation (Hubbard 1963); (Mazumdar and Soos 1981)
also gives parallel bands. Those bands, however, carry a spin index
while charge-spin separation is most essential in one dimension.
Furthermore, the width of those bands is only {\em half\/}
of the exact band width~$W$.
Our band structure picture has to be used carefully if there are
more than one double occupancy or hole. Figure~\ref{HarrisLangedis} suggests
that there are~$L$ states available {\em both\/} in the upper {\em and\/}
in the lower Hubbard band, altogether~$2L$ independent states.
However, this cannot be the case because for~$N_d=N_h=L/2$
we would have~${L \choose L/2}{L \choose L/2}$ states in the band picture
while the Hilbert space actually has only the dimension~${L \choose L/2}$.
The exact solution shows how an appropriate exclusion
principle between particles in the upper Hubbard band
and holes in the lower Hubbard band can be formulated.
For fixed spin background and fixed $k_1$, $k_2$
there are four exact eigenstates with two charges at~$k_1\neq k_2$.
They all have the kinetic energy~$T=\epsilon(k_1)+\epsilon(k_2)$.
They correspond to four different charge excitations in the band picture:
(i)~two particles at momenta~$k_1$, $k_2$ in the upper Hubbard band,
(ii)~two holes at momenta~$\pi/a-k_1$, $\pi/a-k_2$ in the
lower Hubbard band, (iii)~a particle at momentum~$k_1$
in the upper Hubbard band
and a hole at momentum~$\pi/a-k_2$ in the lower Hubbard band,
(iv)~a particle at momentum~$k_2$ in the upper Hubbard band
and a hole at momentum~$\pi/a-k_1$ in the lower Hubbard band.
The condition~$k_1\neq k_2$ is naturally
fulfilled in cases~(i) and~(ii), if we assign a fermionic character to
our particles in the upper and holes in the lower Hubbard band, respectively.
In case~(iii), however, we have to explicitly {\em demand\/}
that the momentum at which we create the hole, $k_h=\pi/a-k_2$,
fulfills~$k_1\neq k_2$, i.~e., $k_h\neq \pi/a-k_1$.
This is the same condition which results from case~(iv).
We thus see that a particle in the upper Hubbard band at momentum~$k$
actually blocks the creation of a hole in the lower Hubbard band at
momentum~$\pi/a-k$ (this is probably the simplest example of a ``statistical
interaction'', see (Haldane 1991)).
With this additional rule the counting of states
in the band picture is correct, and the band picture interpretation
gives indeed the {\em exact\/} results for the Harris-Lange model.
The effective Hamiltonian for fermionic particles in the upper
($\hat{u}_k$) and lower ($\hat{l}_k$) Hubbard band thus reads
\begin{mathletters}
\label{effHL}
\begin{eqnarray}
\hat{H}_{\rm HL}^{\rm band} &=& \hat{P}_{ul} \sum_{|k|\leq \pi/a}\left[
(U+\epsilon(k)) \hat{n}_{k}^{u} + \epsilon(k) \hat{n}_{k}^{l} \right]
\hat{P}_{ul}
\\[6pt]
\hat{P}_{ul} &=& \prod_{|k|\leq \pi/a}
\left[ 1 -\left(1- \hat{n}_{\pi/a-k}^{l}\right)
\hat{n}_{k}^{u} \right]
\end{eqnarray}
\end{mathletters}%
with $\hat{n}_{k}^{u}=\hat{u}_{k}^+ \hat{u}_{k}^{\phantom{+}}$,
$\hat{n}_{k}^{l}=\hat{l}_{k}^+ \hat{l}_{k}^{\phantom{+}}$.
The projection operators guarantee that there is no hole in the
lower Hubbard band at momentum~$\pi/a-k$,
if there is already a particle at momentum~$k$ in the upper Hubbard band.
For half-filling at zero temperature the lower Hubbard band is completely
filled.
\subsubsection{Dimerized Harris-Lange model}
The case of the dimerized Hubbard model can be treated accordingly.
The upper and lower Hubbard band now split into two Peierls subbands
with dispersion relations~$\pm E(k)$.
Formally the band structure Hamiltonian for the lower band becomes
(compare eq.~(\ref{Tink}))
\begin{equation}
\hat{T}_{\rm LHB}^{\rm band}(\delta)= \sum_{|k|\leq \pi/(2a)}
\epsilon(k)\left(
\hat{l}_{k}^+\hat{l}_{k}^{\phantom{+}}
-
\hat{l}_{k+\pi/a}^+\hat{l}_{k+\pi/a}^{\phantom{+}}
\right)
-i \Delta(k)
\left( \hat{l}_{k+\pi/a}^+\hat{l}_{k}^{\phantom{+}}
-
\hat{l}_{k}^+\hat{l}_{k+\pi/a}^{\phantom{+}}
\right) \; ,
\label{Tdimnotdiag}
\end{equation}
and a similar expression holds for the upper Hubbard band.
The band picture Hamiltonian can easily be brought into diagonal
form as in the Peierls case. The quasi-particles in the four subbands
are finally described by
\begin{eqnarray}
\hat{H}_{\rm HL}^{\rm dim,\, band} &=& \hat{P}_{u^+l^+}\hat{P}_{u^-l^-}
\sum_{|k|\leq \pi/(2a)}\biggl[
(U+E(k)) \hat{n}_{k,+}^{u} + (U-E(k)) \hat{n}_{k,-}^{u}
\nonumber \\[6pt]
&& \phantom{\hat{P}_{u^+l^+}\hat{P}_{u^-l^-}
\sum_{|k|\leq\pi/(2a)}\biggl[}
+ E(k) \hat{n}_{k,+}^{l} -E(k) \hat{n}_{k,-}^{l}
\biggr] \hat{P}_{u^+l^+}\hat{P}_{u^-l^-}
\label{bandhldim}
\\[6pt]
\hat{P}_{u^{\pm}l^{\pm}} &=& \prod_{|k|\leq\pi/(2a)} \left[ 1- \left(1-
\hat{n}_{-k,\pm}^{l}\right)\hat{n}_{k,\pm}^{u}\right] \nonumber
\end{eqnarray}
with $\hat{n}_{k,\pm}^{u}=\hat{u}_{k,\pm}^+ \hat{u}_{k,\pm}^{\phantom{+}}$,
$\hat{n}_{k,\pm}^{l}=\hat{l}_{k,\pm}^+ \hat{l}_{k,\pm}^{\phantom{+}}$
as the number operators for
the quasi-particles for the upper ($\tau=+$) and lower ($\tau=-$)
Peierls subband in the upper~($u$) and lower~($l$) Hubbard band.
The quasi-particles in each subband obey a fermionic
exclusion principle in the same Hubbard band.
In addition, a particle at momentum~$k$
in the upper Hubbard band in the upper (lower) Peierls subband blocks the
creation of a hole at momentum~$-k$ in the lower Hubbard band in
the upper (lower) Peierls subband.
There is no hole in the lower Hubbard band at momentum~$-k$
in the upper or lower Peierls subband,
if there is already a particle at momentum~$k$ in the upper Hubbard band
in the corresponding Peierls subband.
Note that the reciprocal lattice vector is
now given by~$\pi/a$.
Thereby the projection operators guarantee the proper counting
of states.
The resulting band structure is shown in figure~\ref{HueckelHarrisLangedis}.
The upper and lower Hubbard band are both Peierls split and display
the Peierls gap~$W\delta$ at the zone boundaries~$\pm \pi/(2a)$.
Note that the upper (lower) Peierls subbands
are still parallel.
\subsection{Band picture interpretation of the current operator}
\subsubsection{Translational invariant case}
According to the spectral decomposition of the current-current correlation
function, eq.~(\ref{decomp}), we need to determine the excitation
energy~$E_n-E_0$ of an exact eigenstate~$|n\rangle$
and its oscillator strength~$|\langle 0 | \hat{\jmath}|n\rangle|^2$.
The respective total momenta of these states are~$P_0$ and~$P_n$.
We are interested in optical excitations from a state with
singly occupied sites only. The excited states which can be reached from
this state have one pair of hole and double occupancy, i.~e.,
\begin{mathletters}
\begin{eqnarray}
|0\rangle &=& |(S_1,\ldots S_L)_{k_S}\rangle\\[3pt]
|n\rangle &=& |k_1,k_2; (\bullet \circ)_{k_C=0};
(S_1,\ldots S_{L-2})_{k_S^{\prime}}\rangle
\end{eqnarray}
\end{mathletters}%
where we used the fact that~$\hat{\jmath}$ creates a charge triplet
with~$k_C=0$. Note that~$(k_1,k_2)$ is the same state as~$(k_2,k_1)$.
We denote~$k_1=k+q/2$, $k_2=\pi/a-k+q/2$ since we will finally represent
the state~$|n\rangle$
by a particle in the upper Hubbard band at momentum~$k+q/2$ and a hole
in the lower Hubbard band at momentum~$k-q/2$.
Since~$\hat{\jmath}$ conserves the total momentum we already know
that~$P_0=P_n$ which implies
$k_S=q+k_S^{\prime}(L-2)/L$, see eq.~(\ref{eigenmomHL}).
Hence, $k_S^{\prime}=L(k_S-q)/(L-2)$ has to hold. Recall that~$k_S^{\prime}$
is quantized in units of~$2\pi/((L-2)a)$.
These considerations imply that the charge (spin) sector in~$|n\rangle$
carries momentum~$q$ ($-q$) relative to~$|0\rangle$.
In the thermodynamical limit the excitation energy is given by
\begin{equation}
E(k,q)=U + \epsilon(k+q/2) - \epsilon(k-q/2)= U + 4t\sin(ka)\sin(qa/2)\; .
\label{EKQ}
\end{equation}
Note that the excitation energy does not depend on the spin configuration.
For this reason it is possible to find a formally equivalent band picture
for the charge sector alone
that gives the same optical absorption as the original model.
Since~$\hat{\jmath}$ itself carries all the information on the
conservation laws
(momentum, charge, and spin quantum numbers) we may equally well
work with the (normalized) states
\begin{eqnarray}
|k+\frac{q}{2};\frac{\pi}{a}-k+\frac{q}{2}\rangle &=&
\frac{1}{L} \sum_{l_1<l_2}
\left( e^{i(k+q/2)l_1a}e^{i(\pi/a-k+q/2)l_2a}
-e^{i(k+q/2)l_2a}e^{i(\pi/a-k+q/2)l_1a} \right)
\nonumber \\
%
&& \phantom{\frac{1}{L} \sum_{l_1<l_2}}
(-1)^{l_1} |S_1^{\prime},\ldots S_{l_1-1}^{\prime},\bullet_{l_1},
S_{l_1}^{\prime},\ldots S_{l_2-2}^{\prime}, \circ_{l_2},
S_{l_2-1}^{\prime},\ldots S_{L-2}^{\prime} \rangle
\end{eqnarray}
rather than the exact eigenstates of eq.~(\ref{eigenstates}).
This will simplify the notation since we do not have to take
any summation restrictions into account.
For fixed~$(k,q)$ and fixed spin configuration~$(S_1^{\prime},\ldots
S_{L-2}^{\prime})$ we calculate
\begin{eqnarray}
\langle 0 |\hat{\jmath}_{\rm inter,-}^{\rm H}
|k+\frac{q}{2};\frac{\pi}{a}-k+\frac{q}{2}\rangle &=&
-iea\epsilon(k) e^{iqa/2} \\
&& \frac{1}{L} \sum_l e^{iqla}
\langle 0 |S_1^{\prime},\ldots S_{l-1}^{\prime},
\left( \uparrow_{l}\downarrow_{l+1}-\downarrow_{l}\uparrow_{l+1}\right),
S_{l}^{\prime},\ldots S_{L-2}^{\prime} \rangle \; .
\nonumber
\end{eqnarray}
We define the operators~$\hat{x}_q^+$ and $\hat{x}_q^{\phantom{+}}$ via their
product
\begin{eqnarray}
\hat{x}_q^+ \hat{x}_q^{\phantom{+}} &=&
\sum_{S_1^{\prime},\ldots S_{L-2}^{\prime}}
\frac{1}{L^2} \sum_{l,r} e^{iq(l-r)a}
\langle 0 | S_1^{\prime},\ldots S_{l-1}^{\prime},
\left( \uparrow_{l}\downarrow_{l+1}-\downarrow_{l}\uparrow_{l+1}\right),
S_{l}^{\prime},\ldots S_{L-2}^{\prime} \rangle \label{xqxq}\\[6pt]
&&
\phantom{\sum_{S_1^{\prime},\ldots S_{L-2}^{\prime}}
\frac{1}{L^2} \sum_{l,r} e^{iq(l-r)} }
\langle S_{L-2}^{\prime},\ldots S_{r}^{\prime},
\left( \downarrow_{r+1} \uparrow_{r}-\uparrow_{r+1}\downarrow_{r}\right),
S_{r-1}^{\prime},\ldots S_{1}^{\prime} | 0 \rangle \; . \nonumber
\end{eqnarray}
Summed over all intermediate spin configurations
the oscillator strength for fixed $(k,q)$ now becomes
\begin{equation}
\Bigl|\langle 0 |\hat{\jmath}_{\rm inter,-}^{\rm H}
|k+\frac{q}{2};\frac{\pi}{a}-k+\frac{q}{2}\rangle\Bigr|^2 =
\Bigl|-iea \epsilon(k)\Bigr|^2 \hat{x}_q^+\hat{x}_q^{\phantom{+}} \; .
\end{equation}
It is clear that we have hidden a very difficult many-body problem in
the operators~$\hat{x}_q$.
Nevertheless we are now in the position to identify the interband
current operator in the band picture.
It is given by
\begin{equation}
\hat{\jmath}_{\rm inter}^{\rm band}=
\sum_{|k|,|q|\leq \pi/a}
iea\epsilon(k) \left( \hat{u}_{k+q/2}^+\hat{l}_{k-q/2}^{\phantom{+}}
\hat{x}_{q}^{\phantom{+}} - \hat{l}_{k-q/2}^+\hat{u}_{k+q/2}^{\phantom{+}}
\hat{x}_{q}^+ \right)\; .
\label{jhlbandpicture}
\end{equation}
This operator acts in the same space as the band Hamiltonian of
section~\ref{bandpictureinterpretation}.
It is seen that the condition~$k+q/2 \neq \pi/a-(k-q/2)$ is automatically
fulfilled since~$\epsilon(\pi/(2a))=0$. Consequently,
the projection operators
in eq.~(\ref{effHL}) can again
be ignored for the case of linear optical absorption.
\subsubsection{Dimerized Harris-Lange model}
For a distorted lattice
the current operator can also modify the momentum of the state
by $\pi/a$.
Thus we address {\em four\/} possible states for fixed~$(k,q)$ from
the reduced Brillouin zone,
$|k+q/2;\pi/a-k+q/2\rangle$,
$|k+q/2;-k+q/2\rangle$,
$|\pi/a+k+q/2;\pi/a-k+q/2\rangle$, and
\hbox{$|\pi/a+k+q/2;-k+q/2\rangle$}.
The same analysis as in the previous subsection leads us to the
definition of the operators~$\hat{x}_q^{+}(\delta,\eta)$,
$\hat{x}_q^{\phantom{+}}(\delta,\eta)$ with the property
\newpage
\typeout{forced newpage}
{\arraycolsep=0pt\begin{eqnarray}
\hat{x}_q^{+}(\delta,\eta)\hat{x}_{q'}^{\phantom{+}}(\delta,\eta)
&=&
\sum_{S_1^{\prime},\ldots S_{L-2}^{\prime}}
\frac{1}{L^2} \sum_{l,r} e^{i(ql-q'r)a}
\bigl(1+\eta\delta +(-1)^l(\delta+\eta)\bigr)
\bigl(1+\eta\delta +(-1)^r(\delta+\eta)\bigr)
\nonumber \\[6pt]
&& \phantom{\sum_{S_1^{\prime},\ldots S_{L-2}^{\prime}}
\frac{1}{L^2} \sum_{l,r} e^{i(ql-q'r)} }
\langle 0 | S_1^{\prime},\ldots S_{l-1}^{\prime},
\left( \uparrow_{l}\downarrow_{l+1}-\downarrow_{l}\uparrow_{l+1}\right),
S_{l}^{\prime},\ldots S_{L-2}^{\prime} \label{xqxqprime} \rangle
\\[6pt]
&& \phantom{\sum_{S_1^{\prime},\ldots S_{L-2}^{\prime}}
\frac{1}{L^2} \sum_{l,r} e^{i(ql-q'r)} }
\langle S_{L-2}^{\prime},\ldots S_{r}^{\prime},
\left( \downarrow_{r+1} \uparrow_{r}-\uparrow_{r+1}\downarrow_{r}\right),
S_{r-1}^{\prime},\ldots S_{1}^{\prime} | 0 \rangle \; .
\nonumber
\end{eqnarray}}%
In practice, $q'=q$ or $q'=q+\pi/a$.
The interband current operator becomes
\begin{eqnarray}
\hat{\jmath}_{\rm inter}^{\rm band}&=&
\hat{\jmath}_{\rm inter,+}^{\rm band}+
\hat{\jmath}_{\rm inter,-}^{\rm band} \nonumber
\\[6pt]
\hat{\jmath}_{\rm inter,+}^{\rm band}&=&
\left(\hat{\jmath}_{\rm inter,-}^{\rm band}\right)^+
\nonumber \\[6pt]
\hat{\jmath}_{\rm inter,-}^{\rm band}&=&
\sum_{|q|,|k|\leq \pi/(2a)}\biggl\{ -iea \epsilon(k) \Bigl[
\hat{l}_{k-q/2}^+ \hat{u}_{k+q/2}^{\phantom{+}}
- \hat{l}_{k-q/2+\pi/a}^+ \hat{u}_{k+q/2+\pi/a}^{\phantom{+}}
\Bigr] \hat{x}_{q}^+ \label{jinterband}\\[9pt]
&& \phantom{\sum_{|q|,|k|\leq \pi/(2a)}\biggl\{ }
+ea \frac{\Delta(k)}{\delta} \Bigl[
\hat{l}_{k-q/2}^+ \hat{u}_{k+q/2+\pi/a}^{\phantom{+}}
- \hat{l}_{k-q/2+\pi/a}^+ \hat{u}_{k+q/2}^{\phantom{+}}
\Bigr] \hat{x}_{q+\pi/a}^+ \biggr\} \nonumber \; .
\end{eqnarray}
Again the conditions~$k+q/2 \neq \pi/a-(k-q/2)$ and
$k+q/2 \neq -(k-q/2)$ will not be violated
since~$\epsilon(\pi/(2a))=0$ and
$\Delta(0)=0$, respectively.
Consequently, the projection operators
in eq.~(\ref{bandhldim})
can be ignored for the case of linear optical absorption.
In semiconductor physics one prefers to work with
the dipole operator rather than the current operator
to set up the perturbation theory in the electrical field
(Haug {\em et al.} 1990).
The corresponding expressions for the dipole operator
for Hubbard interband transitions are derived in appendix~\ref{appdipoleHL}.
\section{Optical absorption in the Harris-Lange model}
\label{optabsHL}
\subsection{Spin average}
All states with no double occupancy are possible ground states in the
Harris-Lange model at half-filling.
Instead of looking at the optical absorption for a specific state~$|0\rangle$
it is more reasonable to calculate the {\em average\/}
absorption, i.~e.,
\begin{equation}
\overline{{\rm Im}\{\chi(\omega)\}} =
\frac{1}{2^L} \sum_{|0\rangle} {\rm Im}\{\chi_{|0\rangle}(\omega)\} \; .
\end{equation}
For the Hubbard model this corresponds to
temperatures $k_{\rm B}T \gg J={\cal O}(W^2/U)$ (``hot-spin case'').
The calculation is performed in appendix~\ref{spiav}.
We find the result
\begin{mathletters}
\begin{eqnarray}
\langle \hat{x}_q^+\hat{x}_{q'}\rangle &=&
\frac{1}{2L} \biggl\{
\delta_{q,q'}
\left[(1+\delta\eta)^2 g(q)+(\delta+\eta)^2 g(q+\frac{\pi}{a})\right]
\nonumber \\[3pt]
&& \phantom{\frac{1}{2L} \biggl\{ }
+ \delta_{q,q'+\frac{\pi}{a}} (1+\delta\eta)(\delta+\eta)
\left[g(q)+ g(q+\frac{\pi}{a})\right]
\biggr\}\label{averagexq}\\[3pt]
g(q) &=& \frac{3}{5+4\cos(qa)} \; .
\end{eqnarray}\end{mathletters}%
The spin problem could thus be traced out
completely.
It is seen that $\hat{x}_q$ keeps its operator character
until we have expressed the current operator in terms of
the Fermion operators for the four Peierls subbands.
\subsection{Optical absorption}
\subsubsection{Translational invariant case}
The real part of the average optical conductivity becomes
\begin{equation}
{\rm Re}\{\overline{\sigma(\omega >0)} \}
= \frac{\pi {\cal N}_{\perp}}{2 L^2 a \omega} \sum_{|q|,|k|\leq \pi/a}
(ea \epsilon(k))^2 g(q) \delta(\omega - E(k,q))
\label{elliptic}
\end{equation}
with $E(k,q)$ from equation~(\ref{EKQ}).
The above expression can be written as
\begin{equation}
\overline{\sigma_{\rm red}(\omega >0)} =
\frac{1}{4\pi}\int_{|u|}^1\frac{dx}{x^2} \frac{\sqrt{x^2-u^2}}{\sqrt{1-x^2}}
\frac{3}{9-8x^2} \label{what}
\end{equation}
for the reduced optical conductivity with $u=|\omega-U|/W\leq 1$.
This integral can be expressed as a sum over elliptic integrals
but we rather prefer to discuss some special cases.
The optical absorption is restricted to $|\omega-U|\leq W$.
Near the band edges the absorption increases linearly
which can be seen from equation~(\ref{what}) by a transformation~$x\to
1-y/|u|$.
The more interesting case is $\omega=U$.
Now the integrand displays a $1/x$ singularity for~$|u|\to 0$.
The {\em parallel\/} Hubbard bands give rise to a logarithmic divergence,
$\sigma(\omega\to U) \sim |\ln(\omega-U)|$, since their large
joint density of states for $\omega=U$ survives even in the presence
of a spinon bath that provides any momentum to the charge sector.
The overall behavior of the optical absorption
is shown in figure~\ref{hl00}.
The same absorption curve has been obtained earlier in
(Lyo {\em et al.} 1977) for their ``random'' spin background.
The result for their ``ferromagnetic'' spin background
follows when we put~$g(q)\equiv 1$, as expected.
We will comment on their N\'{e}el state results in
(Gebhard {\em et al.} III 1996).
\subsubsection{Dimerized Harris-Lange model}
We have to diagonalize the interband current operator of eq.~(\ref{jinterband})
in terms of the Peierls operators for the lower Hubbard band
\begin{mathletters}
\label{trafoforl}
\begin{eqnarray}
\hat{l}_{k}&=&\alpha_{k}\hat{l}_{k,-}+\beta_k\hat{l}_{k,+}\\[3pt]
\hat{l}_{k+\pi/a}&=&-i\beta_{k}\hat{l}_{k,-}+i\alpha_k\hat{l}_{k,+}
\end{eqnarray}
\end{mathletters}%
for $|k|\leq \pi/(2a)$.
The transformation for the upper Hubbard band is analogous.
With this definition the Hamiltonian in the band picture interpretation
became diagonal, see eq.~(\ref{bandhldim}).
The interband current operator becomes
\begin{equation}
\hat{\jmath}_{\rm inter,-}^{\rm band} =\sum_{\tau,\tau'=\pm 1}
\sum_{|k|,|q| \leq \pi/(2a)}
\lambda_{\tau,\tau'} (k,q)
\hat{l}_{k-q/2,\tau}^+\hat{u}_{k+q/2,\tau'}^{\phantom{+}}
\label{jinterdimHL}
\end{equation}
with
\begin{mathletters}
\label{thelambdas}
\begin{eqnarray}
\lambda_{+,+}(k,q) &=& iea \left[ \epsilon(k)
(\alpha_{+}^{\phantom{*}}\alpha_{-}^*
-\beta_{+}^{\phantom{*}}\beta_{-}^*) \hat{x}_q^+
+\frac{\Delta(k)}{\delta}
(\alpha_{+}^{\phantom{*}}\beta_{-}^*
+\beta_{+}^{\phantom{*}}\alpha_{-}^*)
\hat{x}_{q+\pi/a}^+ \right] \\[6pt]
\lambda_{+,-}(k,q) &=& iea \left[ - \epsilon(k)
(\alpha_{+}^{\phantom{*}}\beta_{-}^*
+\beta_{+}^{\phantom{*}}\alpha_{-}^*) \hat{x}_q^+
+\frac{\Delta(k)}{\delta}
(\alpha_{+}^{\phantom{*}}\alpha_{-}^*
-\beta_{+}^{\phantom{*}}\beta_{-}^*)
\hat{x}_{q+\pi/a}^+ \right]
\end{eqnarray}
\end{mathletters}%
and $\lambda_{-,-}(k,q)= -\lambda_{+,+}(k,q)$,
$\lambda_{-,+}(k,q)=\lambda_{+,-}(k,q)$.
Here we used the short-hand notation
$\alpha_{\pm}=\alpha_{k\pm q/2}$ etc. Note that these quantities can be
complex for~$q \neq 0$.
The average optical conductivity becomes
\begin{mathletters}
\label{monstersigma}
\begin{equation}
{\rm Re}\{\overline{\sigma(\omega >0, \delta,\eta)} \}
= \frac{\pi {\cal N}_{\perp}}{L a \omega}
\sum_{\tau,\tau'=\pm 1} \sum_{|q|,|k| \leq \pi/(2a)}
\left| \lambda_{\tau,\tau'} (k,q)\right|^2
\delta(\omega - E_{\tau,\tau'}(k,q))
\end{equation}
with the absorption energies between the respective Peierls subbands
\begin{equation}
E_{\tau,\tau'}(k,q) = U +\tau' E(k+q/2)-\tau E(k-q/2) \; ,
\label{Etautauprime}
\end{equation}
see eq.~(\ref{bandhldim}) and figure~\ref{HueckelHarrisLangedis}.
The transition matrix elements are given by
\begin{eqnarray}
\left| \lambda_{\tau,\tau'} (k,q)\right|^2 &=&
\frac{(ea)^2}{2L} \Biggl\{
g(q) \left| (1+\delta\eta) \epsilon(k)f_{\tau,\tau'}
+\tau\tau' (\delta+\eta) \frac{\Delta(k)}{\delta}f_{\tau,-\tau'}\right|^2
\nonumber \\[3pt]
&& \phantom{ \frac{(ea)^2}{2L} \Biggl\{ }
+ g(q+\frac{\pi}{a}) \left| (\delta+\eta) \epsilon(k)f_{\tau,\tau'}
+\tau\tau' (1+\delta\eta) \frac{\Delta(k)}{\delta}f_{\tau,-\tau'}\right|^2
\Biggr\}
\label{thelambdassquared}
\end{eqnarray}
\end{mathletters}%
with the help functions
\begin{mathletters}
\label{thefs}
\begin{eqnarray}
f_{+,+}(k,q) = f_{-,-}(k,q) &=&
\alpha_{k+q/2}^{\phantom{*}}\alpha_{k-q/2}^*
- \beta_{k+q/2}^{\phantom{*}}\beta_{k-q/2}^*
\\[3pt]
f_{+,-}(k,q) = f_{-,+}(k,q) &=&
\alpha_{k+q/2}^{\phantom{*}}\beta_{k-q/2}^*
+ \beta_{k+q/2}^{\phantom{*}}\alpha_{k-q/2}^*
\end{eqnarray}
\end{mathletters}%
where $\alpha_k$, $\beta_k$ are given in eq.~(\ref{alphabeta}).
It can easily be checked that the case~$\delta=\eta=0$ is reproduced.
For $\delta=1$, $\eta=0$ one recovers the result for
the average optical conductivity of $L/2$ independent
two-site systems since $E(k)=2t$,
$\lambda_{+,-}(k,q)=0$, and $|\lambda_{+,+}(k,q)|^2
= (2tea)^2(g(q)+g(q+\pi/a))/(2L)$:
\begin{equation}
{\rm Re}\{\overline{\sigma(\omega >0, \delta=1,\eta=0)} \}
= \frac{L}{2} \frac{{\cal N}_{\perp}}{La\omega} \pi \delta(\omega -U)
\frac{(Wea)^2}{4}
\end{equation}
where we used $\int_{-\pi}^{\pi}dq/(2\pi)\, g(qa)=1$.
For the direct calculation we have to recall that
only the singlet of the four spin states contributes,
and the hopping between the two sites is~$2t$.
For general~$\delta$, $\eta$ it is necessary to evaluate
the optical conductivity in eq.~(\ref{monstersigma}) numerically.
An example is shown in figure~\ref{hl10}.
It is seen that now there are two side-peaks in the optical absorption
spectrum.
The new peaks due to the Peierls distortion are weaker than the one
at~$\omega=U$ and vanish for $\delta\to 1$.
For large lattice distortions the dominant contribution to the side peaks
comes from the small-$q$ transitions between different Peierls
subbands. Their oscillator strength is maximum for $\omega=U\pm W\sqrt{\delta}$
which determines the position of the peaks for large $\delta$.
The Peierls gap between the bands shows up in the
optical spectrum. For small lattice distortions
all $(k,q)$ contribute. The signature of the
Peierls gap is smeared out and the position of the side peaks cannot be
expressed in terms of a simple function of~$\delta$.
\section{Optical absorption in the extended Harris-Lange model}
\label{optabsHubbard}
\subsection{Extended dimerized Harris-Lange model}
Strongly isotropic, almost ideal
one-dimensional systems like polymers or charge-transfer salts
are not properly described by the Hubbard model of eq.~(\ref{Hubb-Model})
for two reasons:
(i)~the Peierls distortion is not taken into account, and
(ii)~the residual
Coulomb interaction between charges beyond the Hubbard on-site interaction
is neglected.
The exponential decay of the Wannier wave functions naturally
allows to limit the interactions to on-site and nearest-neighbor Hubbard
terms. In the ``Zero Differential Overlap Approximation''
it is further {\em assumed\/} that only the direct Coulomb term has to be taken
into account for the nearest-neighbor Coulomb interaction
(Kivelson, Su, Schrieffer, and Heeger 1987); (Wu, Sun, and Nasu 1987);
(Baeriswyl, Horsch, and Maki 1988); (Gammel and Campbell 1988);
(Kivelson, Su, Schrieffer, and Heeger 1988); (Campbell {\em et al.} 1988);
(Painelli and Girlando 1988); (Painelli and Girlando 1989);
(Campbell, Gammel, and Loh 1990).
Optical absorption spectra for the extended dimerized Hubbard model
could only be calculated numerically for small system sizes.
Within such an approach the Hamiltonian is
explicitly diagonalized
(Soos and Ramesesha 1984); (Tavan and Schulten 1986);
(Guo, Mazumdar, Dixit, Kajzar, Jarka, Kawabe, and Peyghambarian 1993);
(Guo, Guo, and Mazumdar 1994).
Since the dimension of the Hilbert space increases
exponentially (${\rm dim} \hat{H} = 4^L$) the numerical analysis is
restricted to short chains ($L \leq 12$)
due to the limited computer power.
Therefore, it is natural to analytically
investigate the (dimerized) Harris-Lange model
with an additional nearest-neighbor interaction.
We will show below that the optical spectrum can still be calculated
for this model
which is equivalent to the extended dimerized Hubbard model
to order $t (t/U)^{-1}$, $t (t/U)^0$, and $t(V/U)^0$.
The dimerized extended Harris-Lange model reads
\begin{mathletters}\begin{eqnarray}
\hat{H}_{\rm HL}^{\rm dim, ext} &=&
\hat{T}_{\rm LHB}(\delta) + \hat{T}_{\rm UHB}(\delta)
+ U \hat{D} + V \hat{V} \\[6pt]
\hat{V} &=& \sum_l (\hat{n}_l-1)(\hat{n}_{l+1}-1)
\; .
\end{eqnarray}\end{mathletters}%
For half-filling the ground state of the extended dimerized
Harris-Lange model is still $2^L$-fold spin
degenerate because every site is singly
occupied for $| V | < U/2$. The energy of these states is zero, $E_0=0$,
irrespective of the dimerization
value~$\delta$.
The double occupancy and the hole in the excited states
now experience a nearest-neighbor attraction while the spin sector
remains unchanged. Thus we may immediately translate~$\hat{V}$
into our band picture as
\begin{equation}
\hat{V}^{\rm band} = - \frac{2}{L} \sum_{|q|\leq \pi/a}\cos(qa)
\sum_{|k|,|p|\leq \pi/a}
\hat{u}_{k+q}^+\hat{u}_{k}^{\phantom{+}}
\hat{l}_{p}^{\phantom{+}} \hat{l}_{p-q}^+
\end{equation}
which describes the scattering of a hole in the lower Hubbard band
with a particle in the upper Hubbard band. Again, the projection operators
can be disregarded for the optical absorption.
\subsection{Equation of motion technique}
In the presence of the nearest-neighbor interaction it becomes
increasingly tedious to separately calculate the exact eigenenergies
and oscillator strengths.
We rather prefer to directly calculate the optical conductivity from
an equation of motion approach.
\subsubsection{Translational invariant case}
Since we are interested in the real part of the optical conductivity
we can concentrate on the particle current density,
\begin{equation}
\langle \hat{\jmath}_t \rangle = \frac{{\cal N}_{\perp}}{La} \left(
\langle 0(t) | \hat{\jmath} | \Psi(t)\rangle + {\rm h.c.}\right)
\end{equation}
where $|0(t)\rangle$ and $|\Psi(t)\rangle$ are the time evolution
of the ground state with and without the external perturbation
and thus obey the corresponding Schr\"{o}dinger equations.
We have already used the fact that we want to calculate the linear absorption.
We write
\begin{eqnarray}
\langle 0(t) | \hat{\jmath} | \Psi(t)\rangle
&=& \sum_{k,q} -iea\epsilon(k)\hat{x}_{q}^+
\langle 0(t) | \hat{l}_{k-q/2}^+\hat{u}_{k+q/2}^{\phantom{+}} | \Psi(t)\rangle
\nonumber \\[3pt]
& \equiv &\sum_{k,q} \lambda(k,q) j_{k;q}(t) \; .
\end{eqnarray}
Upon Fourier transformation we obtain
\begin{equation}
\langle \hat{\jmath}_{\omega} \rangle =
\frac{{\cal N}_{\perp}}{La} \left(
\sum_{k,q} \lambda(k,q) j_{k;q}(\omega) + \lambda^+(k,q) j_{k;q}^*(-\omega)
\right) \; .
\end{equation}
Since we are interested in the optical conductivity for positive frequencies
(optical absorption) we may disregard the second term which contributes
to~$\omega < 0$.
The equation of motion for~$j_{k;q}(t)$ becomes
\begin{mathletters}
\begin{eqnarray}
i \frac{\partial j_{k;q}(t) }{\partial t} &=&
\langle 0(t) | \left[
\hat{l}_{k-q/2}^+\hat{u}_{k+q/2}^{\phantom{+}}, \hat{H}_{\rm HL}^{\rm band}
\right]_{-} | \Psi(t)\rangle
- \frac{{\cal A}(t)}{c}
\langle 0(t) |
\hat{l}_{k-q/2}^+\hat{u}_{k+q/2}^{\phantom{+}}
\hat{\jmath} | 0(t)\rangle
\\[6pt]
\omega j_{k;q}(\omega) &=& E(k,q) j_{k;q}(\omega)
- 2V\left( \cos(ka) j_q^c(\omega) +\sin(ka)j_q^s(\omega)\right)
- \lambda^+(k,q) \frac{{\cal A}(\omega)}{c}
\end{eqnarray}
\end{mathletters}%
where we kept the expansion linear in the external perturbation,
and performed the Fourier transformation.
Furthermore, we introduced the abbreviations
\begin{equation}
j_q^{c,s}(\omega) = \frac{1}{L}
\sum_k \left( {\cos (ka) \atop \sin (ka)}\right) j_{k;q}(\omega) \; .
\end{equation}
For our calculations we only need~$j_q^c(\omega)$ since our current
operator preserves parity.
The particle current density for positive frequencies becomes
\begin{equation}
\langle \hat{\jmath}_{\omega>0}\rangle = \frac{{\cal N}_{\perp}}{a}(2tiea)
\sum_q \hat{x}_q^+ j_{q}^c(\omega)
\end{equation}
which is proportional to the external field.
We introduce the function
\begin{equation}
F(q) = \frac{2}{L} \sum_{|k|\leq \pi/a} \frac{(\cos ka)^2}{\omega - E(k,q)}
\end{equation}
which allows us to finally express the optical conductivity as
\begin{equation}
{\rm Re} \{ \overline{\sigma(\omega>0,V)} \}
= - \frac{(Wea)^2{\cal N}_{\perp}}{16 a\omega} \frac{1}{L}
\sum_{|q|\leq \pi/a} g(q) {\rm Im}\left\{ \frac{F(q)}{1+VF(q)} \right\} \; .
\label{ResigmaHLV}
\end{equation}
The result will be discussed in the next subsection.
\subsubsection{Extended dimerized Harris-Lange model}
The same procedure can be applied to the dimerized case where it is best
to start from the diagonalized Hamiltonian in the form of
eq.~(\ref{effHL}),
and the current operator in the form of eq.~(\ref{jinterdimHL}).
The calculations are outlined in appendix~\ref{appc}.
We introduce the three functions $F_{1,2,3}$ as
\begin{mathletters}
\label{capitalF}
\begin{eqnarray}
F_{1} (q) &=& \frac{2}{L} \sum_{|k| \leq \pi/(2a)}
\cos^2(ka) \Biggl[ |f_{+,+}|^2
\left( \frac{1}{\omega-E_{-,-}} +\frac{1}{\omega-E_{+,+}} \right)
\nonumber \\[3pt]
&& \phantom{\frac{2}{L} \sum_{|k| \leq \pi/(2a)}
\cos^2(ka) \Biggl[ }
+|f_{+,-}|^2
\left(\frac{1}{\omega-E_{-,+}}+\frac{1}{\omega-E_{+,-}}\right)
\Biggr] \\[6pt]
F_{2} (q) &=& \frac{2}{L} \sum_{|k| \leq \pi/(2a)}
\sin^2(ka) \Biggl[ |f_{+,-}|^2
\left( \frac{1}{\omega-E_{-,-}} +\frac{1}{\omega-E_{+,+}} \right)
\nonumber \\[3pt]
&& \phantom{\frac{2}{L} \sum_{|k| \leq \pi/(2a)}
\sin^2(ka) \Biggl[ }
+|f_{+,+}|^2
\left(\frac{1}{\omega-E_{-,+}}+\frac{1}{\omega-E_{+,-}}\right)
\Biggr]
\\[6pt]
F_{3} (q) &=& \frac{2}{L} \sum_{|k| \leq \pi/(2a)}
\cos(ka)\sin(ka) \Biggl\{ f_{+,+}^{\phantom{*}} f_{+,-}^* \left[
\frac{1}{\omega-E_{+,+}}+ \frac{1}{\omega-E_{-,-}} \right]
\nonumber \\[3pt]
&& \phantom{\frac{2}{L} \sum_{|k| \leq \pi/(2a)}
\cos(ka)\sin(ka) \biggl\{ }
- f_{+,+}^* f_{+,-}^{\phantom{*}}
\left[ \frac{1}{\omega-E_{-,+}}+ \frac{1}{\omega-E_{+,-}} \right]\Biggr\}
\end{eqnarray}
\end{mathletters}%
where $f_{\tau,\tau'}\equiv f_{\tau,\tau'}(k,q)$
and $E_{\tau,\tau'}=E_{\tau,\tau'}(k,q)$
were introduced in eq.~(\ref{thefs}) and eq.~(\ref{Etautauprime}).
Furthermore, we abbreviate~$A_j=(1+\delta\eta)F_j-(\eta+\delta)F_3$,
$B_j=(\delta+\eta)F_j-(1+\delta\eta)F_3$ ($j=1,2$),
$C_1=(1+\delta\eta)^2 F_1+(\delta+\eta)^2 F_2
-2(1+\delta\eta)(\delta+\eta)F_3$, and
$C_2=(1+\delta\eta)^2 F_2+(\delta+\eta)^2 F_1
-2(1+\delta\eta)(\delta+\eta)F_3$.
The real part of the average optical conductivity can then be expressed as
\begin{eqnarray}
{\rm Re}\{\overline{\sigma(\omega >0, V, \delta,\eta)} \}
&=&
{\rm Re}\{\overline{\sigma(\omega >0,\delta,\eta)} \}
\nonumber \\[6pt]
&&
+ \frac{V{\cal N}_{\perp}(Wea)^2}{16 a\omega L} {\rm Im}\Biggl\{
\sum_{|q| \leq \pi/(2a)}
\frac{1}{(1+VF_1)(1+VF_2)-(VF_3)^2}
\label{thefinalresultHL}
\\[6pt]
&&
\phantom{+ \frac{V{\cal N}_{\perp}(Wea)^2}{16 a\omega L} {\rm Im}\Biggl\{ }
\biggl\{
g(q) \left[ A_1^2 +B_2^2 +V(F_1F_2-F_3^2) C_1 \right] \nonumber \\
&& \phantom{+ \frac{V{\cal N}_{\perp}(Wea)^2}{16 a\omega L} {\rm Im}\Biggl\{
\biggl\{ }
+g(q+\frac{\pi}{a}) \left[ A_2^2 +B_1^2 +V(F_1F_2-F_3^2) C_2 \right]
\biggr\} \Biggr\} \; .
\nonumber
\end{eqnarray}
The result for~$V=0$ is given in eq.~(\ref{monstersigma}).
In the following
we will discuss the results for the average optical absorption in the
presence of a nearest-neighbor interaction.
\subsection{Optical absorption}
\subsubsection{Translational invariant case}
The help function~$F(q)$ can be calculated analytically
with the help of eqs.~(2.267,1), (2.266), and~(2.261)
of (Gradshteyhn and Ryzhik 1980).
The result is
\begin{equation}
F(q) \! = \! \frac{2}{[4t\sin(qa/2)]^2}
\! \left\{
\!
\begin{array}{lcl}
\omega-U -\sqrt{(\omega-U)^2- \left(4t\sin(qa/2)\right)^2\, }
& {\rm for} & |\omega-U| \geq |4t\sin(qa/2)|
\\[6pt]
-i \sqrt{ \left(4t\sin(qa/2)\right)^2 - (\omega-U)^2\, }
& {\rm for} & |\omega-U| < |4t\sin(qa/2)|
\end{array}
\right.
\, .
\end{equation}
The result for~$V=0$, eq.~(\ref{what}),
follows after the substitution of~$x=\sin(qa/2)$ into
equation~(\ref{ResigmaHLV}).
The total optical absorption is shown in figure~\ref{hl01}.
For arbitrarily small~$V>0$ there is a bound exciton which
is the standard situation for one-dimensional short-range attractive
potentials between a positive (hole) and negative charge (double occupancy).
This is evident from the form of~$F(q)$ which allows for excitons
with momenta~$q$ if $|\omega -U| \geq |4t\sin(qa/2)|$
which is fulfilled for~$q=0$ for all~$V>0$.
When the attraction between the two opposite charges is strong,
the full exciton band with width~$W_{\rm exc}=4t^2/V$ is formed.
This can be seen from the zeros of the denominator in eq.~(\ref{ResigmaHLV})
in the region~$|\omega-(U-V-W_{\rm exc}/2)|\leq W_{\rm exc}/2$.
One finds from $1+VF(q)=0$ that
\begin{eqnarray}
\omega= U-V-\frac{W_{\rm exc}}{2} + \frac{W_{\rm exc}}{2} \cos qa \; .
\end{eqnarray}
This is precisely the dispersion relation for bound pairs in one dimension
with nearest-neighbor hopping of strength~$t_{\rm exc}=t (t/V)$:
at large~$V$ the excitons are essentially nearest-neighbor pairs
of opposite charges which {\em coherently\/} move with the hopping
amplitude~$t_{\rm exc}$. Note that this motion requires an intermediate
(``virtual'') configuration where the two charges are not nearest neighbors.
Consequently, the hopping integral of the individual constituents,~$t$,
is reduced by the factor~$t/V$ for the motion of the pair.
The full band becomes apparent when $W_{\rm exc}+V > W$ or $V > W/2$,
see figure~\ref{hl01}. Recall that the spin sector provides {\em any\/}
momentum to the charge sector. The momentum transfer, however, is
modulated by the function~$g(q)$ which is maximum at $q=\pi/a$
and reflects the fact that states with antiferromagnetic spin correlations
are best suited for optical absorptions since they contain many neighboring
singlet pairs. Hence, the $q=\pi/a$-exciton dominates over
the $q=0$-exciton for $V>W/2$.
It is amusing to see that the optical absorption of a Peierls insulator
and a Mott-insulator (extended Harris-Lange model: $U \gg W$, $V> W/2$,
$J=0$) can look very similar,
compare figure~2 of~I and figure~\ref{hl01}.
This has already been noted long time ago by Simpson
(Simpson 1951); (Simpson 1955); (Salem 1966); (Fave 1992)
who explained the optical absorption spectra of short polyenes in the
above exciton model.
It is seen that Simpson's model is naturally included in our strong-correlation
approach.
For real polymers, however, Simpson's original approach is not satisfactory.
A fully developed exciton band
only exists in the presence of an
incoherent spin background. Now that
even the spin-Peierls effect is excluded one can by no means
explain the Peierls distortion of the lattice
as an electronic effect.
This does not exclude other, e.g., extrinsic, explanations for a
lattice distortion.
\subsubsection{Extended dimerized Harris-Lange model}
The full spectrum has to be determined numerically. An example
for various values of~$V/t$ is shown in figure~\ref{hl11}.
For large~$V/t$ we have a fully developed
exciton band
which is itself Peierls-split into two branches.
Thus one obtains four van-Hove singularities in the optical
absorption spectrum.
The Peierls gap is given by~$\Delta_{\rm exc}^{\rm P}=
\delta W_{\rm exc}= 4t^2\delta/V$.
Even for~$V=W$ it is smeared out since $V/t$ is not too large yet
and the phenomenological damping~$\gamma$ is already of the order
of the gap.
For small~$V/t$ we obtain the signature
of the $q=0$ exciton for~$\delta=0$, $V>0$.
For intermediate~$V$ this peak develops into a van-Hove singularity
of the upper exciton subband. The signatures of the second
van-Hove singularity of the upper band are clearly visible for~$V=W/2$.
The peaks of the Peierls subbands for~$V=0$, $\delta \neq 0$
are both red-shifted. The peak at
lower energy increases in intensity and finally forms
the lower exciton subband while the peak at higher energy
quickly looses its oscillator strength.
\section{Summary and Outlook}
In this paper we addressed the optical absorption of
the half-filled Harris-Lange model which
is equivalent to the Hubbard model at strong correlations
and temperatures large compared to the spin energy scale.
It is extremely difficult to analytically
calculate optical properties of interacting electrons in one dimension.
For strong coupling when the Hubbard interaction is large compared
to the band-width matters considerably simplified since
the energy scales for the charge and spin excitations are well separated.
We were able to derive an {\em exactly\/} equivalent
band structure picture for the charge degrees
of freedom and found that the upper and lower Hubbard band are actually
{\em parallel\/} bands with the band structure of free Fermions.
We have taken special
care of the spin background which can act as a momentum
reservoir for the charge system.
Since we can exactly integrate out the
spin degrees of freedom for the Harris-Lange model
we were able to solve the problem even in the presence
of a lattice dimerization and a nearest-neighbor interaction between
the electrons.
For a vanishing nearest-neighbor interaction
we found a prominent absorption peak at~$\omega=U$ and additional side peaks
in the absorption bands $|\omega-U| \leq W$
in the presence of a lattice distortion~$\delta$.
When a further nearest-neighbor interaction between the charges
was included, we found the formation of
Simpson's exciton band of band-width $W_{\rm exc.}=4t^2/V$ for $V>W/2$
which is eventually Peierls-split.
As usual the excitons draw almost all oscillator strength from the
band.
It should be clear that the Harris-Lange model with its highly degenerate
ground state is not a suitable model for the study of real materials.
The results presented here are relevant to systems for
which the temperature is much larger than the spin exchange energy.
Real experiments are not carried out in this ``hot-spin'' regime
but at much lower temperatures for which the system
is in an unique ground state with antiferromagnetic correlations.
Unfortunately, this problem cannot be solved analytically.
In the third and last paper of this series
(Gebhard {\em et al.} III 1996) we will
employ the analogy to an ordinary semiconductor (electrons and holes
in a phonon bath) to design a ``no-recoil'' approximation
for the chargeons in a spinon bath.
It will allows us to determine the coherent absorption features
of the Hubbard model at large~$U/t$.
\section*{Acknowledgments}
We thank H.~B\"{a}\ss ler, A.~Horv\'{a}th,
M.~Lindberg, S.~Mazumdar, M.~Schott, and
G.~Weiser for useful discussions.
The project was supported in part by the
Sonderforschungsbereich~383
``Unordnung in Festk\"{o}rpern
auf mesoskopischen Skalen'' of the Deutsche Forschungsgemeinschaft.
\newpage
\begin{appendix}
\section{The Harris-Lange model}
\label{appeigen}
\subsection{Sum rules}
\label{appsumrule}
We briefly account for the sum rules. We have
\begin{equation}
\int_{0}^{\infty} d\omega\ {\rm Im}\{\chi(\omega)\}
= \pi \frac{{\cal N}_{\perp}}{La} \sum_n \left| \langle 0 |
\hat{\jmath}^2|n\rangle\right|^2 = \pi \frac{{\cal N}_{\perp}}{La}
\langle 0 | \hat{\jmath}^2|0\rangle \; .
\end{equation}
It is a standard exercise to show that
\begin{equation}
\langle 0 | \hat{\jmath}^2 | 0 \rangle =
(2tea)^2 \sum_l \left(1+(-1)^l\delta\right)^2\left(1+(-1)^l\eta\right)^2
\langle 0 | \left( \frac{1}{4}
-\hat{\rm\bf S}_l\hat{\rm\bf S}_{l+1}\right) | 0 \rangle \; .
\end{equation}
We define the positive quantities
\begin{equation}
C_S^{\rm even, odd}= \frac{1}{L} \sum_l \frac{1\pm (-1)^l}{2}
\langle 0 | \left( \frac{1}{4}
-\hat{\rm\bf S}_l\hat{\rm\bf S}_{l+1}\right) | 0 \rangle
\label{CSevenCSodd}
\end{equation}
and may then write
\begin{equation}
\int_{0}^{\infty} d\omega\ {\rm Im}\{\chi(\omega)\}
= \pi {\cal N}_{\perp}a (2te)^2 \left[
\left(1+\delta\right)^2\left(1+\eta\right)^2
C_S^{\rm even}
+
\left(1-\delta\right)^2\left(1-\eta\right)^2
C_S^{\rm odd}
\right] \; .
\end{equation}
If we average over all possible states~$|0\rangle$ we obtain
$C_S^{\rm even, odd}=1/8$ since only the singlet configuration contributes.
Hence,
\begin{equation}
\int_{0}^{\infty} d\omega \overline{{\rm Im}\{\chi(\omega)\}}
=
\pi {\cal N}_{\perp}a (te)^2 \left[(1+\delta\eta)^2 +(\delta+\eta)^2\right]
\; .
\end{equation}
The area under the curves for~$\sigma_{\rm red}(\omega)$, eq.~(\ref{sigmared}),
are thus given by
\begin{mathletters}
\begin{eqnarray}
\int_0^{\infty} \frac{d\omega}{W} \sigma_{\rm red}(\omega) &=&
\frac{\pi}{4}
\left[
\left(1+\delta\right)^2\left(1+\eta\right)^2 C_S^{\rm even}
+
\left(1-\delta\right)^2\left(1-\eta\right)^2 C_S^{\rm odd}
\right]
\label{sumrulesigmared}
\\[6pt]
\int_0^{\infty} \frac{d\omega}{W} \overline{\sigma_{\rm red}(\omega)} &=&
\frac{\pi}{16} \left[(1+\delta\eta)^2 +(\delta+\eta)^2\right]
\; .
\end{eqnarray}
\end{mathletters}%
\subsection{Momentum of eigenstates}
\label{appmomentum}
First we will assume that the number of sites and the
number of particles is even. $N_C$ and~$N_S$ will thus also be even.
Let~$\hat{{\cal T}}$ be the translation operator by one site,
i.~e., $\hat{{\cal T}} \hat{c}_{l,\sigma}\hat{{\cal T}}^{-1}=
\hat{c}_{l+1,\sigma}$.
We have to show that
\begin{equation}
\hat{{\cal T}} |\Psi\rangle = e^{-iPa}|\Psi\rangle
\end{equation}
holds. We have to distinguish two cases: (i)~a spin is at site~$L$
in $|\Psi\rangle$ or (ii)~a charge is at site~$L$
in $|\Psi\rangle$.
\paragraph{case~(i):}
the operator~$\hat{{\cal T}}$ shifts the spin from site~$L$ to the first site.
This results in a phase factor~$(-1)$ since one has to commute the Fermion
operator~$(N-1)$-times to obtain the proper order in~$|\Psi\rangle$.
Furthermore, the states in~$|\Psi\rangle$ have, relative to those in
$\hat{{\cal T}}|\Psi\rangle$,
\begin{enumerate}
\item shifted the spin sequence by one unit. This results in a phase factor
$\exp(ik_Sa)$;
\item
an additional factor $(-1)$ for each doubly occupied site.
This gives a phase factor~$\exp(i\pi N_d)$;
\item
a Slater determinant in which each site index is shifted by one.
This results in a phase factor
$\exp(i\sum_{j=1}^{N_C} (k_j+\Phi_{CS})a)$.
\end{enumerate}
In sum we obtain
\begin{eqnarray}
(-1) &=& e^{-iPa} e^{ik_Sa} e^{i\pi N_d} e^{i\sum_{j=1}^{N_C} (k_j+\Phi_{CS})a}
\nonumber \\[6pt]
P&=& k_S +(\pi/a) (N_d-1)+ \sum_{j=1}^{N_C} (k_j+\Phi_{CS})
\quad {\rm mod\ } 2\pi/a \; .
\end{eqnarray}
\paragraph{case~(ii):}
the operator~$\hat{{\cal T}}$ shifts the charge from site~$L$ to the first
site.
The states in~$|\Psi\rangle$ have, relative to those in
$\hat{{\cal T}}|\Psi\rangle$,
\begin{enumerate}
\item shifted the charge sequence by one unit. This results in a phase factor
$\exp(ik_Ca)$;
\item
an additional factor $(-1)$ for each doubly occupied site.
This gives a phase factor~$\exp(i\pi N_d)$;
\item
a Slater determinant in which
\begin{enumerate}
\item each site index is shifted by one.
This results in a phase factor
$\exp(i\sum_{j=1}^{N_C} (k_j+\Phi_{CS})a)$;
\item
the last row and the first row are interchanged.
This gives an additional factor $(-1)^{N_C-1}=-1$;
\item
the first row is $(1,\ldots 1)$ instead of
$\left(\exp(i(k_1+\Phi_{CS})La),\ldots\exp(i(k_{N_C}+\Phi_{CS})La)\right)$
$= (1,\ldots 1)\exp(i(k_C-k_C)a)$.
This gives an additional phase factor~$\exp(-i(k_C-k_S)a)$.
\end{enumerate}
\end{enumerate}
In sum we obtain
\begin{eqnarray}
1 &=& e^{-iPa} e^{ik_C a} e^{i\pi N_d} e^{i\sum_{j=1}^{N_C} (k_j+\Phi_{CS})a}
(-1) e^{-i(k_C-k_S)a}
\nonumber \\[6pt]
P&=& k_S +(\pi/a) (N_d-1)+ \sum_{j=1}^{N_C} (k_j+\Phi_{CS}) \quad {\rm mod\ }
2\pi/a
\end{eqnarray}
as before. If the number of particles~$N$ is odd, the momentum is shifted
by another factor of~$\pi/a$, if one repeats the above arguments.
This proves that the total momentum of the state~$|\Psi\rangle$
is indeed given by eq.~(\ref{eigenmomHL}).
\subsection{Energy of eigenstates}
\label{appenerg}
We want to prove that
\begin{equation}
\hat{H}_{\rm HL} |\Psi\rangle = E|\Psi\rangle
\quad ; \quad
E = \sum_{j=1}^{N_C} \epsilon(k_j+\Phi_{CS}) +U N_d \; .
\label{showit}
\end{equation}
Again we restrict ourselves to even~$N$.
The bulk terms are simple since there is no hopping across the boundary.
A hopping process of a double occupancy and a hole
are equivalent:
\begin{mathletters}
\begin{eqnarray}
(-1)^l | \ldots \bullet_l \sigma_{l+1} \ldots\rangle &\mapsto&
- (-1)^l | \ldots \sigma_{l} \bullet_{l+1} \ldots\rangle
= (-1)^{l+1} | \ldots \sigma_{l} \bullet_{l+1} \ldots\rangle \\[3pt]
| \ldots \circ_l \sigma_{l+1} \ldots\rangle &\mapsto&
| \ldots \sigma_{l} \circ_{l+1} \ldots\rangle \; .
\end{eqnarray}
\end{mathletters}%
The extra minus sign which appears when a double occupancy moves
has been taken care of in the wave function
by the phase factor~$(-1)^l$ for a double
occupancy at site~$l$. Now that there is no difference in the motion of
double occupied sites and holes they dynamically behave as
spinless Fermions. Since the Slater determinant
is the proper phase factor for non-interacting Fermions
eq.~(\ref{showit}) holds for the bulk terms.
This also shows that only the Harris-Lange model with
hopping amplitudes~$|t_{\rm LHB}|=|t_{\rm UHB}|$ can be
solved. Another integrable but trivial
case is~$t_{\rm LHB}=0$, $t_{\rm UHB}\neq 0$
and vice versa.
We now address the boundary terms.
A typical configuration for which transport across the boundary is
possible is $|S_1, \ldots C_L\rangle$.
The phase of the configuration is given by a Slater determinant
in which the last row has the entry
$\left(\exp(i(k_1+\Phi_{CS})La),\ldots\exp(i(k_{N_C}+\Phi_{CS})La)\right)$
$= (1,\ldots 1)\exp(i(k_C-k_S)La)$.
The action of~$\hat{H}_{\rm HL}$ moves the charge from site~$L$
to the first position by which an extra minus sign occurs
since the electron operator for the spin
had to be commuted with~$(N-1)$ other electron operators.
These phase factors have to be compared to the corresponding
configuration in~$E|\Psi\rangle$.
Relative to the configuration in~$\hat{H}_{\rm HL}|\Psi\rangle$
it has
\begin{enumerate}
\item shifted the spin sequence by minus one unit.
This results in a phase factor~$\exp(-ik_Sa)$;
\item shifted the charge sequence by one unit.
This results in a phase factor~$\exp(ik_Ca)$;
\item a Slater determinant which has
$\left(\exp(i(k_1+\Phi_{CS})a),\ldots\exp(i(k_{N_C}+\Phi_{CS})a)\right)$
in the first row.
\end{enumerate}
The boundary terms should give the same result as the bulk terms.
This leads to the condition
\begin{equation}
- e^{i(k_C-k_S)La} = e^{-ik_Sa} e^{ik_C a} (-1)^{N_C-1}
\end{equation}
which is obviously fulfilled. The proof for odd~$N$ is analogous,
and eq.~(\ref{showit}) holds for all~$N$.
\subsection{Electrical dipole operator for the Harris-Lange model}
\label{appdipoleHL}
We can derive the electrical
dipole operator for the Harris-Lange model from its definition in
equation~(A.22) of~I. We use the Hamilton operator in the band picture
interpretation, eq.~(\ref{effHL}), and the corresponding current operator,
eq.~(\ref{jhlbandpicture}). Since~$\hat{x}_q$ can be replaced by its
average value~$\sqrt{g(q)/(2L)}$ one easily sees that
the dipole operator becomes
\begin{mathletters}
\begin{eqnarray}
\hat{P}_{\rm inter}^{\rm HL}&=&\sum_{|k|,|q|\leq \pi/a}
\mu_{\rm inter}^{\rm HL}(k,q)
\left( \hat{u}_{k+q/2}^+\hat{l}_{k-q/2}^{\phantom{+}} + \hat{l}_{k-q/2}^+
\hat{u}_{k+q/2}^{\phantom{+}} \right)
\\[3pt]
\mu_{\rm inter}^{\rm HL}(k,q)&=& i \frac{\lambda(k,q)}{E(k,q)}=
ea \sqrt{\frac{g(q)}{2L}} \frac{\epsilon(k)}{E(k,q)}
\end{eqnarray}
\end{mathletters}%
with~$E(k,q)=U+\epsilon(k+q/2)-\epsilon(k-q/2)$.
One sees that the dipole matrix element is of the order~$t/U$ as it should
be for interband transitions.
Furthermore, for small momentum transfer we obtain
\begin{equation}
\mu_{\rm inter}^{\rm HL}(k;q\to 0) \sim \epsilon(k) \; .
\end{equation}
This is the correct form since we create a neighboring
hole and double occupancy which corresponds to an electric dipole between
nearest neighbors.
The procedure is readily generalized to the dimerized Harris-Lange model.
The interband current operator in terms of the Fermion operators
for the four Peierls subbands is given in eq.~(\ref{jinterdimHL}),
and the diagonalized Hamiltonian can be found in eq.~(\ref{bandhldim}).
One readily finds
\begin{mathletters}
\begin{eqnarray}
\hat{P}_{\rm inter}^{\rm dim.\ HL}&=& \sum_{\tau,\tau'=\pm 1}
\sum_{|k|,|q|\leq \pi/(2a)}
\mu_{\rm inter;\tau,\tau'}^{\rm dim.\ HL}(k,q)
\left( \hat{u}_{k+q/2,\tau'}^+\hat{l}_{k-q/2,\tau}^{\phantom{+}} +
\hat{l}_{k-q/2,\tau}^+\hat{u}_{k+q/2,\tau'}^{\phantom{+}} \right)
\\[3pt]
\mu_{\rm inter;\tau,\tau'}^{\rm HL}(k,q)&=&
i \frac{\lambda_{\tau,\tau'}(k,q)}{E_{\tau,\tau'}(k,q)}
\end{eqnarray}
\end{mathletters}%
with $\lambda_{\tau,\tau'}(k,q)$ as the root
of eq.~(\ref{thelambdassquared}), and
$E_{\tau,\tau'}(k,q)=U+\tau' E(k+q/2)-\tau E(k-q/2)$, see
eq.~(\ref{Etautauprime}).
The dipole matrix elements again simplify for small~$q$.
Note that one obtains both contributions from $q\to 0$ and $q\to \pi/a$.
After some calculations one obtains
\begin{mathletters}
\label{lambdatautauprimeqzero}
\begin{eqnarray}
\left|\lambda_{+,+}(k;q=0)\right|^2 &=& \frac{1}{2L} \Biggl\{
\frac{1}{3} \left[ea\left( E(k)+\delta\eta\frac{(2t)^2}{E(k)}\right)
\right]^2 \nonumber \\[6pt]
&& \phantom{ \frac{1}{4L} \biggl[ }
+ 3 \left[ea \left(\eta E(k)-\delta\frac{(2t)^2}{E(k)}\right)
\right]^2 \Biggr\} \\[6pt]
\left|\lambda_{+,-}(k;q= 0)\right|^2 &=& \frac{1}{2L}
\left( ea \frac{\epsilon(k)\Delta(k)(1-\delta^2)}{\delta E(k)}\right)^2
\left( \frac{\eta^2}{3} +3\right)
\; .
\end{eqnarray}
\end{mathletters}%
Note that the dipole matrix elements $\left|\lambda_{+,-}(k;q=0)\right|^2$
contain the contributions from~$q= \pi/a$ for $\delta=\eta=0$.
Eqs.~(\ref{lambdatautauprimeqzero})
have to be compared to the corresponding expressions for the
Peierls chain. It is seen that the expressions display some similarities
but they show subtle differences. Even for~$q=0$ the
expressions~(\ref{lambdatautauprimeqzero}) could not have been guessed.
The corresponding dipole matrix elements become
\begin{mathletters}
\label{mutautauprimeqzero}
\begin{eqnarray}
\left|\mu_{+,+}(k;q\to 0)\right|^2 &=& \frac{1}{U^2}
\left|\lambda_{+,+}(k;q\to 0)\right|^2\\[6pt]
\left|\mu_{+,-}(k;q\to 0)\right|^2 &=& \frac{1}{(U-2E(k))^2}
\left|\lambda_{+,-}(k;q\to 0)\right|^2 \\[6pt]
\left|\mu_{-,+}(k;q\to 0)\right|^2 &=& \frac{1}{(U+2E(k))^2}
\left|\lambda_{+,-}(k;q\to 0)\right|^2 \; .
\end{eqnarray}
\end{mathletters}%
The dipole matrix elements between the same Peierls subbands are
always strong, irrespective of~$k$ or $\delta$.
However, the dipole matrix elements for transitions between different
subbands are small for strong dimerization.
Furthermore, they are small in the vicinity of
the center and the edge of the reduced Brillouin zone.
\subsection{Spin average in the Harris-Lange model}
\label{spiav}
We need to calculate
\begin{equation}
\langle \hat{x}_q^+\hat{x}_{q'}\rangle =
\frac{1}{2^L}\sum_{|0\rangle} \hat{x}_q^+\hat{x}_{q'}
\end{equation}
where
{\arraycolsep=0pt\begin{eqnarray}
\hat{x}_q^{+}(\delta,\eta)\hat{x}_{q'}^{\phantom{+}}(\delta,\eta)
&=&
\sum_{S_1^{\prime},\ldots S_{L-2}^{\prime}}
\frac{1}{L^2} \sum_{l,r} e^{i(ql-q'r)a}
\bigl(1+\eta\delta +(-1)^l(\delta+\eta)\bigr)
\bigl(1+\eta\delta +(-1)^r(\delta+\eta)\bigr)
\nonumber \\[6pt]
&& \phantom{\sum_{S_1^{\prime},\ldots S_{L-2}^{\prime}}
\frac{1}{L^2} \sum_{l,r} e^{i(ql-q'r)} }
\langle 0 | S_1^{\prime},\ldots S_{l-1}^{\prime},
\left( \uparrow_{l}\downarrow_{l+1}-\downarrow_{l}\uparrow_{l+1}\right),
S_{l}^{\prime},\ldots S_{L-2}^{\prime}\rangle \nonumber
\\[6pt]
&& \phantom{\sum_{S_1^{\prime},\ldots S_{L-2}^{\prime}}
\frac{1}{L^2} \sum_{l,r} e^{i(ql-q'r)} }
\langle S_{L-2}^{\prime},\ldots S_{r}^{\prime},
\left( \downarrow_{r+1} \uparrow_{r}-\uparrow_{r+1}\downarrow_{r}\right),
S_{r-1}^{\prime},\ldots S_{1}^{\prime} | 0 \rangle \; .
\nonumber \\
&& \label{xqxqprimeapp}
\end{eqnarray}}%
Since the set of spin states $|0\rangle$ is complete we
may exactly trace it out and are left with the
calculation of the spin matrix elements
\begin{eqnarray}
&&M(l,r)=\frac{1}{2^L}\sum_{S_1^{\prime},\ldots S_{L-2}^{\prime}}
\\[3pt]
&& \langle S_{L-2}^{\prime},\ldots S_{r}^{\prime},
\left( \downarrow_{r+1} \uparrow_{r}-\uparrow_{r+1}\downarrow_{r}\right),
S_{r-1}^{\prime},\ldots S_{1}^{\prime}
|
S_1^{\prime},\ldots S_{l-1}^{\prime},
\left( \uparrow_{l}\downarrow_{l+1}-\downarrow_{l}\uparrow_{l+1}\right),
S_{l}^{\prime},\ldots S_{L-2}^{\prime} \rangle \; . \nonumber
\end{eqnarray}%
The value of all spins between the sites~$l$ and~$r$ is fixed by
the singlet operators at $(l, l+1)$ and $(r, r+1)$.
We find
\begin{equation}
M(l,r) = 2(-1)^{r-l} 2^{-|r-l|-2}
\end{equation}
which shows that the correlation function for finding
two singlet pairs at distance~$n=|r-l|$ exponentially decays
with correlation length~$\xi_{\rm S}=1/\ln(2)$.
Since $M(l,r)$ only depends on the distance between the two sites
we may carry out one of the lattice sums in
equation~(\ref{xqxqprimeapp}). This gives
\begin{eqnarray}
\langle \hat{x}_q^+\hat{x}_{q'}\rangle =
\frac{1}{2L}\biggl\{
&&
\delta_{q,q'} \sum_{n=-L/2}^{L/2-1}
e^{inqa}2^{-|n|}(-1)^n \left[(1+\delta\eta)^2+(-1)^n(\delta+\eta)^2\right]
\\[3pt] \nonumber
&& + \delta_{q,q'+\frac{\pi}{a}} \sum_{n=-L/2}^{L/2-1}
e^{inqa}2^{-|n|} (1+\delta\eta)(\delta+\eta)(1+(-1)^n)\biggr\} \; .
\end{eqnarray}
The sum over~$n$ is readily taken and gives the final result
\begin{mathletters}
\begin{eqnarray}
\langle \hat{x}_q^+\hat{x}_{q'}\rangle &=&
\frac{1}{2L} \biggl\{
\delta_{q,q'}
\left[(1+\eta\delta)^2 g(q)+(\delta+\eta)^2 g(q+\frac{\pi}{a})\right]
\nonumber \\[3pt]
&& \phantom{\frac{1}{2L} \biggl\{ }
+ \delta_{q,q'+\frac{\pi}{a}} (1+\eta\delta)(\delta+\eta)
\left[g(q)+ g(q+\frac{\pi}{a})\right]
\biggr\}
\end{eqnarray}
with the help function
\begin{equation}
g(q) = \frac{3}{5+4\cos(qa)} \; .
\end{equation}\end{mathletters}%
In particular, for the translational invariant case we find
\begin{equation}
\langle \hat{x}_q^+\hat{x}_{q'}\rangle =
\delta_{q,q'}\frac{g(q)}{2L} \; .
\end{equation}
This shows that $\hat{x}_q$ can be replaced by $\sqrt{g(q)/(2L)}$
in the translational invariant case.
\section{Equation of motion technique for the extended dimerized
Harris-Lange model}
\label{appc}
Here we briefly outline the calculations for the most general case~$V\neq 0$,
$\delta\neq 0$.
We use the diagonalized band picture Hamiltonian in the form of
eq.~(\ref{effHL}),
and the band picture current operator in the form of eq.~(\ref{jinterdimHL}).
This has the advantage that the equation of motion can directly be inverted
and the contribution for~$V=0$ can immediately be separated.
The equations of motions give for the four currents
\begin{equation}
j_{\tau,\tau'}(k,q;\omega) = -\frac{
({\cal A}(\omega)/c)\lambda_{\tau,\tau'}^+
+ 2V
\left( \cos(ka) X_{\tau,\tau'}^c +\sin(ka) X_{\tau,\tau'}^s
\right)
}
{\omega- E_{\tau,\tau'}}
\label{C1}
\end{equation}
where~$E_{\tau,\tau'}\equiv E_{\tau,\tau'}(k,q)=
U+\tau' E(k+q/2)-\tau E(k-q/2)$,
$\lambda_{\tau,\tau'}\equiv\lambda_{\tau,\tau'}(k,q)$, and
$X_{\tau,\tau'}^{c,s}\equiv X_{\tau,\tau'}^{c,s}(k,q)$.
The difficult terms~$X^{c,s}\equiv X_{+,+}^{c,s}=-X_{-,-}^{c,s}$,
$Y^{c,s}\equiv X_{+,-}^{c,s}=X_{-,+}^{c,s}$,
come from the nearest-neighbor interaction which mixes
excited pairs in the different Peierls subbands.
With the help of eq.~(\ref{C1}), $\lambda_{+,+}=-\lambda_{-,-}$,
and $\lambda_{+,-}=\lambda_{-,+}$ we can immediately write
\begin{eqnarray}
\langle \hat{\jmath}_{\omega >0}\rangle(V)
&=&
\langle \hat{\jmath}_{\omega >0}\rangle(V=0)
\nonumber
\\[6pt]
&& - \frac{2V{\cal N}_{\perp}}{La} \sum_{{|q| \leq \pi/(2a)} \atop
{|k| \leq \pi/(2a)} }
\left( {\cos(ka) \atop \sin(ka)} \right)
\Biggl\{
\lambda_{+,+}X^{c,s}
\left( \frac{1}{\omega-E_{+,+}} + \frac{1}{\omega-E_{-,-}} \right)
\\[6pt]
\label{C2}
&&
\phantom{
- \frac{2V{\cal N}_{\perp}}{La} \sum_{|q| \leq \pi/(2a)}
\left( {\cos(ka) \atop \sin(ka)} \right)
\Biggl\{
}
+
\lambda_{+,-}Y^{c,s}
\left( \frac{1}{\omega-E_{+,-}} + \frac{1}{\omega-E_{-,+}} \right)
\Biggr\} \; .
\nonumber
\end{eqnarray}
To determine~$X^{c,s}$ and $Y^{c,s}$ we have to evaluate
$\hat{V} \hat{u}_{k+q/2,\tau'}^+ \hat{l}_{k-q/2,\tau}^{\phantom{+}} |0\rangle$.
Since $\hat{V}$ is simple in terms of the original
operators~$\hat{u}_{k}$, $\hat{l}_{k}$ we first have to
apply the inverse transformation of eq.~(\ref{trafoforl}).
As next step we let~$\hat{V}$ act and then re-transform into
the operators for the Peierls subbands in the last step.
The calculation shows that only four combinations of currents
occur in~$X^{c,s}$ and $Y^{c,s}$, namely,
\begin{mathletters}
\begin{eqnarray}
X^{c,s}(k,q)&=&
- f_{+,+}^*(k,q) J_1^{c,s}(q)
+ f_{+,-}^*(k,q)J_2^{c,s}(q) \\[6pt]
Y^{c,s}(k,q)&=&
f_{+,-}^*(k,q) J_1^{c,s}(q)+
f_{+,+}^*(k,q) J_2^{c,s}(q)
\end{eqnarray}
\end{mathletters}%
with
\begin{mathletters}
\label{C5}
\begin{eqnarray}
J_1^{c,s}(q) &=& \frac{1}{L} \sum_{|p| \leq \pi/(2a)}
\left( {\cos(pa) \atop \sin(pa)}\right)
\left[ f_{+,+}(j_{-,-}-j_{+,+})+f_{+,-}(j_{+,-}+j_{-,+})\right]
\\[6pt]
J_2^{c,s}(q) &=& \frac{1}{L} \sum_{|p| \leq \pi/(2a)}
\left( {\cos(pa) \atop \sin(pa)}\right)
\left[ f_{+,+}(j_{+,-}+j_{-,+})-f_{+,-}(j_{-,-}-j_{+,+})\right] \; .
\end{eqnarray}
\end{mathletters}%
Since the interaction is restricted to nearest-neighbors
the global, i.~e., only $q$-dependent currents~$J_{1,2}^{c,s}(q)$ appear
in the problem.
Equation~(\ref{C2}) becomes
\begin{equation}
\langle \hat{\jmath}_{\omega >0}\rangle(V)
- \langle \hat{\jmath}_{\omega >0}\rangle(V=0)
=
- \frac{2V{\cal N}_{\perp}}{a} \sum_{|q| \leq \pi/(2a) }
\left[ J_1^{c,s}(q)G_1^{c,s}(q)+
J_2^{c,s}(q)G_2^{c,s}(q)\right]
\label{C4}
\end{equation}
with~$G_{1,2}^{c,s}(q)$ given by
\begin{eqnarray}
G_{1,2}^{c,s} (q) &=& \frac{1}{L} \sum_{|k|\leq \pi/(2a)}
\left( {\cos(ka) \atop \sin(ka)}\right) \biggl[
\left( {-f_{+,+}^* \atop f_{+,-}^*}\right)\lambda_{+,+}
\left(\frac{1}{\omega-E_{-,-}}+\frac{1}{\omega-E_{+,+}}\right)
\nonumber \\[6pt]
&& \phantom{ \frac{1}{L} \sum_{|k|\leq \pi/(2a)}
\left( {\cos(ka) \atop \sin(ka)}\right) \biggl[ }
+\left( {f_{+,-}^* \atop f_{+,+}^*} \right) \lambda_{+,-}
\left(\frac{1}{\omega-E_{-,+}}+\frac{1}{\omega-E_{+,-}}\right)
\biggr] \; .
\label{capitalG}
\end{eqnarray}
The quantities~$G_{1,2}^{c,s} (q)$ are still operator valued
objects since they contain $\lambda_{\tau,\tau'}$.
Nevertheless these quantities are known. We insert eq.~(\ref{thelambdas})
and use the fact that $|f_{+,+}(-k,q)|^2=|f_{+,+}(k,q)|^2$,
$|f_{+,-}(-k,q)|^2=|f_{+,-}(k,q)|^2$,
and $f_{+,+}^*(-k,q)f_{+,-}^{\phantom{*}}(-k,q)=
f_{+,+}^*(k,q)f_{+,-}^{\phantom{*}}(k,q)$
to show that $G_1^s(q)=G_2^c(q)=0$.
We set $G_1^c(q)\equiv G_1(q)$,
$G_2^s(q)\equiv G_2(q)$, $J_1^c(q)\equiv J_1(q)$,
$J_2^s(q)\equiv J_2(q)$.
They can be expressed in terms of the functions~$F_{1,2,3}(q)$
of eq.~(\ref{capitalF}) as
\begin{mathletters}
\label{CapitalGred}
\begin{eqnarray}
G_1(q)&=&
itea \left[ \hat{x}_q^+ F_1(q) - \hat{x}_{q+\pi/a}^+ F_3(q)\right]\\[6pt]
G_2(q)&=&
itea \left[ -\hat{x}_q^+ F_3(q) + \hat{x}_{q+\pi/a}^+ F_2(q)\right]\; .
\end{eqnarray}
\end{mathletters}%
It remains to determine~$J_{1,2}(q)$.
They can be obtained from their definitions
in eq.~(\ref{C5}) and the result from the equations of motion, eq.~(\ref{C1}),
\begin{mathletters}
\label{C6}
\begin{eqnarray}
J_{1}(q)+\frac{{\cal A}(\omega)}{c}G_1^{+}(q)
&=&
V \left( - J_1(q)F_1(q) + J_2(q) F_3 (q)\right)
\\[6pt]
J_{2}(q)+ \frac{{\cal A}(\omega)}{c}G_2^{+}(q)
&=&
V\left( - J_2(q) F_2(q) + J_1(q)F_3(q) \right)\; .
\end{eqnarray}
\end{mathletters}%
It is not difficult to invert these equations to obtain the
currents explicitly.
The result for the real part of the optical conductivity becomes
{\arraycolsep=0pt
\begin{eqnarray}
{\rm Re}\{\overline{\sigma(\omega >0, V, \delta,\eta)} \}
&=&
{\rm Re}\{\overline{\sigma(\omega >0,\delta,\eta)} \}
+ \frac{2V{\cal N}_{\perp}}{a\omega} \label{mostimportant}
\\[6pt]
&&{\rm Im}\Biggl\{ \sum_{|q| \leq \pi/(2a) }
\frac{1}{(1+VF_1)(1+VF_2)-(VF_3)^2}
\biggl[G_1^{\phantom{+}}G_1^+ + G_2^{\phantom{+}}G_2^+
\nonumber
\\[6pt]
&& \phantom{ {\rm Im}\biggl\{ \sum_{|q| \leq \pi/(2a) } }
+V \left(
G_1^{\phantom{+}}G_1^+F_2 + G_2^{\phantom{+}}G_2^+F_1
+ (G_1^{\phantom{+}}G_2^+ + G_2^{\phantom{+}}G_1^+)F_3
\right)\biggr]\Biggr\} \; .
\nonumber
\end{eqnarray}}
As a last step we have to factorize the products over the
functions~$G_{1,2}(q)$.
For the Harris-Lange model we use eq.~(\ref{averagexq}).
With the help of eq.~(\ref{CapitalGred}) it is not difficult
to derive the final result for the average optical conductivity
in the Harris-Lange model, eq.~(\ref{thefinalresultHL}).
\end{appendix}
\newpage
\begin{center} {\bf REFERENCES} \end{center}
\begin{itemize}
\item M.~Abramovitz and I.~A.~Stegun, {\sl Handbook of Mathematical Functions},
(9th~printing, Dover Publications, New York, (1970)).
\item L.~Alc\'{a}cer, and A.~Brau and J.-P.~Farges,
in: {\sl Organic Conductors}, ed. by J.-P.~Farges (Marcel Dekker, New York,
(1994)).
\item A.~A.~Aligia and L.~Arrachea, Phys.~Rev.~Lett.~{\bf 73}, 2240 (1994).
\item N.~Andrei, {\sl Integrable models in condensed matter physics},
in {\sl Proceedings of the summer school on field theoretical methods
(24~August -- 4~September 1992)}, ed.\ by Yu~Lu, (World Scientific, Singapore,
(1995)).
\item D.~Baeriswyl, P.~Horsch, and K.~Maki,
Phys.~Rev.~Lett.~{\bf 60} (C), 70 (1988).
\item P.-A.~Bares and F.~Gebhard, Europhys.~Lett.~{\bf 29}, 573, (1995).
\item P.-A.~Bares and F.~Gebhard, Journ.\ Low Temp.~Phys.~{\bf 99}, 565 (1995).
\item P.-A.~Bares and F.~Gebhard, J.~Phys.~Cond.~Matt.~{\bf 7}, 2285 (1995).
\item G.~Beni, P.~Pincus, and T.~Holstein, Phys.~Rev.~B~{\bf 8}, 312 (1973).
\item J.~Bernasconi, M.~J.~Rice, W.~R.~Schneider, and
S.~Str\"{a}\ss ler, Phys.~Rev.~B~{\bf 12}, 1090 (1975).
\item J.~D.~Bjorken and S.~D.~Drell, {\sl Relativistic Quantum Mechanics},
(McGraw-Hill, New York, (1964)).
\item J.~de Boer, V.~E.~Korepin, and A.~Schad\-schneider,
Phys.~Rev.~Lett.~{\bf 74}, 789 (1995).
\item D.~K.~Campbell, J.~T.~Gammel, and E.~Y.~Loh,
Phys.~Rev.~B~{\bf 38}, 12043 (1988).
\item D.~K.~Campbell, J.~T.~Gammel, and E.~Y.~Loh,
Int.~J.~Mod.~Phys.~B~{\bf 3}, 2131 (1989).
\item D.~K.~Campbell, J.~T.~Gammel, and E.~Y.~Loh,
in: {\sl Interacting Electrons in Reduced Dimensions},
ed.~by D.~Baeriswyl and D.~K.~Campbell (NATO ASI Series~B~{\bf 213},
Plenum Press, New York, (1989)), p.~171.
\item D.~K.~Campbell, J.~T.~Gammel, and E.~Y.~Loh,
Phys.~Rev.~B~{\bf 42}, 475 (1990).
\item P.~G.~J.~van Dongen, Phys.~Rev.~B~{\bf 49}, 7904 (1994).
\item P.~G.~J.~van Dongen, Phys.~Rev.~B~{\bf 50}, 14016 (1994).
\item F.~H.~L.~E\ss ler and V.~E.~Korepin (ed.),
{\sl Exactly Solvable Models of Strongly Correlated Electrons},
(World Scientific, Singapore, (1994)).
\item J.-P. Farges (ed.), {\sl Organic Conductors}, (Marcel Dekker, New York,
(1994)).
\item J.~L.~Fave, in: {\sl Electronic Properties of Polymers},
ed.~by H.~Kuzmany, M.~Mehring, and S.~Roth,
(Springer Series in Solid State Sciences~{\bf 107}, Springer, Berlin (1992)).
\item H.~Frahm and V.~E.~Korepin, Phys.~Rev.~B~{\bf 42}, 10553 (1990).
\item H.~Frahm and V.~E.~Korepin, Phys.~Rev.~B~{\bf 43}, 5653 (1991).
\item A.~Fritsch and L.~Ducasse, J.~Physique~I~{\bf 1}, 855 (1991).
\item R.~M.~Fye, M.~J.~Martins, D.~J.~Scalapino, J.~Wagner, and W.~Hanke,
Phys.~Rev.~B~{\bf 45}, 7311 (1992).
\item J.-P.~Galinar, J.~Phys.~C~{\bf 12}, L335 (1979).
\item J.~T.~Gammel and D.~K.~Campbell, Phys.~Rev.~Lett.~{\bf 60} (C),
71 (1988).
\item F.~Gebhard, K. Bott, M.~Scheidler, P.~Thomas, and S.~W.~Koch,
previous article, referred to as~I.
\item F.~Gebhard, K. Bott, M.~Scheidler, P.~Thomas, and S.~W.~Koch,
forthcoming article, referred to as~III.
\item F.~Gebhard, A.~Girndt, and A.~E.~Ruckenstein,
Phys.~Rev.~B~{\bf 49}, 10926 (1994)
\item F.~Gebhard and A.~E.~Ruckenstein, Phys.~Rev.~Lett.~{\bf 68}, 244 (1992).
\item I.~S.~Gradshteyhn and I.~M.~Ryzhik, {\sl A Table of Integrals, Series,
and Products}, (Academic Press, New York, (1980)).
\item D.~Guo, S.~Mazumdar, S.~N.~Dixit, F.~Kajzar, F.~Jarka, Y.~Kawabe,
and N.~Peyghambarian, Phys.~Rev.~B.~{\bf 48}, 1433 (1993).
\item F.~Guo, D.~Guo, and S.~Mazumdar, Phys.~Rev.~B.~{\bf 49}, 10102 (1994).
\item F.~D.~M.~Haldane, Phys.~Rev.~Lett.~{\bf 67}, 937 (1991).
\item E.~R.~Hansen, {\sl A Table of Series and Products},
(Prentice-Hall, Englewood Cliffs, (1975)).
\item A.~B.~Harris and R.~V.~Lange, Phys.~Rev.~{\bf 157}, 295 (1967).
\item H.~Haug and S.~W.~Koch, {\sl Quantum Theory of the Optical and
Electronic Properties of Semiconductors},
(World Scientific, Singapore, (1990)).
\item J.~Hubbard, Proc.~R.~Soc. London, Ser.~A~{\bf 276}, 238 (1963).
\item N.~Kawakami and S.-K.~Yang, Phys.~Lett.~{\bf A~148}, 359 (1990).
\item N.~Kawakami and S.-K.~Yang, Phys.\ Rev.\ Lett.~{\bf 65}, 3063 (1990).
\item N.~Kawakami and S.-K.~Yang, Phys.~Rev.~B~{\bf 44}, 7844 (1991).
\item S.~Kivelson, W.-P.~Su, J.~R.~Schrieffer, and A.~J.~Heeger,
Phys.~Rev.~Lett.~{\bf 58}, 1899 (1987).
\item S.~Kivelson, W.-P.~Su, J.~R.~Schrieffer, and A.~J.~Heeger,
Phys.~Rev.~Lett.~{\bf 60} (C), 72 (1988).
\item D.~J.~Klein, Phys.~Rev.~B~{\bf 8}, 3452 (1973).
\item W.~Kohn, Phys.~Rev.~{\bf 133}, A171 (1964).
\item E.~H.~Lieb and F.~Y.~Wu, Phys.~Rev.~Lett.~{\bf 20}, 1445 (1968).
\item S.~K.~Lyo and J.-P.~Galinar, J.~Phys.~C~{\bf 10}, 1693 (1977).
\item S.~K.~Lyo, Phys.~Rev.~B~{\bf 18}, 1854 (1978).
\item G.~D.~Mahan, {\sl Many-Particle Physics}, (2nd~edition, Plenum
Press, New York (1990)).
\item P.~F.~Maldague, Phys.~Rev.~B~{\bf 16}, 2437 (1977).
\item S.~Mazumdar and S.~N.~Dixit, Phys.~Rev.~B~{\bf 34}, 3683 (1986).
\item S.~Mazumdar and Z.~G.~Soos, Phys.~Rev.~B~{\bf 23}, 2810 (1981).
\item F.~Mila, Phys.~Rev.~B~{\bf 52}, 4788 (1995).
\item M.~Ogata and H.~Shiba, Phys.~Rev.~B~{\bf 41}, 2326 (1990).
\item A.~A.~Ovchinnicov, Zh.~Eksp.~Teor.~Fiz.~{\bf 57}, 2137
(1969) (Sov.~Phys.\ JETP~{\bf 30}, 1160 (1970)).
\item A.~Painelli and A.~Girlando, Synth.~Met.~{\bf 27}, A15 (1988).
\item A.~Painelli and A.~Girlando, in: {\sl Interacting Electrons in
Reduced Dimensions}, ed.~by D.~Baeriswyl and D.~K.~Campbell,
(NATO ASI Series~B~{\bf 213}, Plenum Press, New York, (1989)), p.~165.
\item A.~Painelli and A.~Girlando, Phys.~Rev.~B~{\bf 39}, 2830 (1989).
\item A.~Parola and S.~Sorella, Phys.~Rev.~Lett.~{\bf 60}, 1831 (1990).
\item L.~Salem, {\sl Molecular Orbital Theory of Conjugated Systems},
(Benjamin, London, (1966)).
\item A.~Schad\-schneider, Phys.~Rev.~B~{\bf 51}, 10386 (1995).
\item H.~J.~Schulz, Phys.~Rev.~Lett.~{\bf 64}, 2831 (1990).
\item H.~J.~Schulz, in: {\sl Strongly Correlated Electron Systems~II},
(Proceedings of the Adriatico Research Conference and Miniworkshop, Trieste,
Italy, 18~June -- 27~July~1990), ed.~by G.~Baskaran, A.~E.~Ruckenstein,
E.~Tosatti, and Y.~Lu, (World Scientific, Singapore, (1991)).
\item B.~S.~Shastry, S.~S.~Jha, and V.~Singh (ed.),
{\sl Exactly Solvable Problems in Condensed Matter and Relativistic
Field Theory}, (Lecture Notes in Physics~{\bf 242}, Springer, Berlin, (1985)).
\item B.~S.~Shastry and B.~Sutherland, Phys.~Rev.~Lett.~{\bf 65}, 243 (1990).
\item W.~T.~Simpson, J.~Am.~Chem.~Soc.~{\bf 73}, 5363 (1951).
\item W.~T.~Simpson, J.~Am.~Chem.~Soc.~{\bf 77}, 6164 (1955).
\item Z.~G.~Soos and S.~Ramesesha, Phys.~Rev.~B~{\bf 29}, 5410 (1984).
\item C.~A.~Stafford and A.~J.~Millis, Phys.~Rev.~B~{\bf 48}, 1409 (1993).
\item C.~A.~Stafford, A.~J.~Millis, and B.~S.~Shastry,
Phys.~Rev.~B~{\bf 43}, 13660 (1991).
\item P.~Tavan and K.~Schulten, J.~Chem.~Phys.~{\bf 85}, 6602 (1986).
\item C.-Q.~Wu, X.~Sun, and K.~Nasu, Phys.~Rev.~Lett.~{\bf 59}, 831 (1987).
\end{itemize}
\begin{figure}[th]
\caption{Band structure interpretation of the exact eigenenergies
of the Harris-Lange model for U=2W.}
\label{HarrisLangedis}
\end{figure}
\typeout{figure captions}
\begin{figure}[th]
\caption{Band structure interpretation of the exact eigenenergies
of the dimerized Harris-Lange model for U=2W, $\delta=0.2$.}
\label{HueckelHarrisLangedis}
\end{figure}
\begin{figure}[th]
\caption{Reduced average optical conductivity,
$\overline{\sigma_{\rm red}(\omega >0)}$,
in the Harris-Lange model for $U=2W$.
A broadening of~$\gamma=0.01W$ has been included.}
\label{hl00}
\end{figure}
\begin{figure}[th]
\caption{Reduced average optical conductivity,
$\overline{\sigma_{\rm red}(\omega >0,\delta,\eta)}$,
in the dimerized Harris-Lange model for $U=2W$,
$\delta=0.2$ ($\delta=0.6$),
and $\eta=-0.06$. A broadening of~$\gamma=0.01W$ has been included.}
\label{hl10}
\end{figure}
\begin{figure}[th]
\caption{Reduced average optical conductivity,
$\overline{\sigma_{\rm red}(\omega >0,V)}$,
in the extended Harris-Lange model for $U=2W$
and $V=0, W/2, W$.
A broadening of~$\gamma=0.01W$ has been included.}
\label{hl01}
\end{figure}
\begin{figure}[th]
\caption{Reduced average optical conductivity,
$\overline{\sigma_{\rm red}(\omega >0,V,\delta,\eta)}$,
in the extended dimerized Harris-Lange model for $U=2W$,
$\delta=0.2$, $\eta=-0.06$, $V=0,W/2,W$.
A broadening of~$\gamma=0.01W$ has been included.}
\label{hl11}
\end{figure}
\end{document}
| proofpile-arXiv_065-472 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{INTRODUCTION}
The analysis of primordial nucleosynthesis provides valuable limits on
cosmological and particle physics parameters through a comparison
between the predicted and inferred primordial abundances of D,
\he3, \he4, and \li7. For standard homogeneous big bang nucleosynthesis (HBBN)
the predicted primordial abundances of these
light-elements are in accord with the value inferred from observation
provided that baryon-to-photon ratio ($\equiv \eta$) is between about
$2.5 \times 10^{-10}$ and $6 \times 10^{-10}$. This corresponds to an
allowed range for the baryon fraction of the universal closure density
$\Omega_b^{\rm HBBN}$ (\cite{walker91}; \cite{smith93};
\cite{copi95}; \cite{ScMa95}),
\begin{equation}
0.04 \mathrel{\mathpalette\fun <} \hbox{$\Omega_b$}^{\rm HBBN}\,h_{50}^{2} \mathrel{\mathpalette\fun <} 0.08 ,
\label{eq:1}
\end{equation}
where $\eta = 6.6 \times 10^{-9} \Omega_b\,h_{50}^2$.
The lower limit on $\hbox{$\Omega_b$}^{\rm HBBN}$ arises mainly from the
upper limit on the deuterium plus $^3$He abundance (\cite{yang84};
\cite{walker91}; \cite{smith93}), and the upper limit to $\hbox{$\Omega_b$}$
arises from the upper limit on the \he4 mass fraction $\hbox{$Y_{\rm p}$}$ and/or the
deuterium abundance D/H $\ge 1.2 \times 10^{-5}$ (\cite{linsky93},\
\cite{linsky95}). Here, $h_{50}$ is the Hubble constant in units of 50 km s
$^{-1}$ Mpc$^{-1}$. The fact that this range for $\hbox{$\Omega_b$}\,h_{50}^2$ is
so much greater than the current upper limit to the contribution from
luminous matter $\hbox{$\Omega_b$}^{Lum} \mathrel{\mathpalette\fun <} 0.01$ (see however \cite{jedamzik95})
is one of the strongest arguments for the existence of baryonic dark matter.
Over the years HBBN has provided strong support for the standard,
hot big bang cosmological model as mentioned above. However, as the
astronomical data have become more precise in recent years, a possible
conflict between the predicted abundances of the light element isotopes
from HBBN and the abundances inferred from observations has been suggested
(\cite{olivestei95}; \cite{steigman96a}; \cite{turner96}; \cite{hata96};
see also \cite{hata95}).
There is now a good collection of abundance
information on the \he4 mass fraction, \hbox{$Y_{\rm p}$}, O/H, and N/H in over 50
extragalactic HII regions (\cite{pagel92}; \cite{pagel93}; \cite{izatov94};
\cite{skillman95}). In an extensive study based upon these observations,
the upper limit to $\eta$ from the observed \he4 abundance was found to be
$\sim 3.5 \times 10^{-10}$ (\cite{olivestei95}; \cite{olive96})
when a systematic
error in $\hbox{$Y_{\rm p}$}$ of $\Delta Y_{sys} = 0.005$ is adopted. Recently, it has been
recognized that the $\Delta Y_{sys}$ may even be factor of 2 or 3 larger
(\cite{thuan96}; \cite{copi95}; \cite{ScMa95}; \cite{sasselov95}), making
the upper limit to $\eta$ as large as $7 \times 10^{-10}$.
On the other hand, the lower bound to $\eta$ has been derived
directly from the upper
bound to the combined abundances of D and \he3. This is because
it is believed that deuterium is largely converted into \he3 in stars;
the lower bound then applies if, as has generally been assumed,
a significant fraction of \he3 survives stellar processing (\cite{walker91}).
However,
there is mounting evidence that low mass stars destroy \he3
(\cite{wasserburg95}; \cite{charbonnel95}),
although it is possible that massive stars produce \he3.
Therefore, the uncertainties of chemical evolution models render it difficult
to infer the primordial deuterium and \he3 abundances by using observations
of the present interstellar medium (ISM) or from the solar meteoritic
abundances. Recent data and analysis lead to a lower bound of
$\eta\,\mathrel{\mathpalette\fun >}\,3.5\,\times\,10^{-10}$ on the basis of D and \he3
(\cite{dearborn96}; \cite{hata96}; \cite{steigman96a}; \cite{steigman95}),
if the fraction of \he3 that survives stellar processing in the course of
galactic evolution exceeds $1/4$. This poses a potential conflict
between the observation ($\hbox{$Y_{\rm p}$}$ with low $\Delta Y_{sys}$, D) and HBBN.
In this context, possible detections (\cite{songaila94};
\cite{carswell94}; \cite{carswell96}; \cite{tytler94};
\cite{tytler96}; \cite{rugers96a},1996b; \cite{wampler96}) of an
isotope-shifted Lyman-$\alpha$ absorption line at high redshift ($z\,
\mathrel{\mathpalette\fun >}\,3$) along the line of sight to quasars are of considerable
interest. Quasar absorption systems can sample low metallicity gas at early
epochs where little destruction of D should have occurred. Thus, they should
give definitive measurements of the primordial cosmological D abundance.
A very recent high resolution detection by Rugers \& Hogan (1996a) suggests
a ratio D/H of
\begin{equation}
\rm{D}/\rm{H} = 1.9 \pm 0.4 \times 10^{-4}.
\label{eq:2}
\end{equation}
This result is consistent with the estimates made by Songaila
et al.~(1994) and Carswell et al.~(1994), using lower resolution.
It is also similar to that found recently in another absorption system by
Wampler et al.~(1996), but it is inconsistent with high resolution studies
in other systems at high redshift (Tytler, Fan \& Burles 1996; Burles \&
Tytler 1996) and with the local observations of
D and \he3 in the context of conventional models of stellar and
Galactic evolution (\cite{edmunds95}; \cite{gloeckler96}).
If the high value of D/H is taken to be the primordial
abundance, then the consistency between the observation and HBBN is
recovered and the allowed range of $\hbox{$\Omega_b$}$ inferred from HBBN changes
to $\hbox{$\Omega_b$}^{\rm HBBN}\,h_{50}^{2} = 0.024 \pm 0.002$
(\cite{jedamzik94a}; \cite{krauss94a}; \cite{vangioni95}).
In this case, particularly if $h_{50}$ is greater than $\sim 1.5$, the
big bang prediction could be so close to the baryonic density in luminous
matter that little or no baryonic dark matter is required
(\cite{persic92}; \cite{jedamzik95}).
This could be in contradiction with observation, particularly if the recently
detected microlensing events (Alcock et al.~1993, 1994, 1995abc; \cite{aubourg93})
are shown to be baryonic. This low baryonic density limit would also
be contrary to evidence (\cite{white93}; \cite{white95})
that baryons in the form of hot X-ray gas
may contribute a significant fraction of the closure density
The observations by Tytler, et al.~(1996) and Burles \&
Tytler (1996) yield a low value of D/H. Their average abundance is
\begin{equation}
\rm{D}/\rm{H} = 2.4 \pm 0.9 \times 10^{-5},
\label{eq:3}
\end{equation}
with $\pm 2\sigma$ statistical error and
$\pm 1\sigma$ systematic error. This value is consistent with the
expectations of local galactic chemical evolution.
However, this value would imply
an HBBN helium abundance of $\hbox{$Y_{\rm p}$} = 0.249 \pm 0.003$ which is only marginally
consistent with the observationally inferred $\hbox{$Y_{\rm p}$}$ even if the high $\Delta Y_{sys}$ is
adopted.
With this in mind, it is worthwhile to consider alternative
cosmological models. One of the most widely investigated
possibilities is that of an inhomogeneous density distribution
at the time of nucleosynthesis. Such studies were
initially motivated by speculation (\cite{witten84}; \cite{applegate85})
that a first order quark-hadron phase transition (at $T \sim 100
\mbox{~MeV}$) could produce baryon inhomogeneities as baryon number was
trapped within bubbles of shrinking quark-gluon plasma. In previous
calculations using the baryon inhomogeneous big bang nucleosynthesis
(IBBN) model, it has been usually assumed that the geometry of baryon density
fluctuations is approximated by condensed spheres. Such geometry might be
expected to result from a first order QCD phase transition in the limit that
the surface tension dominated the evolution of shrinking bubbles of
quark-gluon plasma. However, the surface tension may not be large
(\cite{kajantie90}, 1991, 1992) during the QCD transition, which could lead to a "shell"
geometry or the development of dendritic fingers (Freese \& Adams 1990).
Furthermore, such fluctuations might have been produced by a number
of other processes operating in the early universe (cf. \cite{malaney93}),
for which other geometries may be appropriate, e.g. strings, sheets, etc.
Thus, the shapes of any cosmological baryon inhomogeneities must be
regarded as uncertain.
The purpose of this paper is, therefore, to explore the sensitivity of the
predicted elemental abundances in IBBN models to the geometry of the
fluctuations. We consider here various structures and profiles for the
fluctuations in addition to condensed spheres. Mathews et al.~
(1990, 1994, 1996) found that placing the fluctuations in spherical
shells rather than condensed spheres allowed for lower calculated
abundances of \he4
and \li7 for the same $\hbox{$\Omega_b$}$, and that a condensed
spherical geometry is not necessarily the optimum. Here we show that
a cylindrical geometry also allows for an even higher baryonic contribution
to the closure density than that allowed by the usually adopted
condensed sphere. It appears to be a general result that shell geometries
allow for a slightly higher baryon density. This we attribute
to the fact that, for optimum parameters, shell geometries involve
a larger surface area to volume ratio and hence more efficient neutron
diffusion.
An important possible consequence of baryon inhomogeneities at the time
of nucleosynthesis may be the existence of unique nucleosynthetic
signatures. Among the possible observable signatures of baryon
inhomogeneities already pointed out in previous works are the high
abundances of heavier elements such as beryllium and boron
(\cite{boyd89};
\cite{kajino90a}; \cite{malaney89a}; Terasawa \& Sato 1990;
\cite{kawano91}), intermediate mass elements (\cite{kajino90b}), or
heavy elements (\cite{malaney88}; \cite{applegate88}; \cite{rauscher94}).
Such possible
signatures are also constrained, however, by the light-element
abundances. It was found in several previous calculations that the possible
abundances of synthesized heavier nuclei was quite small
(e.g., \cite{alcock90}; Terasawa \& Sato 1990; \cite{rauscher94}).
We find, however, that substantial production of heavier elements
may nevertheless be possible
in IBBN models with cylindrical geometry.
\section{BARYON DENSITY INHOMOGENEITIES}
After the initial suggestion (Witten 1985) of QCD motivated baryon inhomogeneities
it was quickly realized (\cite{applegate85}; \cite{applegate87}) that
the abundances of primordial nucleosynthesis could be affected.
A number of papers have addressed this point (\cite{alcock87};
\cite{applegate87}, 1988; \cite{fuller88};
\cite{kurki88}, 1990; \cite{terasawa89a}, \cite{terasawa89b},
\cite{terasawa89c}, \cite{terasawa90}; \cite{kurki89}, \cite{kurki90a};
\cite{mathews90}, \cite{mathews93b}; \cite{mathews96};
\cite{jedamzik94a}; \cite{jedamzik95}; \cite{thomas94}; \cite{rauscher94}).
Most recent studies in which the coupling between the baryon diffusion and
nucleosynthesis has been properly accounted for (e.g.,
\cite{terasawa89a},
\cite{terasawa89b}, \cite{terasawa89c}, \cite{terasawa90}; \cite{kurki89},
\cite{kurki90a}; \cite{mathews90}, \cite{mathews93b};
\cite{jedamzik94a}; \cite{thomas94}) have concluded that the upper limit on
$\hbox{$\Omega_b$}\,h^2$ is virtually unchanged when compared to the upper
limit on $\hbox{$\Omega_b$}\,h^2$ derived from standard HBBN. It is also
generally believed (e.g. \cite{vangioni95}) that the same holds true
if the new high D/H abundance is adopted.
However, in the previous studies, it
was usually assumed that a fluctuation geometry of centrally condensed
spheres produces the maximal impact on nucleosynthesis.
Here we emphasize that condensed spheres are not necessarily
the optimal nor the most physically motivated
fluctuation geometry.
Several recent lattice QCD calculations (\cite{kajantie90}, 1991,
1992; \cite{brower92}) indicate that the surface tension of nucleated
hadron bubbles is relatively low. In this case, after the hadron bubbles
have percolated, the structure of the regions remaining in the quark phase
may not form spherical droplets but
rather sheets or filaments. We do note that the significant effects on
nucleosynthesis may require a relatively strong first order phase transition
and sufficient surface tension to generate an optimum separation distance
between baryon fluctuations (\cite{fuller88}). However, even if the surface
tension is low, the dynamics of the coalescence of hadron droplets may
lead to a large separation between regions of shrinking
quark-gluon plasma. Furthermore, even though lattice QCD has not
provided convincing evidence for a strongly first order QCD phase
transition (e.g., \cite{fukugita91}), the order of the transition must
still be considered as uncertain (\cite{gottlieb91}; \cite{petersson93}).
It depends sensitively upon the number of light quark flavors. The
transition is first order for three or more light flavors and second
order for two. Because the $s$ quark mass is so close to the transition
temperature, it has been difficult to determine the order of transition.
At least two recent calculations (\cite{iwasaki95}; \cite{kanaya96}) indicate a clear
signature of a first order transition when realistic $u, d, s$ quark
masses are included, but others indicate either second order or no
phase transition at all.
In addition to the QCD phase transition, there remain a number of
alternative mechanisms for generating baryon inhomogeneities
prior to the nucleosynthesis epoch (cf.~\cite{malaney93}), such as
electroweak baryogenesis (\cite{fuller94}), inflation-generated
isocurvature fluctuations (\cite{dolgov93}), and kaon condensation
(\cite{nelson90}). Cosmic strings might also
induce baryon inhomogeneities through electromagnetic (\cite{malaney89b})
or gravitational interactions.
Since the structures, shapes, and origin of any baryon
inhomogeneities are uncertain,
a condensed spherical geometry is not necessarily the most
physically motivated choice. Indeed, we will show that a condensed spherical geometry
is also not necessarily the optimum to allow for the highest values
for $\hbox{$\Omega_b$}$ while still satisfying the light-element abundance
constraints. Here we consider the previously unexplored cylindrical geometry.
String geometries may naturally result from various baryogenesis scenarios such as
superconducting axion strings or cosmic strings. Also, the fact that
QCD is a string theory may predispose QCD-generated fluctuations
to string-like geometry (\cite{kajino93}; \cite{tassie93}).
Hence, cylindrical fluctuations may be a natural choice.
\section{OBSERVATIONAL CONSTRAINTS}
We adopt the following constraints on the
observed helium mass fraction $\hbox{$Y_{\rm p}$}$ and \li7 taken from
Balbes et al.~(1993), Schramm \& Mathews (1995), Copi et al.~(1995) and Olive (1996):
\begin{equation}
0.226 \leq \hbox{$Y_{\rm p}$} \leq 0.247,
\label{eq:4}
\end{equation}
\begin{equation}
0.7 \times 10^{-10} \leq \li7/\rm{H} \leq 3.5 \times 10^{-10}.
\label{eq:5}
\end{equation}
This primordial $\he4$ abundance constraint includes a
statistical uncertainty of $\pm 0.003$ and possible systematic errors
as much as $+0.01/-0.005$ with central value of $0.234$.
A recent reinvestigation (with new data) of the linear regression
method for estimating the primordial $\he4$ abundance has called into
question the systematic uncertainties assigned to $\hbox{$Y_{\rm p}$}$
(\cite{izatov96}). Our adopted upper limit to $\hbox{$Y_{\rm p}$}$ of Eq.~(\ref{eq:4}) is
essentially equal to the limit derived in their study with $1\sigma$
statistical error.
The upper limit to the lithium abundance adopted here includes the
systematic increase from the model atmospheres of Thorburn (1994)
and the possibility of as much as a factor of 2 increase due to stellar
destruction. This is consistent with the recent observations of \li6 in
halo stars (\cite{smith92}; \cite{hobbs94}).
We note that recent discussion of model atmospheres (\cite{kurucz95})
suggests that as much as an order magnitude upward shift in the
primordial lithium abundance could be warranted due to the tendency
of one-dimensional models to underestimate the ionization of lithium.
Furthermore, a
recent determination of the lithium abundance
in the globular cluster M92 having the metallicity [Fe/H] = -2.25
has indicated that at least one star out of seven shows
[Li] =\,12\,+\,log(Li/H) $\approx 2.5$ (\cite{boesgaard96b}).
Since the abundance measurement of the globular cluster stars is
more reliable than that of field stars, this detection along with the
possible depletion of lithium in stellar atmospheres suggests
that a lower limit to the primordial abundance is
$ 3.2 \times 10^{-10} \leq \li7/\rm{H}$.
There also remains the question as to why several stars which are
in all respects similar to the other stars in the Population~II
`lithium plateau', are so lithium rich or lithium deficient
(\cite{deliyannis96}; \cite{boesgaard96a}, 1996b).
Until this is clarified, it may be premature to
assert that the Population~II abundance of lithium reflects the primordial
value. The primordial abundance may instead correspond to the much higher
value observed in Population~I stars which has been depleted down to the
Population~II lithium plateau. The observational evidence (Deliyannis,
Pinsonneault \& Duncan 1993) for a $\pm\,25\,\%$
dispersion in the Population~II lithium plateau is consistent with
this hypothesis (\cite{deliyannis93}; \cite{charbonnel95}; \cite{steigmanli7}).
Rotational depletion was studied in detail by Pinsonneault et al.~
(1992) who note that the depletion factor could have been as large
as $10$. Chaboyer and Demarque (1994) also demonstrated that models
incorporating both rotation and diffusion provide a good match to
the observed \li7 depletion with decreasing temperature in
Population~II stars and their model indicated that the initial
lithium abundance could have been as high as
$\li7/\rm{H} = 1.23 \pm\,0.28 \times 10^{-9}$.
A recent study (\cite{ryan96}), which includes new data on 7 halo
dwarfs, fails to find evidence of significant depletion through
diffusion, although other mechanisms are not excluded. For example,
stellar wind-driven mass loss could deplete a high primordial lithium
abundance of down to the Population~II value [Eq.~(\ref{eq:5})] in a
manner consistent with \li6 observations (\cite{vauclair95}).
Furthermore, it could be possible (\cite{yoshii95}) that some of the \li6
is the result of more recent accretion of interstellar material that
could occur as halo stars episodically plunge through the disk. Such a
process could mask the earlier destruction of lithium.
For comparison, therefore, we adopt a conservative upper limit
on the primordial lithium abundance of
\begin{equation}
\li7/\rm{H} < 1.5 \times 10^{-9}.
\label{eq:6}
\end{equation}
Finally, the primordial abundance of deuterium is even harder to
clarify since it is easily destroyed in stars (at temperatures
exceeding about $6 \times 10^{5}$K). Previously, limits on the
deuterium (and\ also the\ \he3) abundances have been inferred from
their presence in
presolar material (e.g.,~\cite{walker91}). It is also inferred from
the detection in the local interstellar medium (ISM) through its
ultraviolet absorption lines in stellar spectra (\cite{mccullough92};
\cite{linsky93},\ 1995). The limit from ISM data is consistent with
that from abundances in presolar material. It has been argued that
there are no important astrophysical sources of deuterium
(\cite{epstein76}) and ongoing observational attempts to detect
signs of deuterium synthesis in the Galaxy are so far consistent with
this hypothesis (see~\cite{pasachoff89}). If this is indeed so, then the
lowest D abundance observed today should provide a lower bound to the
primordial abundance. Recent precise measurements by Linsky et al.~
(1995, 1993) using the {\it Hubble Space Telescope} implies
\begin{equation}
\rm{D}/\rm{H} > 1.2 \times 10^{-5}.
\label{eq:7}
\end{equation}
We adopt this as a lower limit to the primordial deuterium abundance
for the purposes of exploring the maximal cosmological impact from IBBN.
In addition, we consider the two possible detections of the
deuterium abundance along the line of sight to high red shifted quasars,
Eqs.~(\ref{eq:2}) and (\ref{eq:3}) as possible limits.
In order to derive a lower limit to $\hbox{$\Omega_b$}\,h_{50}^2$, it is useful
to consider the sum of deuterium plus \he3. In the context of a
closed-box instantaneous recycling approximation, it is straightforward
(\cite{olive90}) to show that the sum of primordial deuterium and \he3
can be written
\begin{equation}
y_{23p} \le A_\odot^{(g_3 - 1)}y_{23\odot} \biggl({ X_\odot
\over X_p} \biggr)
\label{eq:8}
\end{equation}
where $A_\odot$ is the fraction of the initial primordial deuterium still
present when the solar system formed, $g_3$ is the fraction of \he3
that survives incorporation into a single generation of stars, $y_{23\odot}$
is the presolar value of [D$+$\he3]/H inferred from the
gas rich meteorites, and $X_\odot/X_p$ is the ratio of the
presolar hydrogen mass fraction to the primordial value.
These factors together imply an upper limit (\cite{walker91};
\cite{copi95}) of
\begin{equation}
y_{23p} \le 1.1 \times 10^{-4}.
\label{eq:9}
\end{equation}
\section{CALCULATIONS}
The calculations described here are based upon the coupled diffusion and
nucleosynthesis code of Mathews et al.~(1990), but with a number of
nuclear reaction rates updated and the numerical diffusion scheme
modified to accommodate cylindrical geometry. We also have implemented an
improved numerical scheme which gives a more accurate description of
the effects of proton and ion diffusion, and Compton drag
at late times. Although our approach is not as sophisticated as that
of Jedamzik et al.~(1994a), it produces essentially the same
results for the parameters employed here. We have also included all of the
new nuclear reaction rates summarized in Smith et al.~(1993) as
well as those given in Thomas et al.~(1993).
We obtain the same result as Smith et al.~(1993)
using these rates and homogeneous conditions in our IBBN model
Calculations were performed in a cylindrical geometry both with the high
density regions in the center (condensed cylinders), and with the high
density regions in the outer zone of computation (cylindrical shells).
Similarly, calculations were made in a spherical geometry with the high density
regions in the center (condensed spheres) and with the high density region
in the outer zones of computation (spherical shells).
In the calculations, the fluctuations are resolved into 16 zones of
variable width as described by Mathews et al.~(1990). We assumed
three neutrino flavors and an initially
homogeneous density within the fluctuations. Such
fluctuation shapes are the most likely to emerge, for example, after
neutrino-induced expansion (\cite{jedamzik94}). We use a neutron
mean life-time of $\tau_{n} = 887.0$ (\cite{particle94}). In addition to
the cosmological parameter, $\hbox{$\Omega_b$}$ and fluctuation geometry, there
remain three parameters to specify the baryon inhomogeneity. They are:
$R$, the density contrast between the high and low-density regions;
$f_{v}$, the volume fraction of the high-density region; and $r$, the
average separation distance between fluctuations.
\section{RESULTS}
The parameters $R$ and $f_{v}$ were optimized to allow for the highest
values for $\hbox{$\Omega_b$}\,h_{50}^{2}$ while still satisfying the
light-element abundance constraints. For fluctuations represented by
condensed spheres, optimum parameters are $R \sim 10^{6}$ and
$f_v^{1/3} \sim 0.5$ (\cite{mathews96}). For other fluctuation
geometries, we have found that optimum parameters are:
\begin{mathletters}
\begin{eqnarray*}
R &\sim & 10^{6}; \qquad \qquad \mbox{for all fluctuation geometries}
\\
\\
f_{v}^{1/3} &\sim & 0.19; \qquad \quad \mbox{~~} \mbox{for spherical shells}
\\
\\
f_{v}^{1/2} &\sim &\left\{
\begin{array}{rl}
0.5;& \quad \quad \mbox{for condensed cylinders} \\
0.15;& \quad \quad \mbox{for cylindrical shells},
\end{array}\right.
\end{eqnarray*}
\mbox{although there is not much sensitivity to $R$ once
$R\,\mathrel{\mathpalette\fun >}\, 10^3$.}
\end{mathletters}
Regarding $f_v$, we have written
the appropriate length scale of high density regions, i.e.
$f_v^{1/3}$ and $f_v^{1/2}$ for the spherical and cylindrical
fluctuation geometries, respectively.
The variable parameters in the calculation are then the fluctuation
cell radius $r$, and the total baryon-to-photon ratio $\eta$ (or
$\hbox{$\Omega_b$}\,h_{50}^{2}$).
\subsection{Constraints on \hbox{$\Omega_b$}$h_{50}^2$}
Figures~\ref{fig:1}, \ref{fig:2}, \ref{fig:3a}, and \ref{fig:4a} show contours
of allowed parameters
in the $r$ versus $\eta$ and $r$ versus $\hbox{$\Omega_b$}\,h_{50}^{2}$ plane for
the adopted light-element abundance constraints of Eqs.~(\ref{eq:4}) -
(\ref{eq:6}) and for a possible Lyman-$\alpha$ D/H of Eqs.~(\ref{eq:2})
and (\ref{eq:3}), for the condensed sphere, spherical shell,
condensed cylinder, and cylindrical shell fluctuation geometries,
respectively.
The fluctuation cell radius $r$ is given in units of meters for a comoving
length scale fixed at a temperature of $kT = 1 \mbox{~MeV}$.
Both of the possible
\li7 limits, Eqs.~(\ref{eq:5}) and (\ref{eq:6}) which we have discussed above,
are also drawn as indicated.
In order to clearly distinguish the two abundance constraints,
we use the single and double-cross hatches for the regions allowed
by the adopted lower (Eq.~(\ref{eq:5})) and higher (Eq.~(\ref{eq:6}))
limits to the \li7 primordial abundance.
Even in the IBBN scenario, if the low D/H of Eq.~(\ref{eq:3})
(\cite{burles96}) is adopted as primordial, this range for D/H appears
to be compatible with the \li7 abundance only when a higher (Population~I)
primordial \li7 abundance limit is adopted,
except for a very narrow region of $\eta \sim 6 \times 10^{-10}$ and
$r \leq 10^{2}\,m$.
This conclusion remains unchanged for any other fluctuation
geometries. Therefore, the acceptance of the low
(Burles \& Tytler 1996) value of D/H would
strongly suggest that significant depletion of \li7 has occurred.
In contrast, adoption of the high D/H of Eq.~(\ref{eq:2})
(\cite{rugers96a}) as primordial allows the concordance of all
light-elements. The upper limits to $\eta$ and $\hbox{$\Omega_b$}\,h_{50}^{2}$
are largely determined by D and $\li7$. The concordance range for the
baryon density is comparable to that for HBBN for small separation distance
$r$. However, there exist other regions of the parameter space with optimum
separation distance, which roughly corresponds to the neutron diffusion length
during nucleosynthesis (\cite{mathews90}), with an increased maximum allowable
value of the baryonic contribution to the closure density to
$\hbox{$\Omega_b$}\,h_{50}^{2} \leq 0.05$ for the cylindrical geometry,
as displayed in Fig.~\ref{fig:4a}.
This is similar to the value for spherical shells as shown in
Mathews et al.~(1996) and also in Fig.~\ref{fig:2} in the present work.
The condensed sphere limits, however, are
essentially unchanged from those of the HBBN model.
If the primordial \li7 abundance could be as high as the upper limit
of $\mbox{Li/H} \leq 1.5 \times 10^{-9}$, the maximum allowable
value of the baryonic content for the condensed sphere would increase to
$\hbox{$\Omega_b$}\,h_{50}^{2} \leq 0.08$, with similar values
for the spherical shell (\cite{mathews96}). For both the condensed
cylinders and cylindrical shells, the upper limits could be as high as
$\hbox{$\Omega_b$}\,h_{50}^{2} \leq 0.1$ as shown in Figs.~\ref{fig:3a} and
\ref{fig:4a}. These higher upper limits relative to those of the HBBN are of interest
since they are consistent with the inferred baryonic mass in the form of
hot X-ray gas (\cite{white93}; \cite{white95}) in dense galactic clusters.
The acceptance of this consistence, as noted above, requires the significant
stellar depletion of \li7.
In Figures~\ref{fig:3b} and \ref{fig:4b}, we also show contours for the
condensed cylinder and cylindrical shell geometries, respectively, but this time
with the conventional light-element constraints of
Eqs.~(\ref{eq:4}), (5), (7), and (9)
as indicated. Since the results for the condensed sphere and
spherical shell geometries with this set of the conventional abundance
constraints have already been discussed by
Mathews et al.~(1996), we do not show those contours here.
The cylindrical shell geometry of the present work gives the highest
allowed value of $\hbox{$\Omega_b$}\,h_{50}^{2}$.
Figure~\ref{fig:4b} shows that the
upper limits to $\eta$ and $\hbox{$\Omega_b$}\,h_{50}^{2}$ are largely determined
by $\hbox{$Y_{\rm p}$}$ and $\li7$.
The upper limits for a cylindrical shell geometry could be as high as
$\hbox{$\Omega_b$}\,h_{50}^{2} \leq 0.13$ with similar results for the spherical
shell geometry (\cite{mathews96}). A high primordial lithium abundance
would increase the allowable baryonic content to as high as
$\hbox{$\Omega_b$}\,h_{50}^{2} \leq 0.2$. The reason that shell geometries allow for
higher baryon densities we attribute to more efficient neutron diffusion which
occurs when the surface area to volume area is increased. This
allows for more initial diffusion to produce deuterium, and
more efficient back diffusion to avoid over producing \li7.
\subsection{Observational Signature}
The production of beryllium and boron
as well as lithium in IBBN models can be sensitive
to neutron diffusion. Therefore, their predicted abundances
are sensitive to not only the fluctuation parameter $r$, $R$, and
$f_{v}$ but also the fluctuation geometry (\cite{boyd89};
\cite{malaney89a}; \cite{kajino90a}; Terasawa \& Sato 1990).
Figures~\ref{fig:5} - \ref{fig:7} show the contours of the
calculated abundances for lithium, beryllium and boron, respectively
in the $r$ versus $\eta$ (and $r$ versus $\hbox{$\Omega_b$}~h_{50}^{2}$) plane.
the shaded region depict is allowed values of $r$ and $\eta$
from the light element
abundance constraints [cf. Fig.~\ref{fig:4b}] for a cylindrical
shell fluctuation geometry.
The contour patterns of lithium (Fig.~\ref{fig:5}) and boron
(Fig.~\ref{fig:7})
abundances are very similar, whereas there is no similarity found
between lithium (Fig.~\ref{fig:5}) and beryllium (Fig.~\ref{fig:6}) abundances.
In order to understand the similarities and differences among
these three elemental abundances, we show in
Figs.~\ref{fig:8} and \ref{fig:9} the decompositions of
the A = 7 abundance into \li7 and \be7 and the boron abundance
into $^{10}$B and $^{11}$B.
These Figures show also the
dependence of the predicted LiBeB abundances in IBBN on the scale of
fluctuations for a cylindrical shell geometry with fixed
$\hbox{$\Omega_b$}\,h_{50}^{2} = 0.1$. This value of
$\hbox{$\Omega_b$}\,h_{50}^{2}$ corresponds to a typical value in the allowable
range of $\eta$ in Fig.~\ref{fig:4b}, which optimizes
the light element abundance constraints, even
satisfying the lower \li7 abundance limit of Eq.~(\ref{eq:5}).
The fluctuation parameters $f_{v}$ and $R$ are
the same as in Fig.~\ref{fig:4b}.
Once the baryonic content $\hbox{$\Omega_b$}$ is fixed, the only variable parameter
is the separation distance, $r$.
As can be seen in Fig.~\ref{fig:8}, as the separation $r$ increases,
neutron diffusion plays an increasingly important role in the production
of $t$ and, by the \he4($t, \gamma$)\li7 reaction.
It works maximally around
$r \sim 10^{4}$~m, which is the typical length scale of neutron
diffusion at $kT = 1 \mbox{~MeV}$.
A similar behavior is observed
in the \li7($t, n$)\be9 reaction. This reaction produces
most of the \be9
in neutron rich environments where $t$ and \li7 are abundant, as
was first pointed out by Boyd and Kajino (1989).
At other separation distances $r$ in a $\hbox{$\Omega_b$}\,h_{50}^{2} = 0.1$
model, most of the A = 7 nuclides are created
as \be7 by the \he4(\he3, $\gamma$)\be7 reaction.
In the limit of $r$ = horizon scale, the nucleosynthesis
products are approximately equal to the sum of those produced
in the proton-rich and neutron-rich zones separately
(\cite{jedamzik94b}).
The predominant contribution from the proton-rich
zones makes the \be7 abundance almost constant at larger $r$,
while both \li7 and \be9 decrease as $r$ increases toward
the horizon at any separation distance.
Figure~\ref{fig:9} shows that $^{11}$B is a predominant component of the total
boron abundance at any separation distance. This is true for
almost all values $\hbox{$\Omega_b$}\,h_{50}^{2}$.
It has been pointed out (\cite{malaney88}; \cite{applegate88};
\cite{kajino90a}) that most
$^{11}$B is produced by the \li7($n, \gamma$)\li8($\alpha, n$)$^{11}$B
reaction sequence in neutron-rich environments at relatively
early times when most of the other
heavier nuclides are made. Recent measurements of the
previously unmeasured
\li7($\alpha$,n)$^{11}$B reaction cross section (\cite{boyd92};
\cite{gu95}; \cite{boyd96}) at the energies of cosmological interest have
removed the significant ambiguity
in the calculated $^{11}$B abundance due to this reaction.
The factor of two discrepancy among several different measurements of the
reaction cross section for \li7($n, \gamma$)\li8 was also
resolved by the new measurement (\cite{nagai91}).
The \li7($\alpha, \gamma$)$^{11}$B reaction
also makes an appreciable but weaker contribution to the production of
$^{11}$B in the neutron-rich environment.
In the proton-rich environment, on the other hand, the
\be7($\alpha, \gamma$)$^{11}$C reaction contributes largely
to the production of $^{11}$C which beta decays to $^{11}$B in 20.39 min.
These facts explain why the contour patterns of the lithium
and boron abundances in Figs.~\ref{fig:5} and \ref{fig:7} look very similar.
It is conventional in the literature to quote the beryllium and boron
abundance relative to H $=10^{2}$. Hence, one defines the quantity
[X] $= 12+\mbox{log(X/H)}$.
In cylindrical shell fluctuation geometry
the beryllium abundance can take the
value of $\mbox{[Be]} \sim -3$ while still satisfying all of the
light-element abundance constraints and the Population~II lithium
abundance constraint
(Figs.~\ref{fig:5} and \ref{fig:6}). This abundance is higher by three orders
magnitude than
that produced in the HBBN model with conventional light-element
abundance constraints. This result is contrary to a recent result
with the condensed sphere geometry and for a more
restricted parameter space (\cite{thomas94}).
Recent beryllium observation of Population~II stars (\cite{rebolo88};
\cite{ryan90},\ 1992; \cite{ryan96a}; \cite{gilmore92a},\ 1992b;
\cite{boesgaard93}; \cite{boesgaard94},\ 1996a,b) have placed the upper
limit on the primordial \be9 abundance to $\mbox{[Be]} \sim -2$,
one order magnitude greater than the beryllium abundance
in the IBBN cylindrical model.
The calculated boron abundance at the optimum separation distance is
essentially equal to the value of the HBBN model.
However, a high primordial lithium
abundance would increase the upper limit to $\hbox{$\Omega_b$}\,h_{50}^{2}$.
In this case, the boron
abundance could be one or two orders magnitude larger than that
of the HBBN model (Fig.~\ref{fig:7}).
\section{CONCLUSIONS}
We have reinvestigated the upper limit to $\eta$ and
$\hbox{$\Omega_b$}\,h_{50}^{2}$ in inhomogeneous primordial nucleosynthesis
models. We have considered effects of various geometries. In particular,
for the first time we consider cylindrical geometry. We have also
incorporated recently revised light-element abundance constraints including
implications of the possible detection
(\cite{songaila94}; \cite{carswell94}; \cite{carswell96}; \cite{tytler94};
\cite{tytler96}; \cite{rugers96a}, 1996b; \cite{wampler96}) of
a high deuterium
abundance in Lyman-$\alpha$ absorption systems. We have shown that
with low primordial deuterium (\cite{tytler94}; \cite{tytler96}),
significant depletion of \li7 is required to obtain
concordance between predicted light-element
abundance of any model of BBN and the observationally inferred primordial
abundance. If high primordial deuterium (\cite{rugers96a}) is adopted
(Eq.~(\ref{eq:2})),
there is a concordance range which is largely determined by D/H, and the
upper limit to $\hbox{$\Omega_b$}\,h_{50}^{2}$ is 0.05.
However, with the presently adopted (Eqs.~(\ref{eq:4}), (\ref{eq:6}),
(\ref{eq:7}), (\ref{eq:9}))
light-element abundance constraints (\cite{ScMa95}; \cite{copi95};
\cite{olive96}), values of $\hbox{$\Omega_b$}\,h_{50}^{2}$ as large as 0.2 are
possible in IBBN models with cylindrical-shell fluctuation geometry.
We have also found that significant beryllium and boron production is possible
in IBBN models without violating the light element abundance constraints.
The search for the primordial abundance of these elements in low metallicity
stars could, therefore, be a definitive
indicator of the presence or absence of cylindrical baryon inhomogeneities in the
early universe.
| proofpile-arXiv_065-473 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
Recently much attention has been given to lower dimensional gauge theories.
Such remarkable results as the chiral symmetry breaking \cite{1}, quantum
Hall effect \cite{2}, spontaneously broken Lorentz invariance by the
dynamical generation of a magnetic field \cite{3}, and the connection
between non-perturbative effects in low-energy strong interactions and QCD$%
_{2}$ \cite{4}, show the broad range of applicability of these theories.
In particular, 2+1 dimensional gauge theories with fractional statistics
-anyon systems \cite{4a}- have been extensively studied. One main reason for
such an interest has been the belief that a strongly correlated electron
system in two dimensions can be described by an effective field theory of
anyons \cite{5}, \cite{5a}. Besides, it has been claimed that anyons could
play a basic role in high-T$_{C}$ superconductivity \cite{5a}-\cite{6b}. It
is known \cite{a} that a charged anyon system in two spatial dimensions can
be modeled by means of a 2+1 dimensional Maxwell-Chern-Simons (MCS) theory.
An important feature of this theory is that it violates parity and
time-reversal invariance. However, at present no experimental evidences of P
and T violation in high-T$_{C}$ superconductivity have been found. It should
be pointed out, nevertheless, that it is possible to construct more
sophisticated P and T invariant anyonic models\cite{6a}. In any case,
whether linked to high-T$_{C}$ superconductivity or not, the anyon system is
an interesting theoretical model in its own right.
The superconducting behavior of anyon systems at $T=0$ has been investigated
by many authors \cite{6}-\cite{15a}. Crucial to the existence of anyon
superconductivity at $T=0$ is the exact cancellation between the bare and
induced Chern-Simons terms in the effective action of the theory.
Although a general consensus exists regarding the superconductivity of anyon
systems at zero temperature, a similar consensus at finite temperature is
yet to be achieved \cite{8}-\cite{16}. Some authors (see ref. \cite{9}) have
concluded that the superconductivity is lost at $T\neq 0$, based upon the
appearance of a temperature-dependent correction to the induced Chern-Simons
coefficient that is not cancelled out by the bare term. In ref. \cite{11} it
is argued, however, that this finite temperature correction is numerically
negligible at $T<200$ $K$, and that the main reason for the lack of a
Meissner effect is the development of a pole $\sim \left( \frac{1}{{\bf k}%
^{2}}\right) $ in the polarization operator component $\Pi _{00}$ at $T\neq
0 $. There, it is discussed how the existence of this pole leads to a so
called partial Meissner effect with a constant magnetic field penetration
throughout the sample that appreciably increases with temperature. On the
other hand, in ref. \cite{8}, it has been independently claimed that the
anyon model cannot superconduct at finite temperature due to the existence
of a long-range mode, found inside the infinite bulk at $T\neq 0$. The long
range mode found in ref. \cite{8} is also a consequence of the existence of
a pole $\sim \left( \frac{1}{{\bf k}^{2}}\right) $ in the polarization
operator component $\Pi _{00}$ at $T\neq 0$.
The apparent lack of superconductivity at temperatures greater than zero has
been considered as a discouraging property of anyon models. Nevertheless, it
may be still premature to disregard the anyons as a feasible solution for
explaining high -T$_{c}$ superconductivity, at least if the reason
sustaining such a belief is the absence of the Meissner effect at finite
temperature. As it was shown in a previous paper \cite{16}, the lack of a
Meissner effect, reported in ref. \cite{11} for the case of a half-plane
sample as a partial Meissner effect, is a direct consequence of the omission
of the sample boundary effects in the calculations of the minimal solution
for the magnetic field within the sample. To understand this remark we must
take into account that the results of ref. \cite{11} were obtained by
finding the magnetization in the bulk due to an externally applied magnetic
field at the boundary of a half-plane sample. However, in doing so, a
uniform magnetization was assumed and therefore the boundary effects were
indeed neglected. Besides, in ref. \cite{11} the field equations were solved
considering only one short-range mode of propagation for the magnetic field,
while as has been emphasized in our previous letter \cite{16}, there is a
second short-range mode whose qualitative contribution to the solutions of
the field equations cannot be ignored.
In the present paper we study the effects of the sample's boundaries in the
magnetic response of the anyon fluid at finite temperature. This is done by
considering a sample shaped as an infinite strip. When a constant and
homogeneous external magnetic field, which is perpendicular to the sample
plane, is applied at the boundaries of the strip, two different magnetic
responses, depending on the temperature values, can be identified. At
temperatures smaller than the fermion energy gap inherent to the
many-particle MCS model ($T\ll \omega _{c}$), the system exhibits a Meissner
effect. In this case the magnetic field cannot penetrate the bulk farther
than a very short distance ($\overline{\lambda }\sim 10^{-5}cm$ for electron
densities characteristic of the high -T$_{c}$ superconductors and $T\sim 200$
$K$). On the other hand, as it is natural to expect from a physical point of
view, when the temperatures are larger than the energy gap ($T\gg \omega
_{c} $) the Meissner effect is lost. In this temperature region a periodic
inhomogeneous magnetic field is present within the bulk.
These results, together with those previously reported in ref. \cite{16},
indicate that, contrary to some authors' belief, the superconducting
behavior (more precisely, the Meissner effect), found in the charged anyon
fluid at $T=0$, does not disappear as soon as the system is heated.
As it is shown below, the presence of boundaries can affect the dynamics of
the system in such a way that the mode that accounts for a homogeneous field
penetration \cite{8} cannot propagate in the bulk. Although these results
have been proved for two types of samples, the half-plane \cite{16} and the
infinite strip reported in this paper, we conjecture that similar effects
should also exist in other geometries.
Our main conclusion is that the magnetic behavior of the anyon fluid is not
just determined by its bulk properties, but it is essentially affected by
the sample boundary conditions. The importance of the boundary conditions in
2+1 dimensional models has been previously stressed in ref.\cite{b}.
The plan for the paper is as follows. In Sec. 2, for completeness as well as
for the convenience of the reader, we define the many-particle 2+1
dimensional MCS model used to describe the charged anyon fluid, and briefly
review its main characteristics. In Sec. 3 we study the magnetic response in
the self-consistent field approximation of a charged anyon fluid confined to
an infinite-strip, finding the analytical solution of the MCS field
equations that satisfies the boundary conditions. The fermion contribution
in this approximation is given by the corresponding polarization operators
at $T\neq 0$ in the background of a many-particle induced Chern-Simons
magnetic field. Using these polarization operators in the low temperature
approximation ($T\ll \omega _{c}$), we determine the system's two London
penetration depths. Taking into account that the boundary conditions are not
enough to completely determine the magnetic field solution within the
sample, an extra physical condition, the minimization of the system
free-energy density, is imposed. This is done in Sec. 4. In this section we
prove that even though the electromagnetic field has a long-range mode of
propagation in the charged anyon fluid at $T\neq 0$ \cite{8}, a constant and
uniform magnetic field applied at the sample's boundaries cannot propagate
through this mode. The explicit temperature dependence at $T\ll \omega _{c}$
of all the coefficients appearing in the magnetic field solution, and of the
effective London penetration depth are also found. In Sec. 5, we discuss how
the superconducting behavior of the charged anyon fluid disappears at
temperatures larger than the energy gap ($T\gg \omega _{c}$). Sec. 6
contains the summary and discussion.
\section{MCS Many-Particle Model}
The Lagrangian density of the 2+1 dimensional non-relativistic charged MCS
system is
\begin{equation}
{\cal L}=-\frac{1}{4}F_{\mu \nu }^{2}-\frac{N}{4\pi }\varepsilon ^{\mu \nu
\rho }a_{\mu }\partial _{\nu }a_{\rho }+en_{e}A_{0}+i\psi ^{\dagger
}D_{0}\psi -\frac{1}{2m}\left| D_{k}\psi \right| ^{2}+\psi ^{\dagger }\mu
\psi \eqnum{2.1}
\end{equation}
where $A_{\mu }$ and $a_{\mu }$ represent the electromagnetic and the
Chern-Simons fields respectively. The role of the Chern-Simons fields is
simply to change the quantum statistics of the matter field, thus, they do
not have an independent dynamics. $\psi $ represents non-relativistic
spinless fermions. $N\ $ is a positive integer that determines the magnitude
of the Chern-$%
\mathop{\rm Si}%
$mons coupling constant. The charged character of the system is implemented
by introducing a chemical potential $\mu $; $n_{e}$ is a background
neutralizing `classical' charge density, and $m$ is the fermion mass. We
will consider throughout the paper the metric $g_{\mu \nu }$=$(1,-%
\overrightarrow{1})$. The covariant derivative $D_{\nu }$ is given by
\begin{equation}
D_{\nu }=\partial _{\nu }+i\left( a_{\nu }+eA_{\nu }\right) ,\qquad \nu
=0,1,2 \eqnum{2.2}
\end{equation}
It is known that to guarantee the system neutrality in the presence of a
different from zero fermion density $\left( n_{e}\neq 0\right) $,$\ $a
nontrivial background of Chern-Simons magnetic field $\left( \overline{b}=%
\overline{f}_{21}\right) $ is generated. The Chern-Simons background field
is obtained as the solution of the mean field Euler-Lagrange equations
derived from (2.1)
\begin{mathletters}
\begin{equation}
-\frac{N}{4\pi }\varepsilon ^{\mu \nu \rho }f_{\nu \rho }=\left\langle
j^{\mu }\right\rangle \eqnum{2.3}
\end{equation}
\begin{equation}
\partial _{\nu }F^{\mu \nu }=e\left\langle j^{\mu }\right\rangle
-en_{e}\delta ^{\mu 0} \eqnum{2.4}
\end{equation}
considering that the system formed by the electron fluid and the background
charge $n_{e}$ is neutral
\end{mathletters}
\begin{equation}
\left\langle j^{0}\right\rangle -n_{e}\delta ^{\mu 0}=0 \eqnum{2.5}
\end{equation}
In eq. (2.5) $\left\langle j^{0}\right\rangle $ is the fermion density of
the many-particle fermion system
\begin{equation}
\left\langle j^{0}\right\rangle =\frac{\partial \Omega }{\partial \mu },
\eqnum{2.6}
\end{equation}
$\Omega $ is the fermion thermodynamic potential.
In this approximation it is found from (2.3)-(2.5) that the Chern-Simons
magnetic background is given by
\begin{equation}
\overline{b}=\frac{2\pi n_{e}}{N} \eqnum{2.7}
\end{equation}
Then, the unperturbed one-particle Hamiltonian of the matter field
represents a particle in the background of the Chern-Simons magnetic field $%
\overline{b\text{,}}$
\begin{equation}
H_{0}=-\frac{1}{2m}\left[ \left( \partial _{1}+i\overline{b}x_{2}\right)
^{2}+\partial _{2}^{2}\right] \eqnum{2.8}
\end{equation}
In (2.8) we considered the background Chern-Simons potential, $\overline{a}%
_{k}$, $(k=1,2)$, in the Landau gauge
\begin{equation}
\overline{a}_{k}=\overline{b}x_{2}\delta _{k1} \eqnum{2.9}
\end{equation}
The eigenvalue problem defined by the Hamiltonian (2.8) with periodic
boundary conditions in the $x_{1}$-direction: $\Psi \left( x_{1}+L,\text{ }%
x_{2}\right) =$ $\Psi \left( x_{1},\text{ }x_{2}\right) $,
\begin{equation}
H_{0}\Psi _{nk}=\epsilon _{n}\Psi _{nk},\qquad n=0,1,2,...\text{ }and\text{ }%
k\in {\cal Z} \eqnum{2.10}
\end{equation}
has eigenvalues and eigenfunctions given respectively by
\begin{equation}
\epsilon _{n}=\left( n+\frac{1}{2}\right) \omega _{c}\qquad \eqnum{2.11}
\end{equation}
\begin{equation}
\Psi _{nk}=\frac{\overline{b}^{1/4}}{\sqrt{L}}\exp \left( -2\pi
ikx_{1}/L\right) \varphi _{n}\left( x_{2}\sqrt{\overline{b}}-\frac{2\pi k}{L%
\sqrt{\overline{b}}}\right) \eqnum{2.12}
\end{equation}
where $\omega _{c}=\overline{b}/m$ is the cyclotron frequency and $\varphi
_{n}\left( \xi \right) $ are the orthonormalized harmonic oscillator wave
functions.
Note that the energy levels $\epsilon _{n}$ are degenerates (they do not
depend on $k$). Then, for each Landau level $n$ there exists a band of
degenerate states. The cyclotron frequency $\omega _{c}$ plays here the role
of the energy gap between occupied Landau levels. It is easy to prove that
the filling factor, defined as the ratio between the density of particles $%
n_{e}$ and the number of states per unit area of a full Landau level, is
equal to the Chern-$%
\mathop{\rm Si}%
$mons coupling constant $N$. Thus, because we are considering that $N$ is a
positive integer, we have in this MCS theory $N$ completely filled Landau
levels. Once this ground state is established, it can be argued immediately
\cite{6}, \cite{6b}, \cite{10a}, \cite{15}, that at $T=0$ the system will be
confined to a filled band, which is separated by an energy gap from the free
states; therefore, it is natural to expect that at $T=0$ the system should
superconduct. This result is already a well established fact on the basis of
Hartree-Fock analysis\cite{6} and Random Phase Approximation \cite{6b},\cite
{10a}. The case at $T\neq 0$ is more controversial since thermal
fluctuations, occurring in the many-particle system, can produce significant
changes. As we will show in this paper, the presence in this theory of a
natural scale, the cyclotron frequency $\omega _{c}$, is crucial for the
existence of a phase at $T\ll \omega _{c}$, on which the system, when
confined to a bounded region, still behaves as a superconductor.
The fermion thermal Green's function in the presence of the background
Chern-Simons field $\overline{b}$
\begin{equation}
G\left( x,x^{\prime }\right) =-\left\langle T_{\tau }\psi \left( x\right)
\overline{\psi }\left( x^{\prime }\right) \right\rangle \eqnum{2.13}
\end{equation}
is obtained by solving the equation
\begin{equation}
\left( \partial _{\tau }-\frac{1}{2m}\overline{D}_{k}^{2}-\mu \right)
G\left( x,x^{\prime }\right) =-\delta _{3}\left( x-x^{\prime }\right)
\eqnum{2.14}
\end{equation}
subject to the requirement of antiperiodicity under the imaginary time
translation $\tau \rightarrow \tau +\beta $ ($\beta $ is the inverse
absolute temperature). In (2.14) we have introduced the notation
\begin{equation}
\overline{D}_{k}=\partial _{k}+i\overline{a}_{k} \eqnum{2.15}
\end{equation}
The Fourier transform of the fermion thermal Green's function (2.13)
\begin{equation}
G\left( p_{4},{\bf p}\right) =\int\limits_{0}^{\beta }d\tau \int d{\bf x}%
G\left( \tau ,{\bf x}\right) e^{i\left( p_{4}\tau -{\bf px}\right) }
\eqnum{2.16}
\end{equation}
can be expressed in terms of the orthonormalized harmonic oscillator wave
functions $\varphi _{n}\left( \xi \right) $ as \cite{Efrain}
\begin{eqnarray}
G\left( p_{4},{\bf p}\right) &=&\int\limits_{0}^{\infty }d\rho
\int\limits_{-\infty }^{\infty }dx_{2}\sqrt{\overline{b}}\exp -\left(
ip_{2}x_{2}\right) \exp -\left( ip_{4}+\mu -\frac{\overline{b}}{2m}\right)
\rho \nonumber \\
&&\sum\limits_{n=0}^{\infty }\varphi _{n}\left( \xi \right) \varphi
_{n}\left( \xi ^{\prime }\right) t^{n} \eqnum{2.17}
\end{eqnarray}
where $t=\exp \frac{\overline{b}}{m}\rho $, $\xi =\frac{p_{1}}{\sqrt{%
\overline{b}}}+\frac{x_{2}\sqrt{\overline{b}}}{2}$, $\xi ^{\prime }=\frac{%
p_{1}}{\sqrt{\overline{b}}}-\frac{x_{2}\sqrt{\overline{b}}}{2}$ and $%
p_{4}=(2n+1)\pi /\beta $ are the discrete frequencies $(n=0,1,2,...)$
corresponding to fermion fields.
\section{Linear Response in the Infinite Strip}
\subsection{Effective Theory at $\mu \neq 0$ and $T\neq 0$}
In ref.\cite{8} the effective current-current interaction of the MCS model
was calculated to determine the independent components of the magnetic
interaction at finite temperature in a sample without boundaries, i.e., in
the free space. These authors concluded that the pure Meissner effect
observed at zero temperature is certainly compromised by the appearance of a
long-range mode at $T\neq 0$. Our main goal in the present paper is to
investigate the magnetic response of the charged anyon fluid at finite
temperature for a sample that confines the fluid within some specific
boundaries. As we prove henceforth, the confinement of the system to a
bounded region (a condition which is closer to the experimental situation
than the free-space case) is crucial for the realization of the Meissner
effect inside the charged anyon fluid at finite temperature.
Let us investigate the linear response of a charged anyon fluid at finite
temperature and density to an externally applied magnetic field in the
specific case of an infinite-strip sample. The linear response of the medium
can be found under the assumption that the quantum fluctuations of the gauge
fields about the ground-state are small. In this case the one-loop fermion
contribution to the effective action, obtained after integrating out the
fermion fields, can be evaluated up to second order in the gauge fields. The
effective action of the theory within this linear approximation \cite{8},%
\cite{11} takes the form
\begin{equation}
\Gamma _{eff}\,\left( A_{\nu },a_{\nu }\right) =\int dx\left( -\frac{1}{4}%
F_{\mu \nu }^{2}-\frac{N}{4\pi }\varepsilon ^{\mu \nu \rho }a_{\mu }\partial
_{\nu }a_{\rho }+en_{e}A_{0}\right) +\Gamma ^{\left( 2\right) } \eqnum{3.1}
\end{equation}
\[
\Gamma ^{\left( 2\right) }=\int dx\Pi ^{\nu }\left( x\right) \left[ a_{\nu
}\left( x\right) +eA_{\nu }\left( x\right) \right] +\int dxdy\left[ a_{\nu
}\left( x\right) +eA_{\nu }\left( x\right) \right] \Pi ^{\mu \nu }\left(
x,y\right) \left[ a_{\nu }\left( y\right) +eA_{\nu }\left( y\right) \right]
\]
Here $\Gamma ^{\left( 2\right) }$ is the one-loop fermion contribution to
the effective action in the linear approximation. The operators $\Pi _{\nu }$
and $\Pi _{\mu \nu }$ are calculated using the fermion thermal Green's
function in the presence of the background field $\overline{b}$ (2.17). They
represent the fermion tadpole and one-loop polarization operators
respectively. Their leading behaviors for static $\left( k_{0}=0\right) $
and slowly $\left( {\bf k}\sim 0\right) $ varying configurations in the
frame ${\bf k}=(k,0)$ take the form
\begin{equation}
\Pi _{k}\left( x\right) =0,\;\;\;\Pi _{0}\left( x\right) =-n_{e},\;\;\;\Pi
_{\mu \nu }=\left(
\begin{array}{ccc}
{\it \Pi }_{{\it 0}}+{\it \Pi }_{{\it 0}}\,^{\prime }\,k^{2} & 0 & {\it \Pi }%
_{{\it 1}}k \\
0 & 0 & 0 \\
-{\it \Pi }_{{\it 1}}k & 0 & {\it \Pi }_{\,{\it 2}}k^{2}
\end{array}
\right) , \eqnum{3.2}
\end{equation}
The independent coefficients: ${\it \Pi }_{{\it 0}}$, ${\it \Pi }_{{\it 0}%
}\,^{\prime }$, ${\it \Pi }_{{\it 1}}$ and ${\it \Pi }_{\,{\it 2}}$ are
functions of $k^{2}$, $\mu $ and $\overline{b}$. In order to find them we
just need to calculate the $\Pi _{\mu \nu }$ Euclidean components: $\Pi
_{44} $, $\Pi _{42}$ and $\Pi _{22}$. In the Landau gauge these Euclidean
components are given by\cite{11},
\begin{mathletters}
\begin{equation}
\Pi _{44}\left( k,\mu ,\overline{b}\right) =-\frac{1}{\beta }%
\sum\limits_{p_{4}}\frac{d{\bf p}}{\left( 2\pi \right) ^{2}}G\left( p\right)
G\left( p-k\right) , \eqnum{3.3}
\end{equation}
\begin{equation}
\Pi _{4j}\left( k,\mu ,\overline{b}\right) =\frac{i}{2m\beta }%
\sum\limits_{p_{4}}\frac{d{\bf p}}{\left( 2\pi \right) ^{2}}\left\{ G\left(
p\right) \cdot D_{j}^{-}G\left( p-k\right) +D_{j}^{+}G\left( p\right) \cdot
G\left( p-k\right) \right\} , \eqnum{3.4}
\end{equation}
\end{mathletters}
\begin{eqnarray}
\Pi _{jk}\left( k,\mu ,\overline{b}\right) &=&\frac{1}{4m^{2}\beta }%
\sum\limits_{p_{4}}\frac{d{\bf p}}{\left( 2\pi \right) ^{2}}\left\{
D_{k}^{-}G\left( p\right) \cdot D_{j}^{-}G\left( p-k\right)
+D_{j}^{+}G\left( p\right) \cdot D_{k}^{+}G\left( p-k\right) \right.
\nonumber \\
&&\left. +D_{j}^{+}D_{k}^{-}G\left( p\right) \cdot G\left( p-k\right)
+G\left( p\right) \cdot D_{j}^{-}D_{k}^{+}G\left( p-k\right) \right\}
\nonumber \\
&&-\frac{1}{2m}\Pi _{4}, \eqnum{3.5}
\end{eqnarray}
where the notation
\begin{eqnarray}
D_{j}^{\pm }G\left( p\right) &=&\left[ ip_{j}\mp \frac{\overline{b}}{2}%
\varepsilon ^{jk}\partial _{p_{k}}\right] G\left( p\right) , \nonumber \\
D_{j}^{\pm }G\left( p-k\right) &=&\left[ i\left( p_{j}-k_{j}\right) \mp
\frac{\overline{b}}{2}\varepsilon ^{jk}\partial _{p_{k}}\right] G\left(
p-k\right) , \eqnum{3.6}
\end{eqnarray}
was used.
Using (3.3)-(3.5) after summing in $p_{4}$, we found that, in the $k/\sqrt{%
\overline{b}}\ll 1$ limit, the polarization operator coefficients ${\it \Pi }%
_{{\it 0}}$, ${\it \Pi }_{{\it 0}}\,^{\prime }$, ${\it \Pi }_{{\it 1}}$ and $%
{\it \Pi }_{\,{\it 2}}$ are
\[
{\it \Pi }_{{\it 0}}=\frac{\beta \overline{b}}{8\pi {\bf k}^{2}}%
\sum_{n}\Theta _{n},\;\qquad {\it \Pi }_{{\it 0}}\,^{\prime }=\frac{2m}{\pi
\overline{b}}\sum_{n}\Delta _{n}-\frac{\beta }{8\pi }\sum_{n}(2n+1)\Theta
_{n},
\]
\[
{\it \Pi }_{{\it 1}}=\frac{1}{\pi }\sum_{n}\Delta _{n}-\frac{\beta \overline{%
b}}{16\pi m}\sum_{n}(2n+1)\Theta _{n},\qquad {\it \Pi }_{\,{\it 2}}=\frac{1}{%
\pi m}\sum_{n}(2n+1)\Delta _{n}-\frac{\beta \overline{b}}{32\pi m^{2}}%
\sum_{n}(2n+1)^{2}\Theta _{n},
\]
\begin{equation}
\Theta _{n}=%
\mathop{\rm sech}%
\,^{2}\frac{\beta (\epsilon _{n}/2-\mu )}{2},\qquad \Delta _{n}=(e^{\beta
(\epsilon _{n}/2-\mu )}+1)^{-1} \eqnum{3.7}
\end{equation}
The leading contributions of the one-loop polarization operator coefficients
(3.7) at low temperatures $\left( T\ll \omega _{c}\right) $ are
\begin{equation}
{\it \Pi }_{{\it 0}}=\frac{2\beta \overline{b}}{\pi }e^{-\beta \overline{b}%
/2m},\qquad {\it \Pi }_{{\it 0}}\,^{\prime }=\frac{mN}{2\pi \overline{b}}%
{\it \Lambda },\qquad {\it \Pi }_{{\it 1}}=\frac{N}{2\pi }{\it \Lambda }%
,\quad {\it \Pi }_{\,{\it 2}}=\frac{N^{2}}{4\pi m}{\it \Lambda },\qquad {\it %
\Lambda }=\left[ 1-\frac{2\beta \overline{b}}{m}e^{-\beta \overline{b}%
/2m}\right] \eqnum{3.8}
\end{equation}
and at high temperatures $\left( T\gg \omega _{c}\right) $ are
\begin{equation}
{\it \Pi }_{{\it 0}}=\frac{m}{2\pi }\left[ \tanh \frac{\beta \mu }{2}%
+1\right] ,\qquad {\it \Pi }_{{\it 0}}\,^{\prime }=-\frac{\beta }{48\pi }%
\mathop{\rm sech}%
\!^{2}\!\,\left( \frac{\beta \mu }{2}\right) ,\qquad {\it \Pi }_{{\it 1}}=%
\frac{\overline{b}}{m}{\it \Pi }_{{\it 0}}\,^{\prime },\qquad {\it \Pi }_{\,%
{\it 2}}=\frac{1}{12m^{2}}{\it \Pi }_{{\it 0}} \eqnum{3.9}
\end{equation}
In these expressions $\mu $ is the chemical potential and $m=2m_{e}$ ($m_{e}$
is the electron mass). These results are in agreement with those found in
refs.\cite{8},\cite{14}.
\subsection{MCS Linear Equations}
To find the response of the anyon fluid to an externally applied magnetic
field, one needs to use the extremum equations derived from the effective
action (3.1). This formulation is known in the literature as the
self-consistent field approximation\cite{11}. In solving these equations we
confine our analysis to gauge field configurations which are static and
uniform in the y-direction. Within this restriction we take a gauge in which
$A_{1}=a_{1}=0$.
The Maxwell and Chern-Simons extremum equations are respectively,
\begin{equation}
\partial _{\nu }F^{\nu \mu }=eJ_{ind}^{\mu } \eqnum{3.10a}
\end{equation}
\begin{equation}
-\frac{N}{4\pi }\varepsilon ^{\mu \nu \rho }f_{\nu \rho }=J_{ind}^{\mu }
\eqnum{3.10b}
\end{equation}
Here, $f_{\mu \nu }$ is the Chern-Simons gauge field strength tensor,
defined as $f_{\mu \nu }=\partial _{\mu }a_{\nu }-\partial _{\nu }a_{\mu }$,
and $J_{ind}^{\mu }$ is the current density induced by the anyon system at
finite temperature and density. Their different components are given by
\begin{equation}
J_{ind}^{0}\left( x\right) ={\it \Pi }_{{\it 0}}\left[ a_{0}\left( x\right)
+eA_{0}\left( x\right) \right] +{\it \Pi }_{{\it 0}}\,^{\prime }\partial
_{x}\left( {\cal E}+eE\right) +{\it \Pi }_{{\it 1}}\left( b+eB\right)
\eqnum{3.11a}
\end{equation}
\begin{equation}
J_{ind}^{1}\left( x\right) =0,\qquad J_{ind}^{2}\left( x\right) ={\it \Pi }_{%
{\it 1}}\left( {\cal E}+eE\right) +{\it \Pi }_{\,{\it 2}}\partial _{x}\left(
b+eB\right) \eqnum{3.11b}
\end{equation}
in the above expressions we used the following notation: ${\cal E}=f_{01}$, $%
E=F_{01}$, $b=f_{12}$ and $B=F_{12}$. Eqs. (3.11) play the role in the anyon
fluid of the London equations in BCS superconductivity. When the induced
currents (3.11) are substituted in eqs. (3.10) we find, after some
manipulation, the set of independent differential equations,
\begin{equation}
\omega \partial _{x}^{2}B+\alpha B=\gamma \left[ \partial _{x}E-\sigma
A_{0}\right] +\tau \,a_{0}, \eqnum{3.12}
\end{equation}
\begin{equation}
\partial _{x}B=\kappa \partial _{x}^{2}E+\eta E, \eqnum{3.13}
\end{equation}
\begin{equation}
\partial _{x}a_{0}=\chi \partial _{x}B \eqnum{3.14}
\end{equation}
The coefficients appearing in these differential equations depend on the
components of the polarization operators through the relations
\[
\omega =\frac{2\pi }{N}{\it \Pi }_{{\it 0}}\,^{\prime },\quad \alpha =-e^{2}%
{\it \Pi }_{{\it 1}},\quad \tau =e{\it \Pi }_{{\it 0}},\quad \chi =\frac{%
2\pi }{eN},\quad \sigma =-\frac{e^{2}}{\gamma }{\it \Pi }_{{\it 0}},\quad
\eta =-\frac{e^{2}}{\delta }{\it \Pi }_{{\it 1}},
\]
\begin{equation}
\gamma =1+e^{2}{\it \Pi }_{{\it 0}}\,^{\prime }-\frac{2\pi }{N}{\it \Pi }_{%
{\it 1}},\quad \delta =1+e^{2}{\it \Pi }_{\,{\it 2}}-\frac{2\pi }{N}{\it \Pi
}_{{\it 1}},\quad \kappa =\frac{2\pi }{N\delta }{\it \Pi }_{\,{\it 2}}.
\eqnum{3.15}
\end{equation}
Distinctive of eq. (3.12) is the presence of the nonzero coefficients $%
\sigma $ and $\tau $. They are related to the Debye screening in the two
dimensional anyon thermal ensemble. A characteristic of this 2+1 dimensional
model is that the Debye screening disappears at $T=0$, even if the chemical
potential is different from zero. Note that $\sigma $ and $\tau $ link the
magnetic field to the zero components of the gauge potentials, $A_{0}$ and $%
a_{0}$. As a consequence, these gauge potentials will play a nontrivial role
in finding the magnetic field solution of the system.
\subsection{Field Solutions and Boundary Conditions}
Using eqs.(3.12)-(3.14) one can obtain a higher order differential equation
that involves only the electric field,
\begin{equation}
a\partial _{x}^{4}E+d\partial _{x}^{2}E+cE=0, \eqnum{3.16}
\end{equation}
Here, $a=\omega \kappa $, $d=\omega \eta +\alpha \kappa -\gamma -\tau \kappa
\chi $, and $c=\alpha \eta -\sigma \gamma -\tau \eta \chi $.
Solving (3.16) we find
\begin{equation}
E\left( x\right) =C_{1}e^{-x\xi _{1}}+C_{2}e^{x\xi _{1}}+C_{3}e^{-x\xi
_{2}}+C_{4}e^{x\xi _{2}}, \eqnum{3.17}
\end{equation}
where
\begin{equation}
\xi _{1,2}=\left[ -d\pm \sqrt{d^{2}-4ac}\right] ^{\frac{1}{2}}/\sqrt{2a}
\eqnum{3.18}
\end{equation}
When the low density approximation $n_{e}\ll m^{2}$ is considered (this
approximation is in agreement with the typical values $n_{e}=2\times
10^{14}cm^{-2}$, $m_{e}=2.6\times 10^{10}cm^{-1}$ found in high-T$_{C}$
superconductivity), we find that, for $N=2$ and at temperatures lower than
the energy gap $\left( T\ll \omega _{c}\right) $, the inverse length scales
(3.18) are given by the following real functions
\begin{equation}
\xi _{1}\simeq e\sqrt{\frac{m}{\pi }}\left[ 1+\left( \frac{\pi ^{2}n_{e}^{2}%
}{m^{3}}\right) \beta \exp -\left( \frac{\pi n_{e}\beta }{2m}\right) \right]
\eqnum{3.19}
\end{equation}
\begin{equation}
\xi _{2}\simeq e\sqrt{\frac{n_{e}}{m}}\left[ 1-\left( \frac{\pi n_{e}}{m}%
\right) \beta \exp -\left( \frac{\pi n_{e}\beta }{2m}\right) \right]
\eqnum{3.20}
\end{equation}
These two inverse length scales correspond to two short-range
electromagnetic modes of propagation. These results are consistent with
those obtained in ref. \cite{8} using a different approach. If the masses of
the two massive modes, obtained in ref. \cite{8} by using the
electromagnetic thermal Green's function for static and slowly varying
configurations, are evaluated in the range of parameters considered above,
it can be shown that they reduce to eqs. (319), (3.20).
The solutions for $B$, $a_{0}$ and $A_{0}$, can be obtained using eqs.
(3.13), (3.14), (3.17) and the definition of $E$ in terms of $A_{0,}$
\begin{equation}
B\left( x\right) =\gamma _{1}\left( C_{2}e^{x\xi _{1}}-C_{1}e^{-x\xi
_{1}}\right) +\gamma _{2}\left( C_{4}e^{x\xi _{2}}-C_{3}e^{-x\xi
_{2}}\right) +C_{5} \eqnum{3.21}
\end{equation}
\begin{equation}
a_{0}\left( x\right) =\chi \gamma _{1}\left( C_{2}e^{x\xi
_{1}}-C_{1}e^{-x\xi _{1}}\right) +\chi \gamma _{2}\left( C_{4}e^{x\xi
_{2}}-C_{3}e^{-x\xi _{2}}\right) +C_{6} \eqnum{3.22}
\end{equation}
\begin{equation}
A_{0}\left( x\right) =\frac{1}{\xi _{1}}\left( C_{1}e^{-x\xi
_{1}}-C_{2}e^{x\xi _{1}}\right) +\frac{1}{\xi _{2}}\left( C_{3}e^{-x\xi
_{2}}-C_{4}e^{x\xi _{2}}\right) +C_{7} \eqnum{3.23}
\end{equation}
In the above formulas we introduced the notation $\gamma _{1}=\left( \xi
_{1}^{2}\kappa +\eta \right) /\xi _{1}$, $\gamma _{2}=\left( \xi
_{2}^{2}\kappa +\eta \right) /\xi _{2}$.
In obtaining eq. (3.16) we have taken the derivative of eq. (3.12).
Therefore, the solution of eq. (3.16) belongs to a wider class than the one
corresponding to eqs. (3.12)-(3.14). To exclude redundant solutions we must
require that they satisfy eq. (3.12) as a supplementary condition. In this
way the number of independent unknown coefficients is reduced to six, which
is the number corresponding to the original system (3.12)-(3.14). The extra
unknown coefficient is eliminated substituting the solutions (3.17), (3.21),
(3.22) and (3.23) into eq. (3.12) to obtain the relation
\begin{equation}
e{\it \Pi }_{{\it 1}}C_{5}=-{\it \Pi }_{{\it 0}}\left( C_{6}+eC_{7}\right)
\eqnum{3.24}
\end{equation}
Eq. (3.24) has an important meaning, it establishes a connection between the
coefficients of the long-range modes of the zero components of the gauge
potentials $(C_{6}+eC_{7})$ and the coefficient of the long-range mode of
the magnetic field $C_{5}$. Note that if the induced Chern-Simons
coefficient ${\it \Pi }_{{\it 1}}$, or the Debye screening coefficient ${\it %
\Pi }_{{\it 0}}$ were zero, there would be no link between $C_{5}$ and $%
(C_{6}+eC_{7})$. This relation between the long-range modes of $B$, $A_{0}$
and $a_{0}$ can be interpreted as a sort of Aharonov-Bohm effect, which
occurs in this system at finite temperature. At $T=0$, we have ${\it \Pi }_{%
{\it 0}}=0$, and the effect disappears.
Up to this point no boundary has been taken into account. Therefore, it is
easy to understand that the magnetic long-range mode associated with the
coefficient $C_{5}$, must be identified with the one found in ref. \cite{8}
for the infinite bulk using a different approach. However, as it is shown
below, when a constant and uniform magnetic field is perpendicularly applied
at the boundaries of a two-dimensional sample, this mode cannot propagate
(i.e. $C_{5}\equiv 0$) within the sample. This result is crucial for the
existence of Meissner effect in this system.
In order to determine the unknown coefficients we need to use the boundary
conditions. Henceforth we consider that the anyon fluid is confined to the
strip $-\infty <y<\infty $ with boundaries at $x=-L$ and $x=L$. The external
magnetic field will be applied from the vacuum at both boundaries ($-\infty
<x\leq -L$, $\;L\leq x<\infty $).
The boundary conditions for the magnetic field are $B\left( x=-L\right)
=B\left( x=L\right) =\overline{B}$ ($\overline{B}$ constant). Because no
external electric field is applied, the boundary conditions for this field
are, $E\left( x=-L\right) =E\left( x=L\right) =0$. Using them and assuming $%
L\gg \lambda _{1}$, $\lambda _{2}$ ($\lambda _{1}=1/\xi _{1}$, $\lambda
_{2}=1/\xi _{2}$), we find the following relations that give $C_{1,2,3,4}$
in terms of $C_{5}$,
\begin{equation}
C_{1}=Ce^{-L\xi _{1}},\quad C_{2}=-C_{1},\quad C_{3}=-Ce^{-L\xi _{2}},\quad
C_{4}=-C_{3},\quad C=\frac{C_{5}-\overline{B}}{\gamma _{1}-\gamma _{2}}
\eqnum{3.25}
\end{equation}
\section{Stability Condition for the Infinite-Strip Sample}
After using the boundary conditions, we can see from (3.25) that they were
not sufficient to find the coefficient $C_{5}$. In order to totally
determine the system magnetic response we have to use another physical
condition from where $C_{5}$ can be found. Since, obviously, any meaningful
solution have to be stable, the natural additional condition to be
considered is the stability equation derived from the system free energy.
With this goal in mind we start from the free energy of the infinite-strip
sample
\[
{\cal F}=\frac{1}{2}\int\limits_{-L^{\prime }}^{L^{\prime
}}dy\int\limits_{-L}^{L}dx\left\{ \left( E^{2}+B^{2}\right) +\frac{N}{\pi }%
a_{0}b-{\it \Pi }_{{\it 0}}\left( eA_{0}+a_{0}\right) ^{2}\right.
\]
\begin{equation}
\left. -{\it \Pi }_{{\it 0}}\,^{\prime }\left( eE+{\cal E}\right) ^{2}-2{\it %
\Pi }_{{\it 1}}\left( eA_{0}+a_{0}\right) \left( eB+b\right) +{\it \Pi }_{\,%
{\it 2}}\left( eB+b\right) ^{2}\right\} \eqnum{4.1}
\end{equation}
where $L$ and $L^{\prime }$ determine the two sample's lengths.
Using the field solutions (3.17), (3.21)-(3.23) with coefficients (3.25), it
is found that the leading contribution to the free-energy density ${\it f}=%
\frac{{\cal F}}{{\cal A}}$ ,\ (${\cal A}=4LL^{\prime }$ being the sample
area) in the infinite-strip limit $(L\gg \lambda _{1}$, $\lambda _{2}$, $%
L^{\prime }\rightarrow \infty )$ is given by
\begin{equation}
f=C_{5}^{2}-{\it \Pi }_{{\it 0}}\left( C_{6}+eC_{7}\right) ^{2}+e^{2}{\it %
\Pi }_{\,{\it 2}}C_{5}^{2}-2e{\it \Pi }_{{\it 1}}\left( C_{6}+eC_{7}\right)
C_{5} \eqnum{4.2}
\end{equation}
Taking into account the constraint equation (3.24), the free-energy density
(4.2) can be written as a quadratic function in $C_{5}$. Then, the value of $%
C_{5}$ is found, by minimizing the corresponding free-energy density
\begin{equation}
\frac{\delta {\it f}}{\delta C_{5}}=\left[ {\it \Pi }_{{\it 0}}+e^{2}{\it %
\Pi }_{{\it 1}}^{\,}\,^{2}+e^{2}{\it \Pi }_{{\it 0}}{\it \Pi }_{\,{\it 2}%
}\right] \frac{C_{5}}{{\it \Pi }_{{\it 0}}}=0, \eqnum{4.3}
\end{equation}
to be $C_{5}=0$.
This result implies that the long-range mode cannot propagate within the
infinite-strip when a uniform and constant magnetic field is perpendicularly
applied at the sample's boundaries.
We want to point out the following fact. The same property of the finite
temperature polarization operator component $\Pi _{00}$ that is producing
the appearance of a long-range mode in the infinite bulk, is also
responsible, when it is combined with the boundary conditions, for the
non-propagation of this mode in the bounded sample. It is known that the
nonvanishing of ${\it \Pi }_{{\it 0}}$ at $T\neq 0$ (or equivalently, the
presence of a pole $\sim 1/k^{2}$ in $\Pi _{00}$ at $T\neq 0$) guarantees
the existence of a long-range mode in the infinite bulk \cite{8}. On the
other hand, however, once ${\it \Pi }_{{\it 0}}$ is different from zero, we
can use the constraint (3.24) to eliminate $C_{6}+eC_{7}$ in favor of $C_{5%
\text{ }}$ in the free-energy density of the infinite strip. Then, as we
have just proved, the only stable solution of this boundary-value problem,
which is in agreement with the boundary conditions, is $C_{5}=0$.
Consequently, no long-range mode propagates in the bounded sample.
In the zero temperature limit $\left( \beta \rightarrow \infty \right) $,
because ${\it \Pi }_{{\it 0}}=0$, it is straightforwardly obtained from
(3.24) that $C_{5}=0$ and no long-range mode propagates.
At $T\neq 0$, taking into account that $C_{5}=0$ along with eq. (3.25) in
the magnetic field solution (3.21), we can write the magnetic field
penetration as
\begin{equation}
B\left( x\right) =\overline{B}_{1}\left( T\right) \left( e^{-(x+L)\xi
_{1}}+e^{\left( x-L\right) \xi _{1}}\right) +\overline{B}_{2}\left( T\right)
\left( e^{-(x+L)\xi _{2}}+e^{\left( x-L\right) \xi _{2}}\right) \eqnum{4.4}
\end{equation}
where,
\begin{equation}
\overline{B}_{1}\left( T\right) =\frac{\gamma _{1}}{\gamma _{1}-\gamma _{2}}%
\overline{B},\text{ \qquad \quad }\overline{B}_{2}\left( T\right) =\frac{%
\gamma _{2}}{\gamma _{2}-\gamma _{1}}\overline{B} \eqnum{4.5}
\end{equation}
For densities $n_{e}\ll m^{2}$, the coefficients $\overline{B}_{1}$and $%
\overline{B}_{2}$ can be expressed, in the low temperature approximation $%
\left( T\ll \omega _{c}\right) $, as
\begin{equation}
B_{1}\left( T\right) \simeq -\frac{\left( \pi n_{e}\right) ^{3/2}}{m^{2}}%
\left[ 1/m+\frac{5}{2}\beta \exp -\left( \frac{\pi n_{e}\beta }{2m}\right)
\right] \overline{B},\qquad \eqnum{4.6}
\end{equation}
\begin{equation}
B_{2}\left( T\right) \simeq \left[ 1+\frac{5\pi n_{e}}{2m^{2}}\sqrt{\pi n_{e}%
}\beta \exp -\left( \frac{\pi n_{e}\beta }{2m}\right) \right] \overline{B}
\eqnum{4.7}
\end{equation}
Hence, in the infinite-strip sample the applied magnetic field is totally
screened within the anyon fluid on two different scales, $\lambda _{1}=1/\xi
_{1}$ and $\lambda _{2}=1/\xi _{2}$. At $T=200K$, for the density value
considered above, the penetration lengths are given by $\lambda _{1}\simeq
0.6\times 10^{-8}cm$ and $\lambda _{2}\simeq 0.1\times 10^{-4}cm$ .
Moreover, taking into account that $\xi _{1}$ increases with the temperature
while $\xi _{2}$ decreases (see eqs. (3.19)-(3.20)), and that $B_{1}\left(
T\right) <0$ while $B_{2}\left( T\right) >0$, it can be shown that the
effective penetration length $\overline{\lambda }$ (defined as the distance $%
x$ where the magnetic field falls down to a value $B\left( \overline{\lambda
}\right) /\overline{B}=e^{-1}$) increases with the temperature as
\begin{equation}
\overline{\lambda }\simeq \overline{\lambda }_{0}\left( 1+\overline{\kappa }%
\beta \exp -\frac{1}{2}\overline{\kappa }\beta \right) \eqnum{4.8}
\end{equation}
where $\overline{\lambda }_{0}=\sqrt{m/n_{e}e^{2}}$ and $\overline{\kappa }%
=\pi n_{e}/m$. At $T=200K$ the effective penetration length is $\overline{%
\lambda }\sim 10^{-5}cm$.
It is timely to note that the presence of explicit (proportional to $N$) and
induced (proportional to ${\it \Pi }_{{\it 1}}$) Chern-Simons terms in the
anyon effective action (3.1) is crucial to obtain the Meissner solution
(4.4). If the Chern-Simons interaction is disconnected ($N\rightarrow \infty
$ and ${\it \Pi }_{{\it 1}}=0$), then $a=0,$ $d=1+e^{2}{\it \Pi }_{{\it 0}%
}{}^{\prime }\neq 0$ and $c=e^{2}{\it \Pi }_{{\it 0}}\,\neq 0$ in eq.
(3.16). In that case the solution of the field equations within the sample
are $E=0$, $B=\overline{B}$. That is, we regain the QED in (2+1)-dimensions,
which does not exhibit any superconducting behavior.
\section{High Temperature Non-Superconducting Phase}
We have just found that the charged anyon fluid confined to an infinite
strip exhibits Meissner effect at temperatures lower than the energy gap $%
\omega _{c}$. It is natural to expect that at temperatures larger than the
energy gap this superconducting behavior should not exist. At those
temperatures the electron thermal fluctuations should make available the
free states existing beyond the energy gap. As a consequence, the charged
anyon fluid should not be a perfect conductor at $T\gg \omega _{c}$. A
signal of such a transition can be found studying the magnetic response of
the system at those temperatures.
As can be seen from the magnetic field solution (4.4), the real character of
the inverse length scales (3.18) is crucial for the realization of the
Meissner effect. At temperatures much lower than the energy gap this is
indeed the case, as can be seen from eqs. (3.19) and (3.20).
In the high temperature $\left( T\gg \omega _{c}\right) $ region the
polarization operator coefficients are given by eq. (3.9). Using this
approximation together with the assumption $n_{e}\ll m^{2}$, we can
calculate the coefficients $a$, $c$ and $d$ that define the behavior of the
inverse length scales,
\begin{equation}
a\simeq \pi ^{2}{\it \Pi }_{{\it 0}}{}^{\prime }{\it \Pi }_{\,{\it 2}}
\eqnum{5.1}
\end{equation}
\begin{equation}
c\simeq e^{2}{\it \Pi }_{{\it 0}}{} \eqnum{5.2}
\end{equation}
\begin{equation}
d\simeq -1 \eqnum{5.3}
\end{equation}
Substituting with (5.1)-(5-3) in eq. (3.18) we obtain that the inverse
length scales in the high-temperature limit are given by
\begin{equation}
\xi _{1}\simeq e\sqrt{m/2\pi }\left( \tanh \frac{\beta \mu }{2}+1\right) ^{%
\frac{1}{2}} \eqnum{5.4}
\end{equation}
\begin{equation}
\xi _{2}\simeq i\left[ 24\sqrt{\frac{2m}{\beta }}\cosh \frac{\beta \mu }{2}%
\left( \tanh \frac{\beta \mu }{2}+1\right) ^{-\frac{1}{2}}\right]
\eqnum{5.5}
\end{equation}
The fact that $\xi _{2}$ becomes imaginary at temperatures larger than the
energy gap, $\omega _{c}$, implies that the term $\gamma _{2}\left(
C_{4}e^{x\xi _{2}}-C_{3}e^{-x\xi _{2}}\right) $ in the magnetic field
solution (3.21) ceases to have a damping behavior, giving rise to a periodic
inhomogeneous penetration. Therefore, the fluid does not exhibit a Meissner
effect at those temperatures since the magnetic field will not be totally
screened. This corroborate our initial hypothesis that at $T\gg \omega _{c}$
the anyon fluid is in a new phase in which the magnetic field can penetrate
the sample.
We expect that a critical temperature of the order of the energy gap ($T\sim
\omega _{c}$) separates the superconducting phase $\left( T\ll \omega
_{c}\right) $ from the non-superconducting one $\left( T\gg \omega
_{c}\right) $. Nevertheless, the temperature approximations (3.8) and (3.9)
are not suitable to perform the calculation needed to find the phase
transition temperature. The field solutions in this new non-superconducting
phase is currently under investigation. The results will be published
elsewhere.
\section{Concluding Remarks}
In this paper we have investigated the magnetic response at finite
temperature of a charged anyon fluid confined to an infinite strip. The
charged anyon fluid was modeled by a (2+1)-dimensional MCS theory in a
many-particle ($\mu \neq 0$, $\overline{b}\neq 0$) ground state. The
particle energy spectrum of the theory exhibits a band structure given by
different Landau levels separated by an energy gap $\omega _{c}$, which is
proportional to the background Chern-Simons magnetic field $\overline{b}$.
We found that the energy gap $\omega _{c}$ defines a scale that separates
two phases: a superconducting phase at $T\ll \omega _{c}$, and a
non-superconducting one at $T\gg \omega _{c}$.
The total magnetic screening in the superconducting phase is characterized
by two penetration lengths corresponding to two short-range eigenmodes of
propagation of the electromagnetic field within the anyon fluid. The
existence of a Meissner effect at finite temperature is the consequence of
the fact that a third electromagnetic mode, of a long-range nature, which is
present at finite temperature in the infinite bulk \cite{8}, does not
propagate within the infinite strip when a uniform and constant magnetic
field is applied at the boundaries. This is a significant property since the
samples used to test the Meissner effect in high-$T_{c}$ superconductors are
bounded.
It is noteworthy that the existence at finite temperature of a Debye
screening (${\it \Pi }_{{\it 0}}\,\neq 0$) gives rise to a sort of
Aharonov-Bohm effect in this system with Chern-Simons interaction ($N$
finite, ${\it \Pi }_{{\it 1}}\neq 0$). When ${\it \Pi }_{{\it 0}}\,\neq 0$,
the field combination $a_{0}+eA_{0}$ becomes physical because it enters in
the field equations in the same foot as the electric and magnetic fields
(see eq. (3.12)). A direct consequence of this fact is that the coefficient $%
C_{5}$, associated to the long-range mode of the magnetic field, is linked
to the coefficients $C_{6}$ and $C_{7}$ of the zero components of the
potentials (see eq. (3.24)).
When $T=0$, since ${\it \Pi }_{{\it 0}}\,=0$ and ${\it \Pi }_{{\it 1}}\neq 0$%
, eq. (3.24) implies $C_{5}=0$. That is, at zero temperature the long-range
mode is absent. This is the well known Meissner effect of the anyon fluid at
$T=0$. When $T\neq 0$, eq. (3.24) alone is not enough to determine the value
of $C_{5}$, since it is given in terms of $C_{6}$ and $C_{7}$ which are
unknown. However, when eq. (3.24) is taken together with the field
configurations that satisfy the boundary conditions for the infinite-strip
sample (eqs. (3.17), (3.21)-(3.23) and (3.25)), and with the sample
stability condition (4.3), we obtain that $C_{5}=0$. Thus, the combined
action of the boundary conditions and the Aharonov-Bohm effect expressed by
eq. (3.24) accounts for the total screening of the magnetic field in the
anyon fluid at finite temperature.
Finally, at temperatures large enough ($T\gg \omega _{c}$) to excite the
electrons beyond the energy gap, we found that the superconducting behavior
of the anyon fluid is lost. This result was achieved studying the nature of
the characteristic lengths (3.18) in this high temperature approximation. We
showed that in this temperature region the characteristic length $\xi _{2}$
becomes imaginary (eq. (5.5)), which means that a total damping solution for
the magnetic field does not exist any more, and hence the magnetic field
penetrates the sample.
\begin{quote}
Acknowledgments
\end{quote}
The authors are very grateful for stimulating discussions with Profs. G.
Baskaran, A. Cabo, E.S. Fradkin, Y. Hosotani and J. Strathdee. We would also
like to thank Prof. S. Randjbar-Daemi for kindly bringing the publication of
ref. \cite{b} to our attention. Finally, it is a pleasure for us to thank
Prof. Yu Lu for his hospitality during our stay at the ICTP, where part of
this work was done. This research has been supported in part by the National
Science Foundation under Grant No. PHY-9414509.
| proofpile-arXiv_065-474 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
An unbroken hidden supersymmetric gauge sector, supporting a gaugino
condensate
$\vev{\lambda\lambda}\sim (10^{14}\ \mbox{GeV})^3,$ can through its coupling to
the dilaton (and other moduli) serve the related dynamical functions of
providing a superpotential for the dilaton and a mass for the gravitino
\cite{Nil}.
Because of its possibly central role in the understanding of these
phenomena, the dynamics associated with this hidden gauge sector may merit
additional study. As a zero temperature field theory, against a Minkowski
space-time background, this has been done (without dilaton) in the original
paper of Veneziano and Yankielowicz \cite{Ven} (see also Amati et al
\cite{Ama}). In the effective theory, the ground state emerges as containing a
gaugino condensate
$\vev{\lambda\lambda}$ with residual $Z_N$ symmetry. The changes which evolve with the
introduction of supergravity and the dilaton are discussed in \cite{Fer}.
In the context of the standard Robertson-Walker cosmology, it is of
interest to examine the transition from the hot unconfined phase in the hidden
sector to the confined phase.\footnote{It is possible that screening, rather
than confinement, characterizes the phase transition
\cite{Gro}. The only property to be used in this paper is the change in the
degrees of freedom from perturbative non-singlet quanta to gauge-singlet
`hadrons' as one passes through the critical temperature.} Is the transition
first or second order?
If first order, is the transition completed in a manner consistent with the
observed smoothness of the universe? Some possible problems associated with a
transition described by the evolution of the condensate as a field theoretic
order parameter were discussed in \cite{Gol}. However, the results depended
critically on the K\"ahler potential of the effective condensate field, and
this is certainly not known. Thus, a more phenomenological approach is
indicated.
In this work, I present a preliminary study of the transition to the confining
phase of the unbroken hidden Yang-Mills sector, under a few well-defined
assumptions:
\begin{itemize}
\itm{1}There exists a hot, unconfined phase of the hidden sector.
\itm{2}The transition to confining phase is first order, proceeding
through the spontaneous nucleation and expansion of critical bubbles
of confined
phase.
\ei
Neither of these assumptions is necessarily correct. The aim of the present
study
is to simply examine the consequences of taking them to be true
The principal conclusion of this study, under the stated
assumptions, is the following: if the lightest composite particles in the
confined phase (`hidden hadrons') are more massive than $\sim 2\ T_c$ (where
$T_c$ is the critical temperature), then the requirement of sufficient
supercooling in order to complete the phase transition implies a critical
bubble size too small to contain enough ({\em i.e.,}
$>100$) hidden hadrons to allow a meaningful thermodynamic description. This
result is largely traceable to the high value of the transition temperature.
The conclusion will be shown to hold as well in the presence of
(near) massless vector-like matter fields. Thus a first
order transition in a {\em cosmological} context may be unlikely in the hidden
sector.
\section{Review of Homogeneous Nucleation}
In this section, I will briefly sketch the derivation of the relevant formulae,
initially following Fuller, Mathews, and Alcock \cite{Ful} in their
discussion of the quark-hadron transition, with some modification for ease of
application to the present situation.
The formation of the confined phase proceeds through the spontaneous nucleation
of critical bubbles of radius $R_c,$ at which the free energy difference
between confined and deconfined phases (pressures $P_{\rm conf}(T)$ and
$P_{\rm deconf}(T),$ respectively)
\be
\Delta F= -\frac{4\pi}{3}R_c^3\ (P_{\rm conf}-P_{\rm deconf}) + 4\pi\sigmaR_c^2
\lab{delf}
\end{equation}
has a saddle point at
\be
R_c(T) = \frac{2\sigma}{P_{\rm conf}(T)-P_{\rm deconf}(T)}\ \ .
\lab{rc}
\end{equation}
Here $\sigma$ is the surface energy density at the interface between the
phases.
The probability of nucleation of a single bubble is then
\be
p(T)=CT^4\ e^{-\Delta F/T} \ \ .
\lab{p1}
\end{equation}
The critical temperature $T_c$ is define through the coexistence condition in
the infinite volume limit
\be
P_{\rm conf}(T_c)=P_{\rm deconf}(T_c)\ \ .
\lab{coex}
\end{equation}
If the amount of supercooling $\eta\equiv (T_c-T)/T_c$ is small, then one may
expand
\be
P_{\rm conf}(T)-P_{\rm deconf}(T)\simeq L\eta\ \ ,
\lab{delp}
\end{equation}
where
\be
L=T_c\ \frac{\partial}{\partial T}\left.(P_{\rm deconf}-P_{\rm
conf})\right|_{T_c}
\lab{lat}
\end{equation}
is the latent heat
released per unit volume during the phase change. Combining these results, we
have
\be
p(T)\simeq CT_c^4\ e^{-A(T)}\ \ ,
\lab{p2}
\end{equation}
where
\be
A(T)=\frac{16\pi}{3}\ \frac{\sigma^3}{L^2T_c\eta^2}
\lab{a}
\end{equation}
is the 3-dimensional critical bubble action. The rapid increase of $p(T)$ with
increased supercooling is then explicit.
The condition for the complete nucleation of the universe at time $t_f$ is
\be
\int^{t_f}_{t_c}dt^{\, \prime}\ p(T^{\, \prime}(t^{\, \prime}))\ \frac{4\pi}{3}\ v_s^3\
(t_f-t^{\, \prime})^3=1\ \ ,
\lab{nuc1}
\end{equation}
where $v_s\simeq 1/\sqrt{3}$ is the speed of the expanding shock wave of the
confined phase into the metastable unconfined phase.
For small supercooling, one may expand about $t^{\, \prime}=t_f:$ \cite{Ful}
\be
\ln p(T)=\ln p(T_f) + \left.\frac{d\ln p}{dT}\right|_{T_f}\
\left.\frac{dT}{dt}\right|_{t_f}\ (t-t_f)
\lab{lnp}
\end{equation}
with
\be
\left.\frac{dT}{dt}\right|_{t_f}=-T_f\ H(T_f)\simeq -T_c H(T_c)\ \ .
\lab{dtdt}
\end{equation}
Thus
\be
p(t)=p(t_f)\ e^{-\lambda(t_f-t)}
\lab{pt}
\end{equation}
with $\lambda=2A(T_f) H(T_c)/\eta_f.$ The condition \eqn{nuc1}\ then
becomes
\be
\frac{8\pi v_s^3}{\lambda^4}\ p(T_f) =1\ \ ,
\lab{nuc2}
\end{equation}
or
\be
\frac{8\pi C}{3\sqrt{3}}\ \left(\frac{T_c}{H(T_c)}\frac{\alpha}{2}\right)^4 =
A^6\ e^A
\lab{nuc3}
\end{equation}
where
\be
\alpha\equiv \left(\frac{16\pi}{3}\ \frac{\sigma^3}{L^2T_c}\right)^{\half}\ \ ,
\lab{alpha}
\end{equation}
and $A\equiv A(T_f).$ Note that
\be
\eta_f=\frac{\alpha}{\sqrt{A}}
\lab{etaf}
\end{equation}
and the Hubble constant at $T_c$ is
\be
H(T_c)=\frac{1}{M_{\rm Pl}}\ \sqrt{\frac{8\pi\rho(T_c)}{3}}\ \ ,
\lab{htc}
\end{equation}
with
\be
\rho(T_c)=\mbox{total energy density at}\ T_c = \cn_{\rm tot}\frac{\pi^2}{30}\
T_c^4\ \ .
\lab{rho}
\end{equation}
Here $\cn_{\rm tot}$ is an effective statistical weight for the degrees of
freedom at $T_c.$ If the universe happens to be dominated by a vacuum energy
$\rho_0$ at the time of supercooling, then Eq.~\eqn{rho}\ is just a
redefinition of $\rho_0$ in terms of $\cn_{\rm tot}$.
Combining Eqs.~\eqn{alpha}-\eqn{rho}, the condition \eqn{nuc3}
becomes
\be
A^6\ e^A =
\left(\frac{0.45\ C^{1/4}\ \alpha}{\sqrt{\cn_{\rm tot}}}\ \
\frac{M_{\rm Pl}}{T_c}\right)^4\ \ .
\lab{nuc4}
\end{equation}
\section{Example of Pure Non-Supersymmetric \su{3} Yang Mills}
As a preliminary for the hidden sector, consider the case of purely
gluonic \su{3}$_c$ (no quarks). Then $\cn_{\rm tot}=16,\ T_c\simeq 200$ MeV, and a
lattice-based estimate of $\alpha$ is \cite{Iwa}
\be
\alpha\simeq 0.0065 \pm 0.0015\ \ .
\lab{alphx}
\end{equation}
With $C\sim 1,$ Eq.~\eqn{nuc4}\ yields
\be
A=124\ \ ,
\lab{a3}
\end{equation}
and Eq.~\eqn{etaf}\ gives $\eta_f= 6\times 10^{-4}.$ From \eqn{rc},
\eqn{delp}, \eqn{alpha}, and \eqn{etaf}, the radius of a
critical bubble is
\be
R_c=\frac{2\sigma}{L\eta_f}=\sqrt{\frac{A}{4\pi}}\
\left(\frac{T_c^3}{\sigma}\right)^{1/2}\ T_c^{-1}\ \ .
\lab{rcf}
\end{equation}
{}From the data of \cite{Iwa}, the surface energy density is not quite scale
invariant with present statistics; nevertheless, it will suffice for the
present application to take \cite{Bro}
\be
\sigma\simeq 0.025\ T_c^3\ \ ,
\lab{sigma}
\end{equation}
which gives
\be
R_c= 35\ T_c^{-1}\ \ .
\lab{rcx}
\end{equation}
I now come to the point of departure of the present work.
With no quarks, the relevant degrees of freedom in the confined phase are
glueballs with masses $M_i \ \,\raisebox{-0.13cm}{$\stackrel{\textstyle>}{\textstyle\sim}$}\,\ 1\ \mbox{GeV}\gg T_c,$ and number densities
\be
n_i(T_c)\simeq T_c^3\ (2S_i+1) \left(\frac{M_i}{2\piT_c}\right)^{3/2}\
e^{-M_i/T_c}\ \ .
\lab{ni}
\end{equation}
Then the number of glueballs
in the critical bubble is
\be
N_c =\sum n_i(T_c)\ \ \tfrac{4}{3}\piR_c^3\ \ .
\lab{nc1}
\end{equation}
Phenomenological studies \cite{Shu} have suggested that a density of
states
$\tau(M)\propto M^3$ (for each isospin and hypercharge) provides a good fit to
the observed hadron spectrum. I adopt this for the glueball spectrum, and
normalize to 1 state in the interval
$(M_0^2-\thalf\mu^2,M_0^2+\thalf\mu^2),$ where
$M_0$ is the mass of the lightest glueball and $\mu^2$ is the inverse of the
Regge slope. This gives for the spectral density
\be
\tau(M)=(2/M_0^2\mu^2)\ M^3\ \ ,
\lab{tau}
\end{equation}
and for the number density
\begin{eqnarray}
n_c&=&\sum n_i(T_c)=2(2\pi)^{-3/2}M_0^{-2}\mu^{-2}\int_{M_0}^{\infty} dM\ M^3\
(M/T_c)^{3/2}\ e^{-M/T_c}\ \ T_c^3 \nonumber\\
&\equiv& 2(2\pi)^{-3/2}\ (T_c/\mu)^2\ f(M_0/T_c)\ \ T_c^3\ \ .
\lab{nc2}
\end{eqnarray}
Combining \eqn{rcx}, (\ref{eq:nc1}), and (\ref{eq:nc2}), one finds
\be
N_c=22,800\ (T_c/\mu)^2\ f(M_0/T_c)\ \ .
\lab{nc3}
\end{equation}
A thermodynamic description, and
hence a first order phase transition, will be {\em cosmologically} viable only
for
$N_c\gg 1,$ say
$N_c>100.$ Once $(\mu/T_c)$ is specified, this will translate via
(\ref{eq:nc3}) to a bound on
$M_0.$ Since $\mu^2\sim 1 \ \ \mbox{GeV}^2,$ and $T_c\simeq 250\ \mev,$ one finds
\be
M_0\le 8\ T_c\simeq 2000\ \mbox{MeV}\ \ .
\lab{mbound}
\end{equation}
Before continuing, I will comment briefly on the question of
possible temperature
dependence of the glueball masses near $T_c.$ A recent theoretical calculation
\cite{Sug} in the context of the dual Ginzburg-Landau theory shows
that there is
some reduction in mass of the gauge singlet QCD monopole at $T_c,$ although the
effect is totally dependent on an assumed (and unknown) temperature dependence
of the dual Higgs quartic coupling. There is no basis for supposing that the
glueball mass goes to zero (or is even much reduced) at $T_c$ in a strongly
first order phase transition, and I will simply reinterpret
\eqn{mbound} as a bound on an effective glueball mass, with the expectation
that it does not differ significantly from the zero temperature mass.
I now turn to examine the implication of these ideas
when applied to the hidden gauge sector which is relevant to gaugino
condensation and SUSY-breaking.
\section{The Hidden Sector: Pure SUSY Yang-Mills}
There are several significant differences between the hidden sector
SUSY-Yang-Mills theory and the non-SUSY SU(3)$_c$ theory just discussed:
\begin{itemize}
\item The theory is supersymmetric: there are Majorana fermions in the adjoint
representation, and only rudimentary lattice results are available for such a
theory \cite{Mon}.
\item The vacuum properties are totally unlike those in ordinary QCD \cite{Ama}.
\item The gauge group is either much larger than SU(3), or is manifest at a
Kac-Moody level $k\ge 2,$ in order that strong
coupling sets in at a scale $\sim
10^{14}$ GeV.
\ei
A question which immediately arises as a consequence of these differences is
whether the transition is first order. The simplest approach here is to assume
that it is first order as a working hypothesis, and examine the consequences.
Since the interface energy $\sigma$ and the parameter $\alpha$ are
completely unknown for the hidden sector, the nucleation condition
\eqn{nuc4}\ must be rewritten in a suitable manner. I assume that the
specific entropy of the hadronic phase is much less than that of the gauge
phase, so that I take for the latent heat
\be
L=T_c\ \left.\frac{dP_{\rm deconf}}{dT}\right|_{T=T_c}=4\
\frac{\pi^2}{90}\ \cn_{\rm hidden}\ T_c^4\ \ .
\lab{lat1}
\end{equation}
It is also convenient to set
\be
\hat\sigma\equiv \sigma/T_c^3\ \ ,
\lab{sigh}
\end{equation}
so that the condition \eqn{nuc4}\ becomes
\be
\left(\frac{A}{\sh}\right)^6\ e^A=\left(\frac{4.8\ C^{1/4}}{\cn_{\rm hidden}\sqrt{\cn_{\rm tot}}}\
\frac{M_{\rm Pl}}{T_c}\right)^4\ \ .
\lab{cond}
\end{equation}
What is $T_c?$
For zero cosmological constant, the gravitino mass is given in terms of the
effective superpotential $W$ and Kahler $K$ by
\be
m_{3/2}=e^{K/2}|W_{eff}|/M^2\ \ ,
\lab{m32}
\end{equation}
where $M=M_{\rm Pl}/\sqrt{8\pi}.$ In the effective theory, with a simple gauge group
and no matter fields, one obtains after integrating out the gaugino condensate
\cite{deC}
\be
W_{eff}=-\frac{b}{6{\rm e}}\ M_{\rm string}^3\ e^{-3S/2b}\ \ ,
\lab{weff}
\end{equation}
where $b=\beta(g)/g^3=3N/16\pi^2$ for \su{N}, Re $S=1/g_{\rm string}^2\simeq
2.0$ at the correct minimum for the dilaton field $S.$ $M_{\rm string}$
sets the
scale for the logarithmic term in the condensate
superpotential. The \su{N} theory becomes strong $(g^2/4\pi=1)$ at a scale
\be
\Lambda=M_{\rm string}\ e^{-S/2b}\ \ ,
\lab{lambda}
\end{equation}
so that
\be
m_{3/2}=e^{K/2}\ \frac{b}{6{\rm e}}\ \frac{\Lambda^3}{M^2}\ \ .
\lab{m32a}
\end{equation}
For $m_{3/2}\simeq 10^3\ \ \mbox{GeV},$ one obtains
\be
\Lambda=e^{-K/6}\ N^{-1/3}\cdot 1.7\times 10^{14}\ \ \mbox{GeV}\ .
\lab{lambda1}
\end{equation}
As a heuristic example, I will choose as the hidden gauge group SU(6), which is
consistent with Eqs.~\eqn{lambda}\ and \eqn{lambda1}\ for
$M_{\rm string}\simeq 10^{18}\ \ \mbox{GeV}, \ e^{-K/6}\simeq 1.$ With $T_c\simeq
\Lambda\simeq 10^{14}\ \ \mbox{GeV},\
\cn_{\rm hidden}=2\left(\frac{15}{8}\right)\left(6^2-1\right)=131.25,$\ $\cn_{\rm tot}=
{\cal N}(Standard
\ Model)+\cn_{\rm hidden}=213.75+131.25=345,$ and $C^{1/4}\simeq 1,$ the
constraint Eq.~\eqn{cond} becomes
\be
A + 6\ln(A/\sh) = 21\ \ .
\lab{cond1}
\end{equation}
For consistency, we must require that $A$ not be small, so that for $A\ge 1,$
one obtains an upper bound
\be
A/\hat\sigma\le 28\ \ .
\lab{ash}
\end{equation}
As before, I now proceed to calculate the number of (hidden) hadrons in a
critical bubble.
With the same spectrum of glueballs as for the non-SUSY example of the last
section (Eq.~\eqn{nc2} with a factor of 4 included for the supermultiplet),
and Eq.~\eqn{rcf} for the bubble radius, one finds
\be
N_c=(\sqrt{6}/\pi^2)\
(T_c/\mu)^2\ f(M_0/T_c)\ (A/\sh)^{3/2}
\lab{nch}
\end{equation}
where again $1/\mu^2$ is the Regge slope for the hidden sector glueballs.
As previously, the requirement that the cosmological description be
thermodynamically viable requires that $N_c\gg 1,$ which I take to mean
$N_c\,\raisebox{-0.13cm}{$\stackrel{\textstyle>}{\textstyle\sim}$}\, 100.$ With the use of \eqn{ash} this devolves to a constraint on
$f(M_0/T_c),$ and hence on
$M_0:$ for $\mu/T_c\ge 3,$ I find
\be
M_0\le 1.4\ T_c\ \ ,
\lab{mbound1}
\end{equation}
which is of dubious credibility since we expect $M_0>\mu.$ For $\mu/T_c\ge
2,$ the bound is
\be
M_0\le 2.0\ T_c\ \ ,
\lab{mbound2}
\end{equation}
which is marginally possible. Thus, the conclusion at this point is that a
bubble description for the first order transition is barely possibly (for
$M_0\simeq \mu\simeq 2\ T_c),$ but seemingly unlikely.
\section{Hidden Sector with Matter Fields}
Suppose that the hidden sector contains $N_f< N$ flavors of
vector-like pairs of
chiral superfields $Q+\overline{Q}.$ If these are massive ($M_Q\gg \Lambda_N,$
the
\su{N} confining scale), then they effectively decouple from the dynamics
discussed in this paper. If they are massless (or nearly massless), then the
discussion in Refs.~\cite{Ama} and \cite{Aff} is germane: the existence of flat
directions $v_{ir}=v_r\delta_{ir} (i=\mbox{color}, r=\mbox{flavor})$ in the
field space of the $Q+\overline{Q}$ breaks the symmetry to \su{N^{\, \prime}},
$N^{\, \prime}\equiv N-N_f,$ at the scale
$v.$ Between
$v$ and $\Lambda_{N^{\, \prime}},$ the effective massless degrees of freedom are the
Goldstone bosons of the broken flavor symmetry and the \su{N^{\, \prime}} gauge
degrees of freedom. (I assume that it is $\Lambda_{N^{\, \prime}}$ which establishes
the condensate scale of interest in this work.) If the
Goldstones are in thermal
equilibrium with the \su{N^{\, \prime}} fields, then they would contribute to the
pressure of the critical bubble with high number density, and the problems
encountered earlier would be alleviated. I will now show that the
Goldstones are
{\em not} in thermal equilibrium with the \su{N^{\, \prime}} fields, and thus
the bubble
is transparent to their existence.
Thermal equilibrium requires that the ratio $\Gamma/H$ be $>1$ during the era
of interest, where $\Gamma$ is the reaction rate of the Goldstones in the
\su{N^{\, \prime}} plasma. The coupling of a Goldstone to a pair of \su{N^{\, \prime}}
gluons is $\frac{g^2}{32\pi^2v\sqrt{N^{\, \prime}}},$ and the cross section in the
plasma
is easily calculated:
\be
\sigma\sim \frac{g^2}{4\pi}\ \left(\frac{g^2}{32\pi^2v}\right)^2\ \ ,
\lab{cross}
\end{equation}
independent of temperature and $N^{\, \prime}.$ The Hubble constant $H\simeq
\sqrt{\cn_{\rm tot}}\ T^2/M_{\rm Pl},$ so that
\be
\frac{\Gamma}{H}=\frac{\sigma n v_G}{H}\simeq
10^{-8}\ \frac{{\cal N}_{N^{\, \prime}}}{\sqrt{\cn_{\rm tot}}}\ \frac{TM_{\rm Pl}}{v^2}\ \ ,
\lab{gamh1}
\end{equation}
where $n=$ plasma number density, $v_G=$ Goldstone velocity. For the \su{6}
example $(N^{\, \prime}=6),$ with $T=T_c\simeq 10^{14}, {\cal N}_{N^{\, \prime}}=131.25,
\cn_{\rm tot}=345,$ one finds
\be
\Gamma/H \simeq 0.007\ (v/T_c)^{-2}\ll 1\ \
\lab{gamh2}
\end{equation}
for any $v\ge T_c.$ Thus, the Goldstones decouple from the \su{N^{\, \prime}} plasma,
and do not contribute to the bubble dynamics.
\section{Summary and Conclusions}
\begin{itemize}
\itm{a}The field theoretic description of hidden gluino
condensates must imply a parallel thermal/cosmological description of the phase
transition beween the unconfined and confined phases of the
unbroken Yang-Mills
theory. This work has examined the conditions under which a
first order transition in terms of classical bubble nucleation is possible. The
result found is that only if the mass of the confined phase
glueballs (and superpartners) is very near to $ 2 T_c$ , are critical
bubbles large enough to contain an adequate number of quanta of the confined
phase particles to satisfy the thermodynamic conditions for a first order
transition in the expanding universe. This is true whether or not
there are Goldstones associated with massless vector-like matter fields.
The highly restrictive conditions on glueball mass leads one to
question whether a first order
transition is possible in the expanding universe.
\itm{b}If a first order transition is not feasible, then a field theoretic
description of a second (or higher) order transition may be of interest. This
entails some difficulty with the Witten index theorem\cite{Wit}: at the
critical temperature, the order parameter (presumably the gaugino condensate)
must change in a continuous manner from zero to a non-zero value. However, the
index for the final state is $N$ (for \su{N}), whereas for
the initial state it
is (presumably) zero. Resolution of this problem will no doubt involve some
non-trivial input to the effective theory.
\itm{c}The critical input to the present analysis is the ratio $M_0/T_c,$
where $M_0$ is the mass of the lightest glueball. It would be extremely
useful to
have some indication, possibly from a lattice study, of this quantity. A
continued pursuit of SUSY on the lattice, following the initial effort in
Ref.~\cite{Mon} would be very welcome.\ei
\clearpage
\noindent{\bf Acknowledgement}
This work was supported in part by Grant No. PHY-9411546 from the National
Science Foundation.
| proofpile-arXiv_065-475 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
The influence of disorder on the Abrikosov vortex lattice
in the mixed phase of high-temperature
superconductors, such as
${\rm Y}{\rm Ba}_{2}{\rm Cu}_{3}{\rm O}_{7-\delta}$ (YBCO), is
an issue of immediate technological interest because
pinning of the flux lines by disorder opens the
possibility of regaining a dissipation-free current flow in the
mixed phase.
The flux line (FL) array in a high-temperature
superconductor (HTSC) is extremely susceptible to thermal and
disorder-induced fluctuations due to the interplay of several
parameters, namely the high transition temperature $T_{c}$, large
magnetic penetration depths $\lambda$ and short coherence lengths
$\xi$, and a strong anisotropy of the material.
This leads to the existence of
a variety of fluctuation dominated phases of the FL array and
very rich phase diagrams for the HTSC materials~\cite{Bl}.
We want to consider here the pinning of FLs by
point defects such as the oxygen vacancies, which is usually referred to as
point disorder.
It is well-known
that the FL lattice is unstable to point disorder~\cite{LO}.
It has been conjectured that due to a {\em collective pinning} by the
point disorder,
the FL array may form a {\em vortex glass phase} with zero
linear resistivity~\cite{F,FGLV,Na,FFH}.
Although the existence of a vortex glass (VG) phase has
been verified experimentally~\cite{Ko89,Ga91,Saf92,Yeh93,Saf93}, its
large scale properties characterizing the nature of the VG phase
are still under debate~\cite{Bl}.
A possible scenario for a description of the low-temperature
properties of the FL array subject to point disorder
is the existence of a topologically ordered, i.e.,
{\em dislocation-free} VG phase, the
so-called {\em Bragg glass} phase~\cite{GL}
as a thermodynamically stable phase.
In this
glassy phase, a quasi long range
positional order of the FL array is
maintained in spite of the pinning~\cite{Na,GL}.
This
entails the existence of algebraically decaying Bragg peaks in
diffraction experiments on this phase (bearing the name ``Bragg glass''
for this property), which have indeed been observed in neutron
diffraction experiments on
${\rm Bi}_{2}{\rm Sr}_{2}{\rm CaCu}_{2}{\rm O}_{8+\delta}$ (BSCCO)
at low magnetic fields~\cite{Cub93}.
In the Bragg glass phase,
the disordered FL array is modeled
as an elastic manifold in a periodic random potential,
similar to a randomly pinned charge density wave or a XY model in
a random field~\cite{VF,CO,K,GL}.
To give a thermodynamically stable phase, this requires the
persistence of the topological order, or absence of unbound
dislocations,
even in the presence of disorder.
In the neutron diffraction experiments by Cubitt
et al.~\cite{Cub93}, it has been observed that upon
increasing the magnetic field, the Bragg peaks vanish,
indicating an instability of the Bragg glass phase.
Critical current measurements of Khaykovich et al.~\cite{Kha96}
show a sharp drop in the (local) critical current $j_c$
upon decreasing the magnetic induction below a critical value.
This can be attributed to the ``disentanglement'' of FLs in the
absence of dislocations when topological order is regained and
dislocation loops vanish upon
lowering the magnetic field.
Similarly, changes in the I-V characteristics of YBCO based
superlattices below a critical magnetic field can be interpreted
as stemming from a sharp drop of the pinning energy and indicate
a restoration of positional order in the VG~\cite{O}.
The existence of a topological transition has also been demonstrated in
recent numerical studies~\cite{GH,RKD}.
In the closely related 3D XY model in a random field,
vortex loops occur in a topological phase transition at a
critical strength of the random field~\cite{GH}.
In simulations of disordered FL arrays~\cite{RKD}, a proliferation of
dislocation lines has been found at a critical magnetic field in
good agreement with the experimental results in Ref.~\cite{Cub93}.
Recently, the quantitative aspects of this issue have been addressed also
analytically~\cite{KNH,F97,EN,Gold,GL97}.
In Ref.~\cite{KNH} a self-consistent variational calculation and
a scaling argument are presented, which show the topological
stability of the elastic Bragg glass phase
over a finite range of parameters that can be estimated by a
Lindemann-like criterion (\ref{cond}). A more detailed discussion of
the
methods used in Ref.~\cite{KNH} and their limitations
as well as of the Lindemann-like criterion (\ref{cond}),
which provides the basis for the calculations in the present work,
will be given in the next paragraph.
Recently,
Fisher~\cite{F97} has presented
refined scaling arguments further
supporting the existence of a topologically ordered Bragg glass
phase.
In Refs.~\cite{EN,Gold,GL97}, purely phenomenological
Lindemann criteria are used as
starting point for an estimate of the phase boundaries of the
Bragg glass.
Erta\c{s} and Nelson~\cite{EN} and Goldschmidt~\cite{Gold}
use ``cage models'' to mimic the interactions between FLs
which yields an effective theory for a single
FL in a random potential, to which they apply the
conventional phenomenological Lindemann criterion.
Giamarchi and Le Doussal~\cite{GL97} apply a slightly modified
phenomenological Lindemann criterion of the form
$\overline{\langle u^2 \rangle}(l) < c_L^2 l^2$, where
$\overline{\langle u^2 \rangle}(l)$ is the (disorder-averaged)
{\em relative}
mean square displacement of neighbouring FLs separated by the FL spacing
$l$ ($c_L$ is the Lindemann-number).
However, it is well known that phenomenological Lindemann criteria
as used in Refs.~\cite{EN,Gold,GL97}
do not allow
a theoretical description of a phase transition
but can only give estimates of the location of the
transition.
They rely on the assumption that the phase transition reflects
in the short scale behaviour of the system.
Also the variational calculation and the scaling argument
presented in Ref.~\cite{KNH} cannot give a complete description
of the transition as only a detailed renormalization
group (RG) analysis of the problem would allow
which is not yet available.
The refined scaling arguments of Ref.~\cite{F97} represent
a further step towards this goal.
In Ref.~\cite{KNH},
the stability of the elastic Bragg glass phase
has been investigated
for a layered uniaxial geometry, where the
magnetic field is parallel to the CuO-planes,
by means of a self-consistent variational
calculation and a scaling argument
identifying the shear instability due to proliferating
dislocations
by the disorder-induced decoupling of the layers.
For this geometry
a Lindemann-like criterion has been derived, which
is given
below in (\ref{cond}) and relates
the stability of the Bragg glass phase to the
ratio of the positional correlation length and the FL spacing.
These findings are supported by a more rigorous RG analysis
\cite{CGL96,KH96} for a simplified model with only two layers of
FLs in a parallel magnetic field.
The usual experimental situation with the magnetic field perpendicular
to the CuO-layers, which will be considered in the present work, is
theoretically less understood
mainly because displacements of the FLs
have two components (biaxial) instead of one
component (uniaxial).
It is unclear at present
whether topological phase transitions
of the biaxial and uniaxial model share the same universality class
\cite{F97}.
In Ref.~\cite{KNH}, it has been argued that the
scaling argument for the uniaxial geometry
can be generalized to the full, biaxial model
leading to the following
criterion estimating the range of
stability of the Bragg glass phase
with respect to
a spontaneous formation of dislocation loops:
\begin{equation}
\label{cond}
R_l > c^{1/2\zeta} \left(l^2 +\lambda^2\right)^{1/2} \simeq c^{1/2\zeta}
\max{\left\{l,\lambda\right\}}~.
\end{equation}
$l$ is the FL distance and $\lambda$ the magnetic penetration
depth (we consider a magnetic field perpendicular to the CuO-planes
of the HTSC, and will specify $\lambda$ below for such a geometry).
$R_l$ is the (transversal) {\em positional correlation length} of the
disordered FL array, which is defined as the crossover length to
the asymptotic large scale behaviour of the Bragg glass phase,
where the average FL displacement starts to exceed the FL spacing
$l$, see (\ref{Rldef}).
$c$ is a number, which was obtained
in Ref.~\cite{KNH} to be of the order of $c \approx {\cal O}(50)$, and
$\zeta \approx 1/5$ is the roughness exponent of the pre-asymptotic
so-called ``random manifold'' regime, see (\ref{zeta}) below.
At the boundaries of the regime given by (\ref{cond}),
a topological transition occurs, and dislocations proliferate.
Beyond the transition line the FL array
may form an amorphous VG with vanishing shear modulus or a viscous
FL liquid.
This article is divided into three parts.
First, we will review the pre-asymptotic regimes of the
FL array subject to point disorder on scales smaller
than the positional correlation length $R_{l}$. This allows us
to express $R_l$ in terms of the microscopic parameters of the HTSC
and the disorder strength, and to obtain its dependence on
magnetic induction $B$ and temperature $T$.
In the second part, we
will demonstrate the equivalence of the above criterion (\ref{cond}) to the
phenomenological
Lindemann criterion in the form
$\overline{\langle u^2 \rangle}(l) < c_L^2 l^2$
(as for example used in Refs.~\cite{GL97}), where
$\overline{\langle u^2 \rangle}(l)$ is the (disorder-averaged)
{\em relative}
mean square displacement of neighboured
FLs. This yields
a relation $c \approx c_L^{-2}$ between $c$ from (\ref{cond}) and the
Lindemann-number $c_L$, and the value $c \approx {\cal O}(50)$ found by
a variational calculation in Ref.~\cite{KNH} turns out to be in good
agreement with a value $c_L \approx 0.15$ widely used in the literature
for the Lindemann-number.
This equivalence further supports a scenario
where the topological transition of the
FL array subject to point disorder may be described as disorder-induced
melting by unbound dislocations on the shortest scale $l$~\cite{KNH}.
Finally, and most importantly from the experimental point of view,
we estimate the region of the phase diagram of YBCO
in the B-T plane (see Fig.~1) where the Bragg glass phase
is stable and should
be observable experimentally or numerically according to the
Lindemann-like criterion (\ref{cond}).
We find qualitative
agreement with experiments~\cite{O}.
The upper phase boundary of the
Bragg glass, which we obtain using (\ref{cond}), turns
out to be identical to the one obtained by Erta\c{s} and Nelson~\cite{EN}.
\section{Positional Correlation Length}
To relate the positional correlation length
$R_{l}$ to the microscopic parameters of the HTSC and the
disorder strength, we have to
review the crossover between the different pre-asymptotic regimes of
the dislocation-free disordered FL array
preceding the asymptotic Bragg glass phase, and the
associated crossover length scales~\cite{Bl}. These crossovers are
induced by the interplay between the
FL interaction, the periodicity of the FL lattice and the disorder
potential,
which are in addition affected by thermal fluctuations,
and lead to essentially two different pre-asymptotic regimes:
On the shortest scales, we have
the ``Larkin'' or ``random force'' regime of Larkin and Ovchinnikov~\cite{LO},
which crosses over
to the so-called ``random manifold'' regime at the Larkin length,
before the
asymptotic Bragg glass behaviour sets in on the largest scales
exceeding the positional correlation length.
In between this sequence of crossovers, one additional length scale
is set by the FL interaction, which describes a crossover from a
``single vortex'' behaviour to a ``collective'' behaviour.
In the following,
we consider the usual experimental situation ${\bf H}||{\bf c}$
of a magnetic field perpendicular to the CuO-planes of the HTSC.
FL positions are parameterized by the two-component displacement-field
${\bf u}({\bf R},z)={\bf u}({\bf r})$ in a continuum approximation of
the Abrikosov lattice,
where ${\bf R}$ is the vectors in the ${\bf ab}$-plane and $z$ is the
coordinate in the ${\bf c}$-direction, or by the Fourier transform
${\bf \tilde{u}}({\bf K},k_z)={\bf \tilde{u}}({\bf k})$.
Let us adopt the convention to denote scales longitudinal to
the FLs in the z-direction by $L$ and transversal scales in the
${\bf ab}$-plane by $R$.
Moreover, it turns out to be convenient to use the
{\em reduced induction}
$b \equiv B/B_{c2}(T) = 2\pi \xi_{ab}^2/l^2$ to measure the strength
of the magnetic field.
\subsection{Interaction-induced Length Scale $L^*$}
The dislocation- and disorder-free FL array can be described by
elasticity theory (see Ref.~\cite{Bl} for a review) in the displacement
field ${\bf u}$ with the elastic moduli
$c_{11}$, $c_{44}$ and $c_{66}$, which can in general be dispersive
(i.e., k-dependent in Fourier-space) due to the non-locality of the
FL interaction.
Except for extremely low magnetic fields, the FL lattice is essentially
incompressible ($c_{11} \gg c_{66}$), and we can neglect
longitudinal compression
modes to a good approximation.
Note also, that the shear modulus
$c_{66}$ is non-dispersive, because volume-preserving shear modes are not
affected by the non-locality in the FL interaction.
Then, the elastic Hamiltonian in the remaining
transversal part ${\bf \tilde{u}}_T$ of the displacement field is
of the form:
\begin{equation}
\label{Hel}
{\cal H}_{el}[{\bf \tilde{u}}_T]
=\frac{1}{2} \int \frac{d^2{\bf K}}{(2\pi)^2}\frac{dk_z}{2\pi}
\left\{
c_{66}({\bf K}\times{\bf \tilde{u}}_T)^2 +
c_{44}[K,k_z](k_{z} {\bf \tilde{u}}_T)^2 \right\}
\end{equation}
The dispersion-free shear modulus is given by
$c_{66}\approx \epsilon_0/4l^2$ in the dense limit
$l/\lambda_{ab} = (b/2\pi)^{-1/2}/\kappa < 1$
(with $\kappa = \lambda_{ab}/\xi_{ab}$), and
exponentially decaying $c_{66} \propto \exp{(-l/\lambda_{ab})}
\epsilon_0/l^2$ in
the dilute limit $l/\lambda_{ab} >1$.
$\epsilon_0 = (\Phi_0/4\pi\lambda_{ab})^2$ is the basic energy
(per length) scale of the FL.
As estimates for YBCO we use $\xi_{ab}(0)\approx 15\mbox{{\AA}}$,
$\epsilon_0(0)\xi_{ab}(0) \approx 1300\mbox{K}$ and
$\kappa \approx 100$~\cite{Bl}.
The tilt
modulus $c_{44}=c_{44}[K,k_z]= c_{44}^{b}[K]+c_{44}^{s}[k_z]$
is dispersive with the bulk-contribution
\begin{equation}
\label{c44b}
c_{44}^{b}[K] \simeq \frac{\epsilon_0}{l^2}
\frac{K_{BZ}^2\lambda_{ab}^2}{1+ K^2 \lambda_{c}^2+k_z^2\lambda_{ab}^2}
\end{equation}
dominating in the dense limit well within the Brillouin zone (BZ)
$K < K_{BZ} = 2\sqrt{\pi}/l$ (approximated by a circular BZ)
and the
{\em single vortex tilt modulus} $c_{44}^{s}= c_{44}^{s}[k_z]$~\cite{GK}
\begin{eqnarray}
c_{44}^{s}[k_z] &=& c_{44}^{s,J} + c_{44}^{s,em}[k_z]
\nonumber\\
&\simeq& \frac{\epsilon_0}{l^2} \left( \varepsilon^2 +
\frac{1}{\lambda_{ab}^2k_z^2}
\ln{(1+\lambda_{ab}^2k_{z}^2)} \right)
\label{c44s}
\end{eqnarray}
dominating in the dilute limit and
at the BZ boundaries $K \simeq K_{BZ}$ or on scales
$R \simeq l$.
$\varepsilon = \lambda_{ab}/\lambda_{c}$ is the anisotropy
ratio of the HTSC and approximately $\varepsilon \approx 1/5$ in
YBCO~\cite{Bl}.
The single vortex tilt modulus has a strongly dispersive
contribution $c_{44}^{s,em}[k_z]$ from the electromagnetic coupling
and an essentially dispersion-free contribution $c_{44}^{s,J}$ from
the Josephson coupling (where we neglect a logarithmically
dispersive factor in $c_{44}^{s,J}$, which is of the order unity for
the relevant wavevectors $k_z$ and magnetic inductions $b$).
The length scale for the onset of dispersion in the bulk contribution
$c_{44}^{b}$ is $\lambda_{c}$
because elements of tilted FLs lying in the ab-plane will
start to interact on scales $R < \lambda_c$~\cite{BS}.
As length scale for the onset of dispersion, $\lambda_{c}$ occurs as
well in
the Lindemann criterion (\ref{cond}).
The contribution from the electromagnetic coupling
to the single vortex tilt modulus
gives the local result $c_{44}^{s}[0] \simeq \epsilon_0/l^2$ for
$k_z < 1/\lambda_{ab}$, but its
strong dispersion $c_{44}^{s,em}[k_z] \propto k_z^{-2}$ for
$k_z > 1/\lambda_{ab}$
leads to its suppression at small wavelengths
$k_z > 1/\varepsilon\lambda_{ab}$ where $c_{44}^{s} \simeq c_{44}^{s,J}$.
From the competition of tilt and shear energy in (\ref{Hel}),
we can obtain a scaling relation between scales $L$
longitudinal to the FLs and transversal scales $R$ for typical
fluctuations involving elastic deformation:
\begin{equation}
\label{aspect}
L \simeq R \left(\frac{c_{44}[1/R,1/L]}{c_{66}}\right)^{1/2}~.
\end{equation}
The three-dimensional elastic Hamiltonian (\ref{Hel}) is valid only on
scales
$R > l$ or
\begin{equation}
L > L^* \simeq \left(
\frac{c_{44}^{s}[1/L^*]}{c_{66}}\right)^{1/2}
~.
\label{L*def}
\end{equation}
When we consider fluctuations on scales $L < L^*$
or $R < l$, the FL array
breaks up into single FLs described by 1-dimensional elasticity
in the longitudinal coordinate $z$
with a line stiffness $\epsilon_l[k_z] = c_{44}^{s}[k_z]l^2$,
because the shear energy
containing the effects of FL interactions
is always small compared to the tilt energy of the single FL.
Thus the interaction-induced
length scale $L^*$ separates a regime of ``collective'' behaviour
described by 3D elasticity from a ``single vortex'' behaviour
described by 1D elasticity.
$L^*$ starts to increase exponentially in the dilute limit
$l/\lambda_{ab}>1$
due to the exponential decay of $c_{66}$.
For the length scale $L^*$ given by (\ref{L*def}), we use therefore
the local result $c_{44}^{s} \approx c_{44}^{s,em} \simeq
\epsilon_0/l^2$ determined
by the electromagnetic coupling.
In the dense limit $l/\lambda_{ab}<1$, the scale $L^*$ is smaller than
$\varepsilon\lambda_{ab}$, and $c_{44}^{s} \approx c_{44}^{s,J}
\simeq \epsilon_0\varepsilon^2/l^2$, i.e.,
the dispersion-free contribution from the
Josephson coupling dominates.
This yields
\begin{eqnarray}
L^* &\approx&
\left\{
\begin{array}{ll}
l<\lambda_{ab}:&\varepsilon l \\
l>\lambda_{ab}:& l \left( \frac{\lambda_{ab}}{l}\right)^{3/4}
\exp{\left(\frac{l}{2\lambda_{ab}}\right)}
\end{array}
\right.
\label{L*}
\end{eqnarray}
for the interaction induced length scale $L^*$ in the dense and dilute
limits.
As we will show below, the criterion (\ref{cond}) is indeed equivalent
to a Lindemann criterion in a more conventional form where
fluctuations $\overline{\langle u^2 \rangle}$ on the transversal
scale $R \simeq l$ are considered, see (\ref{Linde}).
Therefore, the topological phase transition can be detected by
considering
fluctuations of single FLs on the longitudinal scale $L \simeq L^*$.
We focus in this article on the upper branch of the
topological transition line in
moderately anisotropic compounds as YBCO such that
the electromagnetic coupling can
essentially be neglected.
However, similar to the findings for thermal melting
\cite{BGLN}, the
electromagnetic coupling and its strongly dispersive
contribution to the single vortex tilt modulus plays an important role
for the disorder-induced topological phase transition in very
anisotropic compounds such as BSCCO~\cite{remark}.
We will consider effects from the electromagnetic
coupling in detail in Ref.~\cite{JKF2}. We mention here only
that in moderately anisotropic HTSC compounds with
the upper branch $b_{t,u}(T)$ of the topological
phase transition line will lie entirely in the dense regime
$b> 2\pi/\kappa^2$, but below the so-called
``crossover field'' $b_{cr} \sim (2\pi/\kappa^2)
(\varepsilon\lambda_{ab}^2/d^2)$
above which $L^*< d$ and the layered structure of the
HTSC becomes relevant at the transition line and requires
a discrete description in the ${\bf c}$-direction.
Only in this regime of magnetic inductions, the strongly
dispersive electromagnetic
contribution can be neglected at the topological transition
(because $L^*<\varepsilon\lambda_{ab}$),
while a continuous description
in the ${\bf c}$-direction still applies.
In the very anisotropic Bi-compounds, however, the
upper branch $b_{t,u}(T)$ of the topological
phase transition line typically lies
in the dilute limit $b< 2\pi/\kappa^2$
where the electromagnetic coupling
gives the relevant, strongly dispersive contribution to
the single vortex tilt modulus
$c_{44}^{s} \approx c_{44}^{s,em}[k_z] \propto k_z^{-2}$.
Because of this dispersion, the behaviour of a single vortex
of length $L^*$ changes drastically, and short-scale fluctuations
on the (longitudinal) scale $L \simeq \max{\{\varepsilon\lambda_{ab},d\}}$
give the main contribution~\cite{JKF2}.
In the following, we consider the dense regime of a
moderately anisotropic compounds such as YBCO and
can thus neglect the dispersive electromagnetic contribution and
use the dispersion-free, anisotropic
result $c_{44}^{s} \approx c_{44}^{s,J} \simeq
\epsilon_0 \varepsilon^2/l^2$.
At the lower branch of the topological
transition line in the dilute limit
(where $L^* > \lambda_{ab}$),
we have to take into account the electromagnetic coupling
and use the isotropic contribution
$c_{44}^{s} \approx c_{44}^{s,em}[0] \simeq \epsilon_0/l^2$ in the local
limit.
Furthermore, also in this regime effects
from the strong dispersion of $c_{44}^{s,em}[k_z]$
have to be considered.
The details of the calculation of the lower branch of the topological
transition line
will be given in Ref.~\cite{JKF2}, and we will mention only the main
results below.
\subsection{Larkin Length}
When point disorder is introduced, every vortex at position
${\bf R_{\nu}}$ in the Abrikosov lattice experiences a pinning
potential $V({\bf r})$ with mean zero and short-range correlations
\begin{equation}
\overline{V({\bf r})V({\bf r}')} = \gamma \xi_{ab}^4 \delta^2_{r_T}({\bf R}-
{\bf R}')\delta^{1}_{\xi_{c}}(z-z')~,
\end{equation}
where the overbar denotes an average over the quenched disorder.
The strength of the disorder potential is given by $\gamma= n_{pin}f_{pin}^2$,
where $n_{pin}$ is the density of pinning centers and $f_{pin}$ the
maximum pinning force exerted by one pinning center, and the
effective range of the disorder potential is given by
\begin{equation}
r_T =
\left(\xi_{ab}^2 + \langle u^2 \rangle_{th}(0,L_{\xi})\right)^{1/2}
\label{rT}
\end{equation}
($\langle\ldots\rangle_{th}$ denotes a purely thermal average for a
fixed $V({\bf r})$), which is equal to
the size $\xi_{ab}$ of
the core of a vortex at $T=0$ but broadened by thermal fluctuations
at higher temperatures.
As proposed in Ref.~\cite{Bl}, we introduce
the dimensionless disorder
strength $\delta$ as
\begin{equation}
\delta ~=~ \frac{\gamma\xi_{ab}^3}{(\epsilon_0\xi_{ab})^2}~.
\end{equation}
The interaction with the disorder is described by the Hamiltonian
\begin{equation}
{\cal H}_{dis}[{\bf u}] ~=~ \sum_{\bf \nu} \int dz V({\bf R_{\nu}}+
{\bf u}({\bf R_{\nu}},z),z)~.
\label{Hdis}
\end{equation}
For mean square displacements
\begin{equation}
\label{conv}
u(R,L) \equiv
\overline{\langle ({\bf u}({\bf r} + ({\bf R},L))- {\bf u}({\bf r}))^2
\rangle}^{1/2}
\end{equation}
smaller than the effective scale $r_T$ for variations
of the disorder potential
$V$, the FLs explore only {\em one} minimum of the disorder potential
and perturbation theory in the
displacements is valid. Expanding in (\ref{Hdis}) the disorder potential
$V$ in ${\bf u}$ yields the
{\em random force} theory of Larkin and Ovchinnikov~\cite{LO}.
The (longitudinal) {\em Larkin length}
$L_{\xi}$ is defined as the crossover scale for the random force regime,
at which the average FL displacement becomes of order of the effective
range $r_T$ of the point disorder:
\begin{equation}
\label{Lxidef}
u(0,L_{\xi}) \simeq r_T~.
\end{equation}
It is important to note that
for HTSCs such as YBCO and BSCCO,
the generic disorder strength is such that
\begin{equation}
\label{single}
L_{\xi} < L^*
\end{equation}
in the range of magnetic inductions where the elastic
Bragg glass will turn out to be stable~\cite{remark2}.
Therefore, the random force regime lies entirely in the
single vortex regime defined above, and the
Larkin length $L_{\xi}$
is given by the single vortex result
$L_{\xi}^{s}$, which is at low temperatures~\cite{Bl}
\begin{equation}
\label{Lxi0}
L_{\xi}^{s}(0) \simeq \varepsilon
\left( \frac{(\varepsilon\epsilon_0 \xi_{ab})^2}
{\varepsilon\gamma} \right)^{1/3}
\simeq \varepsilon\xi_{ab}~\left(\frac{\delta}{\varepsilon}\right)^{-1/3}~.
\end{equation}
This result holds as long as $r_T \simeq \xi_{ab}$. However,
above the {\em depinning temperature} $T_{dp}^{s}$ of the single vortex,
$r_T$ grows
beyond $\xi_{ab}$~\cite{Bl}:
\begin{equation}
\label{r_T}
r_T^2 \simeq \xi_{ab}^2
\left(1 +\exp{\left( \left(T/T_{dp}^{s}\right)^3 \right)}\right)~,
\end{equation}
where the depinning temperature $T_{dp}^{s}$ is given by~\cite{Bl}
\begin{equation}
\label{Tdp}
T_{dp}^{s} ~\simeq~
\varepsilon\epsilon_0 \xi_{ab}~\frac{\varepsilon\xi_{ab}}{L_{\xi}^{s}(0)}
~\simeq~
\varepsilon\epsilon_0\xi_{ab}~\left(\frac{\delta}{\varepsilon}\right)^{1/3}~.
\end{equation}
Above $T_{dp}^{s}$, $L_{\xi}^{s}(T)$
increases exponentially with temperature due to the fact that random forces
are only marginally relevant for a single FL with two-component
displacements~\cite{Bl}:
\begin{equation}
\label{LxiT}
L_{\xi}^{s}(T) \simeq L_{\xi}^{s}(0) \left\{ \begin{array}{ll}
T\ll T_{dp}^{s}:~& 1 \\
T>T_{dp}^{s}:~& \left(T/T_{dp}^{s}\right)^{-1}
\exp{\left(\left(T/T_{dp}^{s}\right)^3\right)}
\end{array} \right.
\end{equation}
Let us discuss estimates of the quantities $L_{\xi}$ and
$T_{dp}^{s}$ at this point, which provide alternative measures of
the disorder strength for a HTSC.
In Ref.~\cite{EN}, the disorder strength is given by
$T_{dp}^{s}\approx 10K$ in BSCCO (where $\varepsilon\approx 1/100$),
which leads to $\delta/\varepsilon \approx 1$
with (\ref{Tdp}).
This estimate is considerably higher than typical
values given in Ref.~\cite{Bl} for weak pinning.
Therefore, we will use instead
estimates in the range $\delta/\varepsilon \approx 10^{-3}\ldots 10^{-1}$
for YBCO
in accordance with Ref.~\cite{Bl} which yield
$T_{dp}^{s} \approx 20\ldots 65\mbox{K}$
and values of the
order of $L_{\xi}^{s}(0) \approx 30\ldots 6\mbox{{\AA}}$ for the
(longitudinal) Larkin length in YBCO.
For higher disorder strengths
the $T=0$ Larkin length $L_{\xi}^{s}(0)$ can
become {\em smaller} than the layer spacing $d$.
In YBCO, where $d \approx 12\mbox{{\AA}}$, this happens for
quite strong disorder $\delta/\varepsilon \gtrsim 2 \cdot 10^{-2}$.
Then, each pancake-like FL element of length $d$
is pinned individually and we enter a {\em strong pinning} regime.
This requires
a description of pinning at the scale $d$ as the smallest
physical length scale in the longitudinal direction.
In other words, we have to consider the disorder-induced
relative displacement $r_{d}^2 = \overline{\langle u^2 \rangle}$
of two pancake vortices in adjacent layers. This has been calculated
in Ref.~\cite{KGL} at $T=0$ by means of an Imry-Ma argument (see
also~\cite{engel})
with the result
\begin{equation}
r_{d}^2(0) \approx \frac{d U_p }{\epsilon_0\varepsilon^2
\ln{\left(d^2/\varepsilon^2 r_{d}^2(0)\right)}}
\ln^{-1/2}{\left(
\frac{ r_{d}^2(0) }{2\sqrt{\pi}\xi_{ab}^2}\right)}~,
\label{rd(0)}
\end{equation}
where we introduced the mean-square disorder energy
$U_p^2 := \gamma d\xi_{ab}^2$ of a line-segment of length $L \simeq
d$. The result (\ref{rd(0)}) is valid
for $r_d(0) > \xi_{ab}$, i.e., if the relative displacement exceeds
the correlation length of the disorder potential, which is
the case just for $d> L_{\xi}^{s}(0)$.
The equation (\ref{rd(0)}) has to be solved self-consistently,
but in the following we will use the estimate
obtained in the zeroth iteration
\begin{equation}
r_{d}^{2}(0) \simeq \frac{d U_p }{\epsilon_0 \varepsilon^2}
\simeq \xi_{ab}^2\frac{U_p}{T_{dp}^{s}}
\simeq
\xi_{ab}^2 \left(\frac{d}{L_{\xi}^{s}(0)}\right)^{3/2}~.
\label{rd0(0)}
\end{equation}
The exponent $3/2= 2\zeta(1,0)$, see below (\ref{rough1}),
can be interpreted as the exponent characterizing the end-to-end
displacement of a rigid rod that can tilt in a random
potential.
On scales
$L_{\xi}^{s}(0) < L < d$, each pancake can be treated as
such a rigid rod of length $L$.
Because the pinning is strong, each pancake remains individually
pinned in the presence of thermal fluctuations until the thermal
energy $T$ is greater than the typical pinning energy $U_p$
of each pancake.
Therefore, the result (\ref{rd0(0)}) remains to a good approximation
valid in the whole
temperature range $T \le U_p$:
$r_{d}(T) \simeq r_{d}(0)$.
This can be checked in a variational calculation along the lines of
Ref.~\cite{engel}.
For YBCO with a disorder strength
$\delta/\varepsilon \approx 2\cdot 10^{-1}$
we find $U_p \simeq 70\mbox{K}$.
Although the
thermally increased Larkin length $L_{\xi}^{s}(T)$ becomes equal to the
layer spacing $d$ already at a temperature
\begin{equation}
T_{L_\xi=d} \simeq T_{dp}^{s} \left(1-\frac{L_{\xi}^{s}(0)}{d}\right)
< U_p~,
\label{TLxid}
\end{equation}
[in YBCO, we find $T_{L_\xi=d} \simeq 45\mbox{K}$ for
$\delta/\varepsilon \approx 2\cdot 10^{-1}$],
the crossover from the strong
pinning on the
scale $d$ to collective pinning on the scale of the Larkin length
$L_{\xi}^{s}(T)$ can happen only at $T \simeq U_p$, where
the strongly pinned individual pancakes can thermally depin.
This result can be obtained from Ref.~\cite{engel}, where it is
shown that perturbation theory gives only a {\em locally} stable
solution in a variational treatment of the pinning problem for
two pancake vortices in adjacent layers in the temperature range
$T_{L_\xi=d} < T < U_p$ whereas the result (\ref{rd0(0)}) represents
the {\em globally} stable solution.
\subsection{Positional Correlation Length $R_l$}
On scales exceeding the Larkin length $L_{\xi}^s$, the FLs start to
explore many minima of the disorder potential $V$. However, as long as
$u(R,L)$ is smaller than the FL spacing $l$, FLs are {\em not} competing
for the same minima, and the FLs experience effectively {\em independent}
disorder potentials. This leads to the approximation
${\cal H}_{dis}[{\bf u}] \approx \int d^3{\bf r}\tilde{V}({\bf r},{\bf
u}({\bf r}))$
(on longitudinal scales exceeding the layer spacing $d$),
where $\tilde{V}$ has also short-range correlations in ${\bf u}$.
This regime is referred to as the {\em random manifold} regime~\cite{GL}.
For a d-dimensional (dispersion-free)
elastic manifold with a n-component displacement
field ${\bf u}$, the scaling behaviour of the
$\overline{\langle uu\rangle}$-correlations is known to be
\begin{equation}
\label{rough1}
u(0,L) \sim L^{\zeta(d,n)}
\end{equation}
with a roughness exponent $\zeta(d,n)$.
We are interested here in the case $d=1$, $n=2$, which is realized
on
scales $d,L_{\xi}^{s} < L < L^*$ in the single vortex regime,
where the FLs are
described as 1-dimensional elastic manifolds, and the case $d=3$, $n=2$
on scales $L^* < L <L_{l}$ (or transversal
scales
$l < R < R_{l}$) in the collective regime,
where the FL array is described as 3-dimensional elastic manifold.
$L_{l}$ and $R_{l}$ are the {\em positional correlation lengths},
which are defined
as the crossover scales for the random manifold regime,
at which the average FL displacement becomes of the order of the
FL distance $l$:
\begin{equation}
\label{Rldef}
u(R_{l},0) = u(0,L_l) = l~.
\end{equation}
On scales $R > R_{l}$, where $u(R) > l$, FLs start to
compete for the {\em same} minima, and the periodicity of the
FL lattice becomes crucial~\cite{Na,GL}. The FL array reaches its
asymptotic behaviour of the Bragg glass phase with only logarithmically
diverging $\overline{\langle uu\rangle}$-correlations, i.e.,
quasi long range positional order.
The best estimates available for the roughness exponents are~\cite{HZ}
\begin{equation}
\label{zeta}
\zeta(1,2) \approx 5/8 \qquad\mbox{and}\qquad
\zeta\equiv \zeta(3,2) \approx 1/5~,
\end{equation}
where the latter occurs also in the above Lindemann criterion (\ref{cond}).
In the collective regime the scaling relation (\ref{rough1}) gets
slightly modified by the dispersion (\ref{c44b}) of $c_{44}^{b}$ to
\begin{equation}
\label{rough2}
u^2(R,0) \sim \left(\lambda_{c}^2 + R^2\right)^{\zeta(3,2)}~,
\end{equation}
as can be checked by means of a simple Flory-type
argument, where we equate the typical disorder energy and elastic
energy (\ref{Hel}) on {\em one} dominant scale.
(As suggested by a more elaborate variational calculation as in
Ref.~\cite{GL}
we neglect possible small logarithmic
corrections of order $\ln{(1/\varepsilon)}$ in (\ref{rough2}).)
Note that for $l < R < \lambda_{c}$, the relative displacements
(\ref{rough2}) are only marginally growing due to the dispersion
of $c_{44}^{b}$.
The scaling relations (\ref{rough1},\ref{rough2}) enable us
to obtain the relation between the (transversal) positional
correlation length
$R_l$ and the (longitudinal) Larkin length $L_{\xi}^{s}$, which will
allow us to express $R_{l}$ in terms of microscopic parameters, both
for weak pinning on the scale $L_{\xi}^{s}$ (for
$L_{\xi}^{s}(T)>d$) and
for strong pinning of pancakes
on the scale $d$ (for $L_{\xi}^{s}(T)<d$).
Applying the scaling relation (\ref{rough1}) for the
$\overline{\langle uu\rangle}$-correlations to the single vortex
random manifold regime on longitudinal
scales $L_{\xi}^{s} < L < L^*$, we obtain for the case
$L_{\xi}^{s}(0)>d$ of weak pinning
\begin{equation}
\label{LxiL*}
u_* \equiv u(l,0) \simeq u(0,L^*)
\simeq r_T~\left(\frac{L^*}{L_{\xi}^s(T)}\right)^{\zeta(1,2)}~.
\end{equation}
In the same manner we can use (\ref{rough2}) in the collective
random manifold regime on transversal
scales $l < R < R_{l}$:
\begin{equation}
\label{lRl}
l^2 = u^2(R_{l}) \simeq u_*^2
\left( \frac{\lambda_{c}^2+R_l^2}{\lambda_{c}^2 + l^2} \right)^{\zeta(3,2)}
\simeq u_*^2
\left( \frac{R_l}{\lambda_{c}} \right)^{2\zeta(3,2)}~
\end{equation}
with $R_l \gg \lambda_c \gg l$.
Using (\ref{LxiL*},\ref{lRl}), $R_l$ can be expressed as
\begin{equation}
\label{RlLxi>d}
R_{l}(T) \simeq \lambda_{c}
\left(\frac{l}{{r_T}}\right)^{1/\zeta(3,2)}
\left(\frac{L_{\xi}^s(T)}{L^*}\right)^{\zeta(1,2)/\zeta(3,2)}.
\end{equation}
With the results (\ref{r_T}) for $r_T$, (\ref{L*}) for $L^*$, and
(\ref{LxiT}) for $L_{\xi}^s(T)$ together with $\zeta(3,2) \approx 1/5$
and $\zeta(1,2) \approx 5/8$ (\ref{zeta}), this yields the
desired expression for $R_l$:
\begin{eqnarray}
R_{l}(0) &\approx& \lambda_{c}
\left(\frac{b}{2\pi}\right)^{-{15}/{16}}
\left(\frac{\delta}{\varepsilon}\right)^{-{25}/{24}}
\nonumber \\
R_{l}(T) &\approx& R_{l}(0)
\left\{ \begin{array}{ll}
T\ll T_{dp}^{s}:& 1 \\
T>T_{dp}^{s}:&
\left(T/T_{dp}^{s}\right)^{-{25}/{8}} \exp{\left( \frac{5}{8}
\left(T/T_{dp}^{s}\right)^3\right)}
\end{array} \right. ~.
\label{Rl2Lxi>d}
\end{eqnarray}
The weakening of the pinning by thermal fluctuations leads to an
exponential increase of
$R_l(T)$ for temperatures above the
depinning temperature $T_{dp}^{s}$ similar (and related)
to the behaviour
of the thermally increased Larkin length $L_{\xi}^{s}(T)$.
For inductions $b/2\pi = 10^{-4}\ldots 10^{-2}$
in the dense limit $b/2\pi > 1/\kappa^2$,
a disorder strength
$\delta/\varepsilon \approx 10^{-2}$,
and $\lambda_c(0) \approx 7500\mbox{{\AA}}$, we obtain
(transversal) positional
correlation lengths $R_{l}(0) \approx (10^4\ldots 10^6)\cdot\lambda_c
\approx 7,5\cdot(10^{-1}\ldots 10)\mbox{cm}$, which are extremely large
indicating that over a wide range of length scales the
pre-asymptotic random manifold regimes should be observable rather than
the asymptotic Bragg glass regime.
For the case
$L_{\xi}^{s}(0)<d$ of strong pinning of pancakes on the scale $d$,
we apply (\ref{rough1}) for the
$\overline{\langle uu\rangle}$-correlations to the single vortex
random manifold regime on longitudinal
scales $d < L < L^*$ and obtain for low temperatures
\begin{equation}
\label{dL*}
u_* \equiv u(R=l,0) \simeq u(0,L=L^*)
\simeq r_d(0)~\left(\frac{L^*}{d}\right)^{\zeta(1,2)}~
\end{equation}
instead of (\ref{LxiL*}).
Using this and (\ref{lRl}), $R_l$ can be expressed as
\begin{equation}
\label{RlLxi<d}
R_{l}(0) \simeq \lambda_{c}
\left(\frac{l}{{r_d(0)}}\right)^{1/\zeta(3,2)}
\left(\frac{d}{L^*}\right)^{\zeta(1,2)/\zeta(3,2)}.
\end{equation}
With the results (\ref{rd0(0)}) for $r_d(0)$, (\ref{L*}) for $L^*$,
and $\zeta(3,2) \approx 1/5$
and $\zeta(1,2) \approx 5/8$ (\ref{zeta}), we obtain for
the positional correlation length $R_l$:
\begin{eqnarray}
R_{l}(0) &\approx& \lambda_{c}
\left(\frac{b}{2\pi}\right)^{-15/16}
\left(\frac{\delta}{\varepsilon}\right)^{-5/6}
\left(\frac{\xi_{ab}}{d} \varepsilon\right)^{5/8}
\label{Rl2Lxi<d}
\end{eqnarray}
This result is valid for temperatures $T \le U_p$ and gives
a temperature independent, smaller value than (\ref{Rl2Lxi>d})
in this temperature range.
At $T \simeq U_p$, pancakes can thermally depin for strong pinning,
and we expect a crossover
to the weak pinning result
(\ref{Rl2Lxi>d}) with
a pronounced increase of $R_l(T)$ with temperature.
In YBCO, strong pinning is realized for
$\delta/\varepsilon \gtrsim 2 \cdot 10^{-2}$. For $\delta/\varepsilon
\approx 2\cdot 10^{-1}$ and with the layer spacing
$d \approx 12\mbox{{\AA}}$, we find
$R_{l}(0) \approx 5\cdot(10^2\ldots 10^4)\cdot\lambda_c
\approx 3,8\cdot(10^{-2}\ldots 1)\mbox{cm}$
in the induction range $b/2\pi = 10^{-4}\ldots 10^{-2}$
in the dense limit.
\section{Lindemann Criterion}
Let us now show the equivalence of the Lindemann-like criterion
(\ref{cond}) obtained in Ref.~\cite{KNH} to the
conventional form of the Lindemann criterion generalized to
a disordered system.
The Lindemann criterion
has been proven as a very efficient phenomenological
tool to obtain the thermal melting curves of lattices, e.g.\
the disorder-free FL lattice.
There, it is formulated in its conventional
form
\begin{equation}
\label{Lindemann}
\langle u^2 \rangle_{th} = c_L^2 l^2~,
\end{equation}
with a Lindemann-number $c_L \approx 0.1\ldots 0.2$. For the thermal
melting of the FL array,
the main contributions to the left hand side of (\ref{Lindemann})
come from fluctuations on the {\em shortest} scale, which is
in the transverse direction
the FL spacing $l$, i.e.,
$\langle u^2 \rangle_{th} \approx \langle u^2 \rangle_{th}(l,0)$
(note that we apply again a convention like (\ref{conv})).
Therefore, the straightforward generalization of (\ref{Lindemann}) to the
disorder-induced melting by dislocations is
\begin{equation}
\label{Linde}
\overline{\langle u^2 \rangle}(l,0)
\simeq \overline{\langle u^2 \rangle}(0,L^*) \equiv u_*^2 = c_L^2 l^2~,
\end{equation}
where we consider again fluctuations on the shortest scale $R \simeq l$.
In Ref.~\cite{KNH}, one derivation of the
criterion (\ref{cond}) was based on a variational calculation
for a layered superconductor in a parallel field.
There it was found, that unbound
dislocations proliferate indeed on the shortest scale at the topological
transition described by (\ref{cond}),
i.e., in between every layer and thus with a distance $l$.
This suggests that the use of the conventional phenomenological
Lindemann criterion in the form
(\ref{Linde}) might be one possibility to obtain the
topological transition line.
This can be further justified by showing that the criterion (\ref{cond}),
obtained in Ref.~\cite{KNH} on the basis of a scaling argument
and a variational calculation
for a uniaxial model, is actually {\em equivalent}
to the phenomenological Lindemann criterion (\ref{Linde}):
Considering the relation (\ref{lRl}) between $u_*$ and $l$,
it becomes clear that (\ref{cond}) is the analog of
the Lindemann criterion (\ref{Linde}) formulated in terms of the underlying
transversal scales rather than the corresponding displacements.
Using (\ref{lRl}), the criterion (\ref{cond})
for the stability of the Bragg glass
can be written as
\begin{equation}
\label{Linde2}
u_*^2 < c^{-1} l^2~.
\end{equation}
This is just the above phenomenological Lindemann
criterion (\ref{Linde}), and we can identify
\begin{equation}
\label{ccL}
c \approx c_L^{-2}~.
\end{equation}
We see that the equivalence of the criterion (\ref{cond}) to
the phenomenological Lindemann criterion (\ref{Linde}) includes
the agreement of the appearing numerical factors: The
value for the Lindemann-number
$c_L \approx 0.15$, widely used in the literature, produces a
good agreement in (\ref{ccL}) with the value $c \approx {\cal O}(50)$
obtained by the variational calculation.
This equivalence to a scenario where disorder-induced
fluctuations on the shortest scale ``melt'' the FL array favours a
first order transition scenario for the topological transition,
which could not be excluded
in the experiments~\cite{Kha96}.
As we will see, the quantity $u_*^2$ is equivalent to the
mean square displacement of the ``effective'' FL
studied in the ``cage model'' of Erta\c{s} and Nelson~\cite{EN}.
They apply the Lindemann criterion directly in its phenomenological form
(\ref{Linde}) to the ``caged'' FL.
Using (\ref{ccL},\ref{zeta}),
we can cast the Lindemann-like criterion (\ref{cond})
into the form
\begin{equation}
\label{cond2}
R_l > c_L^{-1/\zeta} \left(l^2 +\lambda_{c}^2\right)^{1/2}~\approx~
c_L^{-5} \max{\left\{l,\lambda_{c}\right\}}~.
\end{equation}
\section{Phase Diagram}
Let us now address the issue of phase boundaries of the
topologically ordered Bragg glass in the B-T plane as they follow from
the Lindemann-like criterion (\ref{cond}) in the above form
(\ref{cond2}).
The results are summarized in Fig.~1.
The boundary of the regime given by (\ref{cond2}) defines a
{\em topological transition line} $B_{t}(T)$, where dislocations
proliferate and the topological order of the Bragg glass phase is
lost.
The upper branch $b_{t,u}(T)$ of this line
can be
obtained by applying the expressions (\ref{Rl2Lxi>d}) or
(\ref{Rl2Lxi<d}) for the positional
correlation length $R_{l}$ in the dense limit $b > 2\pi/\kappa^2$
to the criterion (\ref{cond2}).
For weak pinning or $L_{\xi}^{s}(0)>d$ such that we have
collective pinning on the scale $L_{\xi}^{s}$, this
yields a
condition $b < b_{t,u}(T)$ in the b-T plane with
\begin{eqnarray}
b_{t,u}(0) &\approx& 2\pi
\left(\frac{\delta}{\varepsilon}\right)^{-10/9}c_L^{16/3}
~\approx~ 2\pi
\left(\frac{\varepsilon\epsilon_0\xi_{ab}}{T_{dp}^{s}}\right)^{10/3}
c_L^{16/3}
\nonumber\\
b_{t,u}(T) &\approx& b_{t,u}(0)
\left\{ \begin{array}{ll}
T\ll T_{dp}^{s}:& 1 \\
T>T_{dp}^{s}:&
\left(T/T_{dp}^{s}\right)^{-{10}/{3}}
\exp{\left(\frac{2}{3}
\left(T/T_{dp}^{s}\right)^3\right)}.
\end{array} \right.
\nonumber\\
\label{bupTLxi>d}
\end{eqnarray}
Note that the transition line (\ref{bupTLxi>d}) is identical to the one
obtained by Erta\c{s} and Nelson~\cite{EN} by applying the conventional
phenomenological Lindemann criterion to a
``cage model'' for a single FL (this demonstrates the equivalence of the
displacement $u_*$ as defined in (\ref{Linde}) to the average
displacement of the ``caged'' FL).
Estimates of $b_{t,u}(0)$ strongly depend on the chosen value
for the Lindemann number $c_L \approx 0.1\ldots 0.2$.
A value
$c_L \approx 0.15$
and a disorder strength $\delta/\varepsilon \approx 10^{-2}$ lead to
$b_{t,u}(0) \approx 4\cdot 10^{-2}$ or
$B_{t,u}(0) \approx 6\mbox{T}$ with $B_{c2}(0) \approx 150\mbox{T}$.
For temperatures $T< T_{dp}^{s}$ the transition line is essentially
temperature-independent because the mechanism for the proliferation
of dislocation loops is purely disorder-driven at low
temperatures~\cite{KNH}. For $T > T_{dp}^{s}$ it increases exponentially
due to the very effective weakening of the pinning effects by thermal
fluctuations in the single vortex regime.
For $L_{\xi}^{s}(0)<d$, i.e., strong pinning of pancakes
on the scale $d$
(realized for
$\delta/\varepsilon \gtrsim 2 \cdot 10^{-2}$ in YBCO), we obtain instead
\begin{eqnarray}
b_{t,u}(0) &\approx& 2\pi c_L^{16/3}
\left(\frac{\delta}{\varepsilon}
\right)^{-4/3}
\left(\frac{\xi_{ab}}{d} \varepsilon\right)^{2/3}
\label{bup0Lxi<d}
\end{eqnarray}
at low temperatures.
This result gives a lower induction for the topological
transition than (\ref{bupTLxi>d})
and remains valid up to the temperature $T \simeq U_p$,
where pancakes can thermally depin.
For $T > U_p$ we expect a pronounced increase of $b_{t,u}(T)$
with temperature and a crossover to the
weak pinning result (\ref{bupTLxi>d}),
cf.\ Fig.~1.
Estimates for $b_{t,u}(0)$ are again very susceptible to changes in
the chosen value
for the Lindemann number $c_L$.
For $c_L \approx 0.15$ and $\delta/\varepsilon
\approx 2\cdot 10^{-1}$, we find
$b_{t,u}(0) \approx 8\cdot 10^{-4}$ or
$B_{t,u}(0) \approx 0,12\mbox{T}$ for YBCO.
The estimates for $B_{t,u}(0)$ obtained from (\ref{bupTLxi>d})
and (\ref{bup0Lxi<d}) are in qualitative agreement with the
experimental results for the magnetic field where in
YBCO based superlattices a change in the I-V characteristics has
been observed~\cite{O}.
Both (\ref{bupTLxi>d}) and (\ref{bup0Lxi<d}) show that
the magnetic induction $b_{t,u}$ at the topological transition
decreases
for stronger anisotropy or effectively larger disorder strength
$\delta/\varepsilon$,
and the stability region of the topologically ordered Bragg glass
shrinks.
At some temperature
$T_{x,u}$
above $T_{dp}^{s}$ (and $T_{L_\xi=d}$),
the topological transition line $b_{t,u}(T)$ will terminate in the
upper branch of the melting curve $b_{m,u}(T)$,
which is
\begin{equation}
b_{m,u}(T) ~\approx~
2\pi c_L^4 \left(\frac{\varepsilon \epsilon_0 \xi_{ab}}{T}\right)^2
\label{bmuT}
\end{equation}
in this regime of inductions for the moderately anisotropic
compound YBCO~\cite{BGLN}.
The temperature $T_{x,u}$ can be determined from the
condition that the thermally increased Larkin length $L_{\xi}^{s}(T)$
becomes equal to the scale $L^*$ of the dominant fluctuations at
the melting line and the topological transition line. Because
$\overline{\langle u^2 \rangle}(0,L_{\xi}^{s}(T)) =
\langle u^2 \rangle_{th}(0,L_{\xi}^s(T))$ at the
thermally increased Larkin length, the Lindemann criteria
(\ref{Lindemann}) for thermal melting and (\ref{cond2}) in the form
(\ref{Linde}) for the topological phase
transition are indeed fulfilled {\em simultaneously} if
$L^* = L_{\xi}^{s}(T)$:
\begin{equation}
\overline{\langle u^2 \rangle}(0,L^*) =
\langle u^2 \rangle_{th}(0,L^*) = c_L^2 l^2~.
\end{equation}
This yields
\begin{equation}
T_{x,u} \simeq T_{dp}^{s}
\ln^{1/3}{\left(
\left(\frac{\delta}{\varepsilon}\right)^{2/3}c_L^{-2}\right)}
\label{Txu}
\end{equation}
In YBCO, we find $T_{x,u} \approx 80\mbox{K}$ with
the above estimates $c_L \approx 0.15$ and $\delta/\varepsilon
\approx 2\cdot 10^{-1}$.
For $T>T_{x,u}$ beyond the melting curve $b_{m,u}(T)$,
the FL array melts into a disordered FL liquid, and the Bragg
glass order is destroyed by the thermal fluctuations on small scales
where disorder-induced fluctuations are irrelevant, whereas above
the transition line $b_{t,u}(T)$ the Bragg glass ``melts''
by disorder-induced fluctuations,
when unbound dislocation loops proliferate.
For $T<T_{x,u}$, we find $b_{m,u}(T) > b_{t,u}(T)$, and
the melting curve $b_{m,u}(T)$ lies {\em above} the topological
transition line in the b-T plane.
Therefore, we expect for temperatures $T<T_{x,u}$
that an amorphous, i.e., topologically disordered
vortex glass melts into a vortex liquid
at the thermal melting line $b_{m,u}(T)$
and consequently,
that the melting transition into a vortex
liquid at $b_{m,u}(T)$ is of a different nature below and above $T_{x,u}$.
In the experiments reported in Ref.~\cite{Saf93}, such a change in the
properties of the melting transition has indeed been observed in YBCO
at a temperature around $75\mbox{K}$ which is in fairly good agreement with
our result for $T_{x,u}$.
Let us now give the main results for the lower branch of the topological
transition line $b_{t,l}(T)$ at which the strongly dispersive
contribution
from the electromagnetic coupling
to the single vortex tilt modulus is dominating.
At low inductions in the dilute limit $b \ll 2\pi/\kappa^2$,
the criterion (\ref{cond2}) will be violated due to the exponential
decrease of the shear modulus $c_{66}$, or increase of the
interaction-induced length scale $L^*$ (\ref{L*}).
At low temperatures $T\simeq 0$, the positional correlation length
$R_l(0)$ can be determined also from (\ref{RlLxi>d}) using the
{\em isotropic} single vortex
Larkin length (given by (\ref{Lxi0}) with $\varepsilon=1$)
and the appropriate result for
$L^*$.
Because the isotropic Larkin length is always greater than the layer
spacing, the layered structure is irrelevant for the collective
pinning at low inductions.
The criterion (\ref{cond2}) then yields for the
lower branch of the topological transition line
\begin{equation}
b_{t,l}(0)\approx \frac{2\pi}{\kappa^2}
\ln^{-2}{\left(c_L^{16/5} \kappa^{6/5} \delta(0)^{-2/3}\right)}
\label{blow0}
\end{equation}
Due to the strong dispersion of $c_{44}^{s,em}[k_z]$, the
thermal depinning at higher temperatures is more complex and
involves several crossover temperatures. However, only above the
{\em isotropic} single vortex depinning temperature $T_{dp,i}^{s}$
(given by (\ref{Tdp}) with
$\varepsilon=1$)
the positional correlation length is increasing exponentially
similarly to the thermally increased isotropic Larkin length.
This gives only a weak logarithmic temperature dependence for
$T<T_{dp,i}^{s}$
whereas we find the asymptotics
\begin{equation}
\label{blowT}
b_{t,l}(T) ~\sim~ \frac{25\pi}{2\kappa^2}
\left(\frac{T}{T_{dp,i}^{s}}\right)^{-6}~.
\end{equation}
at temperatures $T \gg T_{dp,i}^{s}$ well above the isotropic
single vortex depinning temperature.
At a temperature $T_{x,l}~(>T_{dp,i})$, $b_{t,l}(T)$ will terminate
in the lower branch of
the melting curve $b_{m,l}(T)$, which increases
logarithmically with temperature~\cite{BGLN}
\begin{equation}
b_{m,l}(T) ~\approx~ \frac{2\pi}{\kappa^2}
\ln^{-2}{\left( \frac{ {c_L}^4\epsilon_0^2 \lambda_{ab}^2}{T^2}
\right)}~.
\label{bmlT}
\end{equation}
Analogously to the findings for the upper branch of the topological
transition line, $T_{x,l}$ can be determined from the condition
that the thermally increased isotropic Larkin length becomes equal to
the scale $L^*$ at the melting line. This yields
\begin{equation}
T_{x,l} \simeq
T_{dp,i}^{s} \ln^{1/3}{\left(c_L^2 \kappa
\varepsilon^{2/3} \left(\frac{\delta}{\varepsilon}\right)^{-1/3}
\right)}
\label{Txl}
\end{equation}
With $c_L \approx 0.15$ and $\delta/\varepsilon \approx 10^{-2}$,
we obtain
$b_{t,l}(0) \approx 0.16 (2\pi/\kappa^2) \approx 1\cdot 10^{-4}$, which is
by a factor of 40 smaller than $b_{t,u}(0)$ and experimentally
hard to verify due to the small inductions $B_{t,u}(0) \approx
150\mbox{G}$.
Furthermore we find $T_{dp,i} \approx 70\mbox{K}$ and
$T_{x,l}\approx 85\mbox{K}$.
From (\ref{blow0}) it is clear
that the transition line $b_{t,u}(T)$ increases with the disorder strength
so that
the stability region of the topologically ordered Bragg glass
shrinks.
\section{Conclusion}
In conclusion, we have obtained the region in the phase diagram
of YBCO in the B-T plane,
where the topologically ordered vortex glass should be observable,
and the
topological transition lines $B_{t,u}(T)$ and
$B_{t,l}(T)$, where dislocation loops proliferate.
The resulting phase diagram, as
given by the formulae (\ref{bupTLxi>d}), (\ref{bup0Lxi<d}),
(\ref{blow0}), and (\ref{blowT}) is
depicted in
Fig.~1. The results are in qualitative agreement with the experimental
data of Ref.~\cite{O} if the observed changes in the I-V
characteristics are attributed to a topological transition of the
disordered vortex array.
The phase diagram is based on the
Lindemann-like criterion (\ref{cond}) or (\ref{cond2}),
which has been obtained by a
variational calculation for a uniaxial model and a scaling argument
presented in Ref.~\cite{KNH}. We have demonstrated the equivalence
to the conventional phenomenological formulation of the
Lindemann criterion (\ref{Linde}) up to the
involved numerical factors, i.e., the Lindemann-number $c_L$.
Our results for the upper branch of the topological transition line
$B_{t,u}(T)$ agree with Ref.~\cite{EN},
where the conventional phenomenological Lindemann-criterion
was applied
to the disorder-induced ``melting'' in the framework of a ``cage model''.
\section{Acknowledgments}
The author thanks T.~Nattermann, T.~Hwa, and A.E.~Koshelev
for discussions and support by the Deutsche Forschungsgemeinschaft
through SFB~341~(B8) and grant \mbox{KI~662/1--1}.
| proofpile-arXiv_065-476 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
A common theme in higher-dimensional algebra is `categorification':
the formation of $(n+1)$-categorical analogs of $n$-categorical
algebraic structures. This amounts to replacing equations between
$n$-morphisms by specified $(n+1)$-isomorphisms, in accord with the
philosophy that any {\it interesting} equation --- as opposed to one
of the form $x = x$ --- is better understood as an isomorphism, or
more generally an equivalence.
In their work on categorification in topological quantum field
theory, Freed \cite{Freed} and Crane \cite{Crane} have, in an
informal way, used the concept of a `2-Hilbert space': a category
with structures and properties analogous to those of a
Hilbert space. Our goal here is to define 2-Hilbert spaces
precisely and begin to study them. We concentrate on the
finite-dimensional case, as the infinite-dimensional case
introduces extra issues that we are not yet ready to handle. We
must start by categorifying the various ingredients in the
definition of Hilbert space. These are: 1) the zero element, 2)
addition, 3) subtraction, 4) scalar multiplication, and 5) the
inner product. The first four have well-known categorical
analogs.
1) The analog of the zero vector is a `zero object'. A zero
object in a category is an object that is both initial and
terminal. That is, there is exactly one morphism from it to any
object, and exactly one morphism to it from any object. Consider
for example the category ${\rm Hilb}$ having finite-dimensional
Hilbert spaces as objects, and linear maps between them as
morphisms. In ${\rm Hilb}$, any zero-dimensional Hilbert space is a
zero object.
2) The analog of adding two vectors is forming the direct sum, or
more precisely the `coproduct', of two objects. A coproduct of the
objects $x$ and $y$ is an object $x \oplus y$, equipped with
morphisms from $x$ and $y$ to it, that is universal with respect
to this property. In ${\rm Hilb}$, for example, any Hilbert space equipped
with an isomorphism to the direct sum of $x$ and $y$ is a coproduct of
$x$ and $y$.
3) The analog of subtracting vectors is forming the `cokernel' of
a morphism $f \colon x \to y$. This makes sense only in a
category with a zero object. A cokernel of $f \colon x \to y$
is an object ${\rm cok} f$ equipped with an epimorphism $g \colon y \to
{\rm cok} f$ for which the composite of $f$ and $g$ factors through
the zero object, that is universal with respect to this property.
Note that while we can simply subtract a number $x$ from a number
$y$, to form a cokernel we need to say how the object $x$ is
mapped to the object $y$. In ${\rm Hilb}$, for example, any space
equipped with an isomorphism to the orthogonal complement of ${\rm im}
f$ in $y$ is a cokernel of $f \colon x \to y$. If $f$ is an
inclusion, so that $x$ is a subspace of $y$, its cokernel is
sometimes written as the `direct difference' $y \ominus x$ to
emphasize the analogy with subtraction.
An important difference between zero, addition and subtraction and
their categorical analogs is that these operations represent extra
{\it structure} on a set, while having a zero object, binary
coproducts or cokernels is merely a {\it property} of a category.
Thus these concepts are in some sense more intrinsic to categories
than to sets. On the other hand, one pays a price for this:
while the zero element, sums, and differences are unique in a Hilbert space,
the zero object, coproducts, and cokernels are typically
unique only up to canonical isomorphism.
4) The analog of multiplying a vector by a complex number is tensoring
an object by a Hilbert space. Besides its
additive properties (zero object, binary coproducts, and cokernels),
${\rm Hilb}$ also has a compatible multiplicative structure, that is,
tensor products and a unit object for the tensor product. In other
words, ${\rm Hilb}$ is a `ring category', as defined by Laplaza and Kelly
\cite{Kelly2,Laplaza}. We expect it to play a role in 2-Hilbert space
theory analogous to the role played by the ring ${\Bbb C}$ of complex numbers
in Hilbert space theory. Thus we expect 2-Hilbert spaces to be `module
categories' over ${\rm Hilb}$, as defined by Kapranov and Voevodsky \cite{KV}.
An important part of our philosophy here is that ${\Bbb C}$ is the primordial
Hilbert space: the simplest one, upon which the rest are modelled. By
analogy, we expect ${\rm Hilb}$ to be the primordial 2-Hilbert space. This
is part of a general pattern pervading higher-dimensional algebra; for
example, there is a sense in which $n{\rm Cat}$ is the primordial
$(n+1)$-category. The real significance of this pattern remains somewhat
mysterious.
5) Finally, what is the categorification of the inner
product in a Hilbert space? It appears to be the `$\hom$ functor'. The
inner product in a Hilbert space $x$ is a bilinear map
\[ \langle\,\cdot\,,\, \cdot\, \rangle \colon \overline x \times x \to {\Bbb C}
\]
taking each pair of elements $v,w \in x$ to the inner product
$\langle v,w\rangle$. Here $\overline x$ denotes the conjugate
of the Hilbert space $x$. Similarly, the $\hom$ functor in a
category $C$ is a bifunctor
\[ \hom(\,\cdot\, , \,\cdot\,)\colon C^{\rm op} \times C \to {\rm Set} \]
taking each pair of objects $c,d \in C$
to the set $\hom(c,d)$ of morphisms from $c$ to $d$.
This analogy clarifies the relation between
category theory and quantum theory that is so important in
topological quantum field theory. In quantum theory the inner product
$\langle v,w \rangle$ is a {\it number} representing the
amplitude to pass from $v$ to $w$, while in category
theory $\hom(c,d)$ is a {\it set} of morphisms passing from $c$ to $d$.
To understand this analogy better, note that
any morphism $f \colon x \to y$ in ${\rm Hilb}$
can be turned around or `dualized' to obtain a morphism $f^\ast \colon y
\to x$. The morphism $f^\ast$ is called the adjoint of $f$, and satisfies
\[ \langle fv,w \rangle = \langle v,f^\ast w \rangle \]
for all $v \in x$, $w \in y$.
The ability to dualize morphisms in this way is crucial to
quantum theory. For example, observables are represented by
self-adjoint morphisms, while symmetries are represented by unitary
morphisms, whose adjoint equals their inverse.
The ability to dualize morphisms in ${\rm Hilb}$ makes this category
very different from the category ${\rm Set}$, in which the only
morphisms $f \colon x \to y$ admitting any natural sort of `dual'
are the invertible ones. There is, however,
duals for certain noninvertible morphisms in ${\rm Cat}$ --- namely,
adjoint functors. The functor $F^\ast \colon D \to C$ is
said to be a right adjoint of the functor $F \colon C \to D$ if
there is a natural isomorphism
\[ \hom(Fc,d) \cong \hom(c,F^\ast d) \]
for all $c \in C$, $d \in D$. The analogy to adjoints of
operators between Hilbert spaces is clear. Our main point here
is that that this analogy relies on the more fundamental analogy
between the inner product and the $\hom$ functor.
One twist in the analogy between the inner product and the $\hom$
functor is that the inner product for a Hilbert space takes values in
${\Bbb C}$. Since we are treating ${\rm Hilb}$ as the categorification of ${\Bbb C}$,
the $\hom$-functor for a 2-Hilbert space should take values in ${\rm Hilb}$
rather than ${\rm Set}$. In technical terms \cite{Kelly}, this
suggests that a 2-Hilbert space should be enriched over ${\rm Hilb}$.
To summarize, we expect that a 2-Hilbert space should be
some sort of category with 1) a zero object, 2) binary coproducts, and 3)
cokernels, which is 4) a ${\rm Hilb}$-module and 5) enriched over ${\rm Hilb}$.
However, we also need a categorical analog for the equation
\[ \langle v,w\rangle = \overline{\langle w,v \rangle} \]
satisfied by the inner product in a Hilbert space. That is, for any two
objects $x,y$ in a 2-Hilbert space there should be a natural isomorphism
\[ \hom(x,y) \cong \overline{\hom(y,x)} \]
where $\overline{\hom(y,x)}$ is the complex conjugate of the Hilbert space
$\hom(y,x)$. (The fact that objects in ${\rm Hilb}$ have complex conjugates
is a categorification of the fact that elements of ${\Bbb C}$ have complex
conjugates.) This natural isomorphism should also satisfy some coherence
laws, which we describe in Section 2. We put these ingredients together
and give a precise definition of 2-Hilbert spaces in Section 3.
Why bother categorifying the notion of Hilbert space? As already
noted, one motivation comes from the study of topological quantum
field theories, or TQFTs. In the introduction to this series
of papers \cite{BD}, we proposed that
$n$-dimensional unitary extended TQFTs should be treated as
$n$-functors from a certain $n$-category $n{\rm Cob}$ to a certain
$n$-category $n{\rm Hilb}$. Roughly speaking, the $n$-category $n{\rm Cob}$
should have 0-dimensional manifolds as objects, 1-dimensional
cobordisms between these as morphisms, 2-dimensional cobordisms
between these as 2-morphisms, and so on up to dimension $n$. The
$n$-category $n{\rm Hilb}$, on the other hand, should have `$n$-Hilbert
spaces' as objects, these being $(n-1)$-categories with structures
and properties analogous to those of Hilbert
spaces. (Note that an ordinary Hilbert space is a `1-Hilbert space', and
is a 0-category, or set, with extra structures and properties.)
An eventual goal of this series is to develop the framework needed to
make these ideas precise. This will require work both on $n$-categories in
general --- especially `weak' $n$-categories, which are poorly
understood for $n > 3$ --- and also on the particular $n$-categories
$n{\rm Cob}$ and $n{\rm Hilb}$. One of the guiding lights of weak $n$-category
theory is the chart shown in Figure 1. This describes `$k$-tuply
monoidal $n$-categories' --- that is, $(n+k)$-categories with only one
$j$-morphism for $j < k$. The entries only correspond to theorems for
$n+k \le 3$, but there is evidence that the pattern continues for
arbitrarily large values of $n,k$. Note in particular how as we descend
each column, the $n$-categories first acquire a `monoidal' or tensor
product structure, which then becomes increasingly `commutative' in
character with increasing $k$, stabilizing at $k = n+2$.
\vskip 0.5em
\begin{center}
{\small
\begin{tabular}{|c|c|c|c|} \hline
& $n = 0$ & $n = 1$ & $n = 2$ \\ \hline
$k = 0$ & sets & categories & 2-categories \\ \hline
$k = 1$ & monoids & monoidal & monoidal \\
& & categories & 2-categories \\ \hline
$k = 2$ &commutative& braided & braided \\
& monoids & monoidal & monoidal \\
& & categories & 2-categories \\ \hline
$k = 3$ &` & symmetric & weakly involutory \\
& & monoidal & monoidal \\
& & categories & 2-categories \\ \hline
$k = 4$ &`' & `' &strongly involutory\\
& & & monoidal \\
& & & 2-categories \\ \hline
$k = 5$ &`' &`' & `' \\
& & & \\
& & & \\ \hline
\end{tabular}} \vskip 1em
1. The category-theoretic hierarchy: expected results
\end{center}
\vskip 0.5em
At least in the low-dimensional cases examined so far, the
$n$-categories of interest in topological quantum field theory have
simple algebraic descriptions. For example, knot theorists are familiar
with the category of framed oriented 1-dimensional cobordisms embedded
in $[0,1]^3$. We would call these `1-tangles in 3 dimensions'. They
form not merely a category, but a braided monoidal category. In fact,
they form the `free braided monoidal category with duals on one object',
the object corresponding to the positively oriented point. More
generally, we expect that $n$-tangles in $n+k$ dimensions form the `free
$k$-tuply monoidal $n$-category with duals on one object', $C_{n,k}$.
By its freeness, we should be able to obtain a representation of
$C_{n,k}$ in any $k$-tuply monoidal $n$-category with duals by
specifying a particular object therein.
When the codimension $k$ enters the stable range $k \ge n + 2$ we hope
to obtain the `free stable $n$-category with duals on one object',
$C_{n,\infty}$. A unitary extended TQFT should be a representation of
this in $n{\rm Hilb}$. If as expected $n{\rm Hilb}$ is a stable $n$-category
with duals, to specify a unitary extended TQFT would then simply be to
specify a particular $n$-Hilbert space. More generally, we expect an
entire hierarchy of $k$-tuply monoidal $n$-Hilbert spaces in analogy
to the category-theoretic hierarchy, as shown in Figure 2. We also
hope that an object in a $k$-tuply monoidal $n$-Hilbert space $H$
will determine a representation of $C_{n,k}$ in $H$,
and thus an invariant of $n$-tangles in $(n+k)$ dimensions.
\vskip 0.5em
\begin{center}
{\small
\begin{tabular}{|c|c|c|c|} \hline
& $n = 1$ & $n = 2$ & $n = 3$ \\ \hline
$k = 0$ & Hilbert & 2-Hilbert & 3-Hilbert \\
& spaces & spaces & spaces \\ \hline
$k = 1$ & H*-algebras & 2-H*-algebras & 3-H*-algebras \\ \hline
$k = 2$ &commutative& braided & braided \\
& H*-algebras & 2-H*-algebras & 3-H*-algebras \\ \hline
$k = 3$ &`' & symmetric & weakly involutory \\
& & 2-H*-algebras & 3-H*-algebras \\ \hline
$k = 4$ &`' & `' &strongly involutory\\
& & & 3-H*-algebras \\ \hline
$k = 5$ &`' &`' & `' \\
& & & \\
& & & \\ \hline
\end{tabular}} \vskip 1em
2. The quantum-theoretic hierarchy: expected results
\end{center}
\vskip 0.5em
We are far from proving general results along these lines! However,
in Section 4 we sketch the structure of $2{\rm Hilb}$ as a strongly
involutory 3-H*-algebra, and in Section 5 we define 2-H*-algebras,
braided 2-H*-algebra, and symmetric 2-H*-algebras, and describe their
relationships to 1-tangles in 2, 3, and 4 dimensions, respectively.
An exciting fact about the quantum-theoretic hierarchy is that it
automatically subsumes various branches of representation theory.
2-H*-algebras arise naturally as categories of unitary
representations of certain Hopf algebras, or more generally `Hopf
algebroids', which are to groupoids as Hopf algebras are to
groups \cite{Lu}. Braided 2-H*-algebras arise in a similar way from
certain quasitriangular Hopf algebroids --- for example, quantum
groups --- while symmetric 2-H*-algebras arise from certain
triangular Hopf algebroids --- for example, groups.
In Section \ref{recon} of this paper we concentrate on the
symmetric case. Generalizing the Doplicher-Roberts theorem
\cite{DR}, we prove that all symmetric 2-H*-algebras are
equivalent to categories of representations of `compact
supergroupoids'. If a symmetric 2-H*-algebra is `purely
bosonic', it is equivalent to a category of representations of a
compact groupoid; if it is `connected', it is equivalent to a
category of representations of a compact supergroup. In
particular, any connected even symmetric 2-H*-algebra is
equivalent to the category ${\rm Rep}(G)$ of continuous
unitary finite-dimensional representations
of a compact group $G$. This is the original Doplicher-Roberts
theorem.
One can view our generalized Doplicher-Roberts theorem as a
categorified version of the Gelfand-Naimark theorem. The
Gelfand-Naimark theorem applies to commutative C*-algebras, but
one can easily deduce a version for commutative H*-algebras.
Roughly speaking, this says that every commutative H*-algebra
$H$ is isomorphic to a commutative H*-algebra of {\it
functions} from some set ${\rm Spec}(H)$ to
${\Bbb C}$. Similarly, our theorem implies that every even symmetric
2-H*-algebra $H$ is equivalent to a symmetric 2-H*-algebra of
{\it functors} from some groupoid
${\rm Spec}(H)$ to ${\rm Hilb}$. The equivalence is given explicitly by a
categorified version of the Gelfand transform. We also construct
a categorified version of the Fourier transform, applicable to
the representation theory of compact abelian groups.
These links between the quantum-theoretic hierarchy and
representation theory give new insight into the
representation theory of classical groups. The designation of a
group as `classical' is more a matter of
tradition than of some conceptual definition, but in practice
what makes a group `classical' is that it has a nice right
universal property. In other words, there is a simple
description of homomorphisms into it. Using the fact that group
homomorphisms from $G$ to $H$ determine symmetric 2-H*-algebra
homomorphisms from ${\rm Rep}(H)$ to ${\rm Rep}(G)$, one can show that for
a classical group $H$ the symmetric 2-H*-algebra ${\rm Rep}(H)$ has
nice left universal property: there is a simple description of
homomorphisms out of it.
For example, the group ${\rm U}(n)$ has a distinguished
$n$-dimensional unitary representation $\rho$, its fundamental
representation on ${\Bbb C}^n$. An $n$-dimensional unitary
representation of any group $G$ is essentially the same as
a homomorphism from $G$ to ${\rm U}(n)$. Using this right universal
property of ${\rm U}(n)$, we
show in Section \ref{recon} that the category of unitary
representations of ${\rm U}(n)$ is the `free symmetric 2-H*-algebra on
one object of dimension $n$'. This statement tersely encodes the
usual description of the representations of ${\rm U}(n)$ in terms of
Young diagrams. We also give similar characterizations of the
categories of representations of other classical groups.
In what follows, we denote the composition of 1-morphisms, the
horizontal composition of a 1-morphism and a 2-morphism (in
either order) and the horizontal composition of 2-morphisms is
denoted by $\circ$ or simply juxtaposition. Vertical composition
of 2-morphisms is denoted by $\cdot\,$. {\it Nota bene}: in
composition we use the ordering in which, for example, the
composite of $f \colon x \to y$ and $g \colon y \to z$ is denoted
$f\circ g$. We denote the identity morphism of an object $x$
either as $1_x$ or, if there is no danger of confusion, simply as
$x$. We refer to our earlier papers on higher-dimensional
algebra as HDA0 \cite{BD} and HDA1 \cite{BN}.
\section{H*-Categories}
Let ${\rm Hilb}$ denote the category whose objects are
finite-dimensional Hilbert spaces, and whose morphisms are arbitrary
linear maps. (Henceforth, all Hilbert spaces will taken as
finite-dimensional unless otherwise specified.) The category ${\rm Hilb}$
is symmetric monoidal, with ${\Bbb C}$ as the unit object, the usual tensor
product of Hilbert spaces as the monoidal structure, and the maps
\[ S_{x,y}(v \otimes w) = w \otimes v \]
as the symmetry, where $x,y \in {\rm Hilb}$, $v \in x$, and $w \in y$.
Using enriched category theory \cite{Kelly} we may thus define
the notion of a category enriched over ${\rm Hilb}$, or ${\rm Hilb}$-category.
Concretely, this amounts to the following:
\begin{defn}\hspace{-0.08in}{\bf .}\hspace{0.1in} A ${\rm Hilb}$-{\rm category} $H$ is a category such that
for any pair of objects $x,y \in H$ the set of morphisms $\hom(x,y)$ is
equipped with the structure of a Hilbert space, and for any
objects $x,y,z \in \H$ the composition map
\[ \circ\; \colon \hom(x,y) \times \hom(y,z) \to \hom(x,z) \]
is bilinear. \end{defn}
We may think of the `\hom' in a ${\rm Hilb}$-category $H$ as a functor
\[ \hom \colon \H^{{\rm op}} \times H \to {\rm Hilb} \]
as follows. An object in $\H^{\rm op} \times H$ is just a pair of objects
$(x,y)$ in $\H$, and the $\hom$ functor assigns to this the object
$\hom(x,y) \in {\rm Hilb}$. A morphism $F \colon (x,y) \to (x',y')$ in
$\H^{\rm op} \times \H$ is just a pair of
morphisms $f \colon x' \to x$, $g \colon y \to y'$ in $\H$, and the
$\hom$ functor assigns to $F$ the morphism $\hom(F) \colon
\hom(x,y) \to \hom(x',y')$ given by
\[ \hom(F)(h) = fhg .\]
As described in the introduction, we may regard ${\rm Hilb}$ as the
categorification of ${\Bbb C}$. A structure on ${\Bbb C}$ which is crucial for
Hilbert space theory is complex conjugation,
\[ {}^{\overline{\hbox{\hskip 0.5em}}}\, \colon {\Bbb C} \to {\Bbb C}. \]
The categorification of this map is a functor
\[ {}^{\overline{\hbox{\hskip 0.5em}}}\, \colon {\rm Hilb} \to {\rm Hilb} \]
called {\it conjugation}, defined as follows.
First, for any Hilbert space $x$, there is a conjugate Hilbert space
$\overline x$. This has the same underlying abelian
group as $x$, but to keep things straight let us temporarily write
$\overline v$ for the element of $\overline x$
corresponding to $v \in x$. Scalar multiplication in $\overline x$ is
then given by
\[ c\overline v = \overline {(\overline c v)} \]
for any $c \in {\Bbb C}$, while the inner product is given by
\[ \langle \overline v, \overline w \rangle = \overline {\langle
v,w\rangle}. \]
Second, for any morphism $f \colon x \to y$ in ${\rm Hilb}$, there is
a conjugate morphism $\overline f \colon \overline x \to \overline y$,
given by
\[ \overline f(\overline v) = \overline{f(v)} \]
for all $v \in x$. One can easily check that with these
definitions conjugation is a covariant functor. Note that the
square of this functor is equal to the identity. Also note that
a linear map $f \colon x \to \overline y$ is the same thing as an
antilinear (i.e., conjugate-linear) map from $x$ to $y$, while a
unitary map $f \colon x \to \overline y$ is the same thing as an
antiunitary map from $x$ to $y$.
Now, just as in a Hilbert space we have the equation
\[ \langle v,w \rangle = \overline {\langle w,v\rangle} \]
for any pair of elements, in a 2-Hilbert space we would like
an isomorphism
\[ \hom(x,y) \cong \overline {\hom(y,x) } \]
for every pair of objects. This isomorphism should be be
`natural' in some sense, but $\hom(x,y)$ is contravariant in $x$
and covariant in $y$, while $\overline {\hom(y,x)}$ is covariant
in $x$ and contravariant in $y$. Luckily ${\rm Hilb}$ is a
$\ast$-category, which allows us to define `antinatural
isomorphisms' between covariant functors and contravariant
functors from any category to ${\rm Hilb}$.
This works as follows. In general, a {\it $\ast$-structure} for a
category $C$ is defined as a contravariant functor $\ast \colon C \to C$
which acts as the identity on the objects of $C$ and satisfies $\ast^2 =
1_C$. A {\it $\ast$-category} is a category equipped with a
$\ast$-structure. For example, ${\rm Hilb}$ is a $\ast$-category
where for any morphism $f\colon x \to y$ we define $f^\ast \colon y \to x$
to be the Hilbert space adjoint of $f$:
\[ \langle fv, w\rangle = \langle v, f^\ast w \rangle \]
for all $v \in x$, $w \in y$.
Now suppose that $D$ is a $\ast$-category and $F \colon C \to D$ is a
covariant functor, while $G \colon C \to D$ is a contravariant functor.
We define an {\it antinatural transformation} $\alpha \colon F \Rightarrow G$
to be a natural transformation
from $F$ to $G \circ \ast$.
Similarly, an antinatural transformation from $G$ to $F$ is defined to
be a natural transformation from $G$ to $F \circ \ast$.
As a step towards defining a 2-Hilbert space we now define an
H*-category.
\begin{defn}\hspace{-0.08in}{\bf .}\hspace{0.1in} An {\rm H*-category} is a ${\rm Hilb}$-category with a
$\ast$-structure that defines an antinatural transformation from
$\hom(x,y)$ to $\overline {\hom(y,x)}$.
\end{defn}
\noindent
This may require some clarification.
Given a $\ast$-structure $\ast
\colon \H \to
\H$, we obtain for any objects $x,y \in \H$ a function $\ast \colon \hom(x,y)
\to \hom(y,x)$. By abuse of notation we may also regard this as a function
\[ \ast \colon \hom(x,y) \to \overline{\hom(y,x)}. \]
We then demand that this define an antinatural
transformation between the covariant functor $\hom \colon H^{\rm op} \times
H \to {\rm Hilb}$ to the contravariant functor sending $(x,y) \in
H^{\rm op} \times H$ to $\overline{\hom(y,x)} \in {\rm Hilb}$.
The following proposition gives a more concrete description of
H*-categories:
\begin{prop}\label{H*1}\hspace{-0.08in}{\bf .}\hspace{0.1in} An H*-category $\H$ is the same as a
${\rm Hilb}$-category equipped with antilinear maps $\ast \colon \hom(x,y) \to
\hom(y,x)$ for all $x,y \in \H$, such that
\begin{enumerate}
\item $f^{\ast\ast} = f$,
\item $(fg)^\ast = g^\ast f^\ast$,
\item $\langle fg,h\rangle = \langle g,f^\ast h \rangle$,
\item $\langle fg,h\rangle = \langle f, hg^\ast \rangle$
\end{enumerate}
whenever both sides of the equation are well-defined.
\end{prop}
Proof - First suppose that $\H$ is an H*-category. By the
antinaturality of $\ast$, for all $x,y \in \H$ there is a linear
map $\ast \colon \hom(x,y) \to \overline{\hom(y,x)}$, which is the same as
an antilinear map $\ast \colon \hom(x,y) \to \hom(y,x)$. The fact that
$\ast$ is a $\ast$-structure implies properties 1 and 2.
As for 3 and 4, suppose $(x,y)$ and
$(x',y')$ are objects in $\H^{\rm op} \times \H$, and let $(f,g)$ be a
morphism from $(x,y)$ to $(x',y')$.
The fact that $\ast$ is an antinatural transformation means that
the following diagram commutes:
\[
\begin{diagram}[\hom(x'y')]
\node{\hom(x,y)} \arrow{e,t}{\ast} \arrow{s,l}{V(f,g)}
\node{\overline{\hom(y,x)}} \arrow{s,r}{W(f,g)^\ast} \\
\node{\hom(x',y')} \arrow{e,t}{\ast} \node{\overline{\hom(y',x')}}
\end{diagram}
\]
where $V$ is the covariant functor
\[
\begin{diagram}[xxxxxxx]
\node{\H^{\rm op} \times \H}
\arrow{e,t}{\hom}
\node{{\rm Hilb}}
\end{diagram}
\]
and $W$ is the contravariant functor
\[
\begin{diagram}[xxxxxxx]
\node{\H^{\rm op} \times \H}
\arrow{e,t}{S_{\H^{\rm op},\H}}
\node{\H \times \H^{\rm op}}
\arrow{e,t}{\hom}
\node{{\rm Hilb}}
\arrow{e,t}{{}^{\overline{\hbox{\hskip 0.5em}}}}
\node{{\rm Hilb},}
\end{diagram}
\]
where in this latter diagram
$S$ denotes the symmetry in ${\rm Cat}$, $\hom$ is regarded as a
contravariant functor from $(\H^{\rm op} \times \H)^{\rm op} \cong
\H \times \H^{\rm op}$ to ${\rm Hilb}$, and the overline denotes conjugation.
This is true if and only if for all $h \in \hom(x,y)$
and $k \in \overline{\hom(y',x')}$,
\[ \langle (V(f,g)h)^\ast ,k \rangle =
\langle W(f,g)^\ast h^\ast, k\rangle \]
or in other words,
\[ \langle (fhg)^\ast, k\rangle = \langle h^\ast, gkf\rangle \]
or
\[ \langle g^\ast h^\ast f^\ast, k\rangle = \langle h^\ast, gkf\rangle
.\]
Here the inner products are taken in $\overline{\hom(y',x')}$, but the
equations also hold with the inner product taken in $\hom(y',x')$.
Taking either $f$ or $g$ to be the identity, we obtain
3 and 4 after some relabelling of variables.
Conversely, given antilinear maps $\ast \colon \hom(x,y) \to
\hom(y,x)$ for all $x,y \in \H$, properties 1 and 2 say that these
define a $\ast$-structure for $\H$, and using 3 and 4 we obtain
\begin{eqnarray*} \langle g^\ast h^\ast f^\ast , k\rangle &=&
\langle g^\ast h^\ast , kf\rangle \\
&=& \langle h^\ast, gkf\rangle ,\end{eqnarray*}
showing that $\ast$ is antinatural.
\hskip 3em \hbox{\BOX} \vskip 2ex
\begin{cor}\label{H*2}\hspace{-0.08in}{\bf .}\hspace{0.1in} If $H$ is an H*-category, for all objects $x,y
\in H$ the map $\ast \colon \hom(x,y) \to \hom(y,x)$ is antiunitary.
\end{cor}
Proof - The map $\ast \colon \hom(x,y) \to \hom(y,x)$ is antilinear, and
by 3 and 4 of Proposition \ref{H*1} we have
\[ \langle f,g \rangle = \langle g^\ast, f^\ast \rangle
= \overline{\langle f^\ast, g^\ast \rangle} \]
for all $f,g \in \hom(x,y)$, so $\ast$ is antiunitary. \hskip 3em \hbox{\BOX} \vskip 2ex
Next we give a structure theorem for H*-categories. This relies
heavily on the theory of `H*-algebras' due to Ambrose
\cite{Ambrose}, so let us first recall this theory. For our
convenience, we use a somewhat different definition of
H*-algebra than that given by Ambrose. Namely, we restrict our
attention to finite-dimensional H*-algebras with multiplicative unit,
and we do not require the inequality $\|ab\| \le \|a\| \, \|b\|$.
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in} An {\it H*-algebra} $A$ is a Hilbert space
that is also an associative algebra with unit, equipped with
an antilinear involution $\ast \colon A \to A$ satisfying
\begin{eqnarray*}
\langle ab,c\rangle &=& \langle b,a^\ast c \rangle \\
\langle ab,c\rangle &=& \langle a, cb^\ast \rangle \end{eqnarray*}
for all $a,b,c \in A$.
An {\rm isomorphism} of H*-algebras is a unitary operator
that is also an involution-preserving algebra isomorphism.
\end{defn}
The basic example of an H*-algebra is the space of
linear operators on a Hilbert space $H$.
Here the product is the usual product of operators, the involution is
the usual adjoint of operators, and the inner product is given by
\[ \langle a,b\rangle = k \,{\rm tr}(a^\ast b) \]
where $k > 0$. We denote this H*-algebra by $L^2(H,k)$.
It follows from the work of Ambrose that all H*-algebras can be built
out of H*-algebras of this form. More precisely, every H*-algebra $A$
is the orthogonal direct sum of finitely many
minimal 2-sided ideals $I_i$, each
of which is isomorphic as an H*-algebra to $L^2(H_i,k_i)$
for some Hilbert space $H_i$ and some positive real number $k_i$.
This result immediately classifies H*-categories with one object. Given
an H*-category with one object $x$, ${\rm end}(x)$ is an H*-algebra, and is
thus of the above form. Conversely, any H*-algebra is isomorphic to
${\rm end}(x)$ for some H*-category with one object $x$.
We generalize this to arbitrary H*-categories as follows.
Suppose first that $H$ is an H*-category with finitely many objects.
Let $A$ denote the orthogonal direct sum
\[ A = \bigoplus_{x,y} \hom(x,y) .\]
Then $A$ becomes an H*-algebra if we define
the product in $A$ of morphisms in $H$ to be their composite when the
composite exists, and zero otherwise, and define the involution in $A$
using the $\ast$-structure of $H$. $A$ is thus the
orthogonal direct sum of finitely many minimal 2-sided ideals:
\[ A = \bigoplus_{i = 1}^n L^2(H_i, k_i)
.\]
For each object $x \in H$, the identity morphism $1_x$ can be regarded as an
element of $A$. This element is a self-adjoint projection, meaning that
\[ 1_x^\ast = 1_x, \qquad 1_x^2 = 1_x .\]
It follows that we may write
\[ 1_x = \bigoplus_{i = 1}^n p_i^x \]
where $p_i^x \in L^2(H_i, k_i)$ is the
projection onto some subspace $H_i^x \subseteq H_i$.
Note that the elements $1_x$, $x \in H$, form a complete orthogonal set of
projections in $A$. In other words, $1_x 1_y = 0$ if $x \ne y$,
and
\[ \sum_{x \in H} 1_x = 1.\]
Thus each Hilbert space $H_i$ is the orthogonal direct sum of the
subspaces $H_i^x$.
This gives the following structure theorem for H*-categories:
\begin{thm}\label{H*3}\hspace{-0.08in}{\bf .}\hspace{0.1in} Let $H$ be an H*-category and $S$ any finite
set of objects of $H$. Then for some $n$, there
exist positive numbers $k_i > 0$ and
Hilbert spaces $H_i^x$ for $i = 1, \dots, n$ and $x \in S$,
such that the following hold:
\begin{enumerate}
\item {\rm For $i = 1, \dots, n$, let
\[ H_i = \bigoplus_{x \in S} H_i^x \]
denote the orthogonal direct sum, and let $p_i^x$ be the
self-adjoint projection from $H_i$ to $H_i^x$. Then
for any objects $x,y \in S$, there is a unitary isomorphism
between the Hilbert space $\hom(x,y)$ and the subspace
\[ \bigoplus_i p_i^x
L^2(H_i,k_i) p_i^y \; \subseteq \; \bigoplus_i L^2(H_i, k_i) ,\]
Thus we may write any morphism $f \colon x \to y$ as
\[ f = \bigoplus_i f_i \]
where $f_i \colon H_i^x \to H_i^y$.
\item Via the above isomorphism, the composition map
\[ \circ\;\colon \hom(x,y) \times \hom(y,z) \to \hom(x,z) \]
is given by
\[ f \circ g = \bigoplus_i f_i g_i. \]
\item Via the same isomorphism, the $\ast$-structure
\[ \ast \colon \hom(x,y) \to \hom(y,x) \]
is given by
\[ f^\ast = \bigoplus_i f_i^\ast .\]
\rm}
\end{enumerate}
\noindent Conversely, given a ${\rm Hilb}$-category $H$ with $\ast$-structure
such that the above holds for any finite subset $S$ of its objects, $H$
is an H*-category.
\end{thm}
Proof - If $H$ has finitely many objects and we take $S$ to be the set
of all objects of $H$, properties 1-3 follow from the remarks preceding
the theorem. More generally, by Proposition \ref{H*1} any full subcategory
of an H*-category is an H*-category, so 1-3 hold for any finite subset $S$
of the objects of $H$.
Conversely, given a ${\rm Hilb}$-category $H$ with a $\ast$-structure, if
every full subcategory of $H$ with finitely many objects is an
H*-category, then $H$ itself is an H*-category. One may check using
Proposition \ref{H*1} that if $S$ is any finite subset of the objects
of $H$, properties 1-3 imply the full subcategory of $H$ with $S$ as
its set of objects is an H*-category. Thus $H$ is an H*-category.
\hskip 3em \hbox{\BOX} \vskip 2ex
The notions of unitarity and self-adjointness
will be important in all that follows.
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in} Let $x$ and $y$ be objects of a $\ast$-category.
A morphism $u \colon x \to y$ is {\rm unitary} if $uu^\ast = 1_x$ and
$u^\ast u = 1_y$. A morphism $a \colon x \to x$ is {\rm
self-adjoint} if $a^\ast = a$. \end{defn}
Note that every unitary morphism is an isomorphism. Conversely, the
following proposition implies that in an H*-category, isomorphic objects
are isomorphic by a unitary.
\begin{prop} \label{H*4} \hspace{-0.08in}{\bf .}\hspace{0.1in} Suppose $f \colon x \to y$ is an isomorphism in
the H*-category $H$. Then $f = au$ where $a \colon x \to x$ is self-adjoint
and $u \colon x \to y$ is unitary.
\end{prop}
Proof - Suppose that $f \colon x \to y$ is an isomorphism. Then applying
Theorem \ref{H*3} to the full subcategory of $H$ with $x$ and $y$ as its
only objects, we have $f = \bigoplus f_i$ with $f_i \colon H_i^x \to
H_i^y$ an isomorphism for all $i$. Using the polar
decomposition theorem we may write $f_i = a_i u_i$, where $a_i \colon
H_i^x \to H_i^x$ is the positive square root of $f_i {f_i}^\ast$, and $u_i
\colon H_i^x \to H_i^y$ is a unitary operator given by $u_i =
a_i^{-1} f_i$. Then defining $a = \bigoplus a_i$ and $u = \bigoplus
u_i$, we have $f = au$ where $a$ is self-adjoint and $u$ is unitary. \hskip 3em \hbox{\BOX} \vskip 2ex
One can prove a more general polar decomposition theorem allowing one
to write any morphism $f \colon x \to y$ in an H*-category as the
product of a self-adjoint morphism $a \colon x \to x$ and a partial
isometry $i \colon x \to y$, that is, a morphism for which $ii^\ast$
and $i^\ast i$ are self-adjoint idempotents. However, we will not
need this result here.
\section{2-Hilbert Spaces}
The notion of 2-Hilbert space is intended to be the categorification of
the notion of Hilbert space. As such, it should be a category having a
zero object, direct sums and `direct differences' of objects, tensor
products of Hilbert spaces with objects, and `inner products' of objects.
So far, with our definition of H*-category, we have formalized the
notion of a category in which the `inner product' $\hom(x,y)$ of any
two objects $x$ and $y$ is a Hilbert space. Now we deal with the rest
of the properties:
\begin{defn}\hspace{-0.08in}{\bf .}\hspace{0.1in} A {\rm 2-Hilbert space} is an abelian H*-category.
\end{defn}
Recall that an abelian category is an ${\rm Ab}$-category (a category
enriched over the category ${\rm Ab}$ of abelian groups) such that
\begin{enumerate}
\item There exists an initial and terminal object.
\item Any pair of objects has a biproduct.
\item Every morphism has a kernel and cokernel.
\item Every monomorphism is a kernel, and every epimorphism is a
cokernel.
\end{enumerate}
Let us comment a bit on what this amounts to.
Since an H*-category is enriched over ${\rm Hilb}$ it is automatically
enriched over ${\rm Ab}$. We call an initial and terminal object a
{\it zero object},
and denote it by $0$. The zero object in a 2-Hilbert space is the
analog of the zero vector in a Hilbert space. We call the biproduct
of $x$ and $y$ the {\it direct sum},
and denote it by $x \oplus y$. Recall that by definition, this is equipped
with morphisms $p_x \colon x \oplus y \to x$, $p_x \colon x \oplus y \to y$,
$i_x \colon x \to x \oplus y$, $i_y \colon y \to x \oplus y$ such
that
\[ i_x p_x = 1_x , \qquad i_y p_y = 1_y, \qquad
p_x i_x + p_y i_y = 1_{x\oplus y} .\]
The direct sums in a 2-Hilbert space are the analog of addition
in a Hilbert space. Similarly,
the cokernels in a 2-Hilbert space are the analogs of differences
in a Hilbert space. Finally, the
ability to tensor objects in a 2-Hilbert space by Hilbert
spaces (the analog of scalar multiplication) will follow from the other
properties, so we do not need to include it in the definition of
2-Hilbert space.
Some aspects of our definition of 2-Hilbert space may seem unmotivated
by the analogy with Hilbert spaces. Why should a 2-Hilbert space have
kernels, and why should it satisfy clause 4 in the definition of abelian
category? In fact, these properties follow from the rest.
\begin{prop}\hspace{-0.08in}{\bf .}\hspace{0.1in} \label{2hilb1} Let $H$ be an H*-category. Then
the following are equivalent:
\begin{enumerate}
\item There exists an initial object.
\item There exists a terminal object.
\item There exists a zero object.
\end{enumerate}
Moreover, the following are equivalent:
\begin{enumerate}
\item Every pair of objects has a product.
\item Every pair of objects has a coproduct.
\item Every pair of objects has a direct sum.
\end{enumerate}
Moreover, the following are equivalent:
\begin{enumerate}
\item Every morphism has a kernel.
\item Every morphism has a cokernel.
\end{enumerate}
Finally, if $H$ has a zero object, every pair of objects in $H$ has
a direct sum, and every morphism in $H$ has a cokernel, then
$H$ is a semisimple abelian category.
\end{prop}
Proof - It is well-known \cite{MacLane} that an initial or terminal
object in an ${\rm Ab}$-category is automatically a zero object.
Alternatively, this is true in every $\ast$-category, using the
bijection $\ast \colon \hom(x,y) \to \hom(y,x)$.
It is also well-known that in an ${\rm Ab}$-category,
a binary product or coproduct is automatically a binary biproduct.
Furthermore, it is easy to check that
in any $\ast$-category, the morphism $j \colon k \to x$ is a kernel of
$f\colon x \to y$ if and only if $j^\ast \colon x \to k$ is a cokernel of
$f^\ast \colon y \to x$. Thus a $\ast$-category has kernels if and only
if it has cokernels.
Now suppose that $H$ is an H*-category with a zero object, direct
sums, and cokernels. Then $H$ has kernels as well, so to show $H$ is
abelian we merely need to prove that every monomorphism is a kernel and
every epimorphism is a cokernel. Let us show a monomorphism $f \colon x
\to y$ is a kernel; it follows using the $\ast$-structure that every
epimorphism is a cokernel.
It suffices to show this result for any full subcategory of
$H$ with finitely many objects, so by Theorem \ref{H*3}
we may write
\[ f = \bigoplus_i f_i \]
where $f_i \colon H_i^x \to H_i^y$ is a linear operator. Let $p
\colon y \to y$ be given by $\bigoplus p_i$ where $p_i$ is the
projection onto the orthogonal complement of the range of $f_i$.
We claim that $f\colon x \to y$ is a kernel of $p$. Since $f_i
p_i = 0$ for all $i$ we have $fp = 0$. We also need to show
that if $f' \colon x' \to y$ is any morphism with $f'p = 0$, then
there is a unique $g \colon x' \to x$ with $f' = gf$. Writing $f'
= \bigoplus f'_i$, the fact that $f'p = 0$ implies that the range
of $f'_i$ is contained in the range of $f_i$. Thus by linear
algebra there exists $g_i \colon H_i^{x'} \to H_i^x$ such that
$f'_i = g_i f_i$. Letting $g = \bigoplus g_i$, we have $f' =
gf$, and $g$ is unique with this property because $f$ is monic.
Finally, note that $H$ is semisimple, i.e., every
short exact sequence splits. This follows from Theorem \ref{H*3}
and elementary linear algebra. \hskip 3em \hbox{\BOX} \vskip 2ex
Given a 2-Hilbert space $H$, the fact that $H$ is semisimple implies
that every object is isomorphic to a direct sum of {\it simple} objects,
that is, objects $x$ for which ${\rm end}(x)$ is isomorphic as an algebra
to ${\Bbb C}$. This fact lets us reason about 2-Hilbert spaces using bases:
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in} Given a 2-Hilbert space $H$, a set of
nonisomorphic simple objects of $H$ is called a {\rm basis} if every
object of $H$ is isomorphic to a finite direct sum of objects
in that set. \end{defn}
\begin{cor} \label{2hilb2} \hspace{-0.08in}{\bf .}\hspace{0.1in}
Every 2-Hilbert space $H$ has a basis, and any two bases of $H$
have the same cardinality. \end{cor}
Proof - The 2-Hilbert space $H$ has a basis because it is
semisimple: given any Given two bases $\{e_\alpha\}$ and $\{f_\beta\}$,
each object $e_\alpha$ is isomorphic to a direct sum of copies of
the objects $e_\beta$, but as the $e_\alpha$ and $f_\beta$ are
simple we must actually have an isomorphism $e_\alpha \cong
f_\beta$ for some $\beta$. This $\beta$ is unique since no
distinct $f_\beta$'s are isomorphic. This sets up a function
from $\{e_\alpha\}$ to $\{f_\beta\}$, and similar reasoning gives
us the inverse function. \hskip 3em \hbox{\BOX} \vskip 2ex
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in} The {\rm dimension} of a 2-Hilbert space is the
cardinality of any basis of it. \end{defn}
Note that every basis $\{e_\alpha\}$ of a 2-Hilbert space
is `orthogonal' in the sense that
\[ \hom(e_\alpha, e_\beta) \cong
\cases{L^2({\Bbb C},k_\alpha) & $\alpha = \beta$ \cr
0 & $\alpha \ne \beta$ } \]
where the isomorphism is one of H*-algebras,
and $k_\alpha$ are certain positive constants.
Moreover, up to reordering, the constants $k_\alpha$ are independent of
the choice of basis. For
suppose $x,y$ are two isomorphic objects in an H*-category. By
Proposition \ref{H*4} there is a unitary isomorphism $f \colon x \to y$.
Then there is an H*-algebra isomorphism $\alpha \colon {\rm end}(x) \to
{\rm end}(y)$ given by $\alpha(g) = f^{-1}gf$.
One would also like to be able to tensor objects in a 2-Hilbert space
with Hilbert spaces, but this is a consequence of the definition we
have given, since one may define the tensor product of an object $x$
in a 2-Hilbert space with an $n$-dimensional Hilbert space to be the
direct sum of $n$ copies of $x$. In fact, ${\rm Hilb}$ has a structure
analogous to that of an algebra, with tensor product and direct sum
playing the roles of multiplication and addition. In the terminology
we introduce in Section \ref{2-H*-algebras}, one says that ${\rm Hilb}$ is
a `2-H*-algebra'. One can develop a theory of modules of
2-H*-algebras following the ideas of Kapranov and Voevodsky \cite{KV}
and Yetter \cite{Yetter}. Every 2-Hilbert space $H$ is then a module
over ${\rm Hilb}$. We will not pursue this further here.
\section{2Hilb as a 2-Category}\label{2Hilb}
We now investigate a certain 2-category $2{\rm Hilb}$ of 2-Hilbert spaces.
To keep things simple we take as its objects only finite-dimensional
2-Hilbert spaces. Nonetheless we prove theorems more generally whenever
possible.
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in} A {\rm morphism} $F \colon H \to H'$ between 2-Hilbert
spaces $H$ and $H'$ is an exact functor such that
$F \colon \hom(x,y) \to \hom(F(x), F(y))$ is linear and $F(f^\ast) =
F(f)^\ast$ for all $f \in \hom(x,y)$. \end{defn}
Recall that an exact functor is one preserving short exact
sequences. Exactness is an natural sort of condition for functors
between abelian categories. Similarly, the requirement that
$F\colon \hom(x,y) \to \hom(F(x), F(y))$ be linear is a natural condition
for functors between
${\rm Hilb}$-categories; one calls such a functor a {\it ${\rm Hilb}$-functor}.
Finally, $F(f^\ast) = F(f)^\ast$ is a natural condition
for functors between $\ast$-categories, and functors satisfying it are
called {\it $\ast$-functors}.
The following fact is occaisionally handy:
\begin{prop} \hspace{-0.08in}{\bf .}\hspace{0.1in} Let $F \colon H \to H'$ be a functor between 2-Hilbert
spaces such that for all $x,y \in H$,
$F \colon \hom(x,y) \to \hom(F(x), F(y))$ is linear.
Then the following are equivalent:
\begin{enumerate}
\item $F$ is exact.
\item $F$ is left exact.
\item $F$ is right exact.
\item $F$ preserves direct sums.
\end{enumerate}
\end{prop}
Proof - Following Yetter \cite{Yetter}, we use the
fact that every short exact sequence splits. \hbox{\hskip 30em} \hskip 3em \hbox{\BOX} \vskip 2ex
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in} A {\rm 2-morphism} $\alpha \colon F \Rightarrow F'$ between
morphisms $F,F' \colon H \to H'$ between 2-Hilbert spaces $H$ and $H'$ is
a natural transformation. \end{defn}
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in}
We define the 2-category $2{\rm Hilb}$ to be that for which
objects are finite-dimensional 2-Hilbert spaces, while morphisms
and 2-morphisms are defined as above.
\end{defn}
Now, just as in some sense ${\Bbb C}$ is the primordial Hilbert space and
${\rm Hilb}$ is the primordial 2-Hilbert space, $2{\rm Hilb}$ should be the
primordial 3-Hilbert space. The study of $2{\rm Hilb}$ should thus shed
light on the properties of the still poorly understood 3-Hilbert spaces.
However, note that ${\Bbb C}$ is not merely a Hilbert space, but also a
commutative monoid, in fact a commutative H*-algebra. Similarly,
${\rm Hilb}$ is not merely a 2-Hilbert space, but also a symmetric monoidal
category when equipped with its usual tensor product. Indeed, in
Section \ref{2-H*-algebras} we show that ${\rm Hilb}$ is a `symmetric
2-H*-algebra'. Likewise, we expect $2{\rm Hilb}$ to be not only a
3-Hilbert space, but also a strongly involutory monoidal 2-category, in
fact a `strongly involutory 3-H*-algebra'. As sketched in HDA0,
commutative monoids, symmetric monoidal categories, and strongly
involutory monoidal 2-categories are all examples of `stable'
$n$-categories. In general we expect $n{\rm Hilb}$ to be a `stable
$(n+1)$-H*-algebra.' The results below offer some support for this
expectation.
We begin with a study of duality in $2{\rm Hilb}$, as this is the most
distinctive aspect of Hilbert space theory. Note that every element $x
\in {\Bbb C}$ has a kind of `dual' element, namely, its complex conjugate
$\overline x$. Similarly, the category ${\rm Hilb}$ has duality both for
objects and for morphisms. At the level of morphisms, each linear map
$f \colon x \to y$ between Hilbert spaces has a dual $f^\ast \colon y \to
x$, the usual Hilbert space adjoint of $f$. This defines a
$\ast$-structure on $H$. Duality at the level of objects can be
regarded either as a contravariant functor assigning to each each
Hilbert space $x$ its dual $x^\ast$, or as a covariant functor assigning
to each Hilbert space $x$ its conjugate $\overline x$. These
two viewpoints become equivalent if we take advantage of duality at the
morphism level, since $x^\ast$ and $\overline x$ are antinaturally
isomorphic.
Similarly, $2{\rm Hilb}$ has duality for objects, morphisms, and 2-morphisms.
As in ${\rm Hilb}$, we can use duality at a given level to reinterpret
dualities at lower levels in various ways. This recursive process can
become rather confusing unless we choose by convention to take certain
dualities as `basic' and others as derived. Here we follow the
philosophy of HDA0: any 2-morphism $\alpha \colon F \Rightarrow G$ has a dual
$\alpha^\ast \colon G \Rightarrow F$, any morphism $F \colon H \to H'$ has a dual
$F^\ast \colon H' \to H$, and every object $H$ has a dual $H^\ast$.
(Our notation differs from HDA0 in that we use the same symbol
to denote all these different levels of duality.)
\subsection{Duality for 2-morphisms}
Duals of 2-morphisms are the easiest to define. It pays to do so in
the greatest possible generality:
\begin{defn} \label{star} \hspace{-0.08in}{\bf .}\hspace{0.1in} Given a category $C$ and a $\ast$-category $D$,
the {\rm dual} $\alpha^\ast$ of a natural transformation $\alpha \colon F
\Rightarrow G$ is the natural transformation with $(\alpha^\ast)_c =
(\alpha_c)^\ast$ for all $c \in C$. \end{defn}
It is easy to check that $\alpha^\ast$ is a natural transformation when
$\alpha$ is, and that
\[ (\alpha^\ast)^\ast = \alpha, \qquad 1^\ast = 1.\]
The vertical composite of natural transformations satisfies
\[ (\alpha\cdot \beta)^\ast = \beta^\ast\cdot \alpha^\ast \]
when this is defined. When $D$ is a $\ast$-category, the horizontal
composite of a functor $F \colon B \to C$ and a natural
transformation $\alpha \colon G \Rightarrow H$ with $G,H \colon C \to D$ satisfies
\[ (F\alpha)^\ast = F \alpha^\ast .\]
Similarly, when $F \colon C \to D$ is a $\ast$-functor and $\alpha
\colon G \Rightarrow H$ is a natural transformation between $G,H \colon B
\to C$, we have
\[ (\alpha F)^\ast = \alpha^\ast F.\]
In particular, taking $C,D$ to be 2-Hilbert spaces, we obtain
the definition of the dual of a 2-morphism in $2{\rm Hilb}$.
We also obtain the notion of `unitary' and `self-adjoint' natural
transformations:
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in}
Given a category $C$, a $\ast$-category $D$, and functors $F,G \colon C
\to D$, a natural transformation $\alpha \colon F \Rightarrow G$ is {\rm unitary} if
\[ \alpha \alpha^\ast = 1_F , \qquad \alpha^\ast \alpha = 1_G .\]
A natural transformation $\alpha \colon F \Rightarrow F$ is {\rm self-adjoint} if
\[ \alpha^\ast = \alpha .\]
\end{defn}
Equivalently, $\alpha$ is unitary if $\alpha_c$ is a unitary morphism in
$D$ for all objects $c \in C$, and self-adjoint if $\alpha_c$ is
self-adjoint for all $c \in C$.
Note that every unitary natural transformation is a natural isomorphism.
Conversely:
\begin{prop} \label{unitary.nat} \hspace{-0.08in}{\bf .}\hspace{0.1in} Suppose $F,G \colon H \to H'$ are
morphisms between 2-Hilbert spaces and $\alpha \colon F \Rightarrow G$ is a natural
isomorphism. Then $\alpha = \beta\cdot\gamma$ where $\beta \colon F \Rightarrow
F$ is self-adjoint and $\gamma \colon F \Rightarrow G$ is unitary. \end{prop}
Proof - By Proposition \ref{H*4}, for any $x \in H$ we can write the
isomorphism $\alpha_x \colon F(x) \to G(x)$ as the composite $\beta_x
\gamma_x$, where $\beta_x \colon F(x) \to F(x)$ is self-adjoint and $\gamma_x
\colon F(x) \to G(x)$ is unitary. More importantly, the
polar decomposition gives a
natural way to construct $\beta_x$ and $\gamma_x$ from $\alpha_x$: we
take $\beta_x$ to be the positive square root of $\alpha_x
{\alpha_x}^\ast$, and take $\gamma_x = \beta^{-1}_x \alpha_x$.
Since $\alpha \alpha^\ast$ is a natural transformation from $F$ to
itself, if we define $P(\alpha \alpha^\ast)_x = P(\alpha_x
{\alpha_x}^\ast)$ for any polynomial $P$, we have
\[ P(\alpha_x {\alpha_x}^\ast) F(f) = F(f) P(\alpha_y {\alpha_y}^\ast)
\]
for any morphism $f \colon x \to y$. By the finite-dimensional
spectral theorem, we can find a sequence of polynomials $P_i$ such
that $P_i(\alpha_x {\alpha_x}^\ast) \to \beta_x$ and $P_i(\alpha_y
{\alpha_y}^\ast) \to \beta_y$. Thus
\[ \beta_x F(f) = F(f) \beta_y , \]
so $\beta$ is a natural transformation from $F$ to itself.
It follows that $\gamma = \beta^{-1}\cdot \alpha$ is a natural transformation
from $F$ to $G$. Clearly $\beta$ is self-adjoint and $\gamma$ is
unitary. \hskip 3em \hbox{\BOX} \vskip 2ex
\subsection{Duality for morphisms}
Duals of morphisms in $2{\rm Hilb}$ are just adjoint functors.
Normally one needs to distinguish between left and right adjoint
functors, but duality at the 2-morphism level allows us to turn left
adjoints into right adjoints, and vice versa:
\begin{prop}\hspace{-0.08in}{\bf .}\hspace{0.1in} \label{2cat1} Suppose $F \colon H \to H'$, $G \colon H'
\to H$ are
morphisms in $2{\rm Hilb}$. Then $F$ is left adjoint to $G$ with unit
$\iota \colon 1_H \Rightarrow FG$ and counit $\epsilon \colon GF \Rightarrow 1_{H'}$
if and only if $F$ is right adjoint to $G$ with unit $\epsilon^\ast \colon 1_{H'}
\Rightarrow GF$ and counit $\iota^\ast \colon FG \Rightarrow 1_H$. \end{prop}
Proof - The triangle equations for $\iota$ and $\epsilon$:
\[ (\iota F) \cdot (F \epsilon) = 1_F, \qquad
(G \iota) \cdot (\epsilon G) = 1_G ,\]
become equivalent to those for $\epsilon^\ast$ and $\iota^\ast$:
\[ (\epsilon^\ast G) \cdot (G \iota^\ast) = 1_G, \qquad
(F \epsilon^\ast) \cdot (\iota^\ast F) = 1_F, \]
by taking duals. \hskip 3em \hbox{\BOX} \vskip 2ex
\noindent As noted by Dolan \cite{Dolan}, it is probably quite generally
true in $n$-categories that duality for $j$-morphisms allows us to turn
`left duals' of $(j-1)$-morphisms into `right duals' and vice versa.
This should give the theory of $n$-Hilbert spaces
quite a different flavor from general $n$-category theory.
Every morphism in $2{\rm Hilb}$ has an adjoint. We prove this
using bases and the concept of a skeletal 2-Hilbert space.
\begin{defn}\hspace{-0.08in}{\bf .}\hspace{0.1in} A category is {\rm skeletal} if all isomorphic
objects are equal. \end{defn}
\begin{defn}\hspace{-0.08in}{\bf .}\hspace{0.1in} A {\rm unitary equivalence} between 2-Hilbert spaces
$H$ and $H'$ consists of morphisms $U \colon H \to H'$, $V \colon H' \to
H$ and unitary natural transformations $\iota \colon 1_H \Rightarrow UV$, $\epsilon
\colon VU \Rightarrow 1_{H'}$ forming an adjunction. If there exists a unitary
equivalence between $H$ and $H'$, we say they are {\rm unitarily
equivalent}. \end{defn}
\begin{prop}\hspace{-0.08in}{\bf .}\hspace{0.1in} \label{2cat2} Any 2-Hilbert space is
unitarily equivalent to a skeletal 2-Hilbert space.
\end{prop}
Proof - Let $\{e_\lambda\}$ be a basis for the 2-Hilbert space $H$.
For any nonnegative integers
$\{n^\lambda\}$ with only finitely many nonzero, make a choice of direct sum
\[ \bigoplus_\lambda n^\lambda e_\lambda ,\]
where $n^\lambda e_\lambda$ denotes the direct sum of $n^\lambda$ copies of
$e_\lambda$. (Recall that the direct sum is an object equipped with
particular morphisms; it is only unique up to isomorphism, but here we
fix a particular choice.) Let $H_0$ denote the full subcategory of
$H$ with only these direct sums as objects. Note that $H_0$ inherits
a 2-Hilbert space structure from $H$, and it is skeletal. Let
$V \colon H_0 \to H$ denote the inclusion functor.
For any $x \in H$ there is a unique object $U(x) \in H_0$ for which
$V(U(x))$ is isomorphic to $x$.
By Proposition \ref{H*4}, we may choose a unitary
isomorphism
\[ \iota_x \colon x \to V(U(x)). \]
For $x = V(y)$ we have $U(x) = y$, so we choose $\iota_x$ to be
the identity in this case.
For each morphism $f\colon x
\to y$ define $U(f) \colon U(x) \to U(x')$ so that the following diagram
commutes:
\[ \begin{diagram}[V(U(x))]
\node{x} \arrow{e,t}{f} \arrow{s,l}{\iota_x} \node{y}
\arrow{s,r}{\iota_{y}} \\
\node{V(U(x))} \arrow{e,b}{V(U(f))} \node{V(U(y))}
\end{diagram}
\]
It follows that $U \colon H \to H_0$ is a functor.
One may check that $U$ and $V$ are actually morphisms of 2-Hilbert
spaces. Moreover, one may check that there is a natural isomorphism
\[ \hom(Ux,y) \cong \hom(x,Vy) \]
given by
\[ f \mapsto \iota_x V(f). \]
It follows that $U$ is left adjoint to $V$. The unit of this
adjunction is $\iota$, while the counit is the identity.
These are both unitary natural transformations. \hskip 3em \hbox{\BOX} \vskip 2ex
Just as with Hilbert spaces, phrasing definitions and theorems about
2-Hilbert spaces in terms of a basis is usually a mistake, since
they should be manifestly invariant under unitary equivalence. In
comparison, the use of bases to prove theorems is at worst a minor
lapse of taste, and sometimes convenient. This is facilitated by the
use of skeletal 2-Hilbert spaces.
\begin{prop}\hspace{-0.08in}{\bf .}\hspace{0.1in} \label{2cat3} Let $F \colon H \to H'$ be a morphism in
$2{\rm Hilb}$. Then there is a morphism $F^\ast \colon H' \to H$ that is left
and right adjoint to $F$. \end{prop}
Proof - Here we opt for a lowbrow proof using bases, to illustrate the
analogy between an adjoint functor and the adjoint of a matrix.
By Proposition \ref{2cat2} it suffices to
consider the case where $H$ and $H'$ are skeletal. Let
$\{e_\lambda\}$ be a basis for $H$ and $\{e'_\mu\}$ a basis for $H'$.
Write
\[ F(e_\lambda) = \bigoplus_\mu F_{\lambda \mu} e'_\mu
\]
where $F_{\lambda\mu}$ are nonnegative integers and $F_{\lambda\mu}
e'_\mu$ denotes the direct sum of $F_{\lambda\mu}$ copies of
$e'_\mu$. Let
\[ {F_{\mu\lambda}}^\ast = F_{\lambda\mu}. \]
Defining
\[ F^\ast(e'_\mu) = \bigoplus_{\mu} F^\ast_{\mu\lambda}
e_\lambda ,\]
one may check that $F^\ast$ extends uniquely to a morphism from $H'$
to $H$. Note that both $\hom(F e_\lambda,
e'_\mu)$ and $\hom(e_\lambda, F^\ast e'_\mu)$ may be naturally
identified with a direct sum of $F_{\lambda\mu}$ copies of ${\Bbb C}$, which
sets up an isomorphism $\hom(F e_\lambda, e'_\mu) \cong \hom(e_\lambda,
F^\ast e'_\mu)$. One can check that this extends uniquely to a natural
isomorphism
\[ \hom(Fx,y) \cong \hom(x,F^\ast y), \]
so $F^\ast$ is a right adjoint, and by Proposition \ref{2cat1} also a left
adjoint, of $F$. \hskip 3em \hbox{\BOX} \vskip 2ex
A basic fact in Hilbert space theory is that two objects in ${\rm Hilb}$ are
isomorphic if and only if there is a unitary morphism between them. The
same is true of objects in any other 2-Hilbert space, by Proposition
\ref{H*4}. Similarly, two morphisms in $2{\rm Hilb}$ are isomorphic if and
only if there is a unitary natural transformation between them, by
Proposition \ref{unitary.nat}. Below we show a similar result for
objects in $2{\rm Hilb}$. In general, we expect a recursively defined
notion of `equivalence' of $j$-morphisms in an $n$-category: two
$n$-morphisms are equivalent if they are equal, while two
$(j-1)$-morphisms $x,y$ are equivalent if there exist $f \colon x \to
y$ and $g \colon y \to x$ with $gf$ and $fg$ equivalent to the identity
on $x$ and $y$, respectively. In an $n$-Hilbert space we also expect
a similar notion of `unitary equivalence': two $(n-1)$-morphisms are
unitarily equivalent if they are equal, while two $(j-1)$-morphisms
$x,y$ are unitarily equivalent if there exists $u \colon x \to y$ with
$uu^\ast$ and $u^\ast u$ unitarily equivalent to $1_x$ and $1_y$,
respectively. Our results so far lead us to suspect that, quite
generally, equivalent $j$-morphisms in an $n$-Hilbert space will be
unitarily equivalent.
\begin{defn}\label{equivalence}\hspace{-0.08in}{\bf .}\hspace{0.1in} An {\rm equivalence} between
2-Hilbert spaces $H$ and $H'$ is an pair of morphisms $F \colon H \to
H'$, $G \colon H' \to H$ together with natural isomorphisms $\alpha \colon
1_H \Rightarrow FG$, $\beta \colon GF \Rightarrow 1_{H'}$. If there is an equivalence
between $H$ and $H'$, we say they are {\rm equivalent}. \end{defn}
Note that a unitary equivalence is automatically an equivalence.
Conversely:
\begin{prop} \label{unitary.eq} \hspace{-0.08in}{\bf .}\hspace{0.1in} Suppose $H$ and $H'$ are 2-Hilbert
spaces and the morphisms $F \colon H \to H'$, $G \colon H' \to H$ can be
extended to an equivalence between $H$ and $H'$. Then $F$ and $G$ can
be extended to a unitary equivalence between $H$ and $H'$.
\end{prop}
Proof - Suppose $\alpha \colon 1_H \Rightarrow FG$, $\beta \colon GF \Rightarrow 1_{H'}$
are natural isomorphisms. By Proposition \ref{unitary.nat} we can find
unitary natural transformations $\gamma \colon 1_H \Rightarrow FG$, $\delta
\colon GF \Rightarrow 1_{H'}$. We may then obtain
an adjunction by replacing $\gamma$ with the composite $\gamma'$ given by
\[ \begin{diagram} [FG = F1_{H'}G]
\node{1_H} \arrow{e,t}{\gamma}
\node{FG = F1_{H'}G} \arrow{e,t}{F\delta^{-1}G}
\node{FGFG} \arrow{e,t}{\gamma^{-1}FG}
\node{FG}
\end{diagram} \]
Checking that this is an adjunction is a lengthy but straightforward
calculation. Noting that $\gamma'$ is unitary, we conclude that
$(F,G,\iota,\epsilon)$ is a unitary equivalence. \hskip 3em \hbox{\BOX} \vskip 2ex
\noindent When we are being less pedantic, we call a 2-Hilbert
space morphism $F \colon H \to H'$ an {\it equivalence} if it can
be extended to an equivalence in the sense of Definition
\ref{equivalence}.
Just as Hilbert spaces are classified by their dimension, we have:
\begin{cor} \label{2cat4} \hspace{-0.08in}{\bf .}\hspace{0.1in} Two 2-Hilbert spaces are equivalent
if and only if they have the same dimension.
\end{cor}
Proof - Since an equivalence between $H$ and $H'$ carries a basis
of $H$ to a basis of $H'$, Proposition \ref{2hilb2} implies that
dimension is preserved by equivalence. By Proposition
\ref{2cat2} it thus suffices to show two skeletal 2-Hilbert spaces are
equivalent if they have the same dimension. Let
$\{e_\lambda\}$ be a basis of $H$ and $\{e'_\lambda\}$ a corresponding
basis of $H'$. Then there is a unique 2-Hilbert space morphism with
$F(e_\lambda) = e'_\lambda$, and the adjunction constructed as in the
proof of Proposition \ref{2cat3} is a unitary equivalence. \hskip 3em \hbox{\BOX} \vskip 2ex
\subsection{Duality for objects} \label{dualityforobjects}
Finally, duals of objects in $2{\rm Hilb}$ are defined using an `internal
hom'. Given 2-Hilbert spaces $H$ and $H'$, let $\hom(H,H')$ be the
category having 2-Hilbert space morphisms $F\colon H \to H'$ as objects
and 2-morphisms between these as morphisms.
\begin{prop} \label{2cat5}\hspace{-0.08in}{\bf .}\hspace{0.1in} Suppose $H$ is a finite-dimensional
2-Hilbert space and $H'$ is a 2-Hilbert space. Then the category
$\hom(H,H')$ becomes a
${\rm Hilb}$-category if for any $F,G \in \hom(H,H')$
we make $\hom(F,G)$ into a Hilbert space with the
obvious linear structure and the inner product given by
\[ \langle \alpha, \beta \rangle = \sum_{\lambda}
\langle \alpha_{e_\lambda} , \beta_{e_\lambda} \rangle \]
for any basis $\{e_\lambda\}$ of $H$.
Moreover, $\hom(H,H')$ becomes a 2-Hilbert space if we define the dual
of $\alpha \colon F \Rightarrow G$ by $(\alpha^\ast)_x = (\alpha_x)^\ast$. \end{prop}
Proof - Note first that $\hom(F,G)$ becomes a vector space if we define
\[ (\alpha + \beta)_x = \alpha_x + \beta_x, \qquad
(c\alpha)_x = c(\alpha_x) \]
for any $\alpha, \beta \colon F \Rightarrow G$ and $c \in {\Bbb C}$. Note also
that the inner product described above is nondegenerate, since if
$\alpha_{e_\lambda} = 0$ for all objects $e_\lambda$ in a basis,
then $\alpha = 0$. Finally, note that the inner product is
independent of the choice of basis: if $\{e'_\lambda\}$ is
another basis we may assume after reordering that $e_\lambda \cong
e'_\lambda$, and by Proposition \ref{H*4} we may choose unitary
isomorphisms $u_\lambda \colon e_\lambda \to e'_\lambda$, so that
\[ \alpha_{e'_\lambda} = F(u_\lambda)^\ast \alpha_{e_\lambda}
G(u_\lambda) \]
and similarly for $\beta$. It follows that
\begin{eqnarray*} \langle \alpha_{e'_\lambda} , \beta_{e'_\lambda} \rangle &=&
\langle F(u_\lambda)^\ast \alpha_{e_\lambda} G(u_\lambda) ,
F(u_\lambda)^\ast \beta_{e_\lambda} G(u_\lambda) \rangle \\
&=& \langle \alpha_{e_\lambda} , \beta_{e_\lambda} \rangle .\end{eqnarray*}
Since composition of morphisms in $\hom(H,H')$ is bilinear, it becomes a
${\rm Hilb}$-category.
It is easy to check that defining $(\alpha^\ast)_x =
(\alpha_x)^\ast$ makes $\hom(H,H')$ into a $\ast$-category, and
using Proposition \ref{H*1} one can also check it is an H*-category.
To check that it is a 2-Hilbert space it suffices by Proposition
\ref{2hilb1} to check that it has a zero object, direct sums and
kernels. Any functor $0 \colon H \to H'$ mapping all objects in
$H$ to zero objects in $H'$ is initial in $\hom(H,H')$. Given
$F,F' \in \hom(H,H')$, we may take as the direct sum $F \oplus
F'$ any functor with $(F \oplus F')(x) = F(x) \oplus F(x')$ for
any object $x \in H$ and $(F \oplus F')(f) = F(f) \oplus F(f')$
for any morphism $f$. Similarly, given $\alpha \colon F \to F'$,
we may construct $\ker \alpha \in \hom(H,H')$ by letting $(\ker
\alpha)(x) = \ker \alpha_x$ for any object $x$ and defining
$(\ker \alpha)(f)$ for any morphism using the universal property
of the kernel. \hbox{\hskip 30em} \hskip 3em \hbox{\BOX} \vskip 2ex
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in} Given a finite-dimensional 2-Hilbert space $H$, the
{\rm dual} $H^\ast$ is the 2-Hilbert space $\hom(H,{\rm Hilb})$. \end{defn}
The following is an analog of the Riesz representation theorem for
finite-dimensional 2-Hilbert spaces. In its finite-dimensional form,
the Riesz representation theorem says if $x$ is a Hilbert space, any
morphism $f \colon x \to {\Bbb C}$ is equal to one of the form
\[ \langle v, \cdot \rangle \]
for some $v \in H$. This determines an isomorphism $\overline x \cong
x^\ast$. Similarly, given a 2-Hilbert space $H$, we say a
morphism $F \colon H \to {\rm Hilb}$ is {\it representable} if it is naturally
isomorphic to one of the form
\[ \hom(x, \cdot) \]
for some $x \in H$. The essence of the Riesz
representation theorem for 2-Hilbert spaces is that every morphism
$F \colon H \to {\rm Hilb}$ is representable. This yields an
equivalence between $H^{\rm op}$ and $H^\ast$.
\begin{prop} \hspace{-0.08in}{\bf .}\hspace{0.1in} For any finite-dimensional 2-Hilbert space $H$, the
morphism \break $U \colon H^{\rm op} \to H^\ast$ given by
\[ U(x) = \hom(x,\cdot) , \qquad U(f) = \hom(f,\cdot) \]
is an equivalence between $H^{\rm op}$ and $H^\ast$.
\end{prop}
Proof - It suffices to show that $U$ is fully faithful and
essentially surjective. We can check both of these using a
basis $\{e_\lambda\}$ of $H$. We leave the full faithfulness to the
reader. Checking that $U$ is essentially surjective amounts to checking
that any $F \in H^\ast$ is representable. Note there is a `dual
basis' of 2-Hilbert space morphisms $f^\lambda \in \hom(H,{\rm Hilb})$ with
\[ f^\mu(e_\lambda) \cong \cases{{\Bbb C} & $\lambda = \mu$ \cr
0 & $\lambda \ne \mu$ } \]
Since any morphism $F \colon H \to {\rm Hilb}$ is determined up
to natural isomorphism by its value on the basis $\{e_\lambda\}$, any $F \in
H^\ast$ is isomorphic to a direct sum of the $\{f^\lambda\}$. But
$f^\lambda$ is isomorphic to $U(e_\lambda)$, so $U$ is essentially
surjective. \hskip 3em \hbox{\BOX} \vskip 2ex
\subsection{The tensor product}\label{tensorproduct}
Next we develop the tensor product of 2-Hilbert spaces. For this we need
the analog of a bilinear map:
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in} Given 2-Hilbert spaces $H,H',K$, a functor
$F \colon H \times H' \to K$ is a {\rm bimorphism} of 2-Hilbert spaces
if for any objects $x \in H$, $x' \in H'$ the functors $F(x \otimes
\,\cdot\,) \colon H' \to K$ and $F(\,\cdot \,\otimes x') \colon H \to K$
are 2-Hilbert space morphisms. We write ${\rm bihom}(H\times H', K)$ for
the category having bimorphisms $F \colon H \times H' \to K$ as objects
and natural transformations between these as morphisms.
\end{defn}
\begin{prop} \hspace{-0.08in}{\bf .}\hspace{0.1in} Suppose $H$ and $H'$ are finite-dimensional
2-Hilbert spaces and $K$ is a 2-Hilbert space. Then ${\rm bihom}(H\times H',K)$
becomes a ${\rm Hilb}$-category if for any $F,G \in {\rm bihom}(H \times H',K)$
we make $\hom(F,G)$ into a Hilbert space with the
obvious linear structure and the inner product given by
\[ \langle \alpha, \beta \rangle = \sum_{\lambda,\mu}
\langle \alpha_{(e_\lambda,f_\mu)} , \beta_{(e_\lambda,f_\mu)} \rangle \]
for any bases $\{e_\lambda\}$ of $H$ and $\{f_\mu\}$ of $H'$.
Moreover, ${\rm bihom}(H \times H',K)$ becomes a 2-Hilbert space if we define
the dual of $\alpha \colon F \Rightarrow G$ by $(\alpha^\ast)_x =
(\alpha_x)^\ast$. \end{prop}
Proof - The proof is analogous to that of Proposition \ref{2cat5}. \hskip 3em \hbox{\BOX} \vskip 2ex
Given 2-Hilbert spaces $H,H'$ and $L$, note that a bimorphism
$T \colon H \times H' \to L$ induces a morphism
\[ T^\ast \colon \hom(L,K) \to {\rm bihom}(H \times H',K). \]
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in} Given 2-Hilbert space $H,H'$, a {\rm tensor product}
of $H$ and $H'$ is a bimorphism $T \colon H \times H' \to L$ together
with a choice for each 2-Hilbert space $K$
of an equivalence of 2-Hilbert spaces extending $T^\ast \colon
\hom(L,K) \to {\rm bihom}(H \times H',K)$. \end{defn}
\noindent In the above situation, by abuse of language we
may say simply that $T \colon H \times H'
\to L$ is a tensor product of $H$ and $H'$.
\begin{prop} \label{tensor} \hspace{-0.08in}{\bf .}\hspace{0.1in} Given finite-dimensional
2-Hilbert spaces $H$ and $H'$, there
exists a tensor product $T \colon H \times H' \to L$. Given another
tensor product $T' \colon H \times H' \to L'$, there is an equivalence
$F \colon L \to L'$ for which the following diagram commutes up to
a specified natural isomorphism:
\[
\begin{diagram}[H \times H']
\node[2]{H \times H'} \arrow{sw,t}{T} \arrow{se,t}{T'} \\
\node{L} \arrow[2]{e,b}{F} \node[2]{L'}
\end{diagram}
\]
\end{prop}
Proof - Let $\{e_\lambda\}$ be a basis for $H$, and
$\{f_\mu\}$ a basis for $H'$. Let $L$ be the skeletal
2-Hilbert space with a basis of objects denoted by $\{e_\lambda \otimes
f_\mu\}$, and with
\[ \hom(e_\lambda \otimes f_\mu,e_\lambda \otimes f_\mu) =
\hom(e_\lambda,e_\lambda) \otimes \hom(f_\mu,f_\mu) \]
as H*-algebras (using the obvious tensor product of H*-algebras).
There is a unique bimorphism $T \colon H \times H' \to L$ with
$T(e_\lambda,f_\mu) = e_\lambda \otimes f_\mu$. Given a 2-Hilbert
space $K$ one may check that $T^\ast \colon \hom(L,K) \to {\rm bihom}(H
\times H',K)$ extends to an equivalence. Choosing such an equivalence
for every $K$ we obtain a tensor product of $H$ and $H'$.
Given two tensor products as in the statement of the proposition, let
$F\colon L \to L'$ be the image of $T'$ under the chosen equivalence
${\rm bihom}(H \times H', L') \simeq \hom(L,L')$. One can check that $L$
is an equivalence and that the above diagram commutes up to a
specified natural isomorphism, much as in the usual proof that the tensor
product of vector spaces is unique up to a specified isomorphism.
\hskip 3em \hbox{\BOX} \vskip 2ex
Given a tensor product of the 2-Hilbert spaces $H$ and $H'$, we often
write its underlying 2-Hilbert space as $H \otimes H'$. This notation
may tempt one to speak of `the' tensor product of $H$ and $H'$, which is
is legitimate if one uses the generalized
`the' as advocated by Dolan \cite{Dolan}. In a set, when we
speak of `the' element with a given property, we implicitly mean that
this element is unique. In a category, when we speak of
`the' object with a given property, we merely mean that this object is
unique up to isomorphism --- typically a specified isomorphism.
Similarly, in a 2-category, when we speak of `the' object with a given
property, we mean that this object is unique up to equivalence ---
typically an equivalence that is specified up to a specified
isomorphism. This is the sense in which we may refer to `the' tensor
product of $H$ and $H'$. The generalized `the' may be extended in an
obvious recursive fashion to $n$-categories.
Suppose that $H$ and $H'$ are finite-dimensional 2-Hilbert spaces.
Then for any pair of objects $x \in H$, $x' \in H'$,
we can use the bimorphism $T \colon H \times H' \to H \otimes H'$
to define an object $x \otimes x' = T(x,x')$ in $H \otimes H'$.
Similarly, given a morphism $f \colon x \to y$ in
$H$ and a morphism $f' \colon x' \to y'$, we obtain a morphism
\[ f \otimes f'\, \colon x \otimes x' \to y \otimes y'\]
in $H \otimes H'$. We usually write
\[ f \otimes x' \, \colon x \otimes x' \to y \otimes x' \]
for the morphism $f \otimes 1_{x'}$, and
\[ x \otimes f' \, \colon x \otimes x' \to x \otimes y' \]
for the morphism $1_x \otimes f'$.
We expect that $2{\rm Hilb}$ has the structure of a monoidal 2-category with
the above tensor product as part of the monoidal structure. Kapranov
and Voevodsky \cite{KV} have defined the notion of a weak monoidal
structure on a strict 2-category, which should be sufficient for the
purpose at hand. On the other hand the work of Gordon, Power and Street
\cite{GPS} gives a fully general notion of weak monoidal 2-category,
namely a 1-object tricategory. This should also be suitable for
studying the tensor product on $2{\rm Hilb}$, though it might be considered
overkill. Both these sorts of monoidal 2-category involve various
extra structures besides the tensor product of objects in $2{\rm Hilb}$.
Most of these should arise from the universal property of the tensor
product.
For example, suppose we are given a morphism $F \colon H \to H'$ and an
object $K$ in $2{\rm Hilb}$. Thus we have bimorphisms $T \colon H \times
K \to H \otimes K$ and $T' \colon H \times K' \to H \otimes K'$, and
$T^\ast$ has some morphism
\[ S \colon {\rm bihom}(H \times K, H' \otimes K) \to \hom(H \otimes
K,H' \otimes K) \]
as inverse up to natural isomorphism.
Applying $S$ to the bimorphism given by the composite
\[
\begin{diagram} [H \otimes K]
\node{H \times K} \arrow{e,t}{F \times 1_K} \node{H' \times K} \arrow{e,t}{T'}
\node{H' \otimes K}
\end{diagram}
\]
we obtain a morphism we denote by
\[ F \otimes K \colon H \otimes K \to H' \otimes K .\]
Similarly, given an object $H \in 2{\rm Hilb}$ and a morphism $G \colon K \to
K'$, we obtain a morphism
\[ H \otimes G \colon H \otimes K \to H \otimes K'. \]
Moreover, we have:
\begin{prop} \label{tensorator} \hspace{-0.08in}{\bf .}\hspace{0.1in} Let $F \colon H \to H'$ and $G \colon K
\to K'$ be morphisms in $2{\rm Hilb}$. Then the following diagram
\[
\begin{diagram}
\node{H \otimes K}
\arrow{e,t}{F \otimes K} \arrow{s,l}{H \otimes G}
\node{H' \otimes K}
\arrow{s,r}{H' \otimes G} \\
\node{H \otimes K'} \arrow{e,b}{F \otimes K'} \node{H' \otimes K'}
\end{diagram}
\]
commutes up to a specified natural isomorphism
\[ {\bigotimes}_{F,G} \colon (F \otimes K)(H' \otimes G) \Rightarrow
(H \otimes G)(F \otimes K') . \]
\end{prop}
Proof - Here we have fixed tensor products of all the 2-Hilbert spaces
involved, so we have bimorphisms
\[ T_{H,K} \colon H \times K \to H \otimes K \]
and so on. Applying the equivalence
\[ {\rm bihom}(H \times K, H' \otimes K') \simeq \hom(H \otimes
K,H' \otimes K') \]
coming from the definition of tensor product to the bimorphism given by
the composite
\begin{equation}
\begin{diagram} [H \otimes K]
\node{H \times K} \arrow{e,t}{F \times G} \node{H' \times K'}
\arrow{e,t}{T_{H',K'}} \node{H' \otimes K'}
\end{diagram} \label{bigotimes1}
\end{equation}
we obtain a morphism we denote by
\[ F \otimes G \colon H \otimes K \to H' \otimes K'. \]
We shall construct a natural isomorphism from $(F \otimes K)(H' \otimes G)$
to $F \otimes G$. Composing this with an analogous natural isomorphism
from $F \otimes G$ to $(H \otimes G)(F \otimes K')$ one obtains
$\bigotimes_{F,G}$.
If we precompose $F \otimes G$ with $T_{H,K}$ we obtain a bimorphism
naturally isomorphic to (\ref{bigotimes1}).
If we precompose $(F \otimes K')(H \otimes G)$ with $T_{H,K}$, we obtain
a bimorphism naturally isomorphic to
\begin{equation}
\begin{diagram} [H' \otimes K'.]
\node{H \times K} \arrow{e,t}{F \times K} \node{H' \times K}
\arrow{e,t}{T_{H',K}} \node{H' \otimes K}
\arrow{e,t}{H' \otimes G} \node{H' \otimes K'}
\end{diagram} \label{bigotimes2}
\end{equation}
Note also that in both cases, a {\it specified}
natural isomorphism is given by the definition of tensor product.
Since precomposition with $T_{H,K}$ is an equivalence between
${\rm bihom}(H \times K,H' \otimes K')$ and $\hom(H \otimes K,H' \otimes K'),$
it thus suffices to exhibit a natural isomorphism between
(\ref{bigotimes1}) and (\ref{bigotimes2}).
Factoring these by $F \times K$, it suffices to exhibit a natural
isomorphism between
\[
\begin{diagram} [H' \otimes K]
\node{H' \times K} \arrow{e,t}{H' \times G}
\node{H' \otimes K'} \arrow{e,t}{T_{H',K'}}
\node{H' \otimes K'}
\end{diagram}
\]
and
\[
\begin{diagram} [H' \otimes K'.]
\node{H' \times K}
\arrow{e,t}{T_{H',K}} \node{H' \otimes K}
\arrow{e,t}{H' \otimes G} \node{H' \otimes K'}
\end{diagram}
\]
This arises from the definition of $H' \otimes G$. \hskip 3em \hbox{\BOX} \vskip 2ex
\noindent The 2-morphism $\bigotimes_{F,G}$ is part of the
structure one expects in a monoidal 2-category, and the fact
that the diagram in Proposition \ref{tensorator} does not commute
`on the nose' is one of the key ways in which monoidal
2-categories differ from monoidal categories.
We expect a 2-categorical version of $\hom$-tensor adjointness to hold
for the tensor product defined in this section and the $\hom$ defined
in section \ref{dualityforobjects}. In other words, given
finite-dimensional 2-Hilbert space $H, H',$ and $K$, the obvious
functor from $\hom(H, \hom(H',K))$ to $\hom(H \otimes H', K)$ should
be an equivalence. However, we shall not prove this here.
\subsection{The braiding}
The symmetry in ${\rm Cat}$ gives braiding morphisms in $2{\rm Hilb}$ as follows.
Let $H$ and $H'$ be 2-Hilbert spaces. We may take their tensor product
in either order, obtaining tensor products $T \colon H \times H' \to H
\otimes H'$ and $T' \colon H' \times H \to H' \otimes H$.
By the universal property of the tensor product, the bimorphism given by
the composite
\[
\begin{diagram} [H \otimes K]
\node{H \times H'} \arrow{e,t}{S_{H,H'}} \node{H' \times H} \arrow{e,t}{T'}
\node{H' \otimes H}
\end{diagram}
\]
defines a morphism, the {\it braiding}
\[ R_{H,H'} \colon H \otimes H' \to H' \otimes H .\]
One can check that $R_{H,H'}$ is an equivalence.
We expect that $2{\rm Hilb}$ has the structure of a braided monoidal
2-category with the above braiding morphisms. However, the existing
notion of semistrict braided monoidal 2-category introduced by Kapranov
and Voevodsky \cite{KV} and subsequently refined in HDA1 is
insufficiently general to cover this example, since $2{\rm Hilb}$ is not a
semistrict monoidal 2-category. One should however be able to
strictify $2{\rm Hilb}$, obtaining a semistrict braided monoidal
2-category. Alternatively, the work of Trimble \cite{Trimble} should
give a fully general notion of weak braided monoidal 2-category, namely
a tetracategory with one object and one morphism. This should apply to
$2{\rm Hilb}$ without strictification.
In any event, both semistrict and weak braided monoidal 2-categories
involve various structures in addition to the braiding morphisms. Most
of these should arise from the universal property of the tensor product
together with the properties of the symmetry in ${\rm Cat}$. For example, we
have:
\begin{prop} \label{braiding.naturalizer} \hspace{-0.08in}{\bf .}\hspace{0.1in} Let $F \colon H \to H'$ be a
morphism and let $K$ be an object in $2{\rm Hilb}$. Then the following
diagram
\[
\begin{diagram}
\node{H \otimes K}
\arrow{e,t}{F \otimes K} \arrow{s,l}{R_{H,K}}
\node{H' \otimes K}
\arrow{s,r}{R_{H',K}} \\
\node{K \otimes H} \arrow{e,b}{K \otimes F} \node{K \otimes H'}
\end{diagram}
\]
commutes up to a specified natural isomorphism
\[ R_{F,K} \colon (F \otimes K) R_{H',K} \Rightarrow R_{H,K} (K \otimes F) .\]
Similarly, given an object $H$ and a morphism $G \colon K
\to K'$ in $2{\rm Hilb}$, the following diagram
\[
\begin{diagram}
\node{H \otimes K}
\arrow{e,t}{H \otimes G} \arrow{s,l}{R_{H,K}}
\node{H \otimes K'}
\arrow{s,r}{R_{H,K'}} \\
\node{K \otimes H} \arrow{e,b}{G \otimes H} \node{K' \otimes H}
\end{diagram}
\]
commutes up to a specified natural isomorphism
\[ R_{H,G} \colon (H \otimes G)R_{H,K'} \Rightarrow R_{H,K}(G \otimes H)
.\]
\end{prop}
Proof - We only treat the first case as the second is analogous.
Applying the equivalence
\[ {\rm bihom}(H \times K, K' \otimes H') \simeq \hom(H \otimes
K,K' \otimes H') \]
coming from the definition of tensor product to the bimorphism given by
the composite
\begin{equation}
\begin{diagram} [H \otimes K]
\node{H \times K} \arrow{e,t}{F \times K}
\node{H' \times K} \arrow{e,t}{S_{H',K}}
\node{K \times H'} \arrow{e,t}{T_{K,H'}}
\node{K \otimes H'}
\end{diagram} \label{braiding.nat.1}
\end{equation}
we obtain a morphism we denote by $A \colon H \otimes K \to K' \otimes H$.
We shall construct a natural isomorphism from $(F \otimes K)R_{H',K}$
to $A$.
Using the fact that (\ref{braiding.nat.1}) equals
\[
\begin{diagram} [H \otimes K]
\node{H \times K} \arrow{e,t}{S_{H,K}}
\node{K \times H} \arrow{e,t}{K \times F}
\node{K \times H'} \arrow{e,t}{T_{K,H'}}
\node{K \otimes H'}
\end{diagram}
\]
one can similarly obtain a natural isomorphism from $A$ to
$R_{H,K}(K \otimes F)$. The composite of these is $R_{F,K}$.
If we precompose $A$ with $T_{H,K}$ we obtain a bimorphism
naturally isomorphic to (\ref{braiding.nat.1}).
If we precompose $(F \otimes K)R_{H',K}$ with $T_{H,K}$, we obtain
a bimorphism naturally isomorphic to
\begin{equation}
\begin{diagram} [H' \otimes K.]
\node{H \times K} \arrow{e,t}{F \times K} \node{H' \times K}
\arrow{e,t}{T_{H',K}} \node{H' \otimes K}
\arrow{e,t}{R_{H',K}} \node{K' \otimes H}
\end{diagram} \label{braiding.nat.2}
\end{equation}
In both cases, a
natural isomorphism is given by the definition of tensor product.
It thus suffices to exhibit a natural isomorphism between
(\ref{braiding.nat.1}) and (\ref{braiding.nat.2}). This may be
constructed as in the proof of Proposition \ref{tensorator}. \hskip 3em \hbox{\BOX} \vskip 2ex
\subsection{The involutor}
As indicated in Figure 1, for $2{\rm Hilb}$ to be a stable 2-category it
should possess an extra layer of structure after the tensor product
and the braiding, namely the `involutor'. Also, this structure should
have an extra property making $2{\rm Hilb}$ `strongly involutory'. The
involutor is a weakened form of the equation appearing in the
definition of a symmetric monoidal category. Namely, while the
braiding need not satisfy
\[ R_{H',H} R_{H,H'} = 1_{H \otimes H'} \]
for all objects $H,H' \in 2{\rm Hilb}$, there should be a 2-isomorphism
\[ I_{H,H'} \colon R_{H,H'} R_{H',H} \Rightarrow 1_{H \otimes H'}, \]
the involutor.
We construct the involutor as follows. Choose tensor
products $T \colon H \times H' \to H \otimes H'$ and $T' \colon H' \times
H \to H' \otimes H$.
Then by the universality of the tensor product, the
commutativity of
\[
\begin{diagram} [H \times H']
\node[2]{H' \times H} \arrow{se,t}{S_{H',H}} \\
\node{H \times H'} \arrow[2]{e,b}{1_{H \times H'}}
\arrow{ne,t}{S_{H,H'}} \node[2]{H \times H'} \\
\end{diagram}
\]
implies that
\[
\begin{diagram} [H \times H']
\node[2]{H' \otimes H} \arrow{se,t}{R_{H',H}} \\
\node{H \otimes H'} \arrow[2]{e,b}{1_{H \otimes H'}}
\arrow{ne,t}{R_{H,H'}} \node[2]{H \otimes H'} \\
\end{diagram}
\]
commutes up to a specified natural transformation. This is the involutor
\[ I_{H,H'} \colon R_{H,H'} R_{H',H} \Rightarrow 1_{H \otimes H'}. \]
In addition, for $2{\rm Hilb}$ to be stable, or `strongly involutory', the
involutor should satisfy a special coherence law of its own, in
analogy to how the braiding satisfies a special equation in a
symmetric monoidal category. In HDA0 this equation was described in
terms of $R_{H,H'}$ and a weak inverse thereof, but it turns out
to be easier to give the equation by stating that
the following horizontal composites agree:
\[ I_{H,H'} \circ 1_{R_{H,H'}} \colon R_{H,H'} R_{H',H} R_{H,H'}
\Rightarrow R_{H,H'} \]
and
\[ 1_{R_{H,H'}} \circ I_{H,H'} \colon R_{H,H'} R_{H',H} R_{H,H'}
\Rightarrow R_{H,H'} \]
This is indeed the case, as one can show using the properties of
the tensor product.
\section{2-H*-algebras}\label{2-H*-algebras}
Now we consider 2-Hilbert spaces with extra structure and properties,
as listed in the second column of Figure 2.
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in} A {\rm 2-H*-algebra} $H$ is a 2-Hilbert space
equipped with a {\rm product} bimorphism $\otimes \colon H \times
H \to H$, a {\rm unit} object $1 \in H$, a unitary natural
transformation $a_{x,y,z} \colon (x \otimes y) \otimes z \to x
\otimes (y \otimes z)$ called the {\rm associator}, and unitary
natural transformations $\ell_x \colon 1 \otimes x \to x$, $r_x
\colon x \otimes 1 \to x$ called the {\rm left} and {\rm right
unit laws}, making $H$ into a monoidal category. We require also
that every object $x \in H$ has a left dual. \end{defn}
\noindent Recall that for $H$ to be a monoidal category, one demands that
the following pentagon commute:
\[ \begin{diagram}[((x \otimes y)\otimes z)\otimes w]
\node{((x \otimes y) \otimes z)\otimes w}
\arrow{e,t}{a_{x\otimes y,z,w}}
\arrow{s,l}{a_{x,y,z}\otimes w}
\node{(x\otimes y)\otimes(z\otimes w)}
\arrow{e,t}{a_{x,y,z\otimes w}}
\node{x \otimes (y\otimes(z \otimes w))} \\
\node{(x \otimes (y \otimes z)) \otimes w}
\arrow[2]{e,t}{a_{x,y\otimes z,w}}
\node[2]{x \otimes ((y \otimes z)\otimes w)}
\arrow{n,r}{x \otimes a_{y,z,w}}
\end{diagram} \]
as well as the following diagram involving the unit laws:
\[ \begin{diagram}[(1 \otimes x) \otimes 1]
\node{(1 \otimes x) \otimes 1} \arrow{s,l}{\ell_x \otimes 1}
\arrow[2]{e,t}{a_{1,x,1}} \node[2]{1 \otimes (x\otimes 1)} \arrow{s,r}{1
\otimes r_x} \\
\node{x \otimes 1} \arrow{e,t}{r_x} \node{x} \node{1 \otimes x}
\arrow{w,t}{\ell_x}
\end{diagram} \]
Mac Lane's coherence theorem \cite{MacLane2} says that every monoidal
category is equivalent, as a monoidal category, to a {\it strict}
monoidal category, that is, one for which the associators and unit
laws are all identity morphisms. Sometimes we will use this to
streamline formulas by not parenthesizing tensor products and not
writing the associators and unit laws. Such formulas apply literally
only to the strict case, but one can always use Mac Lane's theorem to
apply them to general monoidal categories. In practice, this amounts
to parenthesizing tensor products however one likes, and inserting
associators and unit laws when needed to make the formulas make sense.
A {\it left dual} of an object $x$ in a monoidal category
is an object $y$ together with morphisms
\[ e \colon y \otimes x \to 1 \]
and
\[ i \colon 1 \to x \otimes y ,\]
called the {\it unit} and {\it counit}, such that the following
diagrams commute:
\[
\begin{diagram} [x \otimes y \otimes x]
\node{x} \arrow[2]{e,t}{1_x} \arrow{se,b}{i \otimes x}
\node[2]{x} \\
\node[2]{x \otimes y \otimes x} \arrow{ne,b}{x \otimes e}
\end{diagram}
\]
\[
\begin{diagram} [x \otimes y \otimes x]
\node{y} \arrow[2]{e,t}{1_y} \arrow{se,b}{y \otimes i}
\node[2]{y} \\
\node[2]{y \otimes x \otimes y} \arrow{ne,b}{e \otimes y}
\end{diagram}
\]
(These diagrams apply literally only when the monoidal category
is strict.) In this situation we also say that $x$ is a {\it right dual}
of $y$, and that $(x,y,i,e)$ is an {\it adjunction}. All adjunctions
having $x$ as right dual are uniquely isomorphic in the following sense:
\begin{prop} \label{2H*1} \hspace{-0.08in}{\bf .}\hspace{0.1in} Given an adjunction $(x,y,i,e)$
in a monoidal category and an isomorphism $f\colon y \to y'$, there is
an adjunction $(x,y',i',e')$ given by:
\[ i' = i(x \otimes f), \qquad e' = (f^{-1} \otimes x)e.\]
Conversely, given two adjunctions $(x,y,i,e)$ and $(x,y',i',e')$,
there is a unique isomorphism $f \colon y \to y'$
for which $i' = i(x \otimes f)$ and $e' = (f^{-1} \otimes x)e$.
This is given in the strict case by the composite
\[ \begin{diagram}[y \otimes (x \otimes y')]
\node{y = y \otimes 1} \arrow{e,t}{y \otimes i'}
\node{y \otimes x \otimes y'} \arrow{e,t}{e \otimes y'}
\node{1 \otimes y' = y'}
\end{diagram} \]
\end{prop}
Proof - This result is well-known and the proof is a simple calculation.
\hskip 3em \hbox{\BOX} \vskip 2ex
\noindent Similarly, any two adjunctions having a given object as
right dual are canonically isomorphic. We may thus
speak of `the' left or right dual of a given object, using the
generalized `the', as described in Section \ref{tensorproduct}.
Note that duality at the morphism level of a 2-H*-algebra allows us to turn
left duals into right duals, and vice versa, at the object level:
\begin{prop}\label{2H*2}\hspace{-0.08in}{\bf .}\hspace{0.1in} Suppose that $H$ is a 2-H*-algebra. Then
$(x,x^\ast,i,e)$ is an adjunction if and only if $(x^\ast,x,e^\ast,i^\ast)$
is an adjunction. \end{prop}
Proof - The proof is analogous to that of Proposition \ref{2cat1}. \hskip 3em \hbox{\BOX} \vskip 2ex
Next we turn to braided and symmetric 2-H*-algebras.
A good example of a braided 2-H*-algebra is the category
of tilting modules of a quantum group when the parameter $q$ is a
suitable root of unity \cite{CP}. Categories very similar to our
braided 2-H*-algebras have been studied by Fr\"ohlich and Kerler
\cite{FK} under the name `C*-quantum categories'; our definitions differ
only in some fine points. A good example of a symmetric 2-H*-algebra
is the category of finite-dimensional continuous unitary
representations of a compact topological group. Doplicher and Roberts
\cite{DR} have studied categories very similar to our symmetric
2-H*-algebras.
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in} A {\rm braided 2-H*-algebra} is a 2-H*-algebra $H$
equipped with a unitary natural isomorphism $B_{x,y} \colon x \otimes y
\to y \otimes x$ making $H$ into a braided monoidal category.
\end{defn}
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in} A {\rm symmetric 2-H*-algebra} is a 2-H*-algebra for
which the braiding is a symmetry. \end{defn}
Recall that for $H$ to be a braided monoidal category, the
following two hexagons must commute:
\[ \begin{diagram}[(x \otimes y) \otimes z)]
\node{x \otimes (y \otimes z)} \arrow{s,l}{B_{x,y \otimes z}}
\arrow{e,t}{a^{-1}_{x,y,z}}
\node{(x \otimes y) \otimes z} \arrow{e,t}{B_{x,y} \otimes z}
\node{(y \otimes x) \otimes z} \arrow{s,r}{a_{y,x,z}} \\
\node{(y \otimes z) \otimes x} \arrow{e,b}{a_{y,z,x}}
\node{y \otimes (z \otimes x)} \arrow{e,t}{y \otimes B_{x,z}}
\node{y \otimes (x \otimes z)}
\end{diagram} \]
\[ \begin{diagram}[(x \otimes y) \otimes z)]
\node{(x \otimes y) \otimes z} \arrow{s,l}{B_{x \otimes y, z}}
\arrow{r,t}{a_{x,y,z}}
\node{x \otimes (y \otimes z)} \arrow{e,t}{x \otimes B_{y,z}}
\node{x \otimes (z \otimes y)} \arrow{s,r}{a_{x,z,y}^{-1}} \\
\node{z \otimes (x \otimes y)} \arrow{e,b}{a^{-1}_{z,x,y}}
\node{(z \otimes x) \otimes y} \arrow{e,t}{B_{x,z} \otimes y}
\node{(x \otimes z) \otimes y}
\end{diagram} \]
as well as the following diagrams:
\[ \begin{diagram}[1 \otimes x]
\node{1 \otimes x} \arrow{se,b}{\ell_x} \arrow[2]{e,t}{B_{1,x}}
\node[2]{x \otimes 1} \arrow{sw,b}{r_x} \\
\node[2]{x}
\end{diagram} \]
\[ \begin{diagram}[1 \otimes x]
\node{x \otimes 1} \arrow{se,b}{r_x} \arrow[2]{e,t}{B_{x,1}}
\node[2]{1 \otimes x} \arrow{sw,b}{\ell_x} \\
\node[2]{x}
\end{diagram} \]
The braiding is a symmetry if $B_{x,y} =
B_{y,x}^{-1}$ for all objects $x$ and $y$.
\subsection{The balancing} \label{balancing}
In the study of braided monoidal categories where objects have duals,
it is common to introduce something called the `balancing'. The
balancing can treated in various ways \cite{FK,JS2,RT}. For example, one
may think of it as a choice of automorphism $b_x \colon x \to x$ for
each object $x$, which is required to satisfy certain laws.
While very important in topology, this extra structure seems somewhat
ad hoc and mysterious from the algebraic point of view. We now
show that braided 2-H*-algebras are automatically equipped with
a balancing. The reason is that not only the objects, but also
the morphisms, have duals. In fact, some of what follows would
apply to any braided monoidal category in which both objects and
morphisms have duals.
In any 2-H*-algebra, Proposition \ref{2H*2} gives a way to make
any object $x$ into the left dual of its left dual $x^\ast$. In a braided
2-H*-algebra, $x$ also becomes the left dual of $x^\ast$ in another
way:
\begin{prop} \label{2H*2.5} \hspace{-0.08in}{\bf .}\hspace{0.1in}
Let $H$ be a braided 2-H*-algebra. Then $(x,x^\ast,i,e)$
is an adjunction if and only if
$(x^\ast,x,iB_{x,x^\ast},B_{x,x^\ast}e)$ is an adjunction. \end{prop}
Proof - The proof is a simple computation. \hskip 3em \hbox{\BOX} \vskip 2ex
It follows from Proposition \ref{2H*1} that these two ways
to make $x$ into the left dual of $x^\ast$ determine an automorphism
of $x$. Simplifying the formula for this automorphism
somewhat, we make the following definition:
\begin{defn}\hspace{-0.08in}{\bf .}\hspace{0.1in} If $H$ is a braided 2-H*-algebra and
$(x,x^\ast,i,e)$ is an adjunction in $H$, the {\rm balancing} of the
adjunction is the morphism $b \colon x \to x$ given in the
strict case by the composite:
\[ \begin{diagram} [(x \otimes x^\ast) \otimes x]
\node{x} \arrow{e,t}{e^\ast \otimes x}
\node{x^\ast \otimes x \otimes x}
\arrow{e,t}{x^\ast \otimes B_{x,x}}
\node{x^\ast \otimes x \otimes x}
\arrow{e,t}{e \otimes x} \node{x}
\end{diagram} \]
\end{defn}
It is perhaps easiest to understand the significance of the balancing
in terms of its relation to topology. We shall be quite
sketchy about describing this, but the reader can fill in the details
using the ideas described in HDA0 and the many references therein.
Especially relevant is the work of Freyd and Yetter \cite{FY}, Joyal
and Street \cite{JS}, and Reshetikhin and Turaev \cite{RT,T}. We
discuss this relationship more carefully in the Conclusions.
\bigskip
\centerline{\epsfysize=1.5in\epsfbox{2dtangle.eps}}
\medskip
\centerline{3. Typical tangle in 2 dimensions}
\medskip
The basic idea is to use tangles to represent certain morphisms in
2-H*-algebras. A typical oriented tangle in 2 dimensions is shown in
Figure 3. If we fix an adjunction $(x,x^\ast,i,e)$ in a strict
2-H*-algebra $H$, any such tangle corresponds uniquely to a morphism
in $H$ as follows. As shown in Figure 4, vertical juxtaposition of
tangles corresponds to the composition of morphisms, while horizontal
juxtaposition corresponds to the tensor product of morphisms.
\vbox{
\bigskip
\centerline{\epsfysize=1.5in\epsfbox{products.eps}}
\medskip
\centerline{4. Composition and tensor product of tangles}
\medskip
}
\noindent
Thus it suffices to specify the morphisms in $H$ corresponding to certain
basic tangles from which all others can be built up by composition and
tensor product. These basic tangles are shown in Figures 5 and 6. A
downwards-pointing segment corresponds to the identity on $x$, while
an upwards-pointing segment corresponds to the identity on $x^\ast$.
\bigskip
\centerline{\epsfysize=0.75in\epsfbox{identities.eps}}
\medskip
\centerline{5. Tangles corresponding to $1_x$ and $1_{x^\ast}$}
\medskip
\noindent
The two oriented forms of a `cup' tangle correspond to the morphisms
$e$ and $i^\ast$, while the two oriented forms of a `cap' correspond
to $i$ and $e^\ast$.
\bigskip
\centerline{\epsfysize=0.75in\epsfbox{unitcounit.eps}}
\medskip
\centerline{6. Tangles corresponding to $e$, $i^\ast$, $i$, and $e^\ast$}
\medskip
\noindent
It then turns out that isotopic tangles correspond to the same
morphism in $H$. The main thing to check is that the isotopic tangles
shown in Figure 7 correspond to the same morphisms. This
follows from the triangle diagrams in the definition of an
adjunction. Similar equations with the orientation of the arrows
reversed follow from Proposition \ref{2H*2}.
\bigskip
\centerline{\epsfysize=1.5in\epsfbox{triangle.eps}}
\medskip
\centerline{7. Tangle equations corresponding to the definition of adjunction}
\medskip
If $H$ is braided, we can also map framed oriented tangles in 3
dimensions to morphisms in $H$. A typical such tangle is shown in
Figure 8. We use the blackboard framing, in which each strand is
implicitly equipped with a vector field normal to the plane in which
the tangle is drawn.
\bigskip
\centerline{\epsfysize=1.5in\epsfbox{3dtangle.eps}}
\medskip
\centerline{8. Typical tangle in 3 dimensions}
\medskip
\noindent We interpret the basic tangles in Figures 5 and 6 as
we did before. Moreover, we let the tangles in Figure 9
correspond to the morphisms $B_{x,x}$, $B_{x^\ast,x}$,
$B_{x,x^\ast}$, and $B_{x^\ast,x^\ast}$, and let the tangles in
Figure 10 correspond to the morphisms $B_{x,x}^{-1}$,
$B_{x^\ast,x}^{-1}$, $B_{x,x^\ast}^{-1}$, and
$B_{x^\ast,x^\ast}^{-1}$.
\bigskip
\centerline{\epsfysize=1in\epsfbox{braid1.eps}}
\medskip
\centerline{7. Tangles corresponding to $B_{x,x}$, $B_{x^\ast,x}$,
$B_{x,x^\ast}$, and $B_{x^\ast,x^\ast}$}
\medskip
\bigskip
\centerline{\epsfysize=1in\epsfbox{braid2.eps}}
\medskip
\centerline{8. Tangles corresponding to $B_{x,x}^{-1}$, $B_{x^\ast,x}^{-1}$,
$B_{x,x^\ast}^{-1}$, and $B_{x^\ast,x^\ast}^{-1}$}
\medskip
Now suppose we wish isotopic framed oriented tangles to correspond to
the same morphism in $H$. Invariance under the 2nd and 3rd
Reidemeister moves follows from the properties of the braiding,
so it suffices to check invariance under the framed version of the 1st
Reidemeister move. For this, note that the tangle shown in
Figure 9 corresponds to the balancing of the adjunction $(x,x^\ast,i,e)$.
This tangle has a $2\pi$ twist in its framing.
\vbox{
\bigskip
\centerline{\epsfysize=1.5in\epsfbox{balancing.eps}}
\medskip
\centerline{9. Tangle corresponding to the balancing $b \colon x \to x$}
\medskip
}
\noindent
The framed version of the 1st Reidemeister move, shown in Figure 10,
represents the cancellation of two opposite $2\pi$ twists in the framing.
Both tangles in this picture correspond to the same morphism in $H$
precisely when the balancing $b \colon x \to x$ is unitary.
\bigskip
\centerline{\epsfysize=1.5in\epsfbox{unitarity.eps}}
\medskip
\centerline{10. Tangle equation corresponding to unitarity of the
balancing}
\medskip
In short, we obtain a map from isotopy classes of framed oriented
tangles in 3 dimensions to morphisms in a braided 2-H*-algebra $H$
whenever we choose an adjunction in $H$ whose balancing is unitary.
This motivates the following definition:
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in} An adjunction $(x,x^\ast,i,e)$ in a braided 2-H*-algebra
is {\rm well-balanced} if its balancing is unitary.
\end{defn}
\noindent
Similarly, given any well-balanced adjunction in a symmetric
2-H*-algebra $H$, we obtain a map from isotopy classes of framed
oriented tangles in 4 dimensions to morphisms in $H$. We may draw
tangles in 4 dimensions just as we draw tangles in 3 dimensions, but
there is an extra rule saying that any right-handed crossing is
isotopic to the corresponding left-handed crossing. One case of this
rule is shown in Figure 11. Invariance under these isotopies follows
directly from the fact that the braiding is a symmetry.
\bigskip
\centerline{\epsfysize=1.0in\epsfbox{symmetry.eps}}
\medskip
\centerline{11. Tangle equation corresponding to symmetry}
\medskip
The important fact is that well-balanced adjunctions exist and
are unique up to a unique unitary isomorphism. Moreover, all of
them have the same balancing:
\begin{thm}\label{2H*3} \hspace{-0.08in}{\bf .}\hspace{0.1in} Suppose $H$ is a braided
2-H*-algebra. For every object $x \in H$ there exists a well-balanced
adjunction $(x,y,i,e)$. Given well-balanced adjunctions $(x,y,i,e)$
and $(x,y',i',e')$, there is a unique morphism $u \colon y \to y'$ such that
\[ i' = i(u \otimes x), \qquad e' = (x \otimes u^{-1})e, \]
and this morphism is unitary. \end{thm}
Proof - To simplify notation we assume without loss of generality that
$H$ is strict. Suppose first that $x \in H$ is simple.
Then for any adjunction $(x,y,i,e)$, the balancing equals $\beta 1_x$
for some nonzero $\beta \in {\Bbb C}$. By Proposition \ref{2H*1} we may
define a new adjunction $(x,y,|\beta|^{1/2}i,|\beta|^{-1/2}e)$.
Since the balancing of this adjunction equals $\beta |\beta|^{-1} 1_x$,
this adjunction is well-balanced.
Next suppose that $x \in H$ is arbitrary. Using Theorem
\ref{H*3} we can write $x$ as an orthogonal direct sum of simple
objects $x_j$, in the sense that there are morphisms
\[ p_j \colon x \to x_j \]
with
\[ p_j^\ast p_j = 1_{x_j}, \qquad \sum_j p_j p_j^\ast =
1_x. \]
Let $y_j$ be a left dual of $x_j$, and
define $y$ to be an orthogonal direct sum
of the objects $y_j$, with morphisms
\[ q_j \colon y \to y_j \]
such that
\[ q_j^\ast q_j = 1_{y_j}, \qquad \sum_j q_j q_j^\ast =
1_y . \]
Since the $x_j$ are simple, there exist adjunctions
$(x_j,y_j,e_j,i_j)$ for which the balancings $b_j \colon x_j \to x_j$
are unitary. Define the adjunction $(x,y,i,e)$ by
\[ i = \sum_j i_j (p_j^\ast \otimes q_j^\ast) , \qquad
e = \sum_j (q_j \otimes p_j) e_j .\]
One can check that this is indeed an adjunction and that
the balancing $b \colon x \to x$ of this adjunction is given by
\[ b = \sum_j p_j b_j p_j^\ast ,\]
and is therefore unitary.
Now suppose that $(x,y',i',e')$ is any other well-balanced adjunction with
$x$ as right dual. Let $b'$ denote the balancing of this adjunction.
We shall prove that $b' = b$.
By Propositions \ref{H*4} and \ref{2H*1} there
exists a unitary morphism $g \colon y \to y'$, and we have
\begin{eqnarray*} b' &=& (e'^\ast \otimes x)(y' \otimes B_{x,x})(e' \otimes x) \\
&=& (e'^\ast(g^\ast \otimes x) \otimes x)(y \otimes B_{x,x})
((g \otimes x)e' \otimes x). \end{eqnarray*}
By Proposition \ref{2H*1}, $(x,y,(g \otimes x)e',i(x \otimes g^{-1}))$ is
an adjunction, so by the uniqueness up to isomorphism of right adjoints
we have $(g \otimes x)e' = (y \otimes f)e$ for some isomorphism
$f \colon x \to x$. We thus have
\begin{eqnarray*} b' &=&
(e^\ast(y \otimes f^\ast) \otimes x)(y \otimes B_{x,x})
((y \otimes f)e \otimes x) \\
&=& fbf^\ast .\end{eqnarray*}
We may write $x$ as an orthogonal direct sum
\[ x = \bigoplus_\lambda x_\lambda \]
where $\{e_\lambda\}$ is a basis of $H$ and $x_\lambda$
is a direct sum of some number of copies of $e_\lambda$.
Then by our previous formula for $b$ we have
\[ b = \bigoplus_\lambda \beta_\lambda 1_{x_\lambda} \]
with $|\beta_\lambda| = 1$ for all $\lambda$.
We also have
\[ f = \bigoplus_\lambda f_\lambda \]
for some morphisms $f_\lambda \colon x_\lambda \to x_\lambda$. It follows that
\[ b' = \bigoplus_\lambda \beta_\lambda f_\lambda {f_\lambda}^\ast \]
Since $b$ and $b'$ are unitary it follows that each morphism
$f_\lambda {f_\lambda}^\ast$ is unitary. Since the only positive
unitary operator is the identity, using Theorem \ref{H*3} it follows that
each $f_\lambda {f_\lambda}^\ast$ is the identity,
so $b' = b$ as desired.
By Proposition \ref{2H*1}, we know there is
a unique isomorphism $u \colon y \to y'$ with
\[ i' = i(u \otimes x) , \qquad e' = (x \otimes u^{-1})e ,\]
and we need to show that $u$ is unitary. Since $b' = b$, we have
\[ (ib' \otimes y)(x \otimes B_{y,y}^{-1})(i^\ast \otimes y)=
(ib \otimes y)(x \otimes B_{y,y}^{-1})(i^\ast \otimes y) ,\]
and if one simplifies this equation using the fact that
\[ b = (e^\ast \otimes x)(y' \otimes B_{x,x})(e \otimes x) \]
and
\[ b' = (e^\ast(x \otimes (u^{-1})^\ast) \otimes x)(y' \otimes B_{x,x})
((x \otimes u^{-1})e \otimes x),\]
one finds that $u$ is unitary. \hskip 3em \hbox{\BOX} \vskip 2ex
\begin{cor} \hspace{-0.08in}{\bf .}\hspace{0.1in} In a braided 2-H*-algebra every well-balanced
adjunction with $x$ as right dual has the same balancing, which
we call {\rm the balancing} of $x$ and denote as $b_x \colon x \to
x$. \end{cor}
Proof - This was shown in the proof above. \hskip 3em \hbox{\BOX} \vskip 2ex
Note that for any simple object $x$ in a braided 2-H*-algebra,
the balancing $b_x$ must equal $1_x$ times some unit complex number, the
{\it balancing phase} of $x$. In physics, the balancing phase describes
the change in the wavefunction of a particle that undergoes a $2\pi$
rotation. Note that in a symmetric 2-H*-algebra
\begin{eqnarray*} b_x &=& (e_x^\ast \otimes 1_x)(1_{x^\ast} \otimes B_{x,x})
(e_x \otimes 1_x) \\
&=& (e_x^\ast \otimes 1_x)(1_{x^\ast} \otimes B^\ast_{x,x})
(e_x \otimes 1_x) \\
&=& b_x^\ast, \end{eqnarray*}
so $b_x^2 = 1_x$. Thus in this case the balancing phase
of any simple object must be $\pm 1$. In physics,
this corresponds to the fact that particles in 4-dimensional spacetime
are either bosons and fermions depending on the phase they acquire when
rotated by $2\pi$, while in 3-dimensional spacetime other
possibilities, sometimes called `anyons', can occur \cite{DR,FK}.
More generally, we make the following definition:
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in} If $H$ is a symmetric 2-H*-algebra, an object $x \in H$ is
{\rm even} or {\rm bosonic} if $b_x = 1$, and {\rm odd} or {\rm
fermionic} if $b_x = -1$. We say $H$ is {\rm even} or {\rm purely
bosonic} if every object $x \in H$ is even. \end{defn}
\noindent Note that if $x \oplus y$ is an orthogonal direct sum,
\[ b_{x \oplus y} = b_x \oplus b_y, \]
so an object in any symmetric 2-H*-algebra
is even (resp.\ odd) if and only if it is
a direct sum of even (resp.\ odd) simple objects. Also, since
\[ b_{x \otimes y} = (b_x \otimes b_y)B_{x,y}B_{y,x}, \]
it follows that the tensor product of two even or two odd objects is even,
while the tensor product of an even and an odd object is odd.
There is a way to turn any symmetric 2-H*-algebra into an even one,
which will be useful in Section \ref{recon}.
\begin{prop} \label{bosonization}
\hspace{-0.08in}{\bf .}\hspace{0.1in} {\rm (Doplicher-Roberts)}\, Suppose $H$ is a symmetric
2-H*-algebra. Then there is a braiding $B'$ on $H$ given on simple
objects $x,y \in H$ by
\[ B^\flat_{x,y} = (-1)^{|x|\,|y|} B_{x,y} \]
where $|x|$ equals $0$ or $1$ depending on whether $x$ is even or odd, and
similarly for $|y|$. Let $H^\flat$ denote $H$ equipped with the
new braiding $B^\flat$. Then $H^\flat$ is an even symmetric
2-H*-algebra, the {\rm bosonization} of $H$. \end{prop}
Proof - This is a series of straightforward computations. One approach
involves noting that for any objects $x,y \in H$,
\[ B^\flat_{x,y} = {1\over 2}\, B_{x,y} (1_x \otimes 1_y + 1_x \otimes
b_y + b_x \otimes 1_y - b_x \otimes b_y) . \]
\hbox{\hskip 30em} \hskip 3em \hbox{\BOX} \vskip 2ex
\noindent The above proposition is essentially due to Doplicher and
Roberts, who proved it in a slightly different context \cite{DR}.
However, the term `bosonization' is borrowed from Majid \cite{Majid},
who uses it to denote a related process that turns a super-Hopf algebra
into a Hopf algebra.
\subsection{Trace and dimension}
The notion of the `dimension' of an object in a braided
2-H*-algebra will be very important in Section \ref{recon}.
First we introduce the related notion of `trace'.
\begin{defn}\hspace{-0.08in}{\bf .}\hspace{0.1in} If $H$ is a braided 2-H*-algebra and $f \colon x \to x$ is a
morphism in $H$, for any well-balanced adjunction $(x,x^\ast,i,e)$
we define the {\rm trace} of $f$, ${\rm tr}(f) \in {\rm end}(1)$, by
\[ {\rm tr}(f) = e (x^\ast \otimes f) e^\ast .\]
\end{defn}
\noindent
The trace is independent of the choice of well-balanced adjunction,
by Theorem \ref{2H*3}. Also, one can show that an obvious
alternative definition of the trace is actually equivalent:
\[ {\rm tr}(f) = i^\ast (f \otimes x^\ast) i .\]
\begin{defn}\hspace{-0.08in}{\bf .}\hspace{0.1in} If $H$ is a braided 2-H*-algebra, we define
the {\it dimension} of $x$, $\dim(x)$, to be ${\rm tr}(1_x)$. \end{defn}
\noindent Note that $x,y$ are objects in a braided 2-H*-algebra, we have
\[ \dim(x \oplus y) = \dim(x) + \dim(y), \qquad
\dim(x \otimes y) = \dim(x) \dim(y), \qquad
\dim(x^\ast) = \dim(x) .\]
Moreoever, we have:
\begin{prop} \label{dim} \hspace{-0.08in}{\bf .}\hspace{0.1in} If $H$ is a symmetric 2-H*-algebra and $x
\in H$ is any object, then the spectrum of $\dim(x)$ is a subset of ${\Bbb N}
= \{0,1,2,\dots\}$.
\end{prop}
Proof - We follow the argument of Doplicher and Roberts \cite{DR}.
For any $n \ge 0$, the group algebra of the symmetric group $S_n$ acts
as endomorphisms of $x^{\otimes n}$, and the morphisms $p_S, p_A$
corresponding to complete symmetrization and complete
antisymmetrization, respectively, are self-adjoint projections in the
H*-algebra ${\rm end}(x^{\otimes n})$. It follows that
${\rm tr}(p_S), {\rm tr}(p_A) \ge 0$. If $x$ is even, a calculation shows that
\[ {\rm tr}(p_A) = {1\over n!}\, \dim(x)(\dim(x) - 1)\cdots (\dim(x) - n +
1) \]
For this to be nonnegative for all $n$, the spectrum of $\dim(x)$ must
lie in ${\Bbb N}$. Similarly, if $x$ is odd, a calculation shows that
\[ {\rm tr}(p_S) = {1\over n!} \, \dim(x)(\dim(x) - 1)\cdots (\dim(x) - n +
1) \]
so again the spectrum of $\dim(x)$ lies in ${\Bbb N}$. In general, any
object $\dim(x)$ is a sum of simple objects, which are either even or
odd, so by the additivity of dimension, the spectrum of $\dim(x)$ again
lies in ${\Bbb N}$. \hskip 3em \hbox{\BOX} \vskip 2ex
For any 2-H*-algebra, the Eckmann-Hilton argument shows that ${\rm end}(1)$
is a commutative H*-algebra, and thus isomorphic to a direct sum of
copies of ${\Bbb C}$. (See HDA0 or HDA1 for a explanation of the Eckmann-Hilton
argument.)
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in} A 2-H*-algebra $H$ is {\rm connected} if the unit
object $1 \in H$ is simple. \end{defn}
\noindent In a connected 2-H*-algebra, ${\rm end}(1) \cong {\Bbb C}$. The
dimension of any object in a connected symmetric 2-H*-algebra
is thus a nonnegative integer.
In addition to the above notion of dimension it is also interesting
to consider the `quantum dimension'. Here our treatment most closely
parallels that of Majid \cite{Majid}.
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in}
If $H$ is a braided 2-H*-algebra and $f \colon x \to x$ is a
morphism in $H$, for any well-balanced adjunction $(x,x^\ast,i,e)$
we define the {\rm quantum trace} of $f$, ${\rm qtr}(f) \in {\rm end}(1)$, by
\[ {\rm qtr}(f) = {\rm tr}(b_x f) .\]
We define the {\rm quantum dimension} of $x$, ${\rm qdim}(x)$, to be
${\rm qtr}(1_x)$. \end{defn}
\noindent In the case of a symmetric 2-H*-algebra, the quantum trace
is also called the `supertrace'. Suppose $H$ is a connected symmetric
2-H*-algebra and $x$ is a simple object. Then ${\rm qdim}(x) \ge 0$ if
$x$ is even and ${\rm qdim}(x) \le 0$ if $x$ is odd. The idea of odd
objects as negative-dimensional is implicit in Penrose's work
on negative-dimensional vector spaces \cite{Penrose}.
\subsection{Homomorphisms and 2-homomorphisms}
There is a 2-category with 2-H*-algebras as objects and `homomorphisms' and
`2-homomorphisms' as morphisms and 2-morphisms, respectively. This
is also true for braided 2-H*-algebras and symmetric 2-H*-algebras.
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in} Given 2-H*-algebras $H$ and $H'$, a {\rm
homomorphism} $F \colon H \to H'$ is a morphism of 2-Hilbert spaces
that is also a monoidal functor. If $H$ and $H'$ are braided, we say
that $F$ is a {\rm homomorphism of braided 2-H*-algebras} if $F$ is
additionally a braided monoidal functor. If $H$ and $H'$ are symmetric,
we say that $F$ is a {\rm homomorphism of symmetric 2-H*-algebras} if $F$ is
a morphism of 2-Hilbert spaces that is also a symmetric monoidal functor.
\end{defn}
Recall that a functor $F \colon C \to C'$ between monoidal categories is
monoidal if it is equipped with a natural isomorphism
$\Phi_{x,y} \colon F(x) \otimes F(y) \to F(x \otimes y)$
making the following diagram commute for any objects $x,y,z \in C$:
\[
\begin{diagram} [F(x) \otimes (F(y) \otimes F(z))]
\node{(F(x) \otimes F(y)) \otimes F(z)}
\arrow{e,t}{\Phi_{x,y}\, \otimes 1_{F(z)}} \arrow{s,l}{a_{F(x),F(y),F(z)}}
\node{F(x \otimes y) \otimes F(z)}
\arrow{e,t}{\Phi_{x\otimes y,z}}
\node{F((x \otimes y)\otimes z)}
\arrow{s,r}{F(a_{x,y,z})} \\
\node{F(x) \otimes (F(y) \otimes F(z))}
\arrow{e,t}{1_{F(x)} \otimes \Phi_{y,z}}
\node{F(x) \otimes F(y \otimes z)}
\arrow{e,t}{\Phi_{x,y\otimes z}}
\node{F(x \otimes (y \otimes z))}
\end{diagram}
\]
together with an isomorphism $\phi \colon 1_{C'} \to
F(1_{C})$ making
the following diagrams commute for any object $x \in C$:
\[
\begin{diagram}[F(1) \otimes F(x)]
\node{1 \otimes F(x)} \arrow{e,t}{\ell_{F(x)}}
\arrow{s,l}{\phi \otimes 1_{F(x)}}
\node{F(x)} \\
\node{F(1) \otimes F(x)} \arrow{e,t}{\Phi_{1,x}}
\node{F(1 \otimes x)} \arrow{n,r}{F(\ell_x)}
\end{diagram}
\]
\[
\begin{diagram}[F(1) \otimes F(x)]
\node{F(x) \otimes 1} \arrow{e,t}{r_{F(x)}}
\arrow{s,l}{1_{F(x)}\otimes \phi}
\node{F(x)} \\
\node{F(x) \otimes F(1)} \arrow{e,t}{\Phi_{x,1}}
\node{F(x \otimes 1)} \arrow{n,r}{F(r_x)}
\end{diagram}
\]
If $C$ and $C'$ are braided, we say that $F$ is braided if additionally
it makes the following diagram commute for all $x,y \in C$:
\[
\begin{diagram}[F(x) \otimes F(y)]
\node{F(x) \otimes F(y)} \arrow{e,t}{B_{F(x),F(y)}}
\arrow{s,l}{\Phi_{x,y}}
\node{F(y) \otimes F(x)}
\arrow{s,r}{\Phi_{y,x}} \\
\node{F(x \otimes y)} \arrow{e,t}{F(B_{x,y})}
\node{F(y \otimes x)}
\end{diagram}
\]
A symmetric monoidal functor is simply a braided monoidal functor that
happens to go between symmetric monoidal categories! No extra condition
is involved here.
Note that if $F \colon H \to H'$ is a homomorphism of braided 2-H*-algebras,
$F$ maps any well-balanced adjunction in $H$ to one in $H'$. Thus it
preserves dimension in the following sense:
\[ \dim(F(x)) = F(\dim(x)) \]
for any object $x \in H$.
In particular, if $H$ and $H'$ are connected, so that we can
identify the dimension of objects in either with numbers, we have
simply $\dim(F(x)) = \dim(x)$.
\begin{defn}\hspace{-0.08in}{\bf .}\hspace{0.1in} If $H$ and $H'$ are 2-H*-algebras, possibly braided or
symmetric, and $F,G \colon H \to H'$ are homomorphisms of the appropriate
sort, a {\rm 2-homomorphism} $\alpha \colon F \Rightarrow G$ is a monoidal
natural transformation. \end{defn}
Suppose that the $(F,\Phi,\phi)$ and $(G,\Gamma,\gamma)$ are monoidal
functors from the monoidal category $C$ to the monoidal category $D$.
Then a natural transformation $\alpha \colon F \to G$ is monoidal if
the diagrams
\[ \begin{diagram}[F(x) \otimes F(y)]
\node{F(x) \otimes F(y)} \arrow{e,t} {\alpha_x \otimes \alpha_y}
\arrow{s,l}{\Phi_{x,y}}
\node{G(x) \otimes G(y)} \arrow{s,r}{\Gamma_{x,y}}
\\ \node{F(x \otimes y)} \arrow{e,t}{\alpha_{x\otimes y}}
\node{G(x \otimes y)}
\end{diagram}
\]
and
\[ \begin{diagram}[F(1)]
\node{1} \arrow{s,l}{\phi} \arrow{se,t}{\gamma} \\
\node{F(1)} \arrow{e,t}{\alpha_1} \node{G(1)}
\end{diagram}
\]
commute.
There are no extra conditions required of `braided monoidal' or
`symmetric monoidal' natural transformations.
Finally, when we speak of two 2-H*-algebras $H$ and $H'$, possibly
braided or symmetric, being {\it equivalent}, we always mean the
existence of homomorphisms $F \colon H \to H'$ and $G \colon H' \to H$ of the
appropriate sort that are inverses up to a 2-isomorphism.
\section{Reconstruction Theorems} \label{recon}
In this section we give a classification of symmetric 2-H*-algebras.
Doplicher and Roberts proved a theorem which implies that connected
even symmetric 2-H*-algebras are all equivalent to categories of compact
group representations \cite{DR,DR2}. Here and in all that follows, by a
`representation' of a compact group we mean a finite-dimensional
continuous unitary representation. Given a compact group $G$, let
${\rm Rep}(G)$ denote the category of such representations of $G$. This
becomes a connected even symmetric 2-H*-algebra in an obvious way.
While Doplicher and Roberts worked using the language of `C*-categories',
their result can be stated as follows:
\begin{thm}\label{dhr1}\hspace{-0.08in}{\bf .}\hspace{0.1in} {\rm (Doplicher-Roberts)} Let $H$ be
a connected even symmetric 2-H*-\break algebra. Then there
exists a homomorphism of symmetric 2-H*-algebras $T \colon H \to
{\rm Hilb}$, unique up to a unitary 2-homomorphism. Let $U(T)$ be the
group of unitary 2-homomorphisms $\alpha \colon T \Rightarrow T$, given
the topology in which a net $\alpha_\lambda \in U(T)$ converges
to $\alpha$ if and only if $(\alpha_\lambda)_x \to \alpha_x$ in
norm for all $x \in H$. Then $U(T)$ is compact, each Hilbert
space $T(x)$ becomes a representation of $U(T)$, and the
resulting homomorphism $\tilde T \colon H \to {\rm Rep}(U(T))$ extends
to an equivalence of symmetric 2-H*-algebras. \end{thm}
Note that any continuous homomorphism $\rho \colon G \to G'$ between
compact groups determines a homomorphism of symmetric 2-H*-algebras,
\[ \rho^\ast \colon {\rm Rep}(G') \to {\rm Rep}(G), \]
sending each representation $\sigma$ of $G'$ to the representation
$\sigma \circ \rho$ of $G$. The above theorem yields a useful converse
to this construction:
\begin{cor}\label{dhr2}\hspace{-0.08in}{\bf .}\hspace{0.1in} {\rm (Doplicher-Roberts)} Let $F \colon H' \to
H$ be a homomorphism of connected even symmetric 2-H*-algebras. Let
$T \colon H \to {\rm Hilb}$ be a homomorphism of
symmetric 2-H*-algebras. Then there exists a continuous group
homomorphism
\[ F^\ast \colon U(T) \to U(FT) \]
such that $F^\ast(\alpha)$ equals the horizontal composite $F\circ \alpha$.
Moreover, $(F^\ast)^\ast$ equals $F$ up to a unitary 2-homomorphism.
\end{cor}
Dolan \cite{Dolan} has noted that a generalization of the
Doplicher-Roberts theorem to even symmetric 2-H*-algebras --- not
necessarily connected --- amounts to a categorification of the
Gelfand-Naimark theorem. The spectrum of a commutative H*-algebra $H$
is a set ${\rm Spec}(H)$ whose points are homomorphisms from $H$ to ${\Bbb C}$.
The Gelfand-Naimark theorem implies that $H$ is isomorphic to the
algebra of functions from ${\rm Spec}(H)$ to ${\Bbb C}$. Similarly, we may
define the `spectrum' of an even symmetric 2-H*-algebra $H$ to be the
groupoid ${\rm Spec}(H)$ whose objects are homomorphisms from $H$ to
${\rm Hilb}$, and whose morphisms are unitary 2-homomorphisms between
these. Moreover, we shall show that $H$ is equivalent to a symmetric
2-H*-algebra whose objects are `representations' of ${\rm Spec}(H)$ ---
certain functors from ${\rm Spec}(H)$ to ${\rm Hilb}$. Indeed, our proof of
this uses an equivalence
\[ \hat{\hbox{\hskip 0.5em}}\colon H \to {\rm Rep}({\rm Spec}(H)) \]
that is just the categorified version of the `Gelfand transform'
for commutative H*-algebras.
In fact, there is no need to restrict ourselves to symmetric
2-H*-algebras that are even. To treat a general symmetric
2-H*-algebra $H$ we need objects of ${\rm Spec}(H)$ to be
homomorphisms from $H$ to a symmetric 2-H*-algebra of
`super-Hilbert spaces'. The spectrum will then be a
`supergroupoid' --- though not the most general sort of thing one
could imagine calling a supergroupoid.
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in} Define ${\rm SuperHilb}$ to be the category whose
objects are $Z_2$-graded (finite-dimensional) Hilbert spaces,
and whose morphisms are linear maps preserving the grading. \end{defn}
The category ${\rm SuperHilb}$ can be made into a symmetric 2-H*-algebra
where the $\ast$-structure is the ordinary Hilbert space adjoint,
the product is the usual tensor product of ${\Bbb Z}_2$-graded
Hilbert spaces, and the braiding is given on homogeneous elements $v \in x$,
$w \in y$ by
\[ B_{x,y}(v \otimes w) = (-1)^{{\rm deg} v\, {\rm \deg} w}
\, w \otimes v. \]
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in} If $H$ is a symmetric 2-H*-algebra, define ${\rm Spec}(H)$
to be the category whose objects are symmetric 2-H*-algebra homomorphisms
$F \colon H \to {\rm SuperHilb}$ and whose morphisms are unitary
2-homomorphisms between these. \end{defn}
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in} A {\rm topological groupoid} is a groupoid for
which the $\hom$-sets are topological spaces and
the groupoid operations are continuous. A {\rm compact groupoid} is
is a topological groupoid with compact Hausdorff $\hom$-sets and finitely
many isomorphism classes of objects. \end{defn}
\begin{defn} \hspace{-0.08in}{\bf .}\hspace{0.1in} A {\rm supergroupoid} is a groupoid $G$ equipped
with a natural transformation $\beta \colon 1_G \Rightarrow 1_G$, the {\rm
balancing,} with $\beta^2 = 1$. A {\rm compact supergroupoid} is a
supergroupoid that is also a compact groupoid. \end{defn}
Let $H$ be a symmetric 2-H*-algebra. Then ${\rm Spec}(H)$ becomes a
topological groupoid if for any $S,T \colon H \to {\rm Hilb}$ we give
$\hom(S,T)$ the topology in which a net $\alpha_\lambda$ converges to
$\alpha$ if and only if $(\alpha_\lambda)_x \to \alpha_x$ in norm for
any $x \in H$. We shall show that ${\rm Spec}(H)$ is a compact groupoid.
Also, ${\rm Spec}(H)$ becomes a supergroupoid if for any object $T \in
{\rm Spec}(H)$ we define $\beta_T \colon T \Rightarrow T$ by
\[ (\beta_T)_x = b_{T(x)} = T(b_x) \]
for any object $x \in H$. One can
check that $\beta \colon 1_{{\rm Spec}(H)} \Rightarrow 1_{{\rm Spec}(H)}$ is a natural
transformation, and $\beta^2 = 1$ because the balancing for $H$ satisfies
$b_x^2 = 1$ for any $x \in H$.
\begin{defn}\hspace{-0.08in}{\bf .}\hspace{0.1in} Given a compact supergroupoid $G$, a (continuous, unitary,
finite-dimensional) {\rm representation} of $G$ is a functor $F \colon G
\to {\rm SuperHilb}$ such that $F(g)$ is unitary for every morphism $g$ in $G$,
$F \colon \hom(x,y) \to \hom(F(x),F(y))$ is continuous for all objects
$x,y \in G$, and
\[ F(\beta_x) = b_{F(x)} \]
for every object $x \in G$. We define ${\rm Rep}(G)$ to be the category having
representations of $G$ as objects and natural transformations between
these as morphisms. \end{defn}
Let $G$ be a compact supergroupoid. Then the category ${\rm Rep}(G)$
becomes an even symmetric 2-H*-algebra in a more or less obvious way
as follows. Given objects $F,F' \in {\rm Rep}(G)$, we make $\hom(F,F')$
into a Hilbert space with the obvious linear structure and the inner
product given by
\[ \langle \alpha, \beta \rangle = \sum_x
{\rm tr}({\alpha_x}^\ast \beta_x) \]
where the sum is taken over any maximal set of nonisomorphic objects of
$G$. This makes ${\rm Rep}(G)$ into a ${\rm Hilb}$-category.
Moreover, ${\rm Rep}(G)$ becomes a 2-Hilbert space if we define the dual
of $\alpha \colon F \Rightarrow F'$ by $(\alpha^\ast)_x = (\alpha_x)^\ast$.
We define the tensor product of objects $F,F' \in {\rm Rep}(G)$ by
\[ (F \otimes F')(x) = F(x) \otimes F'(x), \qquad
(F \otimes F')(f) = F(f) \otimes F'(f) \]
for any object $x \in G$ and morphism $f$ in $G$. It is easy to define a
tensor product of morphisms and associator making ${\rm Rep}(G)$ into a
monoidal category, and to check that ${\rm Rep}(G)$ is then a 2-H*-algebra.
Finally, ${\rm Rep}(G)$ inherits a braiding from the braiding in ${\rm SuperHilb}$,
making ${\rm Rep}(G)$ into a symmetric 2-H*-algebra.
Now suppose $H$ is an even symmetric 2-H*-algebra. Then there is a functor
\[ \hat{\hbox{\hskip 0.5em}}\colon H \to {\rm Rep}({\rm Spec}(H)) , \]
the {\it categorified Gelfand transform}, given as follows.
For every object $x \in H$, $\hat x$ is the representation with
\[ \hat x(T) = T(x) \]
for all $T \in {\rm Spec}(H)$, and
\[ \hat x(\alpha) = \alpha_x \]
for all $\alpha \colon T \Rightarrow T'$, where $T,T' \in {\rm Spec}(H)$. For every
morphism $f \colon x \to y$ in $H$, $\hat f \colon \hat x \Rightarrow \hat y$ is
the natural transformation with
\[ \hat f(T) = T(f) \]
for all $T \in {\rm Spec}(H)$. Our generalized Doplicher-Roberts theorem
states:
\begin{thm} \label{dhr4}\hspace{-0.08in}{\bf .}\hspace{0.1in} Suppose that $H$ is a
symmetric 2-H*-algebra. Then ${\rm Spec}(H)$ is a compact supergroupoid
and $\hat{\hbox{\hskip 0.5em}}\colon H \to {\rm Rep}({\rm Spec}(H))$ extends to
an equivalence of symmetric 2-H*-algebras.
\end{thm}
Proof - We have described how ${\rm Spec}(H)$ is a supergroupoid. To see
that it is compact, note that for any $S,T \in {\rm Spec}(H)$ the $\hom$-set
$\hom(S,T)$ is a compact Hausdorff space, by Tychonoff's theorem. We
also need to show that ${\rm Spec}(H)$ has finitely many isomorphism classes
of objects. The unit object $1_H$ is the direct sum of finitely many
nonisomorphic simple objects $e_i$, the kernels of the minimal
projections $p_i$ in the commutative H*-algebra ${\rm end}(1_H)$. Any
object $x \in H$ is thus a direct sum of objects $x_i = e_i \otimes x$,
and any morphism $f \colon x \to y$ is a direct sum of morphisms $f_i
\colon x_i \to y_i$. In short, $H$ is, in a fairly obvious sense, the
direct sum of finitely many connected symmetric 2-H*-algebras $H_i$.
Any homomorphism $T \colon H \to {\rm SuperHilb}$ induces a homomorphism from
${\rm end}(1_H)$ to ${\rm end}(1_{{\rm SuperHilb}}) \cong {\Bbb C}$, which must annihilate all
but one of the projections $p_i$, so $T$ sends one of the objects $x_i$
to $1_{{\rm SuperHilb}}$ and the rest to $0$. Thus ${\rm Spec}(H)$ is, as a
groupoid, equivalent to the disjoint union of the groupoids
${\rm Spec}(H_i)$, and hence has finitely many isomorphism classes of objects.
To show that the categorified Gelfand transform is an equivalence, first
suppose that $H$ is even and connected. Then the supergroupoid
${\rm Spec}(H)$ has $\beta = 1$, so every representation $F \colon {\rm Spec}(H)
\to {\rm SuperHilb}$ factors through the inclusion ${\rm Hilb} \hookrightarrow
{\rm SuperHilb}$. Moreover, by Theorem \ref{dhr1} all the objects of
${\rm Spec}(H)$ are isomorphic, so ${\rm Spec}(H)$ is equivalent, as a groupoid,
to the group ${\rm U}(T)$ for any $T \in {\rm Spec}(H)$. We thus
obtain an equivalence of symmetric 2-H*-algebras between
${\rm Rep}({\rm Spec}(H))$ and ${\rm Rep}({\rm U}(T))$ as defined in Theorem \ref{dhr1}.
Using this, the fact that $\tilde T \colon H \to {\rm Rep}({\rm U}(T))$ is
an equivalence translates into the fact that $\hat{\hbox{\hskip
0.5em}}\colon H \to {\rm Rep}({\rm Spec}(H))$ is an equivalence.
Next, suppose that $H$ is even but not connected. Then $H$ is a direct
sum of the even connected symmetric 2-H*-algebras $H_i$ as above, and
${\rm Rep}({\rm Spec}(H))$ is similarly the direct sum of the ${\rm Rep}({\rm Spec}(H_i))$.
Because the categorified Gelfand transform $\hat{\hbox{\hskip
0.5em}}\colon H_i \to {\rm Rep}({\rm Spec}(H_i))$ is an equivalence for all $i$,
$\hat{\hbox{\hskip 0.5em}}\colon H \to {\rm Rep}({\rm Spec}(H))$ is an equivalence.
Finally we treat the general case where $H$ is an arbitrary
symmetric 2-H*-algebra. Note that if $H$ and $K$ are
symmetric 2-H*-algebras, a symmetric 2-H*-algebra homomorphism $F
\colon H \to K$ gives rise to a symmetric 2-H*-algebra
homomorphism $F^\flat \colon H^\flat \to K^\flat$ between their
bosonizations, where $F^\flat$ is the same as $F$ on objects and
morphisms. Note also that $F$ is an equivalence of symmetric
2-H*-algebras if and only if $F^\flat$ is. Thus to show that
$\hat{\hbox{\hskip 0.5em}}\colon H \to {\rm Rep}({\rm Spec}(H))$ is an
equivalence, it suffices to show $\hat{\hbox{\hskip 0.5em}}^\flat
\colon H^\flat \to {\rm Rep}({\rm Spec}(H))^\flat$ is an equivalence.
For this, note that any supergroupoid $G$ has a {\it
bosonization} $G^\flat$, in which the underlying compact groupoid
of $G$ is equipped with the trivial balancing $\beta = 1$. Moreover,
there is a homomorphism of symmetric 2-H*-algebras
\[
\begin{diagram}[{\rm Rep}(G)^\flat]
\node{{\rm Rep}(G)^\flat} \arrow{e,t}{X} \node{{\rm Rep}(G^\flat)}
\end{diagram}
\]
sending any representation $F \in {\rm Rep}(G)^\flat$ to the representation
$X(F) \in {\rm Rep}(G^\flat)$ given by the commutative square
\[
\begin{diagram}[{\rm SuperHilb}]
\node{G^\flat} \arrow{e,t}{X(F)} \arrow{s,l}{I} \node{{\rm SuperHilb}} \\
\node{G} \arrow{e,t}{F} \node{{\rm SuperHilb}} \arrow{n,r}{E}
\end{diagram}
\]
Here $I \colon G^\flat \to G$ is the identity on the underlying
groupoids, while the 2-H*-algebra homomorphism $E \colon {\rm SuperHilb} \to
{\rm SuperHilb}$ maps any super-Hilbert space to the even super-Hilbert space
with the same underlying Hilbert space, and acts as the identity on
morphisms. One may check that $X(F)$ is really a compact supergroupoid
representation. Similarly, given a morphism $\alpha \colon F \Rightarrow F'$ in
${\rm Rep}(G)^\flat$, we define $X(\alpha)$ to be the horizontal composite $I
\circ \alpha \circ E$. In fact, $X$ is an equivalence, for given any
representation $F$ of $G^\flat$ we can turn it back into a
representation of $G$ by equipping each Hilbert space $F(x)$, $x \in G$
with the grading $F(\beta_x)$, where $\beta$ is the balancing of $G$.
Similarly, for any symmetric 2-H*-algebra $H$ there is an equivalence
\[
\begin{diagram}[{\rm Spec}(H)^\flat]
\node{{\rm Spec}(H)^\flat} \arrow{e,t}{Y} \node{{\rm Spec}(H^\flat)}
\end{diagram}
\]
sending any object $T \in {\rm Spec}(H)^\flat$ to the object
$Y(T) \in {\rm Spec}(H^\flat)$ given by the commutative square
\[
\begin{diagram}[{\rm SuperHilb}]
\node{H^\flat} \arrow{e,t}{Y(T)} \arrow{s,l}{I} \node{{\rm SuperHilb}} \\
\node{H} \arrow{e,t}{T} \node{{\rm SuperHilb}} \arrow{n,r}{E}
\end{diagram}
\]
where $I \colon H^\flat \to H$ is the identity on the underlying
2-H*-algebras, while $E$ is given as above.
We thus have equivalences
\[ \begin{diagram}[{\rm Rep}({\rm Spec}(H))^\flat]
\node{{\rm Rep}({\rm Spec}(H))^\flat} \arrow{e,t}{\sim}
\node{{\rm Rep}({\rm Spec}(H)^\flat)} \arrow{e,t}{\sim}
\node{{\rm Rep}({\rm Spec}(H^\flat))}
\end{diagram}
\]
and their composite gives a diagram commuting up to natural isomorphism:
\[ \begin{diagram}[{\rm Rep}({\rm Spec}(H))^\flat]
\node{H^\flat} \arrow{e,t}{\hat{\hbox{\hskip 0.5em}}^\flat}
\arrow{se,b}{\hat{\hbox{\hskip 0.5em}}}
\node{{\rm Rep}({\rm Spec}(H))^\flat} \arrow{s}\\
\node[2]{{\rm Rep}({\rm Spec}(H^\flat))}
\end{diagram}
\]
It follows that $\hat{\hbox{\hskip 0.5em}}^\flat
\colon H^\flat \to {\rm Rep}({\rm Spec}(H))^\flat$ is an equivalence, as was to
be shown. \hbox{\hskip 30em} \hskip 3em \hbox{\BOX} \vskip 2ex
Presumably what Theorem \ref{dhr4} is trying to tell us is
that there are 2-functors ${\rm Rep}$ and ${\rm Spec}$ going both ways
between the 2-category of compact supergroupoids and the 2-category
of symmetric 2-H*-algebras, and that these extend to a 2-equivalence
of 2-categories. We shall not try to prove this here. However, it
is worth noting that for any compact supergroupoid $G$, there is
a functor
\[ \check{\hbox{\hskip 0.5em}} \colon G \to {\rm Spec}({\rm Rep}(G)) \]
given as follows. For every object $x \in G$, $\hat x$ is the
object of ${\rm Spec}({\rm Rep}(G))$ with
\[ \hat x(F) = F(x) \]
for all $F \in {\rm Rep}(G)$, and
\[ \hat x(\alpha) = \alpha_x \]
for all $\alpha \colon F \to F'$, where $F,F' \in {\rm Rep}(G)$. For
every morphism $g \colon x \to y$ in $G$, $\check g \colon \check x \Rightarrow
\check y$ is the natural transformation with
\[ \check g(F) = F(g) \]
for all $F \in {\rm Rep}(G)$. Presumably
$\check{\hbox{\hskip 0.5em}} \colon G \to {\rm Spec}({\rm Rep}(G))$ is in some
sense an equivalence of compact supergroupoids.
\subsection{Compact abelian groups}
The representation theory of compact abelian groups is rendered
especially simple by the use of Fourier analysis, as generalized by
Pontryagin. Suppose that $T$ is a compact abelian group. Then its
dual $\hat T$ is defined as the set of equivalence classes of
irreducible representations $\rho$ of $T$. The dual becomes a
discrete abelian group with operations given as follows:
\[ [\rho][\rho'] = [\rho \otimes \rho'], \]
\[ [\rho]^{-1} = [\rho^\ast] \]
Then the Fourier transform is a unitary isomorphism
\[ f \colon L^2(T) \to L^2(\hat T) \]
given by
\[ f(\chi_\rho) = \delta_{[\rho]} \]
where $\chi_\rho$ is the character of the representation $\rho$,
and $\delta_{[\rho]}$ is the function on $\hat T$ which equals
$1$ at $[\rho]$ and $0$ elsewhere.
The Fourier transform has an interesting categorification. Note that
the ordinary Fourier transform has as its domain the
infinite-dimensional Hilbert space $L^2(T)$, which has a basis given
by the characters of irreducible representations of $T$. The
categorified Fourier transform will have as domain the 2-Hilbert space
${\rm Rep}(T)$, which has a basis given by the irreducible representations
themselves. (Taking the character of a representation is a form of
`decategorification'.) Similarly, just as the ordinary Fourier
transform has as its codomain an infinite-dimensional Hilbert space of
${\Bbb C}$-valued functions on $\hat T$, the categorified Fourier transform
will have as its codomain a 2-Hilbert space of ${\rm Hilb}$-valued
functions on $\hat T$.
More precisely, define ${\rm Hilb}[G]$ for any discrete group $G$ to be the
category whose objects are $G$-graded Hilbert spaces for which the
total dimension is finite, and whose morphisms are linear maps
preserving the grading. Alternatively, we can think of ${\rm Hilb}[G]$ as the
category of hermitian vector bundles over $G$ for which the sum of the
dimensions of the fibers is finite. We may write any object $x \in
{\rm Hilb}[G]$ as a $G$-tuple $\{x(g)\}_{g \in G}$ of Hilbert spaces. The
category ${\rm Hilb}[G]$ becomes a 2-H*-algebra in an obvious way with a
product modelled after the convolution product in the group algebra ${\Bbb C}[G]$:
\[ (x \otimes y)(g) = \bigoplus_{\{g',g'' \in G\,\colon\; g'g'' = g\}}
x(g') \otimes y(g'') .\]
If $G$ is abelian, ${\rm Hilb}[G]$ becomes a symmetric 2-H*-algebra.
Now suppose that $T$ is a compact abelian group. Given any object $x
\in {\rm Rep}(T)$, we may decompose $x$ into subspaces corresponding to the
irreducible representations of $T$:
\[ x = \bigoplus_{g \in \hat T} x(g) . \]
We define the {\it categorified Fourier transform}
\[ F \colon {\rm Rep}(T) \to {\rm Hilb}[\hat T] \]
as follows. For any object $x \in {\rm Rep}(T)$, we set
\[ F(x) = \{x(g)\}_{g \in \hat T} . \]
Moreoever, any morphism $f \colon x \to y$ in ${\rm Rep}(T)$ gives rise to
linear maps $f(g) \colon x(g) \to y(g)$ and thus a morphism
$F(f)$ in ${\rm Hilb}[\hat T]$. One can check that $F$ is not only 2-Hilbert
space morphism but actually a homomorphism of symmetric 2-H*-algebras.
This is the categorified analog of how the ordinary Fourier transform
sends pointwise multiplication to convolution. Note that, in analogy
to the formula
\[ f(\chi_\rho) = \delta_{[\rho]} \]
satisfied by the ordinary Fourier transform, for any irreducible
representation $\rho$ of $T$ the categorified Fourier transform
$F(\rho)$ is a hermitian vector bundle that is 1-dimensional at
$[\rho]$ and 0-dimensional elsewhere.
\begin{thm} \hspace{-0.08in}{\bf .}\hspace{0.1in} If $T$ is a compact abelian group, the
categorified Fourier transform $F \colon {\rm Rep}(T) \to {\rm Hilb}(\hat
T)$ is an equivalence of symmetric 2-H*-algebras. \end{thm}
Proof - There is a homomorphism $G \colon {\rm Hilb}[\hat T] \to
{\rm Rep}(T)$ sending each object $\{x(g)\}_{g \in \hat T}$ in
${\rm Hilb}[\hat T]$ to a representation of $T$ which is a direct sum
of spaces $x(g)$ transforming according to the different
isomorphism classes $g \in \hat T$ of irreducible representations
of $T$. One can check that $FG$ and $GF$ are naturally
isomorphic to the identity. \hskip 3em \hbox{\BOX} \vskip 2ex
\subsection{Compact classical groups} \label{classical}
The representation theory of a `classical' compact Lie group has a
different flavor from that of general compact Lie groups. The
representation theory of general compact Lie groups heavily involves
the notions of maximal torus, Weyl group, roots and weights. We hope
to interpret this theory in terms of 2-Hilbert spaces in a future
paper. However, the representation theory of a classical group can
also be studied using Young diagrams \cite{Weyl}. This approach
relies on the fact that its categories of representations have simple
universal properties. These universal properties can be described in
the language of symmetric 2-H*-algebras, and a description along these
lines represents a distilled version of the Young diagram theory.
For example, consider the group ${\rm U}(n)$. The fundamental
representation of ${\rm U}(n)$ on ${\Bbb C}^n$ is the `universal $n$-dimensional
representation'. In other words, for a group to have a
(unitary) representation on ${\Bbb C}^n$ is precisely for it to have a
homomorphism to ${\rm U}(n)$. This universal property can also be
expressed as a universal property of ${\rm Rep}({\rm U}(n))$. Suppose that $G$
is a compact group. Then any $n$-dimensional representation $y \in
{\rm Rep}(G)$ is isomorphic to a representation of the form $\rho \colon G
\to {\rm U}(n)$. The representation $\rho$ gives rise to a homomorphism
\[ \rho^\ast \colon {\rm Rep}({\rm U}(n)) \to {\rm Rep}(G), \]
and letting $x$ denote the fundamental representation of ${\rm U}(n)$, we have
$\rho^\ast(x) = \rho$. Since $\rho$ and $y$ are isomorphic, there is a
unitary 2-homomorphism from $\rho^\ast$ to a homomorphism
\[ F \colon {\rm Rep}({\rm U}(n)) \to {\rm Rep}(G) \]
with $F(x) = y$.
In short, for any $n$-dimensional object $y \in {\rm Rep}(G)$ there is a
homomorphism $F \colon {\rm Rep}({\rm U}(n)) \to {\rm Rep}(G)$ of symmetric
2-H*-algebras with $F(x) = y$. On the other hand, suppose $F' \colon
{\rm Rep}({\rm U}(n)) \to {\rm Rep}(G)$ is any other homomorphism with $F'(x) = y$. We
claim that there is a unitary 2-homomorphism from $F$ to $F'$. By
Corollary \ref{dhr2}, there exists a homomorphism $\rho' \colon G \to
{\rm U}(n)$ with a unitary 2-homomorphism from $F'$ to $\rho'^\ast$. On the
other hand, by construction there is a unitary 2-homomorphism from $F$
to $\rho^\ast$ for some $\rho \colon G \to {\rm U}(n)$. To show there is a
unitary 2-homomorphism from $F$ to $F'$, it thus suffices to show that
$\rho$ and $\rho'$ are isomorphic in ${\rm Rep}(G)$. This holds because
$\rho \cong y = F'(x) \cong \rho'^\ast(x) = \rho'$.
Now, since any connected even symmetric 2-H*-algebra is unitarily
equivalent to ${\rm Rep}(G)$ for some compact $G$ by Theorem \ref{dhr1}, we
may restate these results as follows. Suppose $H$ is a connected even
symmetric 2-H*-algebra and let $y$ be an $n$-dimensional object of $H$.
Then there exists a homomorphism $F \colon {\rm Rep}({\rm U}(n)) \to H$ with $F(x)
= y$. Moreover, this is unique up to a unitary 2-homomorphism.
Furthermore, we can drop the assumption that $H$ is even by working with
the full subcategory whose objects are all the even objects of $H$.
We may thus state the universal property of ${\rm Rep}({\rm U}(n))$ as
follows:
\begin{thm}\label{un}\hspace{-0.08in}{\bf .}\hspace{0.1in} ${\rm Rep}({\rm U}(n))$ is the free connected symmetric
2-H*-algebra on an even object $x$ of dimension $n$. That is, given any
even $n$-dimensional object $y$ of a connected symmetric 2-H*-algebra
$H$, there exists a homomorphism of symmetric 2-H*-algebras $F \colon
{\rm Rep}(U(n)) \to H$ with $F(x) = y$, and $F$ is unique up to a unitary
2-homomorphism.
\end{thm}
Let $\Lambda^n x$ denote the cokernel of $p_A \colon x^{\otimes n} \to
x^{\otimes n}$ (complete antisymmetrization), and let $S^n x$ denote
the cokernel of $p_S \colon x^{\otimes n} \to x^{\otimes n}$ (complete
symmetrization). We can describe the category of representations of
${\rm SU}(n)$ as follows:
\begin{thm}\label{sun} \hspace{-0.08in}{\bf .}\hspace{0.1in} ${\rm Rep}({\rm SU}(n))$ is the free connected symmetric
2-H*-algebra on an even object $x$ with $\Lambda^n x \cong 1$. \end{thm}
Proof - Suppose that $G$ is a compact group and the object $y \in
{\rm Rep}(G)$ has $\Lambda^n y \cong 1$. It follows that $y$ is
$n$-dimensional by the computation in Proposition \ref{dim}, and the
isomorphism $\Lambda^n y \cong 1$ determines a $G$-invariant volume
form on the representation $y$. Thus $y$ is isomorphic to a
representation of the form $\rho \colon G \to {\rm SU}(n)$. The rest of the
proof follows that of Theorem \ref{un}. \hskip 3em \hbox{\BOX} \vskip 2ex
Here we can see in a simple context how our theory is a distillation
of the theory of Young diagrams. (The Young diagram approach to
representation theory is more familiar for ${\rm SU}(n)$ than for ${\rm U}(n)$.)
In heuristic terms, the above theorem says that every representation of
${\rm SU}(n)$ is generated from the fundamental representation $x$ using
the operations present in a symmetric 2-H*-algebra --- the
$\ast$-structure, direct sums, cokernels, tensor products, duals, and
the symmetry --- with no relations other than those implied by the
axioms for a connected symmetric 2-H*-algebra and the fact that
$x$ is even and $\Lambda^n x \cong 1$. The theory of Young diagrams
makes this explicit by listing the irreducible representations of
${\rm SU}(n)$ in terms of minimal projections $p \colon x^{\otimes x} \to
x^{\otimes k}$, or in other words, Young diagrams with $k$ boxes. The
symmetric 2-H*-algebra of representations of a subgroup $G \subset
{\rm SU}(n)$, such as ${\rm SO}(n)$ or ${\rm Sp}(n)$, is a quotient of
${\rm Rep}({\rm SU}(n))$. We may describe this quotienting process by giving
extra relations as in Theorems \ref{spn} and \ref{son} below. These
extra relations give identities saying that different Young diagrams
correspond to the same representation of $G$.
The classical groups $\O(n)$ and ${\rm Sp}(n)$ are related to the concept
of self-duality. Given adjunctions $(x,x^\ast,i_x,e_x)$ and
$(y,y^\ast,i_y,e_y)$ in a monoidal category $C$, for any morphism $f
\colon x \to y$ there is a morphism $f^\dagger \colon y^\ast \to
x^\ast$, given in the strict case by the composite:
\[ \begin{diagram}[y^\ast \otimes x \otimes x^\ast]
\node{y^\ast = y^\ast \otimes 1} \arrow{e,t}{y^\ast \otimes i_x}
\node{y^\ast \otimes x \otimes x^\ast}
\arrow{e,t}{y^\ast \otimes f \otimes x^\ast}
\node{y^\ast \otimes y \otimes x^\ast} \arrow{e,t}{e_y \otimes x^\ast}
\node{1 \otimes x^\ast = x^\ast}
\end{diagram}
\]
(Our notation here differs from that of HDA0.) Since the left dual of
an object in a 2-Hilbert space is also its right dual as in
Proposition \ref{2H*2}, given a morphism $f \colon x \to x^\ast$ we
obtain another morphism $f^\dagger \colon x \to x^\ast$. Using this we
may describe ${\rm Rep}(\O(n))$ and ${\rm Rep}({\rm Sp}(n))$ as certain `free connected
symmetric 2-H*-algebras on one self-dual object':
\begin{thm}\label{on} \hspace{-0.08in}{\bf .}\hspace{0.1in} ${\rm Rep}(\O(n))$ is the free connected symmetric
2-H*-algebra on an even object $x$ of dimension $n$ with an
isomorphism $f \colon x \to x^\ast$ such that $f^\dagger =
f$. \end{thm}
Proof -
Suppose that $G$ is a compact group and the object $y \in
{\rm Rep}(G)$ is $n$-dimensional and equipped with an isomorphism
$f \colon x \to x^\ast$ with $f^\dagger = f$. Then there is a
nondegenerate pairing $F \colon y \otimes y \to 1$ given by $F = (y
\otimes f)i_y^\ast$. A calculation, given in the proof of Proposition
\ref{selfdual}, shows that $F$ is symmetric. It follows that $y$ is
isomorphic to a representation of the form $\rho \colon G \to \O(n)$.
The rest of the proof follows that of Theorem \ref{un}. \hskip 3em \hbox{\BOX} \vskip 2ex
\begin{thm}\label{spn} \hspace{-0.08in}{\bf .}\hspace{0.1in} ${\rm Rep}({\rm Sp}(n))$ is the free connected symmetric
2-H*-algebra on one even object $x$ with $\Lambda^n x \cong 1$
and with an isomorphism $f \colon x \to x^\ast$ such that $f^\dagger =
-f$. \end{thm}
Proof - The proof is analogous to that of Theorem \ref{on}, except
that the pairing $F$ is skew-symmetric. \hskip 3em \hbox{\BOX} \vskip 2ex
Following the proof of Theorem \ref{sun} we may also characterize
${\rm Rep}({\rm SO}(n))$ as follows:
\begin{thm}\label{son} \hspace{-0.08in}{\bf .}\hspace{0.1in} ${\rm Rep}({\rm SO}(n))$ is the free connected symmetric
2-H*-algebra on an even object $x$ with $\Lambda^n x \cong 1$
and with an isomorphism $f \colon x \to x^\ast$ such that $f^\dagger =
f$. \end{thm}
The conditions on the isomorphism $f \colon x \to x^\ast$ in Theorems
\ref{on} and \ref{spn} are quite reasonable, in the following sense:
\begin{prop}\label{selfdual} \hspace{-0.08in}{\bf .}\hspace{0.1in}
Suppose that $x$ is a simple object in a symmetric
2-H*-algebra and that $x$ is isomorphic to $x^\ast$. Then one and only
one of the following is true: either there is an isomorphism
$f \colon x \to x^\ast$ with $f = f^\dagger$, or there is an isomorphism
$f \colon x \to x^\ast$ with $f = -f^\dagger$. \end{prop}
Proof - Note that there is an isomorphism of complex vector spaces
\begin{eqnarray*} \hom(x,x^\ast) &\cong& \hom(x \otimes x,1) \\
f &\mapsto& (1 \otimes f)i_x^\ast, \end{eqnarray*}
and note that
\[ \hom(x \otimes x,1) \cong
\hom(S^2 x,1) \oplus \hom(\Lambda^2 x,1) . \]
Suppose $f \colon x \to x^\ast$ is an
isomorphism and let $F = (1 \otimes f)i_x^\ast$. Since $x$ is simple,
$f$ and thus $F$ is unique up to a scalar multiple, so $F$
must lie either in $\hom(S^2 x,1)$ or $\hom(\Lambda^2 x,1)$.
In other words, $B_{x,x}F = \pm F$. Choose a well-balanced
adjunction for $x$. Assuming without loss of generality that $H$ is
strict, we have
\begin{eqnarray*} f^\dagger
&=& (x \otimes i_x)(x \otimes f\otimes x^\ast)(i_x^\ast \otimes x^\ast) \\
&=& (x \otimes i_x)(F \otimes x^\ast) \\
&=&\pm (x \otimes i_x)(B_{x,x}F \otimes x^\ast) \\
&=&\pm (f \otimes i_x)(B_{x^\ast,x} \otimes x^\ast)(i_x^\ast \otimes x^\ast)\\
&=& \pm f b_{x^\ast} \end{eqnarray*}
Since $b_{x^\ast} = \pm 1_{x^\ast}$ depending on whether $x$,
and thus $x^\ast$, is even or odd, we have $f^\dagger = \pm f$. \hskip 3em \hbox{\BOX} \vskip 2ex
\noindent This result is well-known if $H$ is a category of compact
group representations \cite{FH}. Here one may also think of the morphism
$f \colon x \to x^\ast$ as a conjugate-linear intertwining operator
$j \colon x \to x$. The condition that $f = \pm f^\dagger$ is then
equivalent to the condition that $j^2 = \pm 1_x$.
One says that $x$ is a real representation if $j^2 = 1_x$ and
a quaternionic representation if $j^2 = -1_x$, establishing the
useful correspondence:
\[ {\rm real:complex:quaternionic::orthogonal:unitary:symplectic} \]
The following alternate characterization of ${\rm Rep}({\rm U}(1))$ is
interesting because it emphasizes the relation between duals and
inverses. Whenever $T$ is a compact abelian group and $x \in
{\rm Rep}(T)$, the dual $x^\ast$ is also the inverse of $x$, in the sense
that $x \otimes x^\ast \cong 1$. We have:
\begin{thm}\hspace{-0.08in}{\bf .}\hspace{0.1in} ${\rm Rep}({\rm U}(1))$ is the free connected symmetric 2-H*-algebra
on an even object $x$ with $x \otimes x^\ast \cong 1$. \end{thm}
Proof - By Theorem \ref{un} it suffices to show that an object $x$ in a
connected symmetric 2-H*-algebra is 1-dimensional if and only if $x
\otimes x^\ast \cong 1$. On the one hand, by the multiplicativity of
dimension, $x \otimes x^\ast \cong 1$ implies that $\dim(x) = 1$. On the
other hand, suppose $\dim(x) = 1$. Then we claim $i_x \colon 1 \to x
\otimes x^\ast$ and $i_x^\ast \colon x \otimes x^\ast \to 1$ are inverses.
First, ${i_x}^\ast i_x$ is the identity since $\dim(x) = 1$. Second,
${i_x}^\ast i_x \in {\rm end}(x \otimes x^\ast)$ is idempotent since $\dim(x) =
1$. Since $x \otimes x^\ast$ is 1-dimensional, it is simple (by the
additivity of dimension), so ${i_x}^\ast i_x$ must be the identity. \hskip 3em \hbox{\BOX} \vskip 2ex
Finally, it is interesting to note that ${\rm SuperHilb}$ is the free connected
symmetric 2-H*-algebra on an odd object $x$ with $x \otimes x \cong 1$.
This object $x$ is the one-dimensional odd super-Hilbert space.
\section{Conclusions}
The reader will have noted that some of our results are slight reworkings of
those in the literature. One advantage of our approach is that it
immediately suggests generalizations to arbitrary $n$. While the
general study of $n$-Hilbert spaces will require a deeper
understanding of $n$-category theory, we expect many of the same
themes to be of interest. With this in mind, let us point out some
problems with what we have done so far.
One problem concerns the definition of the quantum-theoretic hierarchy.
A monoid is a essentially a category with one object. More precisely, a
category with one object $x$ can be reconstructed from the monoid
${\rm end}(x)$, and up to isomorphism every monoid comes from a one-object
category in this way. Comparing Figures 1 and 2, one might at first
hope that by analogy an H*-algebra would be a one-dimensional 2-Hilbert
space. Unfortunately, the way we have set things up, this is not the case.
If $H$ is a one-dimensional 2-Hilbert space with basis given by
the object $x$, then ${\rm end}(x)$ is an H*-algebra. However,
${\rm end}(x)$ is always isomorphic to ${\Bbb C}$; one does not get any
other H*-algebras this way. The reason appears to be the
requirement that a 2-Hilbert space has cokernels, so that if
${\rm end}(x)$ has nontrivial idempotents, $x$ has subobjects. If we
dropped this clause in the definition of a 2-Hilbert space, there
would be a correspondence between H*-algebras and 2-Hilbert
spaces, all of whose objects are direct sums of a single object
$x$. Perhaps in the long run it will be worthwhile to modify the
definition of 2-Hilbert space in this way. On the other hand,
an H*-algebra is also an H*-category with one object. An
H*-category has sums and differences of morphisms, but not of
objects, i.e., it need not have direct sums and cokernels.
Perhaps, therefore, a $k$-tuply monoidal $n$-Hilbert space should
really be some sort of `$(n+k)$-H*-category' with one
$j$-morphism for $j < k$, and sums and differences of
$j$-morphisms for $j \ge k$.
A second problem concerns the program of getting an invariant of
$n$-tangles in $(n+k)$-dimensions from an object in a $k$-tuply monoidal
$n$-Hilbert space. Let us recall what is known so far here.
Oriented tangles in 2 dimensions are the morphisms in a monoidal
category with duals, $C_{1,1}$. Here by `monoidal category with duals'
we mean a monoidal $\ast$-category in which every object has a left
dual, the tensor product is a $\ast$-functor, and the associator is a
unitary natural transformation. Suppose that $X$ is any other monoidal
category with duals, e.g.\ a 2-H*-algebra. Then any adjunction
$(x,x^\ast,i,e)$ in $C$ uniquely determines a monoidal $\ast$-functor $F
\colon C_{1,1} \to H$ up to monoidal unitary natural isomorphism. The
functor $F$ is determined by the requirement that it maps the positively
oriented point to $x$, the negatively oriented point to $x^\ast$, and
the appropriately oriented `cup' and `cap' tangles to $e$ and $i$.
According to our philosophy we would prefer $F$ to be determined by an
object $x \in X$ rather than an adjunction. However $F$ is not
determined up to natural transformation by requiring that it map the
positively oriented point to $x$. For example, take $X = {\rm Hilb}$ and let $x
\in X$ be any object. We may let $F$ send the negatively oriented point
to the dual Hilbert space $x^\ast$, and send the cap and cup to the
standard linear maps $e \colon x^\ast \otimes x \to {\Bbb C}$ and $i \colon {\Bbb C}
\to x \otimes x^\ast$. Then $F(ii^\ast) = \dim(x)1_x$. Alternatively
we may let $F$ send the cap and cup to $e' = \alpha^{-1} e$ and $i' =
\alpha i$ for any nonzero $\alpha \in {\Bbb C}$. Then $F(ii^\ast) =
|\alpha|^2 \dim(x)1_x$. The problem is that while adjunctions in $X$
are unique up to unique isomorphism, the isomorphism is not
necessarily unitary.
In HDA0 we outlined a way to deal with this problem by
`strictifying' the notion of a monoidal category with duals.
Roughly speaking, this amounts to equipping each object with a
choice of left adjunction, and requiring the functor $F \colon
C_{1,1} \to X$ to preserve this choice. Then $F$ is determined
up to monoidal unitary natural transformation by the requirement
that it map the positively oriented point to a particular object
$x \in X$. In this paper we have attempted to take the `weak'
rather than the `strict' approach. Our point here is that the
weak approach seems to make it more difficult to formulate the
sense in which $C_{1,1}$ is the `free' monoidal category with
duals on one object.
In higher dimensions the balancing plays an interesting role in this
issue. Framed oriented tangles in 3 dimensions form a braided monoidal
category $C_{1,2}$ with duals. Here by `braided monoidal category with
duals' we mean a monoidal category with duals which is also braided,
such that the braiding is unitary and every object $x$ has a
well-balanced adjunction $(x,x^\ast,i,e)$. For any object $x$ in a
braided monoidal category $X$ with duals, there is a braided monoidal
$\ast$-functor $F \colon C_{1,2} \to X$ sending the positively oriented
point to $x$. Moreover, because well-balanced adjunctions are unique
up to unique {\it unitary} isomorphism, $F$ is unique up to monoidal
unitary natural isomorphism. This gives a sense in which $C_{1,2}$
is the free braided monoidal category with duals on one object.
Similarly, framed oriented tangles in 4 dimensions form a symmetric
monoidal category with duals $C_{1,3}$, i.e., a braided monoidal
category with duals for which the braiding is a symmetry. Again, for
any object $x$ in a symmetric monoidal category $X$ with duals, there is
a symmetric monoidal $\ast$-functor $F \colon C_{1,3} \to X$ sending the
positively oriented point to $x$, and $F$ is unique up to monoidal
unitary natural isomorphism. (For an alternative `strict' approach to
the 3- and 4-dimensional cases, see HDA0.)
In short, we need to understand the notion of $k$-tuply monoidal
$n$-Hilbert spaces more deeply, as well as the notion of `free'
$k$-tuply monoidal $n$-categories with duals.
\subsection*{Acknowledgements}
Many of the basic ideas behind this paper were developed in collaboration
with James Dolan. I would also like to thank Louis Crane, Martin Hyland,
Martin Neuchl, John Power, Stephen Sawin, Gavin Wraith, and David Yetter for
helpful conversations and correspondence. I am grateful to
the Erwin Schr\"odinger Institute and the physics department of
Imperial College for their hospitality while part of this work was
being done.
| proofpile-arXiv_065-477 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
Matter containing a large number of strange quarks may have a lower energy
per baryon than ordinary nuclei and be absolutely stable\cite{1,2}.
This intriguing possibility has generated a great deal of interest across
many subfields of physics. A crucial implicit assumption for the stability
of this new kind of matter, called strange matter, is that the baryon number
$B$ is exactly conserved. Of course, we know nuclei are stable against
$\Delta B \neq 0$ decays. The best mode-independent experimental lower
limit to date is $1.6 \times 10^{25}$ years\cite{3}. However, there may be
effective $\Delta B \neq 0$ interactions which are highly suppressed for
nuclei but not for strange matter. We examine such a hypothesis here and
show that whereas such exotic decays are possible, the relevant lifetime
should be longer than $10^5$ years.
In Sec.~2 we review briefly the present experimental constraints on
$\Delta B \neq 0$ interactions. These come mainly from the nonobservation
of proton decay and of neutron-antineutron oscillation. We then point out
the possible consequences of an effective $\Delta B = 2$, $\Delta N_s = 4$
operator (where $N_s$ denotes the number of $s$ quarks) on the stability of
strange matter. We proceed to discuss some possible theoretical origins of
such an operator in Sec.~3 and Sec.~4. We deal first with the supersymmetric
standard model with $R$-parity nonconserving terms of the form
$\lambda_{ijk} u_i^c d_j^c d_k^c$ and show that their effects are
negligible. We then discuss two possible extended models where the
effects may be large. In Sec.~5 we consider this
operator in conjunction with the usual weak interaction and show that
the suppression coming from the stability of ordinary nuclei translates
to a phenomenological lower limit of $10^5$ years on the lifetime of
strange matter. Finally in Sec.~6, there are some concluding remarks.
\section{Baryon-Number Nonconserving Interactions}
Given the standard $SU(3) \times SU(2) \times U(1)$ gauge symmetry and the
usual quarks and leptons, it is well-known that
the resulting renormalizable Lagrangian conserves both the
baryon number $B$ and the lepton number $L$ automatically.
However, new physics at higher energy scales may induce effective
interactions (of dimension 5 or above) which do not respect these
conservation laws. The foremost example is the possibility of proton
decay. For this to happen, there has to be at least one fermion with a
mass below that of the proton. Since only leptons are known to have this
property, the selection rule $\Delta B = 1$, $\Delta L = 1$ is applicable.
The most studied decay mode experimentally is $p \rightarrow \pi^0 e^+$,
which requires the effective interaction
\begin{equation}
{\cal H}_{int} \sim {1 \over M^2} (uude).
\end{equation}
It is now known\cite{4} that $\tau_{exp} (p \rightarrow \pi^0 e^+) > 9 \times
10^{32}$ years, hence $M > 10^{16}$ GeV. [Here $M$ should be considered
an effective mass, because in some scenarios the denominator of the
right-hand side of Eq.~(1) is really the product of two different masses.]
This means that proton decay probes physics at the grand-unification
energy scale.
The next simplest class of effective interactions has the selection rule
$\Delta B = 2$, $\Delta L = 0$. The most well-known example is of course
\begin{equation}
{\cal H}_{int} \sim {1 \over M^5} (udd)^2,
\end{equation}
which induces neutron-antineutron oscillation and allows a nucleus to decay
by the annihilation of two of its nucleons. Note that since 6 fermions are
now involved, the effective operator is of dimension 9, hence $M$ appears to
the power $-5$. This means that for the same level of nuclear stability,
the lower bound on $M$ will be only of order $10^5$ GeV. The present best
experimental lower limit on the $n - \bar n$ oscillation lifetime is\cite{5}
$8.6 \times 10^7$s. Direct search for $N N \rightarrow \pi \pi$ decay in
iron yields\cite{6} a limit of $6.8 \times 10^{30}$y. Although the above
two numbers differ by 30 orders of magnitude, it is well established by
general arguments as well as detailed nuclear model calculations that\cite{7}
\begin{equation}
\tau (n \bar n) = 8.6 \times 10^7{\rm s} ~ \Rightarrow ~ \tau (N N \rightarrow
\pi \pi) \sim 2 \times 10^{31}{\rm y}.
\end{equation}
Hence the two limits are comparable. To estimate the magnitude of $M$, we
use
\begin{equation}
\tau(n \bar n) = M^5 |\psi(0)|^{-4},
\end{equation}
where the effective wavefunction at the origin is roughly given by\cite{8}
\begin{equation}
|\psi(0)|^{-2} \sim \pi R^3, ~~~ R \sim 1 ~ {\rm fm}.
\end{equation}
Hence we obtain $M > 2.4 \times 10^5$ GeV. Bearing in mind that
${\cal H}_{int}$ is likely to be suppressed also by products of couplings
less than unity, this means that the stability of nuclei is sensitive to
new physics at an energy scale not too far above the electroweak scale of
$10^2$ GeV. It may thus have the hope of future direct experimental
exploration.
Consider next the effective interaction
\begin{equation}
{\cal H}_{int} \sim {1 \over M^5} (uds)^2.
\end{equation}
This has the selection rule $\Delta B = 2$, $\Delta N_s = 2 ~(\Delta S = -2)$.
In this case, a nucleus may decay by the process $N N \rightarrow
K K$\cite{9,10}. Although such decay modes have never been observed, the
mode-independent stability lifetime of $1.6 \times 10^{25}$y mentioned
already is enough to guarantee that it is very small. However, consider now
\begin{equation}
{\cal H}_{int} \sim {1 \over M^5} (uss)^2,
\end{equation}
which has the selection rule
\begin{equation}
\Delta B = 2, ~~~ \Delta N_s = 4.
\end{equation}
To get rid of four units of strangeness, the nucleus must now convert two
nucleons into four kaons, but that is kinematically impossible. Hence the
severe constraint from the stability of nuclei does not seem to apply here,
and the following interesting possibility may occur.
Strangelets ({\it i.e.} stable and metastable configurations of quarks
with large strangeness content) with atomic number $A$ (which is of course
the same as baryon number $B$) and number of strange quarks $N_s$ may now
decay into other strangelets with two less units of $A$ and one to three less
$s$ quarks. For example,
\begin{eqnarray}
(A, N_s) &\rightarrow& (A - 2, N_s - 2) + K K, \\
(A, N_s) &\rightarrow& (A - 2, N_s - 1) + K K K, ~etc.
\end{eqnarray}
For the states of lowest energy, model calculations show\cite{11} that $N_s
\sim 0.8 A$, hence the above decay modes are efficient ways of reducing all
would-be stable strangelets to those of the smallest $A$.
Unlike nuclei which are most stable for $A$ near that of iron, the energy
per baryon number of strangelets decreases with increasing $A$. This has
led to the intriguing speculation that there are stable macroscopic lumps
of strange matter in the Universe. On the other hand, it is not clear how
such matter would form, because there are no stable building blocks such
as hydrogen and helium which are essential for the formation of heavy nuclei.
In any case, the above exotic interaction would allow strange matter to
dissipate into smaller and smaller units, until $A$ becomes too small for
the strangelet itself to be stable. A sample calculation by Madsen\cite{11}
shows that
\begin{equation}
m_0 (A) - m_0 (A - 2) \simeq (1704 + 111 A^{-1/3} + 161 A^{-2/3})~{\rm MeV},
\end{equation}
assuming $m_s = 100$ MeV, and the bag factor $B^{1/4} = 145$ MeV. Hence the
$KK$ and $KKK$ decays would continue until the ground-state mass $m_0$
exceeds the condition that the energy per baryon number is less than 930
MeV, which happens at around $A = 13$.
\section{Supersymmetric Standard Model}
If the standard model of quarks and leptons is extended to include
supersymmetry, the $\Delta B \neq 0$ terms $\lambda_{ijk} u_i^c d_j^c d_k^c$
are allowed in the superpotential. In the above notation, all chiral
superfields are assumed to be left-handed, hence $q^c$ denotes the left-handed
charge-conjugated quark, or equivalently the right-handed quark, and the
subscripts refer to families, {\it i.e.} $u_i$ for $(u, c, t)$ and $d_j$ for
$(d, s, b)$. Since all quarks are color triplets and the interaction must
be a singlet which is antisymmetric in color, the two $d$ quark superfields
must belong to different families. In the minimal supersymmetric standard
model, these terms are forbidden by the imposition of $R$-parity. However,
this assumption is not mandatory and there is a vast body of recent
literature exploring the consequences of $R$-parity nonconservation\cite{12}.
Starting with Yukawa terms of the form $\lambda_{ijk} u_i^c d_j^c d_k^c$,
where two of the fields are quarks and the third is a scalar quark, we can
obtain the effective interaction of Eq.~(7) in several ways\cite{13}. In
Ref.~[10] the $d d \rightarrow \tilde b \tilde b$ box diagram (where
$\tilde q$ denotes the supersymmetric scalar partner of $q$) is considered to
obtain the effective interaction of Eq.~(2) for $n - \bar n$ oscillation
using $\lambda_{udb}$. Here we take
\begin{equation}
s s \rightarrow \tilde b \tilde b
\end{equation}
and use $\lambda_{usb}$ instead. The box diagram involves the exchange of
the $(u,c,t)$ quarks, the $W$ boson, and their supersymmetric partners.
Since only gauge couplings appear, the Glashow-Iliopoulos-Maiani (GIM)
mechanism\cite{14} is operative and this diagram vanishes if the
$(\tilde u,\tilde c,\tilde t)$ scalar quarks have the same mass and that
the $(u,c,t)$ quark masses can be neglected. Thus in Fig.~3 of Ref.~[10],
there is a peak at $M_{\tilde t} = 200$ GeV. However, there are actually
two scalar quarks $(\tilde q_L,\tilde q_R)$ for each quark $q$. Consider
just $t$ and $(\tilde t_L,\tilde t_R)$. The $\tilde t$ mass matrix is
given by
\begin{equation}
{\cal M}^2_{\tilde t} = \left[ \begin{array} {c@{\quad}c} \tilde m_L^2 +
m_t^2 & A m_t \\ A m_t & \tilde m_R^2 + m_t^2 \end{array} \right].
\end{equation}
Since only $\tilde t_L$ couples to $b_L$ through the gauge fermion $\tilde w$
and $\tilde t_L$ is not a mass eigenstate, its approximate effective
contribution is given by
\begin{equation}
M_{\tilde t_L}^{-2} = {{\tilde m_R^2 + m_t^2} \over {(\tilde m_L^2 + m_t^2)
(\tilde m_R^2 + m_t^2) - A^2 m_t^2}}.
\end{equation}
Unless this somehow cancels the $\tilde u$ and $\tilde c$ contributions
accidentally, the box diagram will not vanish. Furthermore, it is often
assumed that the soft supersymmetry-breaking terms $\tilde m_L^2$ and
$\tilde m_R^2$ are universal, in which case this amplitude would be zero
if the $(u,c,t)$ quark masses were neglected. Rewriting Eq.~(7) more
specifically as
\begin{equation}
{\cal H}_{int} = T_1 \epsilon_{\alpha \beta \gamma} \epsilon_{\alpha'
\beta' \gamma'} u_{R \alpha} s_{R \beta} s_{L \gamma} u_{R \alpha'}
s_{R \beta'} s_{L \gamma'},
\end{equation}
where the color indices and chiralities of the quarks are noted,
the contribution of Eq.~(12) is then given by\cite{15}
\begin{equation}
T_1 \simeq {{g^4 \lambda^2_{usb} A^2 m_b^2 m_{\tilde w}} \over {32 \pi^2
\tilde m^8}} V_{ts}^2 F(m^2_{\tilde w}, M_W^2, \tilde m^2, m_t, A),
\end{equation}
where we have assumed universal soft supersymmetry-breaking scalar masses,
$V_{ts}$ is the quark-mixing entry for $t$ to $s$ through the $W$ boson, and
\begin{eqnarray}
F &=& {1 \over 2} J(m^2_{\tilde w}, M_W^2, \tilde m^2 + m_t^2 + A m_t, m_t^2)
+ {1 \over 2} J(m^2_{\tilde w}, M_W^2, \tilde m^2 + m_t^2 - A m_t, m_t^2)
\nonumber \\ &-& J(m^2_{\tilde w}, M_W^2, \tilde m^2, m_t^2) - \{ m_t^2
\rightarrow 0 ~{\rm in~the~last~entry~of~each~term} \},
\end{eqnarray}
with the function $J$ given by\cite{10,13}
\begin{equation}
J(a_1, a_2, a_3, a_4) = \sum_{i=1}^4 {{a_i^2 \ln (a_i)} \over {\prod_{k \neq i}
(a_i - a_k)}}.
\end{equation}
Using the correspondence of $n - \bar n$ oscillation to $N N$ annihilation
inside a nucleus, we estimate the lifetime of strangelets from the above
effective interaction assuming $A = 200$ GeV, $\lambda_{usb} < 1$, and
$\tilde m > 200$ GeV, to be
\begin{equation}
\tau > 10^{22}{\rm y}.
\end{equation}
This tells us that such contributions are negligible from the supersymmetric
standard model.
Recently, another contribution to Eq.~(2) has been identified\cite{13}
involving $\lambda_{tds}$ and $\lambda_{tdb}$ without the GIM suppression
of Eq.~(16). The form of its contribution to Eq.~(7) is
\begin{equation}
{\cal H}_{int} = T_2 \epsilon_{\alpha \beta \gamma} \epsilon_{\alpha'
\beta' \gamma'} u_{L \alpha} s_{L \beta} s_{R \gamma} u_{L \alpha'}
s_{L \beta'} s_{R \gamma'},
\end{equation}
which is not the same as Eq.~(15). The important coupling is now
$\lambda_{tsb}$, but $T_2$ is also suppressed relative to $T_1$ by
$V_{ub}^2$, hence this contribution to Eq.~(7) is even more negligible.
\section{Extended Models Involving Strangeness}
Although the supersymmetric standard model cannot have a sizable contribution
to the $\Delta B = 2$, $\Delta N_s = 4$ operator of Eq.~(7), an extended model
including additional particles belonging to the fundamental {\bf 27}
representation of $E_6$, inspired by superstring theory\cite{16}, may do
better. Consider a slight variation of the model of exotic baryon-number
nonconservation proposed some years ago\cite{17}. In addition to the
usual quark and lepton superfields
\begin{equation}
Q \sim (3, 2, 1/6), ~~~ u^c \sim (\bar 3, 1, -2/3), ~~~ d^c \sim (\bar 3,
1, 1/3);
\end{equation}
\begin{equation}
L \sim (1, 2, -1/2), ~~~ e^c \sim (1, 1, 1), ~~~ N^c \sim (1, 1, 0);
\end{equation}
we have
\begin{equation}
h \sim (3, 1, -1/3), ~~~ h^c \sim (\bar 3, 1, 1/3);
\end{equation}
\begin{equation}
E \sim (1, 2, -1/2), ~~~ \bar E \sim (1, 2, 1/2), ~~~ S \sim (1, 1, 0);
\end{equation}
each transforming under the standard $SU(3) \times SU(2) \times U(1)$ gauge
group as indicated. Imposing the discrete symmetry $Z_2 \times Z_2$ under
which
\begin{eqnarray}
Q, u^c, d^c, N^c &\sim& (+, -), \\ L, e^c &\sim& (-, +), \\ h, h^c, E, \bar E,
S &\sim& (+, +),
\end{eqnarray}
we get the allowed terms
\begin{equation}
Q Q h, ~~ u^c d^c h^c, ~~ d^c h N^c, ~~ N^c N^c,
\end{equation}
in the superpotential, but not $d^c h$. We also assume that $\tilde N^c$
does not develop a vacuum expectation value, so that $B$ is broken
explicitly by the above $N^c N^c$ term only. [Note that without this last
term, we can assign $B = -2/3, 2/3, 1$ to $h, h^c, N^c$ respectively and
$B$ would be conserved.] This model allows us to obtain an effective
$(uss)^2$ operator without going through a loop, as shown in Fig.~1.
Using the formalism of Ref.~[10] for the process
\begin{equation}
(A, N_s) \rightarrow (A-2, N_s-2) + KK,
\end{equation}
we estimate the lifetime to be
\begin{equation}
\tau \sim {{32 \pi m_N^2} \over {9 \rho_N}} \left[ {M \over \tilde \Lambda}
\right]^{10} \sim 1.2 \times 10^{-28} \left[ {M \over \tilde \Lambda}
\right]^{10} {\rm y},
\end{equation}
where $m_N \sim 1$ GeV, $\rho_N \sim 0.25$ fm$^{-3}$ is the nuclear density,
and $\tilde \Lambda \sim 0.3$ GeV is the effective interaction energy scale
corresponding to Eq.~(5). If we assume $M = 1$ TeV, then
\begin{equation}
\tau \sim 2 \times 10^7 {\rm y}.
\end{equation}
In the above, we have assumed that $\lambda_{ush}$ is unconstrained
phenomenologically. However, consider the term $Q_1 Q_2 h$
where $Q_1 = (u, d')$ and $Q_2 = (c, s')$ so that $Q_1 Q_2 = u s' - c d'$.
Hence from this term alone, $\lambda_{udh} = (V_{cd}/V_{cs}) \lambda_{ush}$.
Since $\lambda_{udh}$ may now combine with $\lambda_{s^c h N^c}$ to induce
$N N \rightarrow K K$ decays, its magnitude is seriously suppressed and
$\lambda_{ush}$ would be too small for Eq.~(31) to be valid. However, there
is also a third generation, allowing us the $Q_1 Q_3 = u b' - t d'$ term
which may then be fine-tuned to eliminate the $\lambda_{udh}$ component, thus
saving Eq.~(31). This is of course not a very natural solution, but cannot
otherwise be ruled out.
We may also consider the following tailor-made extension. Let there be a
new exotic scalar multiplet $\tilde Q_6 \sim (\bar 6, 1, 2/3)$ with
$B = -2/3$ and a new exotic fermion multiplet $\psi_8 \sim (8, 1, 0)$
with $B = 1$. Assume the existence of Yukawa terms $s^c s^c \tilde Q_6^*$,
$u^c \psi_8 \tilde Q_6$, and the $B$-nonconserving Majorana mass term
$\psi_8 \psi_8$, then an effective $(uss)^2$ term is possible, as shown in
Fig.~2. Note that the $uss$ combination is now a color octet. This may
in fact be a more efficient way to dissipate strange matter which is
presumably not clumped into color-singlet constituents as in ordinary nuclei.
\section{Stability Limit of Strange Matter}
Since $\tau$ depends on $M/\tilde \Lambda$ to the power 10 in Eq.~(30), it
appears that a much shorter lifetime than that of Eq.~(31) is theoretically
possible. However, there is a crucial phenomenological constraint which
we have yet to consider. Although two nucleons cannot annihilate inside
a nucleus to produce four kaons, they can make three kaons plus a pion.
The effective $(uss)^2$ operator must now be supplemented by a weak
transition $s \rightarrow u + d + \bar u$. We
can compare the effect of this on $N N \rightarrow K K K \pi$ versus that
of the $(uds)^2$ operator on $N N \rightarrow K K$ discussed in Ref.~[10].
First let us look at the phase-space difference. Consider a nucleus with
atomic number $A$ decaying into one with atomic number $A-2$ plus two kaons
with energy-momentum conservation given by $p = p' + k_1 + k_2$. The decay
rate is proportional to
\begin{eqnarray}
F_1 &=& {1 \over {(2 \pi)^5}} \int {{d^3 k_1} \over {2 E_1}} \int {{d^3 k_2}
\over {2 E_2}} \int {{d^3 p'} \over {2 E'}} \delta^4 (p - p' - k_1 - k_2)
\nonumber \\ &\simeq& {1 \over {2 M'}} {1 \over {(2 \pi)^5}} \int {{d^3 k_1}
\over {2 E_1}} \int {{d^3 k_2} \over {2 E_2}} \delta (M - M' - E_1 - E_2)
\nonumber \\ &\simeq& {1 \over {2 M'}} {1 \over {(2 \pi)^3}}
\int_{m_K}^{E_{max}} k_1 k_2 dE_1,
\end{eqnarray}
where $k_1 = (E_1^2 - m_K^2)^{1/2}$, $k_2 = [(2 m_N - E_1)^2 - m_K^2]^{1/2}$,
and $E_{max} = 2 m_N - m_K$.
\newpage
Consider next $A \rightarrow (A-2) + K K K \pi$
with $p = p' + k_1 + k_2 + k_3 + k_\pi$. Because of the limited phase
space, we assume the kaons to be nonrelativistic. Hence
\begin{eqnarray}
F_2 &\simeq& {1 \over {2 M'}} {1 \over {(2 \pi)^{11}}} \int {{d^3 k_1} \over
{2 m_K}} \int {{d^3 k_2} \over {2 m_K}} \int {{d^3 k_3} \over {2 m_K}} \int
{{d^3 k_\pi} \over {2 E_\pi}} \delta (M - M'- 3 m_K - {{k_1^2 + k_2^2 +
k_3^2} \over {2 m_K}} - E_\pi) \nonumber \\ &\simeq& {1 \over {2 M'}}
{1 \over {(2 \pi)^7}} {1 \over m_K^3} \int k_\pi k_1^2 k_2^2 k_3^2 dk_1 dk_2
dk_3,
\end{eqnarray}
where $k_\pi = \{[2m_N - 3m_K - (k_1^2 + k_2^2 + k_3^2)/2m_k]^2 - m_\pi^2
\}^{1/2}$. The above integral can be evaluated by treating $k_{1,2,3}$ as
Cartesian coordinates and then convert them to three-dimensional polar
coordinates. The angular integration over the $k_{1,2,3} > 0$ octant
yields a factor of $\pi/210$ and we get
\begin{equation}
F_2 \simeq {1 \over {2 M'}} {1 \over {(2 \pi)^7}} {1 \over m_K^3} {\pi \over
{210}} \int_0^{k_{max}} k_\pi k^8 dk,
\end{equation}
where $k_{max} = [2 m_K (2 m_N - 3 m_K - m_\pi)]^{1/2}$. The effective
interaction here also differs from that of Eq.~(32) by the appearance of
a third kaon and an extra pion which can be thought of as having been
converted by the weak interaction from a fourth kaon. We thus estimate the
suppression factor to be
\begin{equation}
\xi \sim \left[ {{f_K^2 T_{K \pi}} \over {\tilde \Lambda^6}} \right]^2
{F_2 \over F_1} \sim 10^{-20},
\end{equation}
where $f_K = 160$ MeV, and $T_{K \pi} = 0.07$ MeV$^2$ is obtained using the
symmetric soft-pion reduction\cite{18} of the experimental
$K \rightarrow 2 \pi$ amplitude. We have again invoked the effective
interaction energy scale $\tilde \Lambda = 0.3$ GeV used in Eq.~(30).
Obviously, our estimate depends very sensitively on this parameter, but
that is not untypical of many calculations in nuclear physics. Since the
stability of nuclei is at least $1.6 \times 10^{25}$ years, the reduction
by $\xi$ of the above yields
\begin{equation}
\tau > 10^5 {\rm y}
\end{equation}
for the lifetime of strangelets against $\Delta B = 2$, $\Delta N_s = 4$
decays.
\section{Concluding Remarks}
We have pointed out in this paper that an effective $(uss)^2$ interaction
may cause stable strange matter to decay, but the lifetime is constrained
phenomenologically by the stability of nuclei against decays of the type
$A \rightarrow (A-2) + K K K \pi$ and we estimate it to have a lower limit
of $10^5$ years. This result has no bearing on whether stable or metastable
strangelets can be created and observed in the laboratory, but may be
important for understanding whether there is stable strange bulk matter
left in the Universe after the Big Bang and how it should be searched for.
For example, instead of the usual radioactivity of unstable nuclei, strange
matter may be long-lived kaon and pion emitters. The propagation of these
kaons and pions through the matter itself and their interactions
within such an environment are further areas of possible study. Since
energy is released in each such decay, although the amount is rather
small, it may be sufficient to cause a chain reaction and break up the
bulk matter in a cosmological time scale. Furthermore, if the particles
mediating this effective interaction have masses of order 1 TeV as
discussed, then forthcoming future high-energy accelerators such as the
Large Hadron Collider (LHC) at the European Center for Nuclear Research
(CERN) will have a chance of confirming or refuting their existence.
\vspace{0.3in}
\begin{center} {ACKNOWLEDGEMENT}
\end{center}
This work was supported in part by the U. S. Department of Energy under
Grant No. DE-FG03-94ER40837.
\newpage
\bibliographystyle{unsrt}
| proofpile-arXiv_065-478 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section*{Note Added:}
\noindent After this letter was accepted for publication, we became aware of
the work of Girvin and MacDonald \cite{girvin}, where they
showed that the gauge-transformed Laughlin
wave-function [ eq. (7) of their paper] shows off-diagonal long-range order.
It then immediately follows that the Calogero-Sutherland ground state
wave-function in two dimensions as given by \eq{grst} [which is identical to
eq. (7) of \cite{girvin}] also exhibits off-diagonal long-range order.
| proofpile-arXiv_065-479 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section*{Introduction}
Random sequential adsorption (RSA) \cite{Evans-93} is an irreversible process
which particles are deposited randomly and consecutively on a surface.
The depositing particles, represented by hard-core extended objects,
satisfy the excluded volume condition where they are not allowed to overlap.
The exclusion of certain regions
for further deposition attempts due to the adsorbed particles leads
to a dominant infinite-memory correlation effect where the system
approaches partially covered, fully blocked stage at large times.
However, this picture is altered when the diffusional relaxation
is introduced \cite{Privman-Nielaba-92,Nielaba-Privman-92,%
Wang-group-93}.
Privman and Nielaba \cite{Privman-Nielaba-92} have shown
that the effect of added diffusional relaxation in the deposition of
dimer on a 1D lattice substrate is to allow the full, saturation coverage
via a $\sim t^{-1/2}$ power law at large times, preceded
by a mean-field crossover regime with the
intermediate $\sim t^{-1}$ behavior for fast diffusion.
Series expansion is one of the powerful analytical methods in the
RSA studies \cite{Baram-Kutasov-89,Dickman-Wang-Jensen-91,%
Oliveira-group-92,Bonnier-group-93,%
Baram-Fixman-95,Gan-Wang-96}. Long series in powers
of time $t$ have been obtained,
reminiscent of series expansions in equilibrium statistical mechanics,
by using a computer \cite{Martin-74}.
Recently, the authors \cite{Gan-Wang-96}
have proposed an efficient algorithm for generating
long series for the coverage $\theta$
in powers of time $t$ based on the hierarchical rate equations.
The present work is to study the time-dependent quantity
$\theta$ for one-dimensional models of RSA with
diffusional relaxation, both analytically and numerically.
It will be seen that even though relatively
long series have been obtained, we are still unable to extract
the kinetics of the systems at large times for general $\gamma$
due to long, rich transient crossover regime that the series must
describe.
Extensive computer simulations are performed to confirm the $t^{-1/2}$
power law approach of $\theta$, where we have employed an
efficient event-driven algorithm.
The remainder of this paper is organized as follows.
Section~\ref{sec:model} introduces two related models.
Details of series expansion are explained
in Section~\ref{sec:series_expansion}.
Analyses of the series can be found in
Section~\ref{sec:series_analysis}. Monte Carlo results
are presented in Section~\ref{sec:monte_carlo} and finally
Section~\ref{sec:conclusion}
contains the summary and conclusions.
\section{The models}
\label{sec:model}
Two models have been studied in this work. We start with
an initially empty, infinite linear lattice. Dimers are dropped randomly
and sequentially
at a rate of $k$ per lattice site
per unit time, onto the lattice. Hereafter we set $k$ equal to
unity without loss of generality. If the chosen two
neighbor sites are unoccupied, the dimer is adsorbed on the lattice. If
one of the chosen sites is occupied, the adsorption attempt is rejected.
One of the simplest possibilities of diffusional relaxation in this dimer
adsorption process is that
the adsorbed dimer
is permitted to hop either to left or right by one lattice
constant at a diffusion rate $\gamma$ from the original dimer position,
provided that the diffusion attempt does not violate the excluded volume
condition.
This model
has been initiated and studied by
Privman and Nielaba \cite{Privman-Nielaba-92}.
We refer this model as the dimer RSA with dimer diffusion or
diffusive dimer model.
A second possibility is that an adsorbed dimer
is allowed to dissociate into two independent
monomers; each monomer can diffuse to one of its nearest neighbor sites
with a diffusion rate $\gamma$, provided that
the diffusion attempt does not violate the excluded volume condition.
This model bears a strong resemblance to the
former model and is exactly solvable when $\gamma = 1/2$
\cite{Grynberg-Stinchcombe-95}.
We refer this model as the dimer RSA with monomer diffusion or
diffusive monomer model.
Interestingly enough,
the special case of the diffusive monomer problem
with $\gamma = 1/2$ can be mapped to
the diffusion-limited process
\begin{equation}
{\cal A} + {\cal A} \rightarrow \hbox{inert},
\end{equation} which is known as one-species annihilation
process \cite{Avraham-group-90}.
This model has been solved exactly by a number of researchers
\cite{Lushnikov-87,Spouge-88,Balding-group-88}.
We observe that when $\gamma = 1/2$, the effect of a dimer
deposition attempt in the diffusive monomer model corresponds to
two diffusion attempts of $\cal A$ in an adjacent pair of $\cal A$
of the ${\cal A} + {\cal A} \rightarrow \hbox{inert}$ process.
The time-dependent quantity coverage
$\theta(t)$ (fraction of occupied sites)
for the diffusive monomer model with $\gamma = 1/2$
is given by
\begin{eqnarray}
\theta(t) = 1- \exp(-2t)I_0(2t),
\end{eqnarray}
where $I_n(z) $ is the modified Bessel function of integer order $n$.
\unitlength=6pt
\def\circle{0.6}{\circle{0.6}}
\def\circle*{0.6}{\circle*{0.6}}
\def\o{\begin{picture}(1,1)(-0.5,-0.5)
\put(0,0){\circle{0.6}}
\end{picture}}
\def\x{\begin{picture}(1,1)(-0.5,-0.5)
\put(0,0){\circle*{0.6}}
\end{picture}}
\def\oo{\begin{picture}(2,1)(-0.5,-0.5)
\put(0,0){\circle{0.6}}
\put(1,0){\circle{0.6}}
\end{picture}}
\def\ooo{\begin{picture}(3,1)(-0.5,-0.5)
\put(0,0){\circle{0.6}}
\put(1,0){\circle{0.6}}
\put(2,0){\circle{0.6}}
\end{picture}}
\def\oox{\begin{picture}(3,1)(-0.5,-0.5)
\put(0,0){\circle{0.6}}
\put(1,0){\circle{0.6}}
\put(2,0){\circle*{0.6}}
\end{picture}}
\def\oxo{\begin{picture}(3,1)(-0.5,-0.5)
\put(0,0){\circle{0.6}}
\put(1,0){\circle*{0.6}}
\put(2,0){\circle{0.6}}
\end{picture}}
\def\oooo{\begin{picture}(4,1)(-0.5,-0.5)
\put(0,0){\circle{0.6}}
\put(1,0){\circle{0.6}}
\put(2,0){\circle{0.6}}
\put(3,0){\circle{0.6}}
\end{picture}}
\def\ooox{\begin{picture}(4,1)(-0.5,-0.5)
\put(0,0){\circle{0.6}}
\put(1,0){\circle{0.6}}
\put(2,0){\circle{0.6}}
\put(3,0){\circle*{0.6}}
\end{picture}}
\def\ooxo{\begin{picture}(4,1)(-0.5,-0.5)
\put(0,0){\circle{0.6}}
\put(1,0){\circle{0.6}}
\put(2,0){\circle*{0.6}}
\put(3,0){\circle{0.6}}
\end{picture}}
\def\ooxx{\begin{picture}(4,1)(-0.5,-0.5)
\put(0,0){\circle{0.6}}
\put(1,0){\circle{0.6}}
\put(2,0){\circle*{0.6}}
\put(3,0){\circle*{0.6}}
\end{picture}}
\def\oxxo{\begin{picture}(4,1)(-0.5,-0.5)
\put(0,0){\circle{0.6}}
\put(1,0){\circle*{0.6}}
\put(2,0){\circle*{0.6}}
\put(3,0){\circle{0.6}}
\end{picture}}
\def\oooxx{\begin{picture}(5,1)(-0.5,-0.5)
\put(0,0){\circle{0.6}}
\put(1,0){\circle{0.6}}
\put(2,0){\circle{0.6}}
\put(3,0){\circle*{0.6}}
\put(4,0){\circle*{0.6}}
\end{picture}}
\def\ooxxo{\begin{picture}(5,1)(-0.5,-0.5)
\put(0,0){\circle{0.6}}
\put(1,0){\circle{0.6}}
\put(2,0){\circle*{0.6}}
\put(3,0){\circle*{0.6}}
\put(4,0){\circle{0.6}}
\end{picture}}
\section{Series expansions}
\label{sec:series_expansion}
To illustrate how series expansions are performed, we note
that the first few rate equations for the dimer
and monomer diffusive models are
\begin{eqnarray}
{dP(\o) \over dt} & = & -2 P(\oo), \label{eq:1stdimer}\\
{dP(\oo) \over dt} & = & -P(\oo) - 2P(\ooo) -
2 \gamma P(\ooxx) + 2 \gamma P(\oxxo),
\label{eq:2nddimer} \\
{dP(\ooo)\over dt} & = & -2P(\ooo) - 2P(\oooo) - 2\gamma P(\oooxx)
+2 \gamma P(\ooxxo), \\
\cdots \nonumber
\end{eqnarray}
and
\begin{eqnarray}
{dP(\o) \over dt} & = & -2 P(\oo), \\
{dP(\oo) \over dt} & = & -P(\oo) - 2P(\ooo) - 2\gamma P(\oox)
+ 2\gamma P(\oxo), \\
{dP(\ooo) \over dt} & = & -2P(\ooo) - 2P(\oooo)
- 2\gamma P(\ooox) + 2\gamma P(\ooxo), \\
{dP(\oox)\over dt} & = & -P(\oox) + P(\oooo) - \gamma P(\oox)
+ \gamma P(\oxo) - \nonumber \\
& & \gamma P(\ooxo) + \gamma P(\ooox), \\
\cdots \nonumber
\end{eqnarray}
respectively, where $P(C)$ denotes the probability
of finding a configuration $C$ of sites specified empty
`$\o$' or filled `$\x$'. Unspecified sites can be occupied or
empty. Here we have taken into account the
symmetries of a configuration under all lattice group operations.
For the one-dimensional configurations, we just need to
consider the reflection operation only.
Let $C_o$ denote a particular configuration of interest, and
$P_{{C_o}} \equiv P(C_o)$ the associated configuration probability.
$P_{C_o}$ is expected to be a well behaved function of time $t$,
so one can
obtain the Taylor series expansion with the expansion
point at $t = 0$,
$P_{C_o}(t) = \sum\limits_{n=0}^{\infty} {P_{C_o}}^{(n)}t^n/n! $, with
the $n$th {\it derivative} of $P_{C_o}$ given by
\begin{equation}
{P_{C_0}}^{(n)} = \left. {d^n P_{C_0}(t) \over d t^n}\right|_{t=0}.
\end{equation}
Let $G_i$ denote the set of new configurations generated in
the calculation of the $i$th derivative of $P_{C_0}$, and
$G_i^j$ the corresponding $j$th
derivatives of the set of configurations.
We observe that $G_0^{n-1}$, $G_1^{n-2}$,
$\ldots$, $G_{n-1}^0$ (determined at the $(n-1)$th derivative),
$G_0^{n-2}$, $G_1^{n-3}$, $\ldots$,
$G_{n-2}^0$ (determined at the $(n-2)$th derivative), $\ldots$,
$G_0^0$ are predetermined before calculating the $n$th derivative
of $P_{C_o}$. In the calculation of $n$th derivative of $P_{C_o}$,
we determine systematically $G_0^n$, $G_1^{n-1}$, $\ldots$, $G_{n-1}^1$,
$G_{n}^0$, by recursive use of rate equations. This algorithm is
efficient since each value
in $G_i^{n-i}$, $ 0 \le i \le n$ and
the rate equation for a configuration $C$ is generated once only.
However, this algorithm consumes the memory quickly as a result of
storage of intermediate results.
The computation of the expansion coefficients makes use of the
isomorphism between a lattice configuration and
its binary representation if we map an occupied
(empty) site to 1 (0).
The data structures used to represent Eq.~(\ref{eq:1stdimer})
and Eq.~(\ref{eq:2nddimer}) are depicted in Fig.~\ref{fig:flow}.
A node for a configuration $C$ is characterized by its four
components; (i) the representation of $C$
in the computer, (ii) a pointer
to the derivatives of $P_C$, ${P_C}^{(n)}$ for $n = 1, 2, 3, \ldots$,
(iii) the highest order of derivative $h$ of $P_C$
obtained so far, and (iv) a
pointer to a linked list of nodes of configurations
(`children') which appear in the
right hand side of the rate equation for $P_C$.
The linked list contains the associated coefficients for each
`child'.
The variable $h$ is used so that we know the values of
${P_C}^{(n)}$ where $ 1 \le n \le h$ have already been calculated
and can be retrieved when needed.
All pointers to the configuration nodes generated during the
enumeration process are stored in a hash table or a binary tree
to allow efficient checking of the existence of
any configuration. Use of the algorithm and data structures
allows us to obtain
coefficients up to $t^{31}$ and $t^{27}$
(presented in Appendix
\ref{appendix:series_coefficient})
for $P(\o,t)$ of the diffusive dimer and monomer models, respectively.
\section{Analyses of series}
\label{sec:series_analysis}
Analytically, we are interested in confirming
the power law approach
of $t^{-1/2}$ of the coverage $\theta$ at large times for both
diffusive dimer and monomer models through the
unbiased and biased analyses of the series.
The unbiased analysis does not fix the saturation coverage of the system,
while the biased analysis assumes the saturation coverage
to be the value 1.
For the unbiased analysis, let $\theta(t) = 1 - P(\o, t)$ be
the time dependent coverage. Let assume that at very large times $t$
the coverage $\theta$ satisfies the equation
\begin{eqnarray}
\theta(t) = \theta_c - \frac{A(t)}{t^{\delta}},
\label{eq:power_law}
\end{eqnarray}
where $\theta_c$ is the saturation coverage.
$A(t)$ is assumed to be a
function of $t$, which tends to a constant value as $t\rightarrow\infty$,
and $\delta$ is the exponent that characterizes the saturation
approach (we expect to obtain $\delta = 1/2$ from the
analysis of the series for all $\gamma$ values).
Writing $t = A(t)^{1/\delta}(\theta_c - \theta)^{-1/\delta}$,
we see that if we perform a DLog Pad\'e
\cite{Baker-61,Hunter-Baker-73,Baker-Hunter-73} analysis
to the inverted series $t= t(\theta)$, where
\begin{eqnarray}
\frac{d}{d\theta} \log t(\theta) =
\frac{1}{\delta}\frac{d}{d\theta} \log A(t) -
\frac{1}{\delta}\cdot\frac{1}{\theta-\theta_c},
\end{eqnarray}
then the power law of Eq.~(\ref{eq:power_law})
implies a simple, isolated pole of $\theta_c$ with
an associated residue of $-1/\delta$. Fig.~\ref{fig:explode} shows
the plot of the inverted series $t$ versus $\theta$
for the 28-term series with $\gamma = 1/2$ for the monomer diffusive
model.
For the diffusive dimer problem,
the closest real pole to the value 1 (the expected saturation coverage)
for [16,15], [15,16], [15,15], [16,14], and [14,16]
Pad\'e approximants
are shown in
Fig.~\ref{fig:dimer_theta}, with the corresponding saturation
exponents $\delta$ displayed in Fig.~\ref{fig:dimer_delta}.
Similarly we form
[14,13], [13,14], [13,13], [14,12], [12,14] Pad\'e approximants
for the diffusive monomer problem, where the results are
displayed in Fig.~\ref{fig:monomer_theta}
and Fig.~\ref{fig:monomer_delta}.
Comparing the graphs for these two models, the diffusive dimer
series give a better convergence of $\theta_c$ and $\delta$ against
$\gamma$ than that for the diffusive monomer
series generally, presumably due to the fact that the
coefficients of the series of $P(\o, t)$
alternate in signs in the former model.
For small $\gamma$
values ($\gamma < 5$), the estimates for $\theta_c$ and
$\delta$ are
unstable --- different
Pad\'e approximants do not agree with
one another.
The series with $\gamma = 0$ describes a pure lattice RSA
behavior \cite{Dickman-Wang-Jensen-91}, where
the system approaches the jamming coverage exponentially.
Hence we expect the confirmation for
power law of Eq.~(\ref{eq:power_law}) is interfered by the
exponential behavior of the series when $\gamma$ is small.
For $\gamma > 5$, there are physically
favorable estimates for $\theta_c$ and $\delta $ where
$\theta_c = 1.00 \pm 0.05$ and
$\delta = 1.0\pm 0.1$ for $10 < \gamma < 20$, for both models.
These results are the manifestations of
the transient regime of $t^{-1}$ approach to saturation.
The distribution plot of the poles and zeros in the vicinity of
$(1, 0) $ is displayed in Fig.~\ref{fig:zeropole} for the
$[14, 13]$ Pad\'e approximant for the 28-term series with $\gamma
= 1/2$ for the diffusive monomer model. We see that the
real pole closest to $(1,0)$ is not distinguished and isolated
from the nearby poles and zeros.
This explains
the difficulty of unbiased analysis that
the intermediate crossover effect masks the power law approach
at late stages.
We also perform biased analyses for the series. This
series analysis have been used by Jensen and Dickman
\cite{Jensen-Dickman-93} to extract
critical exponents from series in powers of time $t$. We define
the $F$-transform of $f(t)$ by
\begin{equation}
F[f(t)] = t \frac{d}{dt} \ln f.
\end{equation}
If $f \sim A t^{-\alpha}$ for some constant
$A$, then $F(t) \rightarrow \alpha$ as
$t \rightarrow \infty$. We consider the exponential
transformation
\begin{equation}
\label{eq:transform}
z = \frac{1-e^{-bt}}{b},
\end{equation}
which proved to be very useful in the analysis of RSA series
\cite{Dickman-Wang-Jensen-91,Gan-Wang-96,Jensen-Dickman-93}.
This transformation involves a parameter $b$ which cannot be
fixed a priori is then followed by the construction of various orders
of Pad\'e approximants to the $z$-series.
Crossing region is then searched for in the
graphs of $\alpha$ versus $b$, the transformation parameter.
To illustrate this biased analysis,
we take the saturation coverage $\theta_c$ to be 1 and choose
$f(t)$ to be $P(\o,t)= \theta_c - \theta(t)$. Since we expect
$P(\o,t) \sim t^{-1/2}$ for large times $t$,
specifically we have formed [14,13],
[13,14], [13,13], [14,12], [12,14] Pad\'e approximants to the
$z$-series for the 28-term series with $\gamma = 1/2$ for the
diffusive monomer model. We find that
the estimates for $\delta$ is $0.5061(5)$,
for $ 0.45 < b < 0.50 $,
as we can see from
Fig.~\ref{fig:biased_exact}.
Thus the exact analytical function of $\exp(-2t)I_0(2t)$
serves as a useful guide of
this analysis, where the exponent deviates from
the value $1/2$ by only about 1\%.
Given a value of $\gamma$, we obtain the corresponding
estimates of $\delta$
from the first convergence of all Pad\'e
approximants by locating the crossing region.
The results of $\delta$ estimates for several values of $\gamma$
are presented in Table~\ref{tab:allexp}.
The corresponding uncertainties for $\delta$
which reflect the variation of $\delta$ over a range of $b$ are
shown in the same
table.
For the diffusive dimer model,
we have formed [16,15], [15,16], [15,15], [16,14], [14,16],
[15,14], and [14,15] Pad\'e approximants to the $z$-series.
The corresponding graphs are displayed in
Fig.~\ref{fig:allexp}. It is seen that for small values of
$\gamma$,
we obtain small estimates of $\delta$, while
for large $\gamma$, $ \delta \rightarrow 1$,
suggesting the approach to
the limiting saturation is via a mean-field like result, i.e.
the $t^{-1}$ power law.
Hence we see that even though the exponential transformation
Eq.~(\ref{eq:transform}) works well for the exact series of
of diffusive monomer model when $\gamma = 1/2$, its use for general
$\gamma$ is not very appropriate. We have also tried the
transformation
$ z = 1 - (1 + bt)^{-1/2}$ to the series for both diffusive
models but the convergence is rather poor.
We have tried and used a third
method of extracting the saturation exponent $\delta$.
If we assume that for large enough times $t$,
the saturation coverage $\theta$ assumes a power law
\begin{equation}
1 - \theta \propto t^{-\delta},
\end{equation}
then we expect a plot of $d\ln(1-\theta)/d\ln(t)$ versus
$t$ or $\log_{10}(t)$ should give a plateau
of constant $-\delta$ values. By forming [14,13], [13,14],
[13,13] Pad\'e approximants to the $d\ln(1-\theta)/d\ln(t)$ of the 28-term
series for the diffusive monomer model with $\gamma = 1/2$,
we observe from Fig.~\ref{fig:illus} that the agreement between
different Pad\'e estimates and the exact solution is excellent
for $\log_{10}(t)$ up to around $0.9$.
For diffusive dimer problem, three Pad\'e approximants of
[16,15], [15,16], and [15,15] are formed.
The plots of $d\ln(1-\theta)/d\ln(t)$ versus
$\log_{10}(t)$ for the diffusive dimer and monomer models,
shown in Fig.~\ref{fig:dimerwing} and Fig.~\ref{fig:monomwing},
respectively,
are obtained by taking the average of the 3 different Pad\'e
estimates.
The graphs end before the difference
between at least a pair of Pad\'e estimates is more
that 0.001.
The last estimates
in Fig.~\ref{fig:dimerwing} and Fig.~\ref{fig:monomwing}
are taken as the estimates for $\delta$ and
they are listed in the last two columns of Table~\ref{tab:allexp}.
These estimates for $\delta$ are plotted
in the same graph for the $F$-transformed
analysis for comparisons (Fig.~\ref{fig:allexp}).
It is seen that our last method of extracting the
saturation exponents appears to be better than
the $F$-transform analysis since
it yields almost about the same estimates for $\delta$.
It does not involve any
transformation which is not known in advance that will yield
consistent results \cite{Jensen-Dickman-93}. Looking
at the ends of the curves in Fig.~\ref{fig:dimerwing} and
Fig.~\ref{fig:monomwing}, we are certain that the power law
regime is
still not reached since the $\delta$ estimates do not seem
to converge to
a constant value,
except the case when $\gamma = 1/2$ for the
diffusive monomer model.
From this we know that
our estimates for $\delta$
do not describe the true power law approach at large times $t$.
Such information cannot be found in the $F$-transform analysis.
We note that our last method of analyzing the series
is easy to use compared to the $F$-transform analysis.
\section{Monte Carlo simulations}
\label{sec:monte_carlo}
To study the short and large time
behaviors of the coverage, we have performed extensive and
exhaustive simulations for the diffusive dimer and monomer models.
For both models, we take an initially empty linear lattice with
$N = 20000$ sites with periodic boundary conditions so that
the finite-size effects can be ignored. In each Monte Carlo step,
a pair of adjacent sites is chosen randomly.
The type of attempted process is then decided: deposition
with probability
$p$, where $ 0 < p \le 1$, or diffusion with
probability $(1-p)$.
In the case of
the deposition attempts, if any one of the chosen sites is occupied,
the deposition attempt is rejected (unsuccessful attempt), else
the adsorption attempt is accepted.
In the case of diffusion, yet another decision
is made either to move right or left, with equal probability. If
the selected decision is diffusion to the right,
we check the selected pair of sites are occupied
and
its right nearest neighbor site is unoccupied,
then the dimer is moved by one lattice constant to the right. The
left-diffusion attempts are treated similarly.
In contrast to the diffusive dimer model, the diffusive monomer
model allows monomers to move by one lattice constant.
We define one time unit interval ($\Delta t = 1$)
to be during which a deposition
attempt is performed for each lattice site. Thus for
$N$-site lattice, one unit time corresponds to $N$ deposition
attempts, on average. The diffusion rate $\gamma$ relative to the
deposition rate, is then $\gamma = (1-p)/2p$.
Straightforward simulation procedure, as described above,
encounters a serious drawback in which at late stages,
most adsorption and diffusion attempts are rejected.
In order to study the behavior of the system at large times, we
have used an event-driven algorithm to speed up the dynamics
of the simulations \cite{Brosilow-group-91,Wang-94}.
Let $q$ be the probability that we can make a successful
move, then the probability that the first $(i-1)$th trials is unsuccessful,
and the $i$th trial is successful is
\begin{eqnarray}
P_i = q (1-q)^{i-1}, \hbox{\ \ \ } i = 1, 2, 3, \ldots
\label{eq:event_driven}
\end{eqnarray}
If we restrict all trials to be coming from the successful ones,
then two consecutive trials are in fact separated by a random variable $i$
in Eq.~(\ref{eq:event_driven}). This distribution can be generated
by
\begin{eqnarray}
i = \left\lfloor\frac{\ln \xi}{\ln(1-q)}\right\rfloor + 1,
\end{eqnarray}
where $\xi$ is a uniformly distributed random number between 0 and 1.
In employing this method, we have to keep and update an
active list of successful
moves/attempts, where from its length we can evaluate $q$ at any instance.
Simulations are performed on a cluster of fast workstations. Our numerical
results are obtained for $\gamma = $ 0.05, 0.10, 0.20, $\ldots$,
and 6.40, for $t$ up to $2^{20}$.
Each data set is averaged over 500 runs, and the longest run
take about 150 CPU hours on a HP712/60. The coverage (fraction of
occupied sites), $\theta(t)$, is plotted in Fig.~\ref{fig:dimer_phase}
and Fig.~\ref{fig:monom_phase} for
the diffusive dimer and monomer models, respectively.
We have also performed the simulation at $\gamma = 1/2$ for
the diffusive monomer model in order to compare the simulation
results with the exact
results. It is seen that the agreement between them are so good
that actually an overlapping line is observed in
Fig.~\ref{fig:monom_phase}.
For $\gamma = 0$, we have the exact solution \cite{Dickman-Wang-Jensen-91}
\begin{eqnarray}
\theta(t) = \frac{1-\exp(-2(1-\exp(-t) ))}{2}.
\end{eqnarray}
For extremely fast diffusion case, i.e., $\gamma = \infty$,
exact results have been
obtained, where
\begin{eqnarray}
t(\theta) = \frac{1}{4}\bigl(\frac{1}{1-\theta} - 1\bigr)
- \frac{1}{4} \ln(1-\theta),
\label{eq:dimer_fast_diff}
\end{eqnarray}
for the diffusive dimer model \cite{Privman-Barma-92} and
\begin{eqnarray}
t(\theta) = \frac{1}{2}\bigl(\frac{1}{1-\theta} - 1\bigr).
\label{eq:monom_fast_diff}
\end{eqnarray}
for diffusive monomer model.
The approach of $(\theta_c - \theta) \sim t^{-1} $ at all times is obvious
for these extremely fast diffusion models. We have included the
lines of slope
$-1/2$ to indicate the $t^{-1/2}$ power law clearly.
It is seen from Fig.~\ref{fig:dimer_phase} and
Fig.~\ref{fig:monom_phase} that
for $\gamma \ge 3.20$, the system takes a very long time
($t \approx 10^{4}$) before it can
enter the final $t^{-1/2}$ regime. This explain why we have
difficulty in extracting the actual power law
approach from the primitive expansion in time $t$.
To further confirm that the saturation approach indeed follows a power
law, we have used a scaling analysis.
For large times $t$, let us assume that $(1-\theta)$ has the following
scaling form
\begin{equation}
\label{eq:functional}
1-\theta = (\gamma t)^{-1/2}G(\gamma^b/t) ,
\end{equation}
where $G$ is a scaling function and $b$ is a constant to be determined.
Eq. (\ref{eq:functional}) requires that $G(u)$ tends to a
constant when $u$ tends to 0 \cite{Privman-private}.
This is required because for large times $t$,
$(1-\theta) \sim t^{-1/2}$. Let further assume that for large $u$, $G(u)
\sim u^z$ for some constant $z$.
For extremely large $\gamma$, we have $ (1 -\theta) \sim t^{-1}$
(see Eqs. (\ref{eq:dimer_fast_diff}) and
(\ref{eq:monom_fast_diff}))
hence $z = 1/2$ and $b = 1$.
Writing Eq. (\ref{eq:functional}) as
\begin{eqnarray*}
1-\theta & = &(1/\gamma)(\gamma/t)^{1/2}G(\gamma/t) \\
& \equiv & (1/\gamma)F(t/\gamma),
\end{eqnarray*}
we then make log-log plots of $\gamma (1-\theta)$ versus
$t/\gamma$ for the diffusive dimer and monomer models as shown in
Figs. \ref{fig:dimerscale} and \ref{fig:monomscale},
respectively. It is seen that the data collapse into single
curves at large times $t$. Our numerical data confirm not only
the power law decay but also a scaling behavior for $(1-\theta)$.
\section{Conclusions}
\label{sec:conclusion}
By using an efficient algorithm based on the hierarchical rate
equations, relatively long series are
obtained for two models of 1-d RSA with diffusional relaxation.
Analyses of series are performed, but it is seen that
even though the series are long, we only manage to extract
the behaviors of the systems up to intermediate times only.
To study the power law of a system at large times $t$,
we see that a series which exhibits a continuous crossover
behavior in its short and intermediate times ought to be long
enough so that various orders of Pad\'e approximants can still
converge in the power law regime.
Using these long series, we find that the analysis
of series based on the ratio method by Song and Poland
\cite{Song-Poland-92} was not useful. Specifically,
for the diffusive monomer series when $\gamma = 1/2$ (which
corresponds to the A + A $\rightarrow$ 0 process in
Song and Poland's work), we obtain the saturation
exponent (where they have used the symbol
$\nu$) $\delta$ $=$ $2.0$, $1.2$,
$0.895$, $0.729$, $0.624$, $0.551$, $0.498$, $0.458$,
$0.426$, $0.401$, $0.380$, $0.363$, $0.351$, $\ldots$,
which is not seen
to be converging towards the expected value of $1/2$.
We have also performed extensive computer simulations using
an efficient event-driven algorithm, where it
allows us to use and simulate a larger system to much
larger times $t$ than it was done previously on a supercomputer
\cite{Privman-Nielaba-92}. The $t^{-1/2}$ power law approach
of $\theta$ to its saturation is confirmed numerically
at large times $t$.
\section*{Acknowledgement}
This work was supported in part by
an Academic Research Grant RP950601 of National University of
Singapore. Part of the calculations were performed on the
facilities of the Computation Center of the Institute of Physical
and Chemical Research, Japan. We would like to thank
V. Privman for pointing out Ref. \cite{Grynberg-Stinchcombe-95}
and useful discussion on how to analyse the series.
We also appreciate one of the referees for
suggesting us to make a scaling behavior study.
\pagebreak
\bibliographystyle{plain}
| proofpile-arXiv_065-480 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{\bf INTRODUCTION}
Binary neutron star systems which are spiraling toward their final
coalescence under the dissipative influence of gravitational
radiation reaction forces are the primary targets for detection of
gravitational waves by interferometric gravitational wave
detectors such as LIGO and VIRGO\cite{Abramovici92,Thorne95}.
Extracting
the gravitational waves from the detector noise and
making use of the information encoded in the signals will require a
thorough knowledge of the expected waveforms produced by these
binaries\cite{Thorne95,Cutler93}.
In this paper we explore the effect of the neutron-star equation of
state on the orbital evolution and gravitational wave
emission of binaries just prior to merging.
Specifically, we show that a combination of post-Newtonian
(relativistic) effects and Newtonian tidal effects
(which depend on the equation of state) conspire to induce
a dynamical instability in the orbital motion, which
causes the plunge to final coalescence to begin somewhat
sooner -- and to proceed somewhat faster -- than it would simply under
the influence of the dissipative radiation reaction force.
Thus the motion of the bodies during the late stages of binary
inspiral depends on the structure of the neutron stars.
Consequently, the gravitational waveform emitted during this short
portion of the final coalescence will be imprinted
with information about the nuclear equation of state.
During the final $\sim 10$ minutes or the last $\sim 8000$ orbits of a
neutron star binary inspiral, the orbital frequency increases from
about 5 Hz on up to a cutoff of a few hundreds to a thousand
Hertz (roughly corresponding to the orbital frequency when the
final plunge begins). Thus the gravitational wave frequency (twice the
orbital frequency for the dominant quadrupole radiation) {\it chirps}
through the LIGO detector bandwidth during
this period \cite{Abramovici92}.
The evolution of the binary in these last few minutes
of the inspiral is very sensitive to a number of relativistic
effects, such as gravitational-wave tails and
spin-orbit coupling (dragging of inertial frames).
The gravitational waveform emitted by the binary in this
portion of the coalescence, the adiabatic inspiral, is currently
being extensively studied \cite{bdiww,biww,ww}.
During most of this inspiral phase the neutron stars can be
treated as simple point masses because the effects associated with
the finite stellar size turn out to be small:
(i) The neutron star has too small a viscosity to allow for angular
momentum transfer from the orbit to the stellar spin
via viscous tidal torque\cite{Bildsten92,Kochanek92};
(ii) The effect of the spin-induced quadrupole is negligible
unless the neutron star has rotation rate close to the break-up
limit\cite{Bildsten92,LRS94}; (iii) Resonant excitations of neutron
star internal modes (which occur at orbital frequencies less than
$100$ Hz) produce only a small change in the orbital phase due to the weak
coupling between the modes and the tidal
potential\cite{Lai94,Reisenegger94,Shibata94};
(iv) The correction to the equation of motion from the (static) tidal
interaction is of order $(R_o/r)^5$ (where $R_o$ is the neutron star
radius, $r$ is the orbital separation\cite{notation}),
which is negligible except when $r$ is smaller than a few stellar
radii. Since $R_o\simeq 5M$
for a typical neutron star of mass $1.4M_\odot$ and radius $10$ km,
the tidal effect is essentially a (post)$^5$-Newtonian
correction \cite{quad}.
The expression for the phase error induced by the tidal effect is given in
Ref.\cite{LRS94}.
The fact that the evolution of the binary system as it sweeps through
the low frequency band of the detector is insensitive to
finite-size effects means that the measurement of the
inspiral waveform will allow us to probe cleanly into
the intricate structure of general relativity, and to test
whether general relativity is the correct theory of
gravity \cite{lucsathya,cliffscalar}. Moreover, some of
the parameters of the binary system, such as the masses of the stars, can
be determined with reasonable accuracy\cite{finnchernoff,cutlerflanagan}.
However, the waveform's lack of dependence on the finite size of the
objects during the most of the adiabatic inspiral
also implies that information about the internal structure of the
neutron star is only imprinted on the radiation emitted
just prior to coalescence when the orbital radius is small.
Indeed at small orbital separations,
tidal effects are expected to be very important.
In a purely {\it Newtonian} analysis, the interaction potential
between star $M'$ and the tide-induced quadrupole of $M$,
$V_{tide}\sim -M'^2R_o^5/r^6$, increases with decreasing $r$.
The potential becomes so steep that a dynamical instability
develops, accelerating the coalescence at small orbital
radius. This Newtonian instability has been fully explored
using semi-analytic models in Ref.\cite{LRS94} and
Ref.\cite{LS95} (hereafter referred to as LS).
It has also been examined numerically in Refs.\cite{Rasio,Centrella}.
However, a purely Newtonian treatment of the binary at small separation
is clearly not adequate, as general relativistic effects
will also be important in this regime; and general relativistic
effects can also make the orbit unstable.
For example, a test particle in circular orbit around a
Schwarzschild black hole will experience an ``innermost stable circular
orbit'' at $r_{\rm isco}=6M$ (or $5M$ in harmonic coordinates). This
unstable behavior is caused by higher-order relativistic corrections
included in the Schwarzschild geodesic equations of motion.
For computing the orbital evolution of two neutron stars of
comparable mass near coalescence, the test-mass limit is obviously
inadequate.
In order to explore the orbital instability for
such systems, Kidder, Will and Wiseman
\cite{KWW93} (hereafter referred as KWW)
developed {\it hybrid} equations of motion.
These equations augment the the Schwarzschild geodesic equations
of motion with the finite-mass terms of the (post)$^{5/2}$-Newtonian
equations of motion.
Including these finite mass terms in the equation of motion
moved the innermost-stable-circular-orbit radius farther out (in units
of the total mass).\footnote{See Wex and Sch\"afer\cite{Wex93}
for a critique and an alternative
construction. Their post-Newtonian calculation suggests that
the innermost stable orbit may occur at an even greater separation.}
In this paper, we augment the hybrid equations
with contributions due to the tidal deformation of the stars.
In a nutshell, the work presented here combines the Newtonian
tidal analysis of LS \cite{LS95} with the relativistic point-mass
analysis of KWW \cite{KWW93} to yield a more complete picture of the
neutron-star coalescence prior to merging.
We note, that unlike a test-particle around a Schwarzschild
black hole, the very notion of ``innermost stable circular orbit''
is poorly defined for objects of comparable mass.
After all, in the relativistic regime the binary orbit will be decaying
rapidly due to radiation reaction; thus the orbit
is not circular, but rather a decaying spiral.
In order to give a semi-quantitative definition of ``innermost
stable circular orbit'' we use the artifice of ``shutting off''
all the dissipative terms in the equation of motion and
looking for the point where the solutions of the
remaining non-dissipative equations become dynamically unstable.
The use of hybrid equations of motion
augmented with the tidal terms
allows us to map out the dependence of
the critical radius $r_{\rm isco}$, or the corresponding orbital
frequency $f_{\rm isco}$, for a wide range of allowed neutron star equations
of state (parametrized by radius and effective polytropic index; see
Figure 1). We believe that clearing up such dependence is
important,
and this analysis provides a benchmark with which comparisons can be
made with future numerical results.
{\it Indeed, an important point we wish
to make in this paper is that neither
relativistic (post Newtonian) effects nor Newtonian tidal effects
can be neglected near the instability limit, and the critical
frequency can be much lower than
the value obtained when only one of these effects are included.}
\begin{figure}[t]
\special{hscale=42 vscale=42 hoffset=0.0 voffset=-285.0
angle=0.0 psfile=fig1.eps}
\vspace*{2.8in}
\caption[Fig. 1]{
The critical orbital frequency (at the inner-most stable orbit)
as a function of the ratio of the neutron star radius $R_o$ and
mass $M$. The solid curves show the results including both
relativistic and
tidal effects (the lower curve is for $\Gamma=3$ while the upper one
is for $\Gamma=2$), the dashed curves are the Newtonian limit
given by Eqs.~(8)-(9). The vertical line corresponds to
$R_o/M=9/4$, the minimum value for any physical neutron star.
The insert is a close-up for the nominal range of $R_o/M=4-8$
as given by all the available nuclear EOS's.
Two curves within the insert should bracket all the physical
values of $f_{\rm isco}$.}
\end{figure}
The main results of our analysis are summarized in Figure 2 and 3.
Figure 2 shows that the rate of radial infall for stars
near coalescence is substantially
underestimated if one models the coalescence as
a Newtonian circular-orbit decaying solely under the
influence of radiation reaction (top dotted curve).
In other words, the rate of coordinate infall is substantially
enhanced by the non-dissipative terms.
Somewhat more relevant for observational purposes, Figure 3
shows that the number of orbits (or gravitational wave cycles)
per logarithmic frequency interval is substantially reduced
by the unstable collapse of the orbit.
Both plots show modest sensitivity to the equation of state.
\begin{figure}[t]
\special{hscale=52 vscale=55 hoffset=-42.0 voffset=-365.0
angle=0.0 psfile=fig2.eps}
\vspace*{3.1in}
\caption[Fig. 2]{
The radial infall coordinate velocity during binary coalescence, with
$M=M'=1.4M_\odot$, $R_o/M=5$, $\Gamma=3$, all calculated using
$2.5PN$ radiation reaction.
The solid line is the result including relativistic and tidal effects,
the short-dashed line includes only tidal effects, the long-dashed
line includes only relativistic effects.
The dotted line is the point mass ``Newtonian'' result.
}
\end{figure}
\begin{figure}[t]
\special{hscale=52 vscale=55 hoffset=-50.0 voffset=-360.0
angle=0.0 psfile=fig3.eps}
\vspace*{3.1in}
\caption[Fig. 3]{
The number of orbits the binary spends per logarithmic
frequency. The labels are the same as in Fig.~2.
}
\end{figure}
The remainder of the paper is organized as follows:
In section II we present our equations of motion.
In section III we examine the orbital stability using the non-dissipative
portion of the equations of motion,
and thus identify the location of the ``innermost stable circular orbit''.
In section IV we include the dissipative terms that were omitted in
the analysis of section III, and evolve the full equations of motion.
In section V we briefly discuss the relevance of our results to
numerical hydrodynamic calculations and to gravitational
wave signal analysis.
\section{\bf EQUATIONS OF MOTION INCLUDING
TIDAL AND GENERAL RELATIVISTIC EFFECTS}
Consider a binary containing two neutron stars of mass $M$ and $M'$,
each obeying a polytropic equation of state
$P=K\rho^\Gamma$. We use the compressible ellipsoid model
for binary stars developed in LS\cite{LS95}. Basically,
we model the tidally deformed neutron star
as an ellipsoid, with internal density profile
similar to that of a spherical polytrope.
The dynamics of such a neutron star (so called
Riemann-S ellipsoid) is characterized by the three
principal axes ($a_1,a_2,a_3$ for star $M$ and $a_1',a_2',a_3'$
for star $M'$), the angular velocity ($\Omega$ and $\Omega'$)
of the ellipsoidal figure about a principal axis (perpendicular to
the orbital plane) and the internal motion of the
fluid with uniform vorticity. The non-zero internal fluid motion is
necessary because the binary neutron stars are not expected
to corotate with the orbit due to rapid orbital decay and small
viscosity\cite{Bildsten92,Kochanek92}. Although the Newtonian tidal
interaction between the neutron stars can be treated
exactly in the linear regime using mode decomposition\cite{Lai94},
the ellipsoid model has the advantage that it can be extended to
the nonlinear regime at small orbital radii, when the tidal
deformation of the star becomes significant.
The Newtonian dynamical equations for the binary neutron stars
as derived in LS include the familiar
Newtonian ($1/r^2$) force-law for point-masses orbiting one another;
they further contain Newtonian terms involving finite
size (tidal) effects. A post-Newtonian
treatment of the tidal problem would give the relativistic
corrections to these terms, namely
the standard point-mass, post-Newtonian corrections to the
equations of motion, as well as relativistic corrections
to the quadrupole moment and corrections
due to higher moments the bodies. (See Appendix F of \cite{WW96}.)
To insure that our equations of motion at least agree with
the known post-Newtonian, point-mass equations we augment these
Newtonian equations of motion with the hybrid equations of
KWW. However, we use only the Newtonian equations to describe the
evolution of the neutron stars structure ($a_i$ and $a_i'$) and the
fluid motion (the figure rotation rate and the internal vorticity)
within the stars. These are given by Eqs.~(2.18)-(2.22) of LS.
In other words, we neglect the relativistic corrections to the fluid
motion, self-gravity and tidal interaction.
These corrections are secondary effects and should not modify the
{\it orbital dynamics} appreciably (e.g., the Newtonian tidal
interaction between the two stars scales approximately as
$M^2R_o^5/r^6$, and its relativistic correction is of order $M/r$
smaller). As noted before, the tidal interaction
enters the Newtonian potential as a correction of
$O[(R_o/r)^5]\sim O[(M_t/r)^5]$, in effect, as a (post)$^5$-Newtonian
term. Therefore, by not including relativistic corrections to
these tidal terms, we are merely omiting terms
which are of (post)$^6$-Newtonian order.
In fact, the largest error comes
from neglecting the post-Newtonian correction (of order $M/R_o$) to
the internal stellar structure (see Sec.~III.B for an estimate of its
effect on $f_{\rm isco}$). The relativistic corrections to the
orbital motion, however, are very important.
Our equations of orbital motion can be assembled from
Eqs.~(2.23)-(2.24) of LS and from Eqs.~(1.2)-(1.3) of KWW:
\begin{eqnarray}
\ddot r &=& r{\dot\theta}^2-{M_t\over r^2}(A_H+B_H\dot r)\nonumber\\
&&-{3\kappa_n\over 10}{M_t\over r^4}\left[a_1^2(3\cos^2\alpha-1)
+a_2^2(3\sin^2\alpha-1)-a_3^2\right] \nonumber\\
&&-{3\kappa_n'\over 10}{M_t\over r^4}\left[a_1'^2(3\cos^2\alpha'-1)
+a_2'^2(3\sin^2\alpha'-1)-a_3'^2\right] \nonumber\\
&&-{M_t\over r^2}(A_{5/2}+B_{5/2}\dot r)
-{32\over 5}r\bigl[\Omega^5(I_{11}-I_{22})\sin 2\alpha \nonumber\\
&&+\Omega'^5(I_{11}'-I_{22}')\sin 2\alpha'\bigr],
\\
\ddot\theta &=& -{2\dot r\dot\theta\over r}
-{M_t\over r^2}B_H\dot\theta\nonumber\\
&&-{3\kappa_n\over 10}{M_t\over r^5}(a_1^2-a_2^2)\sin 2\alpha
-{3\kappa_n'\over 10}{M_t\over r^5}(a_1'^2-a_2'^2)\sin 2\alpha' \nonumber\\
&&-{M_t\over r^2}B_{5/2}\dot\theta
-{32\over 5}\bigl[\Omega^5(I_{11}-I_{22})\cos 2\alpha \nonumber\\
&&+\Omega'^5(I_{11}'-I_{22}')\cos 2\alpha'\bigr],
\end{eqnarray}
where $M_t=M+M'$ is the total mass, $\alpha$ ($\alpha'$) is
the misalignment angle between the tidal bulge of $M$ ($M'$)
and the line joing the two masses, $\kappa_n,\kappa_n'$ are
dimensionless structure constants depending on the mass concentration
within the stars.
In Eqs.~(1) and (2) the last two lines contain the ``dissipative''
terms due to gravitational radiation reaction.
The quantities $A_H$, $B_H$, $A_{5/2}$, $B_{5/2}$,
which include the ``hybrid'' corrections to the equation of motion,
are given by
\begin{eqnarray}
A_H &=& {1-M_t/r\over (1+M_t/r)^3}-\left[{2-M_t/r\over
1-(M_t/r)^2}\right]{M_t\over r}\dot r^2+v^2\nonumber\\
&&-\eta\left(2{M_t\over r}-3v^2+{3\over 2}\dot r^2\right)
+\eta\biggl[{87\over 4}\left({M_t\over r}\right)^2\nonumber\\
&&+(3-4\eta)v^4+{15\over 8}(1-3\eta)\dot r^4
-{3\over 2}(3-4\eta)v^2\dot r^2\nonumber\\
&&-{1\over 2}(13-4\eta){M_t\over r}v^2
-(25+2\eta){M_t\over r}\dot r^2\biggr]
\\
B_H &=& -\left[{4-2M_t/r\over 1-(M_t/r)^2}\right]\dot r
+2\eta\dot r-{1\over 2}\eta\dot r\biggl[(15+4\eta)v^2\nonumber\\
&&-(41+8\eta){M_t\over r}-3(3+2\eta)\dot r^2\biggr],
\\
A_{5/2} &=& -{8\over 5}\eta{M_t\over r}\dot r\left(18v^2
+{2\over 3}{M_t\over r}-25\dot r^2\right),
\\
B_{5/2} &=& {8\over 5}\eta{M_t\over r}\left(6v^2
-2{M_t\over r}-15\dot r^2\right) \; ,
\end{eqnarray}
where $v^2=\dot r^2+r^2\dot\theta^2$,
$\eta=\mu/M_t$ and $\mu=MM'/M_t$.
Also the multipoles moments can be expressed as
\begin{equation}
I_{ii}=\kappa_nMa_i^2/5,~~~~~
I_{ii}'=\kappa_n'M'a_i'^2/5 \; .
\end{equation}
Note that in Eqs.~(1)-(2), we have also included the leading order
radiation reaction forces due to tidal deformation.
Admittedly, this is not a consistent post-Newtonian expansion of the
true equations of motion; however it is correct in several important
limiting cases: (i) In the limit that
$a_i \rightarrow 0$ and $a_i' \rightarrow 0$ and the
limit $\eta \rightarrow 0$, we recover
the {\it exact} Schwarzschild equation of motion.
(ii) In the point-mass limit ($a_i\rightarrow 0$ and $a_i'\rightarrow 0$)
we recover the hybrid equations given in KWW.
KWW presented an argument that suggested
that the higher-order, $\eta$-dependent, (post)$^3$-Newtonian
-- as yet uncalculated -- corrections to these equations have only a modest
effect on the equations of motion. See Figure 6 of Ref.\cite{KWW93}.
However, until these terms are calculated it is unclear just how
large an effect they will have on the location of the innermost
stable orbit.
(iii) In the non-relativistic limit we recover the equations of motion
given in LS. These equations contain the dominant contributions to the
equations of motion due to the finite sizes of the objects.
Note that although Eqs.~(1) and (2) make reference to the orbital
radius $r$, we are always aware that this is a gauge dependent quantity
and of little meaning for a distant observer.
Observationally, the more meaningful quantity is the orbital frequency
as measured by distant observers, and we shall use frequency
rather the radius in presenting most of our results.
\section{\bf INSTABILITY OF THE NON-DISIPATIVE EQUATIONS OF MOTION}
\subsection{Method to Determine the Stability Limit}
We now form a set of non-dissipative equations of motion,
by simply discarding the gravitational radiation reaction terms
given in the last two lines of Eq.~(1) and Eq.~(2).
These non-dissipative dynamical equations admit equilibrium solutions,
which are obtained by setting $\dot r=\ddot
r=\ddot\theta=\dot\Omega_{orb}=\alpha=\alpha'=0$ as well as
$\dot a_i=\dot a_i'=0$. For a given $r$, the evolution equations for
the neutron star structure reduce to a set of algebraic equations
for $a_i$ and $a_i'$, while the orbital equation (2) gives the orbital
frequency $\Omega_{orb}$. These equations are solved using a
Newton-Raphson method, yielding an equilibrium binary model. Thus a
sequence of binary models parametrized by $r$ can be constructed.
To determine the stability of the orbit of a binary model,
we simply use the equilibrium parameters as initial conditions
for our non-dissipative equations of motion. We add a small perturbation
to the equilibrium model and let the system evolve.
In this way we locate the critical point of the dynamical equations,
corresponding to the dynamical stability limit of the equilibrium
binary or the inner-most stable circular orbit:
for $r>r_{\rm isco}$, the binary is stable, and the
system oscillates with small amplitude about the initial
configuration; for $r<r_{\rm isco}$, the binary is unstable, and
the perturbation grows, leading to the swift merger of the
neutron stars even in the absence of dissipation.
\subsection{Results}
For concreteness, we present results only for binary neutron stars
with equal masses ($M=M'$), both having zero spin at
large orbital separation, although our equations are adequate
to treat the most general cases\cite{LS95}.
The polytropic relation $P=K\rho^\Gamma$ provides a useful
parametrization to the most general realistic nuclear equation of
state (EOS). Since the radius $R_o$ of the nonrotating neutron star
of mass $M$ is uniquely determined by $K$ and $\Gamma$, we can
alternatively use $R_o/M$ and $\Gamma$ to characterize the EOS.
For a canonical neutron star with mass $M=1.4M_\odot$, all EOS
tabulated in\cite{Arnett77} give $R_o/M$ in the range of $4-8$,
while modern microscopic nuclear calculations typically give
$R_o/M=5$\cite{Wiringa88}. For a given $R_o/M$, the polytropic index
$\Gamma$ specifies the mass concentration within the star. Except for
extreme neutron star masses ($M\lo 0.5M_\odot$ or $M\go 1.8M_\odot$)
typical values of $\Gamma$ lie in the range of $\Gamma=2-3$
\cite{LRS94}.
In Table I, we list the physical properties of
the equilibrium binary neutron stars at the dynamical stability
limit for several values of $R_o/M$ and $\Gamma=3$.
In Figure 1, the orbital frequency $f_{\rm isco}$
is shown as a function of $R_o/M$ for $\Gamma=2$ and
$\Gamma=3$. Clearly, in the limit of $R_o/M\rightarrow 0$,
$f_{\rm isco}$ approachs the point mass result $f_{\rm
isco}=697M_{1.4}^{-1}$
Hz obtained in KWW\footnote{KWW\cite{KWW93}
give a correct expression for $r_{\rm isco}/M_t$, but incorrectly give
$f_{\rm isco}$=710 Hz due to a numerical error.}.
In the non-relativistic limit
we recover the pure Newtonian result\cite{LRS94,LS95}:
\begin{eqnarray}
f_{\rm isco} &=& 657M_{1.4}^{-1}(5M/R_o)^{3/2}~({\rm Hz})
~~~~~(\Gamma=3),\\
f_{\rm isco} &=& 766M_{1.4}^{-1}(5M/R_o)^{3/2}~({\rm Hz})
~~~~~(\Gamma=2).
\end{eqnarray}
{\it For typical neutron star radius $R_o/M=5$, the critical
frequency ranges from $488$ Hz (for $\Gamma=3$)
to $540$ Hz (for $\Gamma=2$), while both the pure Newtonian (with tides)
calculation and the pure point-mass hybrid calculation give a result
$30-40\%$ larger}. There are two physical causes for the reduction
in $f_{\rm isco}$: (i) The binary becomes unstable at larger
orbital separation due to the steepening of the interaction potential
from both tidal and relativistic effects;
(ii) For a given orbital radius (itself a gauge dependent quantity),
the post-Newtonian orbital frequency as measured by an observer at
infinity is smaller than the Newtonian orbital
frequency\footnote{In the case of equal masses,
at first post-Newtonian order
$\Omega_{orb} = \Omega_{Kepler}[1 - (11/8)(M_t/r)]$.
See Ref.\cite{bdiww}.}.
We conclude that to neglect either the
tidal effects or the relativistic effects can lead to large error in
the estimated critical frequency.
Except for the intrinsic uncertainties associated with
the hybrid equations of motion\cite{KWW93,Wex93}, the main
uncertainty in our determination of $f_{\rm isco}$ comes from
neglecting post-Newtonian corrections to (i) the stellar structure and
(ii) the the tidal potential.
The first correction {\it decreases} the tide-induced quadrupole;
the fractional change is of order $-M/R_o$.
The second {\it increases} the quadrupole by a fraction of order
$M'/r$. We can estimate how much $f_{\rm isco}$ is modified by these
two corrections.
In Newtonian theory, $r_{\rm isco}$ is approximately determined by
the condition $MM'/r\sim M'^2R_o^5/r^6$. Including the relativistic
corrections
this condition becomes $MM'/r\sim M'^2R_o^5/r^6(1-\delta)$, where
$\delta\sim [O(M/R_o)-O(M'/r)]\lo 20\%$. Thus the change
in $f_{\rm isco}$ due to these two effects is
$\Delta f_{\rm isco}/f_{\rm isco}\simeq
0.3\,\delta\lo 6\%$, i.e., the critical frequency increases by
a few percent\cite{Wilsonnote}.
As emphasized in Sec.~I, the critical radius (or critical frequency),
at which the non-dissipative equations become dynamically unstable,
is meaningful only in the sense that when $r<r_{\rm isco}$, the binary
will coalesce on dynamical (orbital) timescale even
in the absence of dissipation. In the realistic situation,
the dissipative radiation reaction forces will also be rapidly driving
the binary to coalescence. Therefore to determine the significance of the
dynamical instability we must compute the orbital evolution with
the full equations of motion -- including the radiation reaction.
\section{\bf ORBITAL EVOLUTION PRIOR TO MERGER}
We now include the dissipative radiation reaction forces in our analysis.
In this case the plunge will be driven by both the
dissipative, as well as the non-dissipative effects associated with
the steepening potential (both the tidal potential and the
relativistic potential).
But what effect is dominant?
In order to numerically investigate this question,
we choose a specific system with $M=M'=1.4M_\odot$, $R_o/M=5$ and
$\Gamma=3$. The orbital evolution begins when the stars are well
outside the innermost stable circular orbit limit.
We consider four different inspiral scenarios.
(i) A purely dissipative inspiral: a system of point masses subject only
to a Newtonian ($1/r^2$) force and (post)$^{5/2}$-Newtonian
radiation reaction force. In this case the infall rate is given by
$v_r=\dot r=-(64/5)\eta (M_t/r)^3$.
[Specifically, we set
$A_H = 1$ and $B_H = a_i = a_i' = I_{kk} = I_{kk}' = 0$ in
Eqs.~(1)-(2).]
This is depicted by the dotted
curve in Figure 2 and 3.
(ii) A purely relativistic plunge in which
we neglect the tidal effects
[Specifically we set $a_i=a_i'= I_{kk} = I_{kk}' =0$ in Eqs.~(1)-(2)].
This relativistic case is depicted by the long-dashed curve in Figure
2 and 3.
(iii) A tidally enhanced plunge: we include only the
Newtonian terms in Eqs.~(1)-(2) and the radiation reaction force.
[Specifically we set $A_H = 1$ and $B_H = 0$ in Eqs.~(1)-(2).]
This case is depicted by the short-dashed curve in Figure 2 and 3.
(iv) Finally, we evolve the complete dynamical equations including all
terms in Eqs.~(1)-(2); this is depicted by the solid curve in Figure 2
and 3.
Each intergration is terminated when the surfaces of the stars
touch, i.e., at $r\simeq 2a_1$ (for the point-mass problem,
the calculation is terminated at $r\simeq 2R_o$).
In Figure 2 we clearly see that the non-dissipative effects
-- tidal and relativistic -- substantially increase the rate of
infall. The radial velocity at binary contact is comparable to the
tangential velocity. We also note that the radial coordinate
velocity is a gauge dependent quantity;
therefore our only intent in using it in Figure 2 is to
convey the general trend that the rate of infall is enhanced by the
dynamical instability.
Figure 3 shows the number of orbits the binary spends per logarithmic
frequency. In the simplest point-mass,
Newtonian-plus-radiation-reaction case
[case (i) above], the result can be calculated
analytically
\begin{eqnarray}
dN_{orb}/d\ln f_{orb} &=&(5/192\pi)\mu^{-1}M_t^{-2/3}
(2\pi f_{orb})^{-5/3}\nonumber\\
&=&1.95\times 10^5(M_{1.4}f_{orb}/{\rm Hz})^{-5/3},
\end{eqnarray}
which gives $6$ cycles at $f_{gw}\simeq
2f_{orb}=1000$ Hz. In contrast, the tidal and relativistic effects
reduce this number to less than $2$.
Figure 4 shows the wave energy emitted around a given frequency,
$dE_{gw}/d\ln f_{orb}=(\Omega_{orb}/\dot\Omega_{orb})\dot E_{gw}$,
where $\dot E_{gw}$ is calculated using the simple quadrupole
radiation formula. The Newtonian plus radiation reaction result is
$dE_{gw}/d\ln f_{orb}=1.63\times 10^{-3}(f_{orb}/
{\rm Hz})^{2/3}M^2/R_o$. We see that the radiation power near
contact becomes much smaller.
Note that $dE_{gw}/d\ln f_{orb}$ calcuated in this way is not
exactly the energy power spectrum, which must be obtained
from the Fourier transform of the waveform\cite{Kennefick96};
however, it does provide a semi-quantitative feature
of the full analysis; in particular, the dip in the
$dE_{gw}/d\ln f_{orb}$ curve around $600$ Hz results
from the dynamical instability of the orbit (see also
Refs.\cite{Centrella,Ruffert}, although the calculations presented
there are purely Newtonian).
\begin{figure}[t]
\special{hscale=52 vscale=55 hoffset=-50.0 voffset=-375.0
angle=0.0 psfile=fig4.eps}
\vspace*{3.1in}
\caption[Fig. 4]{
The quadrupolar gravitational energy emitted near a given frequency.
The labels are the same as in Fig.~2.
}
\end{figure}
\section{\bf DISCUSSION}
A number of authors have tried to define and locate the
innermost stable circular orbit for relativistic coalescing
systems of comparable masses
\cite{Wex93,Wilson95,clarkeardley,blackburndetweiler,cook}
in order to characterize the final moments of a binary coalescence
(See \cite{eardleyhirschnamm} for a discussion).
The results of the various analyses are not converging to an agreed
answer. Obviously, the precise nature of the final coalescence of two
neutron stars will only be determined by a numerical simulation
using full general-relativistic hydrodynamics.
However, the present analysis does point to two intersting features
to look for in a full numerical treatment:
(i) To get even a qualitative picture of the
coalescence, it is necessary to begin the numerical evolution
when the stars are still well separated, {\it i.e.} before
the onset of the orbital dynamical instability.
The instability -- the plunge -- causes the coalescence
to proceed much more swiftly than a coalescence driven
solely by radiation reaction; thus the actual coalescence may differ
qualitatively from one computed with a simple
radiation-reaction driven inspiral.
The final coalescence may be more of a splat,
than the slow winding together of the stars\cite{Nakamura}.
(ii) The instability results from both the tidal effects and
the relativistic corrections in the equations
of motion. KWW showed that there
is no instability in the {\it first} post-Newtonian relativistic
equations of motion; the instability does not show up until at least
second post-Newtonian order.
Therefore, in order for numerical simulations to see the
effects of the relativistic unstable plunge,
it will probably require the use of at least second-order, post-Newtonian
hydrodynamic code. KWW also showed that the location of
the dynamic instability does not converge very rapidly as
one increases the post-Newtonian order of the approximation.
[This fact led KWW to the introduce the hybrid equations of motion.]
Thus, to get even a qualitatively accurate evolution of the binary
near coalescence, it may be necessary to use a full general relativistic
hydrodynamic treatment of the coalescence problem, and begin
the evolution when the stars are still well separated.
As we have shown, the dynamical instability in the equation
of motion will, in effect, cut off the chirping waveform.
The frequency of the cut-off is somewhat dependent on the
neutron star equation of state. Only in this late stage of the
evolution does the equation of state leave a tell-tale sign in the
emitted waveform.
However, devising a strategy to dig this information
from the detector output requires further consideration.
Most detection/measurement strategies for coalescing binaries
involve integrating template waveforms against long stretches
(8000 orbits!) of raw output data,
the idea being that one can detect/measure
a relatively low amplitude signal by integrating for a long time.
Looking for the signature of this very late stage
plunge is precisely the opposite:
we are looking at the waveform just before coalescence
when the amplitude is fairly strong,
but the plunge is of fairly short in duration.
So answering questions about the plunge (such as, at what orbital
frequency did it begin?) requires measuring a relatively large
amplitude, but short-duration, effect.
Clearly, analysis of such events
will require a different detection strategy\cite{Kennefick96}.
\acknowledgments
We thank Kip Thorne for useful discussions. This work has been
supported by NSF Grants AST-9417371, PHY-9424337 and
NASA Grant NAGW-2756 to Caltech. DL also
acknowledges support of the Richard C. Tolman Fellowship at Caltech.
| proofpile-arXiv_065-481 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
\bigskip
Over the past decade, beginning with the measurement of nucleon
spin--polarized structure function, $g_{1}(x,Q^2)$ by the
EMC \cite{emc88} at CERN and most recently with the
spin--structure function $g_2(x,Q^2)$ in the E143
experiment \cite{slac96} at SLAC, a wealth of information has been
gathered on the spin--polarized structure functions of the nucleon
and their corresponding sum rules (see in addition
\cite{smc93}, \cite{smc94a}, \cite{smc94b}, \cite{slac93},
\cite{slac95a}, \cite{slac95b}).
Initially the analysis of these experiments cast doubt
on the non--relativistic quark model \cite{kok79}
interpretations regarding the spin content of the proton.
By now it is firmly established
that the quark helicity of the nucleon is much smaller than the
predictions of that model, however, many questions remain
to be addressed concerning the spin structure.
As a result there have been numerous investigations within
models for the nucleon in an effort to determine the manner in which
the nucleon spin is distributed among its constituents.
One option is to study
the axial current matrix elements of the
nucleon such as
$\langle N |{\cal A}_{\mu}^{i}|N\rangle = 2\Delta q_{i}S_{\mu}$,
which, for example, provide information on
the nucleon axial singlet charge
\begin{eqnarray}
g_{A}^{0}&=&\langle N |{\cal A}_{3}^{0}|N\rangle\
= \left( \Delta u + \Delta d + \Delta s\right)=
\Gamma_{1}^{p} (Q^2) + \Gamma_{1}^{n} (Q^2) \ .
\end{eqnarray}
Here $\Delta q$ are the axial charges of the quark constituents and
$\Gamma_{1}^{N} (Q^2)=\int_{0}^{1} dx g_{1}^{N}(x,Q^2)$ is the
first moment of the longitudinal nucleon spin structure function,
$g_1^N\left(x,Q^{2}\right)$.
Of course, it is more illuminating to
directly compute the longitudinal and transverse
nucleon spin--structure functions, $g_{1}\left(x,Q^2\right)$
and $g_{T}(x,Q^2)=g_{1}(x,Q^2)+g_{2}(x,Q^2)$, respectively as
functions of the Bjorken variable $x$.
We will calculate these structure functions within the
Nambu--Jona--Lasinio (NJL) \cite{Na61} chiral soliton model \cite{Re88}.
Chiral soliton models are unique both in being
the first effective models of hadronic physics to shed light
on the so called ``proton--spin crisis" by predicting a singlet
combination in accord with the data \cite{br88}, and in predicting a
non--trivial strange quark content to the axial vector current of the
nucleon \cite{br88}, \cite{pa89}, \cite{jon90}, \cite{blo93};
about $10-30\%$ of the down quarks
(see \cite{wei96a} and \cite{ell96} for reviews).
However, while the leading moments of these
structure functions have been calculated within chiral soliton
models, from the Skyrme model \cite{Sk61}, \cite{Ad83} and its various
vector--meson extensions, to
models containing explicit quark degrees of freedom such as the
(NJL) model \cite{Na61},
the nucleon spin--structure functions
themselves have not been investigated in these models.
Soliton model calculations of structure functions
were, however, performed in
Friedberg-Lee \cite{frie77} and color-dielectric \cite{nil82} models.
In addition, structure functions have extensively been studied
within the framework of effective quark models such as the
bag--model \cite{Ch74}, and the Center of Mass
bag model \cite{So94}.
These models are confining by construction but they
neither contain non--perturbative pseudoscalar fields nor are they
chirally symmetric\footnote{In the
cloudy bag model the contribution of the pions to structure
functions has at most been treated perturbatively
\cite{Sa88}, \cite{Sc92}.}.
To this date it is fair to say that
many of the successes of low--energy effective models rely on
the incorporation of chiral symmetry and its spontaneous
symmetry breaking (see for e.g. \cite{Al96}).
In this article we therefore present our
calculation of the polarized spin structure functions in the NJL
chiral soliton model \cite{Re89}, \cite{Al96}.
Since in particular the static axial properties of the nucleon are
dominated by the valence quark contribution in this model
it is legitimate to focus on the valence quarks in this model.
At the outset it is important to note that a major difference
between the chiral soliton models and models previously employed
to calculate structure functions is the form of the nucleon
wave--function. In the latter the nucleon wave--function is a
product of Dirac spinors while in the former the nucleon appears as
a collectively excited (topologically) non--trivial meson
configuration.
As in the original bag model study \cite{Ja75}
of structure functions for
localized field configurations, the structure functions are most
easily accessible when the current operator is at most quadratic in
the fundamental fields and the propagation of the interpolating
field can be regarded as free.
Although the latter approximation is
well justified in the Bjorken limit the former condition is
difficult to satisfy in soliton models where mesons
are fundamental fields ({\it e.g.} the Skyrme
model \cite{Sk61}, \cite{Ad83},
the chiral quark model of ref. \cite{Bi85} or the chiral bag model
\cite{Br79}).
Such model Lagrangians typically possess all orders of the fundamental
pion field. In that case the current operator is not confined to
quadratic order and the calculation of the hadronic tensor
(see eq. (\ref{deften}) below) requires drastic approximations.
In this respect the chirally invariant NJL model
is preferred because it is entirely defined in terms of quark
degrees of freedom and formally the current
possesses the structure as in a non--interacting
model. This makes the evaluation of the hadronic tensor
feasible.
Nevertheless after bosonization
the hadronic currents
are uniquely defined
functionals of the solitonic meson fields.
The paper is organized as follows: In section 2 we give a brief
discussion of the standard operator product expansion (OPE) analysis
to establish the connection between the effective models for the
baryons at low energies and the quark--parton model description.
In section 3 we briefly review the NJL chiral soliton.
In section 4 we extract the polarized structure
functions from the hadronic tensor, eq. (\ref{had})
exploiting the ``valence quark approximation".
Section 5 displays the results of the spin--polarized
structure functions calculated in the NJL chiral soliton model
within this approximation and compare this result with a recent
low--renormalization point parametrization \cite{Gl95}.
In section 6 we use Jaffe's prescription \cite{Ja80} to impose
proper support for the structure
function within the interval $x\in \left[0,1\right]$.
Subsequently the structure functions
are evolved \cite{Al73}, \cite{Al94}, \cite{Ali91}
from the scale characterizing the NJL--model to the
scale associated with the experimental data. Section 7 serves to
summarize these studies and to propose further explorations.
In appendix A we list explicit analytic expressions for the isoscalar
and isovector polarized structure functions.
Appendix B summarizes details on the evolution of the twist--3
structure function, ${\overline{g}}_2\left(x,Q^2\right)$.
\bigskip
\section{DIS and the Chiral Soliton}
\bigskip
It has been a long standing effort to establish the connection
between the chiral soliton picture of the baryon, which essentially
views baryons as mesonic lumps and the quark parton model which
regards baryons as composites of almost non--interacting, point--like
quarks. While the former has been quite successful in describing
static properties of the nucleon, the latter, being firmly
established within the context of deep inelastic scattering (DIS),
has been employed extensively to calculate the short distance or
perturbative processes within QCD.
In fact this connection can be made through the OPE.
The discussion begins with the hadronic tensor for electron--nucleon
scattering,
\begin{eqnarray}
W_{\mu\nu}(q)=\frac{1}{4\pi}\int d^4 \xi \
{\rm e}^{iq\cdot\xi}
\langle N |\left[J_\mu(\xi),J^{\dag}_\nu(0)\right]|N\rangle\ ,
\label{deften}
\end{eqnarray}
where $J_\mu={\bar q}(\xi)\gamma_\mu {\cal Q} q(\xi)$ is the
electromagnetic
current, ${\cal Q}=\left(\frac{2}{3},\frac{-1}{3}\right)$ is the (two
flavor) quark charge matrix and $|N\rangle$ refers to the
nucleon state. In the DIS regime the OPE enables one to
express the product of these
currents in terms of the forward Compton scattering
amplitude $T_{\mu\nu}(q)$ of a virtual photon
from a nucleon
\begin{eqnarray}
T_{\mu\nu}(q)=i\int d^4 \xi \
{\rm e}^{iq\cdot\xi}
\langle N |T\left(J_\mu(\xi)J^{\dag}_\nu(0)\right)|N\rangle\ ,
\label{im}
\end{eqnarray}
by an expansion on the light cone $\left(\xi^2 \rightarrow 0\right)$
using a set of renormalized local
operators \cite{muta87}, \cite{rob90}. In the Bjorken
limit the influence of these operators is determined
by the twist, $\tau$ or the light cone singularity of their coefficient
functions. Effectively this becomes a power series in the inverse
of the Bjorken variable $x=-q^{2}/2P\cdot q$, with $P_\mu$ being the
nucleon momentum:
\begin{eqnarray}
T_{\mu\nu}(q)\ =\sum_{n,i,\tau}
\left(\frac{1}{x}\right)^{n}\ e_{\mu\nu}^{i}\left(q,P,S\right)\
C^{n}_{\tau,i}(Q^2/\mu^2,\alpha_s(\mu^2)){\cal O}^{n}_{\tau,i}(\mu^2)
(\frac{1}{Q^2})^{\frac{\tau}{2}\ - 1}\ .
\label{series}
\end{eqnarray}
Here the index $i$ runs over all scalar matrix
elements, ${\cal O}_{\tau,i}^{n}(\mu^2)$, with the
same Lorentz structure (characterized by
the tensor, $e_{\mu\nu}^{i}$). Furthermore,
$S^{\mu}$ is the spin
of the nucleon,
$\left(S^2=-1\ , S\cdot P\ =0\right)$ and $Q^2=-q^2 > 0$.
As is evident, higher twist contributions
are suppressed by powers of $1/{Q^2}$.
The coefficient functions,
$C^{n}_{\tau,i}(Q^2/\mu^2,\alpha_s(\mu^2))$
are target independent and in principle include all
QCD radiative corrections. Their $Q^2$
variation is determined from the solution of the renormalization
group equations and logarithmically diminishes at large $Q^2$.
On the other hand the reduced--matrix elements,
${\cal O}_{\tau,i}^{n}(\mu^2)$,
depend only on the renormalization
scale $\mu^2$ and reflect the non--perturbative properties
of the nucleon \cite{ans95}.
The optical theorem states that the hadronic tensor is
given in terms of the imaginary part of the virtual Compton scattering
amplitude, $W_{\mu\nu}=\frac{1}{2\pi}{\rm Im}\ T_{\mu\nu}$.
From the analytic properties of $T_{\mu\nu}(q)$,
together with eq. (\ref{series}) an infinite set of sum rules
result for the form factors, ${\cal W}_{i}\left(x,Q^2\right)$,
which are defined via the Lorentz covariant decomposition
$W_{\mu\nu}(q)=e_{\mu\nu}^{i}{\cal W}_i\left(x,Q^2\right)$.
These sum rules read
\begin{eqnarray}
\int^{1}_{0}dx\ x^{n-1}\ {\cal W}_{i}\left(y,Q^2\right)&=&
\sum _{\tau}\
C^{n}_{\tau,i}\left(Q^2/\mu^2,\alpha_s(\mu^2)\right)
{\cal O}_{\tau,i}^{n}(\mu^2)
(\frac{1}{Q^2})^{\frac{\tau}{2}\ - 1}\ .
\label{ope}
\end{eqnarray}
In the
{\em impulse approximation}
(i.e. neglecting radiative
corrections) \cite{Ja90,Ji90,Ja91}
one can directly sum the OPE gaining direct access
to the structure functions in terms of the reduced matrix elements
${\cal O}_{\tau,i}^{n}(\mu^2)$.
When calculating the renormalization--scale dependent
matrix elements, ${\cal O}_{\tau,i}^{n}(\mu^2)$
within QCD, $\mu^2$ is an arbitrary parameter adjusted to ensure
rapid convergence of the perturbation series.
However, given the difficulties of obtaining a
satisfactory description of the nucleon
as a bound--state in the $Q^2$ regime of DIS processes
it is customary to calculate these
matrix elements in models at a
low scale $\mu^2$ and subsequently evolve these results
to the relevant DIS momentum region of the data
employing, for example, the
Altarelli--Parisi evolution \cite{Al73}, \cite{Al94}.
In this context, the scale, $\mu^2 \sim \Lambda_{QCD}^{2}$,
characterizes the non--perturbative regime
where it is possible to formulate a nucleon
wave--function from which structure functions
are computed.
Here we will utilize the NJL chiral--soliton model to
calculate the spin--polarized nucleon structure functions
at the scale, $\mu^2$, subsequently evolving
the structure functions according to the Altarelli--Parisi scheme.
This establishes the connection between chiral soliton and the
parton models. In addition we compare the structure functions
calculated in the NJL model to a parameterization of spin structure
function \cite{Gl95} at a scale commensurate with our model.
\bigskip
\section{The Nucleon State in the NJL Model}
\bigskip
The Lagrangian of the NJL model reads
\begin{eqnarray}
{\cal L} = \bar q (i\partial \hskip -0.5em / - m^0 ) q +
2G_{\rm NJL} \sum _{i=0}^{3}
\left( (\bar q \frac {\tau^i}{2} q )^2
+(\bar q \frac {\tau^i}{2} i\gamma _5 q )^2 \right) .
\label{NJL}
\end{eqnarray}
Here $q$, $\hat m^0$ and $G_{\rm NJL}$ denote the quark field, the
current quark mass and a dimensionful coupling constant, respectively.
When integration out the gluon fields from QCD a current--current
interaction remains, which is meditated by the gluon propagator.
Replacing this gluon propagator by a local contact interaction and
performing the appropriate Fierz--transformations yields the
Lagrangian (\ref{NJL}) in leading order of $1/N_c$ \cite{Re90},
where $N_c$ refers to the number of color degrees of freedom. It is
hence apparent that the interaction term in eq. (\ref{NJL}) is a
remnant of the gluon fields. Hence gluonic effects are included
in the model described by the Lagrangian (\ref{NJL}).
Application of functional bosonization techniques \cite{Eb86} to the
Lagrangian (\ref{NJL}) yields the mesonic action
\begin{eqnarray}
{\cal A}&=&{\rm Tr}_\Lambda\log(iD)+\frac{1}{4G_{\rm NJL}}
\int d^4x\ {\rm tr}
\left(m^0\left(M+M^{\dag}\right)-MM^{\dag}\right)\ ,
\label{bosact} \\
D&=&i\partial \hskip -0.5em /-\left(M+M^{\dag}\right)
-\gamma_5\left(M-M^{\dag}\right)\ .
\label{dirac}
\end{eqnarray}
The composite scalar ($S$) and pseudoscalar ($P$) meson fields
are contained in $M=S+iP$ and appear as quark--antiquark bound
states. The NJL model embodies the approximate chiral symmetry of QCD
and has to be understood as an effective (non--renormalizable) theory
of the low--energy quark flavor dynamics.
For regularization, which is indicated by the cut--off
$\Lambda$, we will adopt the proper--time scheme \cite{Sch51}.
The free parameters of the model are the current quark mass $m^0$,
the coupling constant $G_{\rm NJL}$ and the cut--off $\Lambda$.
Upon expanding ${\cal A}$ to quadratic order in $M$ these parameters are
related to the pion mass, $m_\pi=135{\rm MeV}$ and pion decay constant,
$f_\pi=93{\rm MeV}$. This leaves one undetermined parameter which we
choose to be the vacuum expectation value $m=\langle M\rangle$. For
apparent reasons $m$ is called the constituent quark mass. It is
related to $m^0$, $G_{\rm NJL}$ and $\Lambda$ via the gap--equation,
{\it i.e.} the equation of motion for the scalar field $S$\cite{Eb86}. The
occurrence of this vacuum expectation value reflects the spontaneous
breaking of chiral symmetry and causes the pseudoscalar fields to
emerge as (would--be) Goldstone bosons.
As the NJL model soliton has exhaustively been discussed in
recent review articles \cite{Al96}, \cite{Gok96}
we only present those features,
which are relevant for the computation of the structure functions
in the valence quark approximation.
The chiral soliton is given by the hedgehog configuration
of the meson fields
\begin{eqnarray}
M_{\rm H}(\mbox{\boldmath $x$})=m\ {\rm exp}
\left(i\mbox{\boldmath $\tau$}\cdot{\hat{\mbox{\boldmath $x$}}}
\Theta(r)\right)\ .
\label{hedgehog}
\end{eqnarray}
In order to compute the functional trace in eq. (\ref{bosact}) for this
static configuration we express the
Dirac operator (\ref{dirac}) as, $D=i\gamma_0(\partial_t-h)$
where
\begin{eqnarray}
h=\mbox{\boldmath $\alpha$}\cdot\mbox{\boldmath $p$}+m\
{\rm exp}\left(i\gamma_5\mbox{\boldmath $\tau$}
\cdot{\hat{\mbox{\boldmath $x$}}}\Theta(r)\right)\
\label{hamil}
\end{eqnarray}
is the corresponding Dirac Hamiltonian. We denote the eigenvalues
and eigenfunctions of $h$ by $\epsilon_\mu$ and $\Psi_\mu$,
respectively. Explicit expressions for these wave--functions are
displayed in appendix A. In the proper time regularization scheme
the energy functional of the NJL model is found to be \cite{Re89,Al96},
\begin{eqnarray}
E[\Theta]=
\frac{N_C}{2}\epsilon_{\rm v}
\left(1+{\rm sgn}(\epsilon_{\rm v})\right)
&+&\frac{N_C}{2}\int^\infty_{1/\Lambda^2}
\frac{ds}{\sqrt{4\pi s^3}}\sum_\nu{\rm exp}
\left(-s\epsilon_\nu^2\right)
\nonumber \\* && \hspace{1.5cm}
+\ m_\pi^2 f_\pi^2\int d^3r \left(1-{\rm cos}\Theta(r)\right) ,
\label{efunct}
\end{eqnarray}
with $N_C=3$ being the number of color degrees of freedom.
The subscript ``${\rm v}$" denotes the valence quark level. This state
is the distinct level bound in the soliton background, {\it i.e.}
$-m<\epsilon_{\rm v}<m$. The chiral angle, $\Theta(r)$, is
obtained by self--consistently extremizing $E[\Theta]$ \cite{Re88}.
States possessing good spin and isospin quantum numbers are
generated by rotating the hedgehog field
\cite{Ad83}
\begin{eqnarray}
M(\mbox{\boldmath $x$},t)=
A(t)M_{\rm H}(\mbox{\boldmath $x$})A^{\dag}(t)\ ,
\label{collrot}
\end{eqnarray}
which introduces the collective coordinates $A(t)\in SU(2)$. The
action functional is expanded \cite{Re89} in the angular velocities
\begin{eqnarray}
2A^{\dag}(t)\dot A(t)=
i\mbox{\boldmath $\tau$}\cdot\mbox{\boldmath $\Omega$} \ .
\label{angvel}
\end{eqnarray}
In particular the valence quark wave--function receives a first
order perturbation
\begin{eqnarray}
\Psi_{\rm v}(\mbox{\boldmath $x$},t)=
{\rm e}^{-i\epsilon_{\rm v}t}A(t)
\left\{\Psi_{\rm v}(\mbox{\boldmath $x$})
+\frac{1}{2}\sum_{\mu\ne{\rm v}}
\Psi_\mu(\mbox{\boldmath $x$})
\frac{\langle \mu |\mbox{\boldmath $\tau$}\cdot
\mbox{\boldmath $\Omega$}|{\rm v}\rangle}
{\epsilon_{\rm v}-\epsilon_\mu}\right\}=:
{\rm e}^{-i\epsilon_{\rm v}t}A(t)
\psi_{\rm v}(\mbox{\boldmath $x$}).
\label{valrot}
\end{eqnarray}
Here $\psi_{\rm v}(\mbox{\boldmath $x$})$ refers to the spatial part
of the body--fixed valence quark wave--function with the rotational
corrections included. Nucleon states $|N\rangle$ are obtained
by canonical quantization of the collective coordinates, $A(t)$. By
construction these states live in the Hilbert space of a rigid rotator.
The eigenfunctions are Wigner $D$--functions
\begin{eqnarray}
\langle A|N\rangle=\frac{1}{2\pi}
D^{1/2}_{I_3,-J_3}(A)\ ,
\label{nwfct}
\end{eqnarray}
with $I_3$ and $J_3$ being respectively the isospin and spin
projection quantum numbers of the nucleon.
\bigskip
\section{Polarized Structure Functions in the NJL model}
\bigskip
The starting point for computing nucleon structure functions
is the hadronic tensor, eq. (\ref{deften}). The polarized structure
functions are extracted from its antisymmetric
piece, $W^{(A)}_{\mu\nu}=(W_{\mu\nu}-W_{\nu\mu})/2i$.
Lorentz invariance implies that the
antisymmetric portion, characterizing polarized
lepton--nucleon scattering, can be decomposed into
the polarized structure functions,
$g_1(x,Q^2)$ and $g_2(x,Q^2)$,
\begin{eqnarray}
W^{(A)}_{\mu\nu}(q)=
i\epsilon_{\mu\nu\lambda\sigma}\frac{q^{\lambda}M_N}{P\cdot q}
\left\{g_1(x,Q^2)S^{\sigma}+
\left(S^{\sigma}-\frac{q\cdot S}{q\cdot p}P^{\sigma}\right)
g_2(x,Q^2)\right\}\ ,
\label{had}
\end{eqnarray}
again, $P_\mu$ refers to the nucleon momentum and $Q^2=-q^2$.
The tensors multiplying the structure functions
in eq. (\ref{had})
should be identified with the Lorentz tensors $e_{\mu\nu}^{i}$
in (\ref{series}).
Contracting $W^{(A)}_{\mu\nu}$ with the longitudinal
$\Lambda^{\mu\nu}_{L}$ and transverse
$\Lambda^{\mu\nu}_{T}$ projection operators \cite{ans95},
\begin{eqnarray}
\Lambda^{\mu\nu}_{L}&=&\frac{2}{b}\left\{2P\cdot qxS_{\lambda}+
\frac{1}{q\cdot S}\left[(q\cdot S)^{2}-
\left(\frac{P\cdot q}{M}\right)^2\right] q_{\lambda}\right\}\ P_\tau \
\epsilon^{\mu\nu\lambda\tau },
\label{proj1}\\
\Lambda^{\mu\nu}_{T}&=&\frac{2}{b}
\left\{\left[\left(\frac{P\cdot q}{M}\right)^2+2P\cdot \
qx\right]S_\lambda + \left(q\cdot S\right)q_\lambda\right\}\ P_\tau \
\epsilon^{\mu\nu\lambda\tau }
\label{projT}
\end{eqnarray}
and choosing the pertinent polarization,
yields the longitudinal component
\begin{eqnarray}
g_L(x,Q^2)=g_1(x,Q^2)\ ,
\end{eqnarray}
as well as the transverse combination
\begin{eqnarray}
g_T(x,Q^2)=g_1(x,Q^2) + g_2(x,Q^2)\ .
\end{eqnarray}
Also, $b=-4M\left\{\left(\frac{P\cdot q}{M}\right)^2 + 2P\cdot{qx}-
\left(q\cdot{S}\right)^2\right\}$. In the Bjorken limit, which
corresponds to the kinematical regime
\begin{eqnarray}
q_0=|\mbox{\boldmath $q$}| - M_N x
\quad {\rm with}\quad
|\mbox{\boldmath $q$}|\rightarrow \infty \ ,
\label{bjlimit}
\end{eqnarray}
the antisymmetric component of the hadronic tensor
becomes \cite{Ja75},
\begin{eqnarray}
W^{(A)}_{\mu\nu}(q)&=&\int \frac{d^4k}{(2\pi)^4} \
\epsilon_{\mu\rho\nu\sigma}\ k^\rho\
{\rm sgn}\left(k_0\right) \ \delta\left(k^2\right)
\int_{-\infty}^{+\infty} dt \ {\rm e}^{i(k_0+q_0)t}
\nonumber \\* &&
\times \int d^3x_1 \int d^3x_2 \
{\rm exp}\left[-i(\mbox{\boldmath $k$}+\mbox{\boldmath $q$})\cdot
(\mbox{\boldmath $x$}_1-\mbox{\boldmath $x$}_2)\right]
\nonumber \\* &&
\times \langle N |\left\{
{\bar \Psi}(\mbox{\boldmath $x$}_1,t){\cal Q}^2\gamma^\sigma\gamma^{5}
\Psi(\mbox{\boldmath $x$}_2,0)+
{\bar \Psi}(\mbox{\boldmath $x$}_2,0){\cal Q}^2\gamma^\sigma\gamma^{5}
\Psi(\mbox{\boldmath $x$}_1,t)\right\}| N \rangle \ ,
\label{stpnt}
\end{eqnarray}
where $\epsilon_{\mu\rho\nu\sigma}\gamma^\sigma \gamma^5$
is the
antisymmetric combination of $\gamma_\mu\gamma_\rho\gamma_\nu$.
The matrix element between the nucleon states is to be taken in
the space of the collective coordinates, $A(t)$ (see eqs.
(\ref{collrot}) and (\ref{nwfct})) as the object in curly brackets
is an operator in this space. In deriving the expression (\ref{stpnt})
the {\it free} correlation function for the intermediate quark
fields has been assumed\footnote{Adopting a dressed correlation will
cause corrections starting at order twist--4 in QCD \cite{Ja96}.}
after applying Wick's theorem to the product of quark currents in eq.
(\ref{deften}). \cite{Ja75}. The use of the {\it free} correlation
function is justified because in the Bjorken limit (\ref{bjlimit})
the intermediate quark fields carry very large momenta and are hence
not sensitive to typical soliton momenta. This procedure reduces the
commutator $[J_\mu(\mbox{\boldmath $x$}_1,t),
J^{\dag}_\nu(\mbox{\boldmath $x$}_2,0)]$ of the quark currents in
the definition (\ref{deften}) to objects which are merely bilinear
in the quark fields. Consequently, in the Bjorken limit
(\ref{bjlimit}) the momentum, $k$, of the intermediate quark state
is highly off--shell and hence is not sensitive to momenta typical for
the soliton configuration. Therefore, the use of the free correlation
function is a good approximation in this kinematical regime.
Accordingly, the intermediate quark states are taken to be massless,
{\it cf.} eq. (\ref{stpnt}).
Since the NJL model is originally defined in terms of quark degrees of
freedom, quark bilinears as in eq. (\ref{stpnt}) can be computed
from the functional
\begin{eqnarray}
\hspace{-1cm}
\langle {\bar q}(x){\cal Q}^{2} q(y) \rangle&=&
\int D{\bar q} Dq \ {\bar q}(x){\cal Q}^{2} q(y)\
{\rm exp}\left(i \int d^4x^\prime\ {\cal L}\right)
\nonumber \\*
&=&\frac{\delta}{i\delta\alpha(x,y)}\int D{\bar q} Dq \
{\rm exp}\left(i\int d^4x^\prime d^4y^\prime
\left[\delta^4(x^\prime - y^\prime ){\cal L} \right. \right.
\nonumber \\* && \hspace{5cm}
\left. \left.
+\ \alpha(x^\prime,y^\prime){\bar q}(x^\prime){\cal Q}^{2}
q(y^\prime)\right] \right)\Big|_{\alpha(x,y)=0}\ .
\label{gendef}
\end{eqnarray}
The introduction of the bilocal source $\alpha(x,y)$ facilitates
the functional bosonization after which eq. (\ref{gendef})
takes the form
\begin{eqnarray}
\frac{\delta}{\delta\alpha(x,y)}{\rm Tr}_{\Lambda}{\rm log}
\left(\delta^4(x-y)D+\alpha(x,y){\cal Q}^{2})\right)
\Big|_{\alpha(x,y)=0}\ \ .
\label{gendef1}
\end{eqnarray}
The operator $D$ is defined in eq. (\ref{dirac}).
The correlation $\langle {\bar q}(x){\cal Q}^2 q(y) \rangle$ depends on
the angle between $\mbox{\boldmath $x$}$ and $\mbox{\boldmath $y$}$.
Since in general the functional (\ref{gendef}) involves quark states
of all angular momenta ($l$) a technical difficulty arises because
this angular dependence has to be treated numerically. The major
purpose of the present paper is to demonstrate that polarized
structure functions can indeed be computed from a chiral soliton.
With this in mind we will adopt the valence quark approximation
where the quark configurations in (\ref{gendef}) are restricted to the
valence quark level. Accordingly the valence quark wave--function
(\ref{valrot}) is substituted into eq. (\ref{stpnt}). Then only quark
orbital angular momenta up to $l=2$ are relevant. From a physical
point of view this approximation is justified for moderate constituent
quark masses ($m\approx400{\rm MeV}$) because in that parameter
region the soliton properties are dominated by their valence quark
contributions \cite{Al96}, \cite{Gok96}. In particular this is the
case for the axial properties of the nucleon.
In the next step the polarized structure functions, $g_1(x,\mu^2)$
and $g_T(x,\mu^2)$, are extracted according to eqs. (\ref{proj1})
and (\ref{projT}). In the remainder of this section we will omit
explicit reference to the scale $\mu^2$.
We choose the frame such that the nucleon is
polarized along the
positive--$\mbox{\boldmath $z$}$ and positive--$\mbox{\boldmath $x$}$
directions in the longitudinal and transverse cases, respectively.
Note also that this implies
the choice ${\mbox{\boldmath $q$}}=q\hat{\mbox{\boldmath $z$}}$.
When extracting the structure functions the integrals
over the time coordinate in eq. (\ref{stpnt}) can readily be done yielding the conservation
of energy for forward and backward moving intermediate quarks. Carrying
out the integrals over $k_0$ and $k=|\mbox{\boldmath $k$}|$ gives for
the structure functions
\begin{eqnarray}
\hspace{-1cm}
g_1(x)&=&-N_C\frac{M_N}{\pi}
\langle N,\frac{1}{2}\hat{\mbox{\boldmath $z$}}|\int d\Omega_{\mbox{\boldmath
$k$}} k^2
\Bigg\{\tilde\psi_{\rm v}^{\dag}(\mbox{\boldmath $p$})
\left(1-\mbox{\boldmath $\alpha$}\cdot
{\hat{\mbox{\boldmath $k$}}}\right)\gamma^5\Gamma
\tilde\psi_{\rm v}(\mbox{\boldmath $p$})
\Big|_{k=q_0+\epsilon_{\rm v}}
\nonumber \\* && \hspace{3cm}
+\tilde\psi_{\rm v}^{\dag}(-\mbox{\boldmath $p$})
\left(1-\mbox{\boldmath $\alpha$}\cdot
{\hat{\mbox{\boldmath $k$}}}\right)\gamma^5\Gamma
\tilde\psi_{\rm v}(-\mbox{\boldmath $p$})
\Big|_{k=q_0-\epsilon_{\rm v}}
\Bigg\} |N,\frac{1}{2}\hat{\mbox{\boldmath $z$}}\rangle\ ,
\label{valg1}\\
\hspace{-1cm}
g_{T}(x)&=&g_1(x)+g_2(x)
\nonumber \\*
&=&-N_C\frac{M_N}{\pi} \langle N,\frac{1}{2}\hat{\mbox{\boldmath $x$}} |
\int d\Omega_{\mbox{\boldmath $k$}} k^2
\Bigg\{\tilde\psi_{\rm v}^{\dag}(\mbox{\boldmath $p$})
\left(\mbox{\boldmath $\alpha$}\cdot
{\hat{\mbox{\boldmath $k$}}}\right)\gamma^5\Gamma
\tilde\psi_{\rm v}(\mbox{\boldmath $p$})
\Big|_{k=q_0+\epsilon_{\rm v}}
\nonumber \\* && \hspace{3cm}
+\tilde\psi_{\rm v}^{\dag}(-\mbox{\boldmath $p$})
\left(\mbox{\boldmath $\alpha$}\cdot
{\hat{\mbox{\boldmath $k$}}}\right)\gamma^5\Gamma
\tilde\psi_{\rm v}(-\mbox{\boldmath $p$})
\Big|_{k=q_0-\epsilon_{\rm v}}
\Bigg\} |N,\frac{1}{2}\hat{\mbox{\boldmath $x$}} \rangle\ ,
\label{valgt}
\end{eqnarray}
where $\mbox{\boldmath $p$}=\mbox{\boldmath $k$}+\mbox{\boldmath $q$}$
and $\Gamma =\frac{5}{18}{\mbox{{\sf 1}\zr{-0.16}\rule{0.04em}{1.55ex}\zr{0.1}}} +\frac{1}{6}D_{3i}\tau_{i}$
with $D_{ij}=\frac{1}{2}\
{\rm tr}\left(\tau_{i}A(t)\tau_{j}A^{\dagger}\right)$ being the
adjoint representation of the collective
rotation {\it cf.} eq. (\ref{collrot}).
The second entry
in the states labels the spin orientation.
$N_C$ appears as a multiplicative factor
because the functional trace (\ref{gendef1}) includes the color
trace as well. Furthermore the Fourier transform of the
valence quark wave--function
\begin{eqnarray}
\tilde\psi_{\rm v}(\mbox{\boldmath $p$})=\int \frac{d^3x}{4\pi}\
\psi_{\rm v}(\mbox{\boldmath $x$})\
{\rm exp}\left(i\mbox{\boldmath $p$}\cdot
\mbox{\boldmath $x$}\right)
\label{ftval}
\end{eqnarray}
has been introduced. Also, note that the wave--function $\psi_{\rm v}$
contains an implicit dependence on the collective coordinates through
the angular velocity $\mbox{\boldmath $\Omega$}$, {\it cf.}
eq. (\ref{valrot}).
The dependence of the wave--function
$\tilde\psi(\pm\mbox{\boldmath $p$})$ on the integration variable
${\hat{\mbox{\boldmath $k$}}}$ is only implicit.
In the Bjorken
limit the integration variables may then be changed to \cite{Ja75}
\begin{eqnarray}
k^2 \ d\Omega_{\mbox{\boldmath $k$}} =
p dp\ d\Phi\ , \qquad p=|\mbox{\boldmath $p$}|\ ,
\label{intdp}
\end{eqnarray}
where $\Phi$ denotes the azimuth--angle between
$\mbox{\boldmath $q$}$ and $\mbox{\boldmath $p$}$.
The lower bound for the $p$--integral is adopted when
$\mbox{\boldmath $k$}$ and $\mbox{\boldmath $q$}$ are anti--parallel;
$p^{\rm min}_\pm=|M_N x\mp \epsilon_{\rm v}|$
for $k=-\left(q_0\pm\epsilon_{\rm v}\right)$,
respectively. Since the wave--function
$\tilde\psi(\pm\mbox{\boldmath $p$})$ acquires its dominant
support for $p\le M_N$ the integrand is different from
zero only when $\mbox{\boldmath $q$}$ and $\mbox{\boldmath $k$}$
are anti--parallel. We may therefore take
${\hat{\mbox{\boldmath $k$}}}=-{\hat{\mbox{\boldmath $z$}}}$.
This is nothing but the light--cone description for computing
structure functions \cite{Ja91}. Although expected, this result is
non--trivial and will only come out in models which have a current
operator which, as in QCD, is formally identical to the one of
non--interacting quarks. The valence quark state possesses positive
parity yielding
$\tilde\psi(-\mbox{\boldmath $p$})=\gamma_0
\tilde\psi(\mbox{\boldmath $p$})$.
With this we arrive at the expression for the isoscalar
and isovector parts of the
polarized structure function in the valence quark approximation,
\begin{eqnarray}
\hspace{-.5cm}
g^{I=0}_{1,\pm}(x)&=&-N_C\frac{5\ M_N}{18\pi}
\langle N,\frac{1}{2}\hat{\mbox{\boldmath $z$}}|
\int^\infty_{M_N|x_\mp|}p dp \int_0^{2\pi}d\Phi\
\nonumber \\* && \hspace{4cm}\times
\tilde\psi_{\rm v}^{\dag}(\mbox{\boldmath $p$}_\mp)
\left(1\pm\alpha_3\right)\gamma^5\tilde\psi_{\rm v}(\mbox{\boldmath $p$}_\mp)
|N,\frac{1}{2}\hat{\mbox{\boldmath $z$}}\rangle
\label{g10} \\
\hspace{-.5cm}
g^{I=1}_{1,\pm}(x)&=&-N_C\frac{M_N}{6\pi}
\langle N,\frac{1}{2}\hat{\mbox{\boldmath $z$}}| D_{3i}
\int^\infty_{M_N|x_\mp|}p dp \int_0^{2\pi}d\Phi\
\nonumber \\* && \hspace{4cm}\times
\tilde\psi_{\rm v}^{\dag}(\mbox{\boldmath $p$}_\mp)\tau_i
\left(1\pm\alpha_3\right)\gamma^5\tilde\psi_{\rm v}(\mbox{\boldmath $p$}_\mp)
|N,\frac{1}{2}\hat{\mbox{\boldmath $z$}}\rangle\ ,
\label{g11}\\
\hspace{-.5cm}
g^{I=0}_{T,\pm}(x)&=&-N_C\frac{5\ M_N}{18\pi}
\langle N,\frac{1}{2}\hat{\mbox{\boldmath $x$}}|
\int^\infty_{M_N|x_\mp|}p dp \int_0^{2\pi}d\Phi\
\nonumber \\* && \hspace{4cm}\times
\tilde\psi_{\rm v}^{\dag}(\mbox{\boldmath $p$}_\mp)
\alpha_3\gamma^5\tilde\psi_{\rm v}(\mbox{\boldmath $p$}_\mp)
|N,\frac{1}{2}\hat{\mbox{\boldmath $x$}}\rangle
\label{gt0}\ , \\
\hspace{-.5cm}
g^{I=1}_{T,\pm}(x)&=&-N_C\frac{M_N}{6\pi}
\langle N,\frac{1}{2}\hat{\mbox{\boldmath $x$}}| D_{3i}
\int^\infty_{M_N|x_\mp|}p dp \int_0^{2\pi}d\Phi\
\nonumber \\* && \hspace{4cm}\times
\tilde\psi_{\rm v}^{\dag}(\mbox{\boldmath $p$}_\mp)\tau_i
\alpha_3\gamma^5\tilde\psi_{\rm v}(\mbox{\boldmath $p$}_\mp)
|N,\frac{1}{2}\hat{\mbox{\boldmath $x$}}\rangle\ ,
\label{gt1}
\end{eqnarray}
where $x_{\pm}=x\pm\epsilon_{\rm v}/{M_N}$ and
${\rm cos}(\Theta^\pm_p)={M_N}x_\pm/{p}$.
The complete structure functions are given by
\begin{eqnarray}
g_{1}(x)&=&g^{I=0}_{1,+}(x)+g^{I=1}_{1,+}(x)
-\left(g^{I=0}_{1,-}(x)-g^{I=1}_{1,-}(x)\right)
\label{gone} \\* \hspace{-1cm}
g_{T}(x)&=&g^{I=0}_{T,+}(x)+g^{I=1}_{T,+}(x)
-\left(g^{I=0}_{T,-}(x)-g^{I=1}_{T,-}(x)\right)\ .
\label{gtran}
\end{eqnarray}
Note also, that we have made explicit the isoscalar
$\left(I=0\right)$
and isovector $\left(I=1\right)$ parts.
The wave--function implicitly depends
on $x$ because
$\tilde\psi_{\rm v}(\mbox{\boldmath $p$}_\pm)=
\tilde\psi_{\rm v}(p,\Theta^\pm_p,\Phi)$
where the polar--angle, $\Theta^\pm_p$, between $\mbox{\boldmath $p$}_\pm$
and $\mbox{\boldmath $q$}$ is fixed for a given value of the Bjorken
scaling variable $x$.
Turning to the evaluation of the nucleon matrix elements defined
above we first note that the Fourier transform of the wave--function
is easily obtained because the angular parts are tensor spherical
harmonics in both coordinate and momentum spaces. Hence, only the
radial part requires numerical treatment.
Performing straightforwardly
the azimuthal integrations in eqs. (\ref{g10}) and (\ref{g11})
reveals that the surviving isoscalar part of the longitudinal structure
function, $g_{1}^{I=0}$, is linear in the angular velocity,
$\mbox{\boldmath $\Omega$}$. It is this part which is associated with the
proton--spin puzzle. Using the standard quantization condition,
$\mbox{\boldmath $\Omega$} =\mbox{\boldmath $J$}/\ \alpha^2$,
where $\alpha^2$ is the moment of inertia of the soliton
and further noting that
the ${\hat{\mbox{\boldmath $z$}}}$--direction is distinct,
the required nucleon matrix elements are
$\langle N,\frac{1}{2}\hat{\mbox{\boldmath $z$}}|
J_{z}|N,\frac{1}{2}\hat{\mbox{\boldmath $z$}}\rangle=\frac{1}{2}$.
Thus, $g_1^{I=0}$ is identical for all nucleon states.
Choosing a symmetric ordering \cite{Al93}, \cite{Sch95} for
the non--commuting operators,
$D_{ia}J_j\rightarrow \frac{1}{2}\left\{D_{ia},J_j \right\}$
we find that the nucleon matrix elements associated with the
cranking portion of the isovector piece, $\langle
N,\pm\frac{1}{2}\hat{\mbox{\boldmath $z$}}|\left\{D_{3y},J_x
\right\}|N,\pm\frac{1}{2}\hat{\mbox{\boldmath $z$}}\rangle$, vanish.
With this ordering we avoid the occurance of PCAC violating pieces in
the axial current.
The surviving terms stem solely from the classical part of the
valence quark wave--function,
$\Psi_{\rm v}\left({\mbox{\boldmath $x$}}\right)$ in
combination with the collective Wigner--D function, $D_{3z}$. Again
singling out the ${\hat{\mbox{\boldmath $z$}}}$--direction,
the nucleon matrix elements become \cite{Ad83}
\begin{eqnarray}
\langle
N,\frac{1}{2}\hat{\mbox{\boldmath $z$}}|D_{3z}|N,\frac{1}{2}\hat{\mbox{\boldmath $z$}}
\rangle = -\frac{2}{3} i_3\ ,
\label{matz}
\end{eqnarray}
where $i_3=\pm\frac{1}{2}$ is the nucleon isospin.
For the transverse structure function, the surviving piece of the
isoscalar contribution is again linear in the angular velocities.
The transversally polarized nucleon gives rise to
the matrix elements,
$\langle N,\frac{1}{2}\hat{\mbox{\boldmath $x$}}
|J_{x}|N,\frac{1}{2}\hat{\mbox{\boldmath $x$}}\rangle=\frac{1}{2}$.
Again choosing symmetric ordering for terms
arising from the cranking contribution, the nucleon matrix elements
$\langle
N,\frac{1}{2}\hat{\mbox{\boldmath $x$}}|\left\{D_{3y},J_y
\right\}|N,\frac{1}{2}\hat{\mbox{\boldmath $x$}}\rangle$
and $\langle
N,\frac{1}{2}\hat{\mbox{\boldmath $x$}}|\left\{D_{33},J_y
\right\}|N,\frac{1}{2}\hat{\mbox{\boldmath $x$}}\rangle$ vanish.
As in the longitudinal case, there is a surviving isovector contribution
stemming solely from the classical part of the
valence quark wave--function, $\Psi_{\rm v}({\mbox{\boldmath $x$}})$
in combination with the collective Wigner--D function, $D_{3x}$.
Now singling out the $\hat{\mbox{\boldmath $x$}}$--direction
the relevant nucleon matrix elements become \cite{Ad83},
\begin{eqnarray}
\langle
N,\frac{1}{2}\hat{\mbox{\boldmath $x$}}|D_{3x}
|N,\frac{1}{2}\hat{\mbox{\boldmath $x$}}\rangle
= -\frac{2}{3} i_3\ .
\label{matx}
\end{eqnarray}
Explicit expressions in terms of the valence quark
wave functions (\ref{gone} and \ref{gtran}) for
$g^{I=0}_{1,\pm}(x)$, $g^{I=1}_{1,\pm}(x)$, $g^{I=0}_{2,\pm}(x)$
and $g^{I=1}_{,\pm}(x)$ are listed in
the appendix A.
Using the expressions given in the appendix A
it is straightforward to verify the Bjorken sum rule \cite{Bj66}
\begin{eqnarray}
\Gamma_1^{p}-\Gamma_1^{n}&=&\int_{0}^{1} dx\ \left(g_{1}^{p}(x)-
g_{1}^{n}(x)\right)=g_{A}/6\ ,
\label{bjs}
\end{eqnarray}
the Burkhardt--Cottingham sum rule \cite{bur70}
\begin{eqnarray}
\Gamma_2^{p}&=&\int_{0}^{1} dx\ g_{2}^{p}(x)=0\ ,
\label{bcs}
\end{eqnarray}
as well as the axial singlet charge
\begin{eqnarray}
\Gamma_1^{p}+\Gamma_1^{n}&=&\int_{0}^{1} dx\ \left(g_{1}^{p}(x)+
g_{1}^{n}(x)\right)=g_A^{0}\ ,
\label{gas}
\end{eqnarray}
in this model calculation when the moment of inertia
$\alpha^2$, as well as the axial charges $g_A^0$ and $g_A$, are
confined to their dominating valence quark pieces.
We have used
\begin{eqnarray}
g_A&=&-\frac{N_C}{3}\int d^3 r
{\bar\psi}_{\rm v}^{\dagger}(\mbox{\boldmath $r$})\gamma_3
\gamma_5\tau_3 \psi_{\rm v}(\mbox{\boldmath $r$})
\label{gaval} \\
g_A^0&=&\frac{N_C}{\alpha_{\rm v}^2}
\int d^3 r{\bar\psi}_{\rm v}^{\dagger}(\mbox{\boldmath $r$})\gamma_3
\gamma_5\psi_{\rm v}(\mbox{\boldmath $r$}) \ .
\label{ga0val}
\end{eqnarray}
to verify the Bjorken Sum rule as well as the axial singlet charge.
This serves as an analytic check on our treatment.
Here $\alpha_{\rm v}^2$ refers to the valence quark contribution
to the moment of inertia, {\it i.e.}
$\alpha_{\rm v}^2=(1/2)\sum_{\mu\ne{\rm v}}
|\langle\mu|\tau_3|{\rm v}\rangle|^2/(\epsilon_\mu-\epsilon_{\rm v})$.
The restriction to the valence quark piece is required by consistency
with the Adler sum rule in the calculation of the unpolarized
structure functions in this approximation \cite{wei96}.
\bigskip
\section{Numerical Results}
\bigskip
In this section we display the results of the spin--polarized
structure functions calculated from eqs. (\ref{g1zro}--\ref{gton})
for constituent quark masses of $m=400{\rm MeV}$ and $450{\rm MeV}$.
In addition to checking the above mentioned sum rules
see eqs. (\ref{bjs})--(\ref{gas}),
we have numerically calculated the
first moment of $g_{1}^{p}(x,\mu^{2})$\footnote{Which in
this case amounts to the Ellis--Jaffe sum rule \cite{Ja74}
since we have omitted the strange degrees of freedom. A careful
treatment of symmetry breaking effects indicates that the role of the
strange quarks is
less important than originally assumed \cite{jon90,Li95}.}
\begin{eqnarray}
\Gamma_1^{p}&=&\int_{0}^{1} dx\ g_{1}^{p}(x)\ ,
\label{ejs}
\end{eqnarray}
and the
Efremov--Leader--Teryaev (ELT) sum rule \cite{Ef84}
\begin{eqnarray}
\Gamma_{\rm ETL}&=&\int_{0}^{1} dx\ x\left(g_{1}^{p}(x)
+2g_{2}^{n}(x)\right)\ .
\label{elts}
\end{eqnarray}
We summarize the results for the sum rules in table 1.
When comparing these results with the experimental data one observes
two short--comings, which are already known from studies of the
static properties in this model. First, the axial charge
$g_A\approx 0.73$ comes
out too low as the experimental value is $g_A=1.25$. It has
recently been speculated that a different ordering of the collective
operators $D_{ai}J_j$ ({\it cf.} section 4) may fill the gap
\cite{Wa93,Gok96}. However, since such an ordering unfortunately gives
rise to PCAC violating contributions to the axial current \cite{Al93}
and furthermore inconsistencies with $G$--parity may occur in
the valence quark approximation \cite{Sch95} we will not pursue
this issue any further at this time. Second, the predicted axial singlet
charge $g_A^0\approx 0.6$ is approximately twice as large
as the number extracted from experiment\footnote{Note
that this analysis assumes $SU(3)$ flavor symmetry, which, of course,
is not manifest in our two flavor model.} $0.27\pm0.04$\cite{ell96}.
This can be
traced back to the valence quark approximation as there are direct
and indirect contributions to $g_A^0$ from both the polarized
vacuum and the valence quark level. Before canonical quantization
of the collective coordinates one finds a sum of valence
and vacuum pieces
\begin{eqnarray}
g_A^0=2\left(g_{\rm v}^0+g_{\rm vac}^0\right)\Omega_3
=\frac{g_{\rm v}^0+g_{\rm vac}^0}
{\alpha^2_{\rm v}+\alpha^2_{\rm vac}} \ .
\label{ga0val1}
\end{eqnarray}
Numerically the vacuum piece is negligible, {\it i.e.}
$g_{\rm vac}^0/g_{\rm v}^0\approx 2\%$. Canonical quantization
subsequently involves the moment of inertia
$\alpha^2=\alpha^2_{\rm v}+\alpha^2_{\rm vac}$, which also has
valence and vacuum pieces. In this case, however, the vacuum
part is not so small: $\alpha^2_{\rm vac}/\alpha^2\approx25\%$.
Hence the full treatment of the polarized vacuum will drastically
improve the agreement with the empirical value for $g_A^0$.
On the other hand our model calculation nicely reproduces the
Ellis--Jaffe sum rule since the empirical value is $0.136$.
Note that this comparison is legitimate since neither the
derivation of this sum rule nor our model imply strange quarks.
While the vanishing Burkhardt--Cottingham sum rule can be
shown analytically in this model, the small value for the
Efremov--Leader--Teryaev sum rule is a numerical prediction.
Recently, it has been demonstrated \cite{So94} that that the ELT
sum rule (\ref{elts}), which is derived within the parton model,
neither vanishes in the Center of Mass bag model\cite{So94}
nor is supported by the
SLAC E143 data \cite{slac96}. This is also the case for our
NJL--model calculation as can be seen from table I.
In figure 1 we display the spin structure functions
$g_{1}^{p}(x,\mu^{2})$ and $g_{2}^{p}(x,\mu^{2})$ along with the
twist--2 piece, $g_{2}^{WW(p)}\left(x,\mu^{2}\right)$ and twist--3
piece, ${\overline{g}}_{2}^{p}\left(x,\mu^{2}\right)$. The actual
value for $\mu^2$ will be given in the proceeding section in the
context of the evolution procedure. We observe that the structure
functions $g_{2}^{p}(x,\mu^{2})$ and $g_{2}^{WW(p)}(x,\mu^{2})$ are
well localized in the interval $0\le x\le1$, while for $g_1^{p}$ about
$0.3\%$ of the first moment,
$\Gamma_1^{p}=\int_{0}^{1} dx\ g_{1}^{p}(x,\mu^2)$
comes from the region, $x > 1$.
The polarized structure function $g_1^{p}(x,\mu^2)$ exhibits a
pronounced maximum at $x\approx0.3$ which is smeared out when the
constituent quark mass increases. This can be understood as follows:
In our chiral soliton model the constituent mass serves as a coupling
constant of the quarks to the chiral field (see eqs. (\ref{bosact})
and (\ref{hamil})).
The valence quark
becomes more strongly bound as the constituent quark mass increases.
In this case the lower components of the valence quark
wave--function increase and relativistic effects
become more important resulting in a broadening of the maximum.
With regard to the Burkhardt--Cottingham sum rule the polarized
structure function $g_2^{p}(x,\mu^2)$ possesses a node. Apparently
this node appears at approximately the same value of the Bjorken
variable $x$ as the maximum of $g_1^{p}(x,\mu^2)$. Note also that
the distinct twist contributions to $g_2^{p}(x,\mu^2)$
by construction diverge as ${\rm ln}\left(x\right)$ as
$x\to0$ while their sum stays finite(see section 6 for details).
As the results displayed in figure 1 are the central issue of
our calculation it is of great interest to compare them with the
available data. As for all effective low--energy models of the
nucleon, the predicted results are at a lower scale $Q^2$ than
the experimental data. In order to carry out a sensible comparison
either the model results have to be evolved upward or the QCD
renormalization group equations have to be used to extract structure
functions at a low--renormalization point. For the combination
$xg_1(x)$ a parametrization of the empirical structure function
is available at a low scale \cite{Gl95}\footnote{These authors also
provide a low scale parametrization of quark distribution functions.
However, these refer to the distributions of perturbatively interacting
partons. Distributions for the NJL--model constituent quarks could
in principle be extracted from eqs. (\ref{g10})--(\ref{gt1}). It is
important to stress that these distributions may not be compared
to those of ref \cite{Gl95} because the associated quarks fields
are different in nature.}. In that study the experimental high $Q^2$
data are evolved to the low--renormalization point $\mu^2$, which is
defined as the lowest $Q^2$ satisfying the positivity constraint
between the polarized and unpolarized structure functions. In a
next--to--leading order calculation those authors found
$\mu^2=0.34{\rm GeV}^2$ \cite{Gl95}. In figure 2 we compare our
results for two different constituent quark masses with that
parametrization. We observe that our predictions reproduce gross
features like the position of the maximum. This agreement is the
more pronounced the lower the constituent quark is, {\it i.e.} the
agreement improves as the applicability of the valence quark
approximation becomes more justified. Unfortunately, such a
parametrization is currently not available for the transverse
structure function $g_T(x)$ (or $g_2(x)$). In order to nevertheless
be able to compare our corresponding results with the (few) available
data we will apply leading order evolution techniques to the structure
functions calculated in the valence quark approximation to the
NJL--soliton model. This will be subject of the following section.
\bigskip
\section{Projection and Evolution}
\bigskip
One notices that our baryon states are not momentum eigenstates
causing the structure functions (see figures 1 and 2)
not to vanish exactly for $x>1$ although the contributions
for $x>1$ are very small. This short--coming is due to the
localized field configuration and thus the nucleon not being a
representation of the Poincar\'{e} group which is common to the
low--energy effective models. The most feasible procedure to cure
this problem is to apply Jaffe's prescription \cite{Ja80},
\begin{eqnarray}
f(x)\longrightarrow \tilde f(x)=
\frac{1}{1-x}f\left(-{\rm log}(1-x)\right)
\label{proj}
\end{eqnarray}
to project any structure function $f(x)$ onto the interval
$[0,1]$. In view of the kinematic regime of DIS this
prescription, which was
derived in a Lorentz invariant fashion within the 1+1 dimensional
bag model, is a reasonable approximation. It is important to
note in the NJL model the unprojected nucleon wave--function
(including the cranking piece\footnote{Which in fact yields the
leading order to the Adler sum rule,
$F_1^{\nu p}\ - F_1^{{\bar \nu}p}$ \cite{wei96} rather than being a
correction.}, see \ref{valrot}) is anything but a product of
Dirac--spinors. In this context, techniques such as
Peierls--Yoccoz\cite{Pei57} (which does not completely enforce
proper support \cite{Sig90}, $0\le x\le1$ nor restore Lorentz
invariance, see \cite{Ard93}) appear to be infeasible. Thus,
given the manner in which the nucleon
arises in chiral--soliton models Jaffe's projection
technique is quite well suited.
It is also important to note
that, by construction, sum rules are not effected by this
projection, {\it i.e.}
$\int_0^\infty dxf(x)=
\int_0^1 dx \tilde f(x)$. Accordingly the sum--rules of the
previous section remain intact.
With regard to evolution of the spin--polarized structure functions
applying the OPE analysis of Section 2, Jaffe and Ji brought to light
that to leading order in $1/Q^{2}$,
$g_1(x,Q^2)$ receives only a leading order twist--2
contribution, while $g_2(x,Q^2)$ possesses contributions
from both twist--2 and twist--3 operators;
the twist--3 portion coming from
spin--dependent gluonic--quark correlations \cite{Ja90},\cite{Ji90}
(see also, \cite{ko79} and \cite{sh82}).
In the {\em impulse approximation}
\cite{Ja90}, \cite{Ji90}
these leading contributions are given by
\begin{eqnarray}
\hspace{-2cm}
\lim_{Q^2\to\infty}
\int_{0}^{1} dx\ x^{n} g_{1}(x,Q^2)&=&\frac{1}{2}\sum _{i}\
{\cal O}_{2,i}^{n}\ \ ,\ \ n=0,2,4,\ldots\ ,
\label{ltc1} \\
\lim_{Q^2\to\infty}
\int_{0}^{1} dx\ x^{n}\ g_{2}(x,Q^2)&=&-\frac{n}{2\ (n+1)}
\sum_{i} \left\{ {\cal O}_{2,i}^{n}
-{\cal O}_{3,i}^{n} \right\},\ n=2,4,\ldots\ .
\label{ltc2}
\end{eqnarray}
Note that there is no sum rule for the first
moment, $\Gamma_{2}(Q^2)=\int_{0}^{1}\ dx g_{2}(x,Q^2)$, \cite{Ja90}.
Sometime ago Wandzura and Wilczek \cite{wan77}
proposed that $g_2(x,Q^2)$ was given in terms of $g_1(x,Q^2)$,
\begin{eqnarray}
g_{2}^{WW}(x,Q^2)=-\ g_{1}(x,Q^2)+\ \int_{x}^{1}\frac{dy}{y}\ g_{1}(y,Q^2)
\label{ww}
\end{eqnarray}
which follows immediately from eqs. (\ref{ltc1}) and (\ref{ltc2})
by neglecting the twist--3 portion in the sum in
(\ref{ltc2}). One may reformulate
this argument to extract the twist--3 piece
\begin{eqnarray}
{\overline{g}}_{2}(x,Q^2)\ =\ g_{2}(x,Q^2)\ -\ g_{2}^{WW}(x,Q^2)\ ,
\end{eqnarray}
since,
\begin{eqnarray}
\int_{0}^{1} dx\ x^{n}\ {\overline{g}}_{2}(x,Q^2)=\frac{n}{2\ (n+1)}
\sum_{i} {\cal O}_{3,i}^{n}\ \ , \ n=2,4,\ \ldots \ .
\end{eqnarray}
In the NJL model as in the bag--model there are no explicit gluon
degrees of freedom, however, in both models twist--3
contributions to $g_2(x,\mu^2)$ exist. In contrast to the bag
model where the bag boundary simulates the quark--gluon and
gluon--gluon correlations \cite{So94} in the NJL model the
gluon degrees of freedom, having been ``integrated" out,
leave correlations characterized by the four--point quark
coupling $G_{\rm NJL}$. This is the source of the twist--3
contribution to $g_2(x,\mu^2)$, which is shown in figure 1.
For $g_{1}\left(x,Q^2\right)$ and the twist--2 piece
$g_2^{WW}\left(x,Q^2\right)$
we apply the leading
order (in $\alpha_{QCD}(Q^2)$) Altarelli--Parisi
equations \cite{Al73} to evolve
the structure functions from the model
scale, $\mu^2$, to that
of the experiment $Q^2$, by iterating
\begin{eqnarray}
g(x,t+\delta{t})=g(x,t)\ +\ \delta t\frac{dg(x,t)}{dt}\ ,
\end{eqnarray}
where $t={\rm log}\left(Q^2/\Lambda_{QCD}^2\right)$.
The explicit expression for the evolution differential
equation is given by the convolution integral,
\begin{eqnarray}
\frac{d g(x,t)}{dt}&=&\frac{\alpha(t)}{2\pi}
g(x,t)\otimes P_{qq}(x)
\nonumber \\* \hspace{1cm}
&=&\frac{\alpha(t)}{2\pi}
C_{R}(F)\int^1_{x}\ \frac{dy}{y}P_{qq}\left(y\right)
g\left(\frac{x}{y},t\right)
\label{convl}
\end{eqnarray}
where the quantity
$P_{qq}\left(z\right)=\left(\frac{1+z^2}{1-z^2}\right)_{+}$
represents the quark probability to emit a gluon such that the
momentum of the quark is reduced by the fraction $z$.
$C_{R}(f)=\frac{{n_{f}^{2}}-1}{2{n_{f}}}$ for $n_f$--flavors,
$\alpha_{QCD}=\frac{4\pi}{\beta\log\left(Q^2/ \Lambda^2\right)}$
and $\beta=(11-\frac{2}{3}n_f)$.
Employing the ``+" prescription\cite{Al94} yields
\begin{eqnarray}
\frac{d\ g(x,t)}{dt}&=&\frac{2C_{R}(f)}{9\ t}
\left\{\ \left(x + \frac{x^{2}}{2}+2\log(1-x)\right)g(x,t)
\right.
\nonumber \\*&& \hspace{1cm}
\left.
+\ \int^{1}_{x}\ dy
\left(\frac{1+y^2}{1-y}\right)
\left[\frac{1}{y}\ g\left(\frac{x}{y},t\right)-g(x,t)\right]\
\right\}\ .
\label{evol}
\end{eqnarray}
As discussed in section 2 the initial value for integrating the
differential equation is given by the scale $\mu^2$ at which the model is
defined. It should be emphasized that this scale essentially is a
new parameter of the model. For a given constituent quark mass we fit
$\mu^2$ to maximize the agreement of the predictions with the
experimental data on previously \cite{wei96} calculated unpolarized
structure functions for (anti)neutrino--proton scattering:
$F_2^{\nu p}-F_2^{\overline{\nu} p}$. For the constituent quark mass
$m=400{\rm MeV}$ we have obtained $\mu^2\approx0.4{\rm GeV}^2$.
One certainly wonders whether for such a low scale the restriction to
first order in $\alpha_{QCD}$ is reliable. There are two answers. First,
the studies in this section aim at showing that the required evolution
indeed improves the agreement with the experimental data and, second,
in the bag model it has recently been shown \cite{St95} that a
second order evolution just increases $\mu^2$ without significantly
changing the evolved data. In figure 3 we compare the
unevolved, projected, structure function
$g_1^{p}\left(x,\mu^{2}\right)$ with the one
evolved from $\mu^{2}=0.4{\rm GeV}^2$ to $Q^2=3.0{\rm GeV}^2$.
Also the data from the E143-collaboration from
SLAC\cite{slac95a} are given. Furthermore in
figure 3 we compare the projected, unevolved structure
function $g_2^{WW(p)}\left(x,\mu^{2}\right)$ as well as the one evolved
to $Q^2=5.0{\rm GeV}^2$ with the data from the recent E143-collaboration
at SLAC\cite{slac96}.
As expected we observe that the evolution pronounces the structure
function at low $x$; thereby improving the agreement with the
experimental data. This change towards small $x$ is a general feature
of the projection and evolution process and presumably not very
sensitive to the prescription applied here. In particular, choosing
an alternative projection technique may easily be compensated by
an appropriate variation of the scale $\mu^2$.
While the evolution of the structure function
$g_{1}\left(x,Q^2\right)$ and the twist--2 piece
$g_2^{WW}\left(x,Q^2\right)$ from $\mu^2$ to $Q^2$ can be performed
straightforwardly using the ordinary Altarelli--Parisi equations
this is not the case with the twist--3 piece
${\overline{g}}_{2}(x,Q^2)$.
As the twist--3 quark and quark--gluon operators mix
the number of independent operators contributing
to the twist--3 piece increases
with $n$, where $n$ refers to the $n^{\underline{\rm th}}$ moment\cite{sh82}.
We apply an approximation (see appendix B) suggested in\cite{Ali91}
where it is demonstrated
that in $N_c\to \infty$ limit the quark
operators of twist--3 decouple from the evolution equation
for the quark--gluon operators of the same twist resulting
in a unique evolution scheme.
This scheme is particularly suited for the NJL--chiral soliton model,
as the soliton picture for baryons is based on $N_c\rightarrow \infty$
arguments\footnote{This scheme has also employed by Song\cite{So94}
in the Center of Mass bag model.}.
In figure 4 we compare the projected unevolved structure function
${\overline{g}}_{2}^{p}(x,\mu^2)$ evolved to $Q^2=5.0{\rm GeV}^2$
using the scheme suggested in \cite{Ali91}. In addition
we reconstruct $g_2^{p}\left(x,Q^2\right)$ at $Q^2=3.0{\rm GeV}^2$ from
$g_2^{WW(p)}\left(x,Q^2\right)$ and ${\overline{g}}_{2}(x,Q^2)$
and compare it with the recent SLAC data\cite{slac96}
for $g_2^{p}\left(x,Q^2\right)$. As is evident our model
calculation of $g_2^{p}\left(x,Q^2\right)$,
built up from its twist--2 and twist--3 pieces,
agrees reasonably well
with the data although the experimental errors are quite large.
\bigskip
\section{Summary and Outlook}
\bigskip
In this paper we have presented the calculation of the polarized
nucleon structure functions $g_1\left(x,Q^2\right)$ and
$g_2\left(x,Q^2\right)$ within a model which is
based on chiral symmetry and its spontaneous breaking. Specifically
we have employed the NJL chiral soliton model which reasonably
describes the static properties of the nucleon \cite{Al96},
\cite{Gok96}. In this model the
current operator is formally identical to the one in an
non--interacting relativistic quark model. While the quark
fields become functionals of the chiral soliton upon bosonization,
this feature enables one calculate the hadronic tensor.
From this hadronic tensor we have then extracted the polarized
structure functions within the valence quark approximation. As the
explicit occupation of the valence quark level yields the
major contribution (about 90\%) to the associated static quantities
like the axial charge this presumably is a justified approximation.
When cranking corrections are included this share may be reduced
depending on whether or not the full moment of inertia is substituted.
It needs to be stressed that in contrast to {\it e.g.} bag models
the nucleon wave--function arises as a collective excitation of
a non--perturbative meson field configuration. In particular, the
incorporation of chiral symmetry leads to the distinct feature that
the pion field cannot be treated perturbatively. Because of the
hedgehog structure of this field one starts with grand spin symmetric
quark wave--functions rather than direct products of spatial-- and
isospinors as in the bag model. On top of these grand spin
wave--functions one has to include cranking corrections to generate
states with the correct nucleon quantum numbers. Not only are these
corrections sizable but even more importantly one would not be able
to make any prediction on the flavor singlet combination of the
polarized structure functions without them. The structure functions
obtained in this manner are, of course, characterized by the scale
of the low--energy effective model. We have confirmed this issue by
obtaining a reasonable agreement of the model predictions for the
structure function $g_1$ of the proton with the low--renormalization
point parametrization of ref \cite{Gl95}. In general this scale of
the effective model essentially represents an intrinsic parameter of
a model. For the NJL--soliton model we have previously determined
this parameter from the behavior of the unpolarized structure
functions under the Altarelli--Parisi evolution \cite{wei96}. Applying
the same procedure to the polarized structure functions calculated in
the NJL model yields good agreement with the data extracted from
experiment, although the error bars on $g_1\left(x,Q^2\right)$ are
still sizable. In particular, the good agreement at low $x$ indicates
that to some extend gluonic effects are already incorporated in the
model. This can be understood by noting that the quark fields, which
enter our calculation, are constituent quarks. They differ from the
current quarks by a mesonic cloud which contains gluonic components.
Furthermore, the existence of gluonic effects in the model would not
be astonishing because we had already observed from the non--vanishing
twist--3 part of $g_2\left(x,Q^2\right)$, which in the OPE is
associated with the quark--gluon interaction, that the model contains
the main features allocated to the gluons.
There is a wide avenue for further studies in this model. Of course,
one would like to incorporate the effects of the polarized vacuum,
although one expects from the results on the static axial properties
that their direct contributions are negligible. It may be more
illuminating to include the strange quarks within the valence quark
approximation. This extension of the model seems to be demanded by
the analysis of the proton spin puzzle. Technically two changes will
occur. First, the collective matrix elements will be more complicated than
in eqs. (\ref{matz}) and (\ref{matx}) because the nucleon wave--functions
will be distorted $SU(3)$ $D$--functions in the presence of flavor
symmetry breaking \cite{Ya88,wei96a}. Furthermore the valence quark
wave--function (\ref{valrot}) will contain an additional correction
due to different non--strange and strange constituent quark masses
\cite{We92}. When these corrections are included direct information
will be obtained on the contributions of the strange quarks to polarized
nucleon structure functions. In particular the previously developed
generalization to three flavors \cite{We92} allows one to
consistently include the effects of flavor symmetry breaking.
\bigskip
\acknowledgements
This work is supported in part by the Deutsche
Forschungsgemeinschaft (DFG) under contract Re 856/2-2.
LG is grateful for helpful comments by G. R. Goldstein.
\bigskip
\bigskip
| proofpile-arXiv_065-482 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
Instantons~\cite{bpst} are well known to represent tunnelling transitions in
non-abelian gauge theories between degenerate vacua of different topology.
These transitions induce processes which are {\it forbidden} in perturbation
theory, but have to exist in general~\cite{th} due to Adler-Bell-Jackiw
anomalies. Correspondingly, these processes imply a violation of
certain fermionic quantum numbers, notably, $B+L$ in the electro-weak gauge
theory and chirality ($Q_5$) in (massless) QCD.
An experimental discovery of such a novel, non-perturbative
manifestation of non-abelian gauge theories would clearly be of basic
significance.
A number of results has revived the interest in instanton-induced processes
during recent years:
\begin{itemize}
\item First of all, it was shown~\cite{r,m} that the generic exponential
suppression of these tunnelling rates, $\propto \exp (-4\pi /\alpha )$, may be
overcome at {\it high energies}, mainly due to multi-gauge boson emission
in addition to the minimally required fermionic final state.
\item A pioneering and encouraging theoretical estimate
of the size of the instanton ($I$) induced contribution to the
gluon structure functions in deep-inelastic scattering
was recently presented in Ref.~\cite{bb}. The summation over the $I$-induced
multi-particle final state was implicitly performed by
starting from the optical theorem
for the virtual $\gamma^\ast g\rightarrow \gamma^\ast g $
forward amplitude. The strategy was then to
evaluate the contribution to the functional integral coming from the
vicinity of the instanton-antiinstanton ($\iai$) configuration in
Euclidean space, to analytically
continue the result to Minkowski space and, finally, to take the imaginary
part. While the instanton-induced contribution to the
gluon structure functions turned
out to be small at larger values of the Bjorken variable $x$,
it was found in Ref.~\cite{bb} to increase dramatically towards
smaller $x$.
\item Last not least, a systematic phenomenological and theoretical study
is under way~\cite{rs,grs,rs1,ggmrs}, which clearly indicates that
deep-inelastic $e p$ scattering at HERA now offers a unique window to
experimentally detect QCD-instanton induced processes through their
characteristic final-state signature. The searches for instanton-induced
events have just started at HERA and a first upper limit of $0.9$ nb
at $95\%$ confidence level for the cross-section of QCD-instanton induced
events has been placed by the H1 Collaboration~\cite{H1}. New, improved
search strategies are being developped~\cite{ggmrs} with the help of a
Monte Carlo generator (QCDINS 1.3)~\cite{grs} for instanton-induced events.
\end{itemize}
The central question is, of course, whether instanton-induced processes
in deep-inelastic scattering can both be reliably computed and
experimentally measured. In particular, whether contributions associated
with the non-perturbative vacuum
structure can be controlled in the same way as perturbative short-distance
corrections, in terms of a hard scale ${\mathcal Q}$.
In the work of Refs.~\cite{bb,bbgg} on deep-inelastic scattering,
the integrals over the instanton size $\rho$ were found to
be infrared (IR) divergent, like in a number of previous
instanton calculations in different areas. Yet, the authors claimed
that this problem does not affect the possibility, to isolate in
deep-inelastic scattering a well-defined, IR-finite and sizable
instanton contribution in the regime of small QCD-gauge coupling, on
account of the (large) photon virtuality $Q^2$. The IR-divergent pieces of
the $I$-size integrals were supposed to be factorizable into the parton
distributions, which anyway have to be extracted from experiment at
some reference scale.
On the other hand, also IR-finite instanton contributions to certain
observables in momentum space have been found in the past~\cite{as,early}.
In this ideal case, the size of the contributing
instantons is limited by the inverse momentum scale ${\mathcal Q}^{-1}$ of
the experimental probe, as one might intuitively expect. No {\it ad hoc}
cutoff or assumption about the behaviour of large, overlapping instantons
need be introduced.
A main issue of the present work is to shed further light on these important
questions around the IR-behaviour associated with the instanton size in
deep-inelastic scattering.
This paper represents the first of several
papers in preparation~\cite{mrs}, containing our theoretical results on
$I$-induced processes in the deep-inelastic regime.
\begin{figure}
\begin{center}
\epsfig{file=ampl_gen.ps,width=13cm}
\caption[dum]{\label{f1}
Instanton-induced {\it chirality-violating} process,\\ $\gamma^{\ast}
+{\rm g}\rightarrow \sum_{\rm flavours}^{n_{f}}\left[ \overline{{\rm
q}_{L}} +{\rm q}_{R}\right] +n_{g}\,{\rm g}$, corresponding to
three massless flavours
($n_{f}=3$).}
\end{center}
\end{figure}
For clarity, let us reduce here the realistic task of evaluating
the $I$-induced cross-sections of the chirality violating multi-particle
processes (illustrated in Fig.~\ref{f1})
\begin{equation}
\gamma^{\ast}+{\rm g}\Rightarrow
\overline{{\rm u}_{L}}+{\rm u}_{R}\,
+ \overline{{\rm d}_{L}}+{\rm d}_{R}\,
+\overline{{\rm s}_{L}}+{\rm s}_{R}\,
+n_g\, {\rm g},
\end{equation}
to the detailed study of the {\it simplest} one,
without additional gluons and with just one massless flavour ($n_f=1$),
\begin{equation}
\gamma^{\ast}(q)+{\rm g}(p)\Rightarrow
\overline{{\rm q}_{L}}(k_{1})+{\rm
q}_{R}(k_{2})\, .
\end{equation}
The price is, of course, that this process only represents a small fraction of
the total $I$-induced contribution to the gluon structure functions.
However, there are also a number of important virtues:
The present calculation provides a clean and explicit discussion of most of
the crucial steps involved in our subsequent task~\cite{mrs} to calculate
the {\it dominant} $I$-induced contributions. While unessential technical
complications have been eliminated here, a generalization
to the realistic case with gluons and more flavours in the final state is
entirely straightforward~\cite{mrs,q96}.
We shall explicitly calculate the corresponding fixed angle cross-section
and the contributions to the gluon structure functions in leading
semi-classical approximation within standard instanton perturbation theory
(Sect.~3). Gauge invariance is kept manifest along the
calculation and we may compare at various stages with the appropriate
chirality-conserving process, calculated in leading order of perturbative QCD.
As a central result of this paper and unlike Ref.~\cite{bb}, we find
{\it no} IR divergencies associated with the integration over the
instanton size $\rho$, which can even be perfomed analytically. We are able
to demonstrate explicitly that the typical hard scale ${\mathcal Q}$
in deep-inelastic scattering provides a {\it dynamical} infrared cutoff for
the instanton size, $\rho \mbox{\,\raisebox{.3ex {\mathcal O}(1/{\mathcal Q})$.
Additional gluons in the final state will not change this conclusion, as is
briefly outlined in Sect.~4. Thus, deep-inelastic
scattering may indeed be viewed as a distinguished process for studying
manifestations of QCD-instantons.
\section{Setting the Stage}
Let us start with the matrix elements
${\mathcal T}^\mu (q,p; k_{1},\ldots ,k_{n})$ of the
general, exclusive photon-parton reactions
\begin{equation}
\gamma^{\ast}(q)+ p\rightarrow k_{1}+\ldots +k_{n}\, ,
\end{equation}
in terms of which we form the inclusive structure tensor
${\mathcal W}^{\mu \nu}_{p}$ of the parton $p$,
\begin{eqnarray}
{\mathcal W}^{\mu \nu}_{ p} (q,p ) & =& \sum_{n=1}^\infty
{\mathcal W}^{\mu \nu\, (n)}_{ p} (q,p ) \, ,
\label{wmunu}
\hspace{24pt} {\rm with} \\[10pt]
{\mathcal W}^{\mu \nu (n)}_{ p} ( q,p )& =&
\frac{1}{4\,\pi}\,
\int dPS^{(n)}\,
{\mathcal T}^\mu (q,p;k_{1},\ldots,k_{n})
{\mathcal T}^{\nu\,\ast}(q,p;k_{1},\ldots ,k_{n} ) ,
\label{wmunu_n}
\end{eqnarray}
and
\begin{eqnarray}
\int dPS^{(n)} = \prod_{j=1}^{n} \int
\frac{d^{4}k_{j}}{(2\,\pi)^{3}}\, \delta^{(+)}\left( k_{j}^{2}\right)
(2\,\pi)^{4}\, \delta^{(4)}\left( q+p-k_{1}-\ldots -k_{n}\right) \, .
\end{eqnarray}
Averaging over colour and spin of the initial state is implicitly understood
in Eq.~(\ref{wmunu_n}); the index $n$ is to label besides the final
state partons also their spin and colour degrees of freedom.
For exclusive $2\rightarrow 2$ processes,
$\gamma^{\ast}(q) +p\rightarrow k_{1}+k_{2}$, the differential cross-section
is then expressed as
\begin{equation}
\label{diffcross}
\frac{d\sigma}{dt}= 8\,\pi^{2}\,\alpha\ \frac{x}{Q^{2}}\,
\left[ -g_{\mu \nu}\, \frac{d{\mathcal W}^{\mu \nu\, (2)}_{p}}{dt}
\right] \, ,
\end{equation}
where $Q^{2}=-q^{2}$ denotes the photon virtuality and
\begin{equation}
\label{t}
t=\left( q-k_{1}\right)^{2}=\left( p-k_{2}\right)^{2}
\, ,
\end{equation}
the momentum transfer squared.
In general, each final state, $k_{1}+\ldots +k_{n}$, contributes
to the structure functions ${\mathcal F}_{i\,{ p}}$
of the parton $p$ via the projections
\begin{eqnarray}
{\mathcal F}^{(n)}_{2\,{ p}} (x ,Q^2) &= & \left[- g_{\nu\mu }\,
+6\,x\,\frac{p_\mu p_\nu}{p\cdot q} \right]
x\,{\mathcal W}_{ p}^{\mu \nu\, (n)} (q,p) \,
,
\label{projF2}
\\[10pt]
{\mathcal F}^{(n)}_{L\,{ p}} (x ,Q^2)
&= & 4\,x^2\,\frac{p_\mu p_\nu}{p\cdot q}
{\mathcal W}_{ p}^{\mu \nu\, (n)} (q,p) \ ,
\label{projFL}
\hspace{24pt} {\rm such\ that} \\[10pt]
{\mathcal F}_{i\,{ p}}(x,Q^{2})&=&\sum_{n=1}^{\infty}
{\mathcal F}^{(n)}_{i\,{ p}}(x,Q^{2})\, ,
\end{eqnarray}
with
\begin{equation}
x\equiv \frac{Q^2}{2\,p\cdot q} \, .
\end{equation}
denoting the Bjorken variable of the photon-parton subprocess.
The spin averaged proton structure functions $F_2$ and $F_L$, appearing
(in the one photon exchange approximation) in the unpolarized inclusive
lepto-production cross-section as
\begin{equation}
\frac{d^2\sigma}{dx_{\rm Bj}\, d\ybj } =
\frac{4\pi\alpha^2}{Sx_{\rm Bj}^2\ybj^2} \left[ \left\{ 1-\ybj +\frac{\ybj^2}{2}
\right\} F_2 (x_{\rm Bj} ,Q^2) - \frac{\ybj^2}{2} F_L (x_{\rm Bj} ,Q^2 ) \right] ,
\end{equation}
are expressed via
a standard convolution in terms of the parton structure functions
${\mathcal F}_{i\,{p}}$ and corresponding parton densities $f_{ p}$,
\begin{equation}
F_i\, (x_{\rm Bj} ,Q^2) = \sum_{p=q,g} \int_{x_{\rm Bj}}^1 \frac{dx}{x}\, f_{ p}\left(
\frac{x_{\rm Bj}}{x}\right) \, \frac{x_{\rm Bj}}{x}\,
{\mathcal F}_{i\,{ p}} (x,Q^2)
\ ,\
\ i=2,L\, .
\label{protonstruc}
\end{equation}
Here, $\sqrt{S}$ is the center-of-mass (c.m.) energy of the
lepton-hadron
system. The corresponding Bjorken variables are defined as usual
\begin{equation}
x_{\rm Bj} \equiv \frac{Q^2}{2\,P\cdot q} ; \hspace{48pt} \ybj \equiv \frac{P\cdot
q}{P\cdot k} ,
\end{equation}
where $P\,(k)$ is the four-momentum of the incoming proton (lepton).
\begin{figure}
\begin{center}
\epsfig{file=ampl_pt.ps,width=9cm}
\caption[dum]{\label{f2}
Perturbative {\it chirality-conserving} process, $\gamma^{\ast}(q)+{\rm
g}(p )\rightarrow \overline{{\rm q}_{L}}(k_{1})+{\rm
q}_{L}(k_{2})$. }
\end{center}
\end{figure}
As outlined in the Introduction, we shall consider in this paper only
the contributions from the simplest
instanton-induced, {\it chirality-violating} photon-gluon process,
corresponding to one massless quark flavour $(n_{f}=1)$,
\begin{equation}
\gamma^{\ast}(q)+{\rm g}(p)\rightarrow
\overline{{\rm q}_{L}}(k_{1})+{\rm q}_{R}(k_{2})\ ;
\hspace{24pt}
\left( \triangle
Q_{5}\equiv \triangle \left( Q_{R}-Q_{L}\right)=2\right) \, .
\label{instproc}
\end{equation}
It will be very instructive to compare with the appropriate leading-order
perturbative QCD amplitudes for the {\it chirality-conserving} process
(Fig.~\ref{f2}),
\begin{equation}
\gamma^{\ast}(q)+{\rm g}(p)\rightarrow
\overline{{\rm q}_{L}}(k_{1})+{\rm q}_{L}(k_{2})\ ;
\hspace{24pt}
\left( \triangle
Q_{5}\equiv \triangle \left( Q_{R}-Q_{L}\right)=0\right) \, .
\label{pertproc}
\end{equation}
at the various stages of the instanton calculation. Therefore, for reference,
let us summarize the well-known perturbative results next
(see any textbook on perturbative QCD, e.g. Ref.~\cite{f}).
Let us use two-component Weyl-notation for the (massless) fermions involved,
in order to facilitate the comparison with the instanton calculation later on.
The leading-order amplitude for the perturbative process (\ref{pertproc})
(Fig.~\ref{f2}) then reads,
\begin{eqnarray}
\label{ampl_pt}
\lefteqn{{\mathcal T}^{A}_{\mu \,\mu ^{\prime}}
\left( \gamma^{\ast}+{\rm g}\rightarrow
\overline{{\rm q}_{L}}+{\rm q}_{L}\right) =}
\\[0.5ex]
\nonumber
&&
e_{q}\,g_s\,t^{A}\, \chi_{L}^{\dagger}(k_{2}) \left[
\overline{\sigma}_{\mu ^{\prime}}\frac{(q-k_{1})}{(q-k_{1})^{2}}
\overline{\sigma}_\mu
-\overline{\sigma}_\mu \frac{(q-k_{2})}{(q-k_{2})^{2}}
\overline{\sigma}_{\mu ^{\prime}} \right]
\chi_{L}(k_{1})\ ,
\end{eqnarray}
where the two-component Weyl-spinors $\chi_{L,\,R}$ satisfy
the Weyl-equations,
\begin{equation}
\overline{k}\,\chi_L(k)=0 \, ;\hspace{24pt}
k\,\chi_R(k) =0 \, ,
\label{weyleq}
\end{equation}
and
\begin{equation}
\chi_L(k)\,\chi_L^\dagger (k) = k \, ; \hspace{24pt}
\chi_R(k)\,\chi_R^\dagger (k) = \overline{k} \, .
\label{compl}
\end{equation}
In Eqs.~(\ref{ampl_pt}-\ref{compl}) and throughout the paper we use the
abbreviations,
\begin{equation}
v \equiv v_\mu \,\sigma^\mu \, ; \hspace{24pt} \overline{v} \equiv v_\mu
\,\overline{\sigma}^\mu \, ,\ \mbox{for any four-vector $v_\mu $},
\end{equation}
where the familiar $\sigma$-matrices\footnote{\label{sigmas}
We use the standard notations, in Minkowski space:
$\sigma_\mu =(1,\vec{\sigma})$, $\overline{\sigma}_\mu =(1,-\vec{\sigma})$,
and in Euclidean space:
$\sigma_\mu =(-\ii\,\vec{\sigma},1)$, $\overline{\sigma}_\mu =
(\ii\,\vec{\sigma},1)$, where $\vec{\sigma}$ are the Pauli
matrices.}
satisfy,
\begin{equation}
\sigma_\mu \overline{\sigma}_\nu +\sigma_\nu \overline{\sigma}_\mu =
2\,g_{\mu \nu} \, .
\label{com}
\end{equation}
Finally, in Eq.~(\ref{ampl_pt}), $t^A,A=1,\ldots ,8$, are the SU(3)
generators, $e_q$ is the quark charge in units of the electric charge $e$, and
$g_s$ is the SU(3) gauge coupling.
With help of Eqs.~(\ref{weyleq}), (\ref{com}) and the on-shell conditions
$k_1^2=k_2^2=0$, the gauge-invariance constraints,
\begin{equation}
q^\mu \,{\mathcal T}^{A}_{\mu \,\mu ^{\prime}}=0\, ;\hspace{24pt} {\mathcal
T}^{A}_{\mu \,\mu ^{\prime}}\,p^{ \mu ^{\prime}}=0\, ,
\label{gaugeinv}
\end{equation}
are easily checked.
Next, we obtain the leading-order contribution
of the process (\ref{pertproc}) to ${\mathcal W}^{\mu \nu\,(2)}_{g}(q,p)$ by
contracting Eq.~(\ref{ampl_pt}) with the gluon
polarization vector $\epsilon_{g}^{\mu ^{\prime}}(p)$ and
taking the traces in Eq.~(\ref{wmunu_n}) by means of
relations (\ref{compl}) and (\ref{com}). Averaging over the initial-state
gluon polarization and colour amounts to an overall factor $1/16$.
The final result for the projections needed in
Eqs.~(\ref{diffcross}), (\ref{projF2}), and (\ref{projFL})
then reads
\begin{eqnarray}
\label{gmunupt}
\lefteqn{ -g_{\mu \nu}\,
\frac{d{\mathcal W}^{\mu \nu\,{(2)}}_{g}}{dt}
\,\left( \gamma^{\ast}+{\rm g}\rightarrow
\overline{{\rm q}_{L}}+{\rm q}_{L}\right)
=}
\\[0.5ex]
\nonumber
&&
e_q^2\, \frac{\alpha_s}{4\,\pi}\,
\frac{x}{2\,Q^2}\, \left[
\frac{u}{t}+\frac{t}{u}-2\,\frac{1-x}{x}\,\frac{Q^4}{tu}
\right]
\, ,
\end{eqnarray}
\begin{equation}
\label{pmupnupt}
\frac{p_\mu \,p_{\nu}}{p\cdot q} \,
\frac{d{\mathcal W}^{\mu \nu\,{(2)}}_{g}}{dt}
\,\left( \gamma^{\ast}+{\rm g}\rightarrow
\overline{{\rm q}_{L}}+{\rm q}_{L}\right)
=
e_q^2\,\frac{\alpha_s}{4\,\pi}\,
\frac{x\,(1-x)}{Q^2}
\, ,
\end{equation}
where $u=-t-Q^2/x$.
Upon integrating Eq.~(\ref{gmunupt}) over $t$, we encounter the familiar
collinear divergencies for $t,u\rightarrow 0$. In order to isolate the hard
contributions to the gluon structure functions, it is adequate to regularize
the collinear singularities by introducing an infrared cutoff scale
$\mu _{c}$ in the integration limits,
$\{-Q^{2}/x+\mu _{c}^{2},-\mu _{c}^{2}\}$. On account of
Eqs.~(\ref{projF2}), (\ref{projFL}), one then
obtains the familiar results\footnote{\label{foot1}When comparing with the
literature, one has to remember that we considered only the production of a
$\overline{{\rm q}_{L}}\,{\rm q}_{L}$ pair (c.\,f. Eq.~(\ref{pertproc})).
In the full ${\mathcal O}(\alpha_{s})$ contribution
to the gluon structure functions, the production of a
$\overline{{\rm q}_{R}}\,{\rm q}_{R}$ pair has also to be included. This
amounts to multiplying Eqs.~(\ref{F2pert}),
(\ref{FLpert}), by a
factor of 2.} in the Bjorken limit,
\begin{eqnarray}
\label{F2pert}
&& {\mathcal F}_{2\,g}^{(\overline{{\rm q}_L}{\rm q}_L)}
\, (x,Q^2;\mu _{c}^{2})=
e_q^2\, \frac{\alpha_s}{2\,\pi}\,\times
\\[0.8ex]
\nonumber
&&
x\,
\left[
P_{qg}(x)\,
\ln\left(\frac{Q^{2}}{\mu _{c}^{2}}\right)
+P_{qg}(x)\,\ln \left(\frac{1}{x}\right)
-\frac{1}{2} +3\,x\,(1-x)
\right]
\left[1+{\mathcal O}\left(\frac{\mu _{c}^{2}}{Q^{2}}\right)\right]
\, ,
\end{eqnarray}
\begin{equation}
{\mathcal F}_{L\,g}^{(\overline{{\rm q}_L}{\rm q}_L)}\,(x,Q^2)=
e_q^2\,\frac{\alpha_s}{\pi}\,
x^{2}\,(1-x)
\, ,
\label{FLpert}
\end{equation}
with the splitting function
\begin{equation}
\label{pqg}
P_{qg}(x) \equiv \frac{1}{2}\,\left(x^{2}+(1-x)^{2}\right)\, .
\end{equation}
Of course, the {\it finite} part of the structure function
${\mathcal F}_{2\,g}^{(\overline{{\rm q}_L}{\rm q}_L)}$,
that is everything except for the large logarithm,
$\ln (Q^{2}/\mu _{c}^{2})$, is scheme dependent.
\section{The Instanton-Induced Process\\ $\gamma^\ast + {\rm g}\rightarrow
\overline{{\rm q}_L} + {\rm q}_R$}
In this section, we turn to the central issue of this paper. We
consider the simplest instanton-induced
exclusive process, $\gamma^\ast + {\rm g}\rightarrow\overline{{\rm q}_L} +
{\rm q}_R$, and compute its contributions to the fixed angle differential
cross-section and the gluon structure functions
${\mathcal F}_{2\,{ g}}$ and
${\mathcal F}_{L\,{ g}}$, in leading semi-classical approximation.
To this end, the respective Green's function is first
set up according to standard instanton-perturbation theory in Euclidean
configuration space~\cite{th,brown,ber,abc,r}, then Fourier transformed to
momentum space, LSZ amputated, and finally continued to Minkowski space.
The basic building blocks are (in Euclidean
configuration space and in the singular gauge):
\bigskip
\noindent
i) The classical instanton gauge field~\cite{bpst}
$A^{(I)}_{\mu ^{\prime}}$,
\begin{eqnarray}
A_{\mu ^{\prime}}^{(I)}(x)&=&-\ii\,\frac{2\,\pi^{2}}{g_{s}}\,
\rho^{2}\,U\,\left(
\frac{
\sigma_{\mu ^{\prime}}\,\overline{x}-x_{\mu ^{\prime}}}{2\,\pi^{2}\,x^{4}}
\right)\,U^{\dagger}
\
\frac{1}{\Pi_{x}}\, ,
\label{igauge}
\\
\Pi_x &\equiv & 1+\frac{\rho^2}{x^2} \, ,
\end{eqnarray}
depending on the various collective coordinates, the instanton size $\rho$
and the colour orientation matrices $U^k_{\ \alpha}$. The $U$ matrices
involve both colour ($k=1,2$) and spinor ($\alpha=1,2$) indices, the
former ranging as usual only in the
$2\times 2$ upper left corner of $3\times 3$ SU(3) colour matrices.
Indices will, however, be suppressed, as long as no confusion can arise.
\bigskip
\noindent
ii) The quark zero modes~\cite{th}, $\kappa$ and $\overline{\phi}$,
\begin{eqnarray}
\kappa^m_{\ \dot\beta}\,(x)
&=& 2\,\pi\,\rho^{3/2}\, \epsilon^{\gamma\delta}\,
\left( U\right)^m_{\ \delta}\,
\frac{\overline{x}_{\dot\beta\gamma}}{2\,\pi^2\,x^4}\
\frac{1}{\Pi_x^{3/2}}\, ,
\label{kappa}
\\
\overline{\phi}^{\dot\alpha}_{\ l}\,(x)
&=&2\,\pi\,\rho^{3/2}\, \epsilon_{\gamma\delta}\,
\left( U^\dagger\right)^\gamma_{\ l}\,
\frac{x^{\delta\dot\alpha}}{2\,\pi^2\,x^4}\
\frac{1}{\Pi_x^{3/2}}\, ,
\label{phibar}
\end{eqnarray}
and
\bigskip
\noindent
iii)
the quark propagators in the instanton background~\cite{brown},
\begin{eqnarray}
\label{si}
&&S^{(I)}(x,y) =
\\[1.6ex]
\nonumber
&&
\frac{1}{\sqrt{\Pi_x\Pi_y}}
\left[
\frac{x-y}{2\pi^2(x-y)^4}
\left( 1 +\rho^2\frac{
\left[ U x \overline{y} U^\dagger\right]}
{x^2 y^2}\right)
+\frac{\rho^2\sigma_\mu }{4\pi^2}
\frac{\left[
U x\,(\overline{x}-\overline{y}) \sigma_\mu \overline{y} U^\dagger
\right]}
{x^2(x-y)^2 y^4\Pi_y}
\right],
\\[2ex]
\label{sibar}
&&\overline{S}^{(I)}(x,y) =
\\[1.6ex]
\nonumber
&&
\frac{1}{\sqrt{\Pi_x\Pi_y}}
\left[
\frac{\overline{x}-\overline{y}}{2\pi^2(x-y)^4}
\left( 1 +\rho^2\frac{
\left[ U x \overline{y} U^\dagger\right]}
{x^2 y^2}\right)
+\frac{\rho^2
\overline{\sigma}_\mu }{4\pi^2}
\frac{\left[
U x \overline{\sigma}_\mu (x-y) \overline{y}
U^\dagger \right]}
{\Pi_x x^4(x-y)^2 y^2 }
\right] .
\end{eqnarray}
\begin{figure}
\begin{center}
\epsfig{file=ampl_ld.ps,width=9cm}
\caption[dum]{\label{f3}
Instanton-induced {\it chirality-violating} process, $\gamma^{\ast}(q)
+{\rm g}(p )\rightarrow \overline{{\rm q}_{L}}(k_{1})+{\rm
q}_{R}(k_{2})$, in leading semi-classical approximation. The corresponding
Green's function involves the products of the appropriate classical fields
(lines ending at blobs) as well as the quark propagator in the instanton
background (quark line with central blob). }
\end{center}
\end{figure}
The relevant diagrams for the exclusive process of interest,
Eq.~(\ref{instproc}), are displayed in Fig.~\ref{f3}, in leading semi-classical
approximation. The amplitude is expressed in terms of an
integral over the collective coordinates $\rho$ and
the colour orientation $U$,
\begin{equation}
{\mathcal T}_{\mu \,\mu ^{\prime}}^{a}
\left( \gamma^\ast + {\rm g}\rightarrow
\overline{{\rm q}_L} + {\rm q}_R \right) = \int\limits_0^\infty
\frac{d\rho}{\rho^5}\,d(\rho ,\mu _{r} )\,\int dU\, {\mathcal A}_{\mu
\,\mu ^{\prime}}^{a}(\rho, U) \,;\ a=1,2,3,
\label{ampl_inst}
\end{equation}
where
\begin{equation}
d(\rho ,\mu _{r} )= d\,\left( \frac{2\,\pi}{\alpha_s(\mu _{r} )}\right)^6\,
{\exp}\left[-\frac{2\,\pi}{\alpha_s(\mu _{r} )}\right] \,
\left(
\rho\,\mu _{r} \right)^{\beta_0+\frac{\alpha_s(\mu _{r} )}{4\,\pi}\,
\left( \beta_1 - 12\,\beta_0 \right)}\, ,
\label{density}
\end{equation}
denotes the instanton density~\cite{th,ber,abc,morretal}, with $\mu _{r}$
being the
renormalization scale. The form (\ref{density}) of the density, with
next-to-leading order expression for $\alpha_{s}(\mu _{r})$,
\begin{equation}
\label{alpha}
\alpha_{s}(\mu _{r})=
\frac{4\,\pi}{\beta_{0}\,\ln \left(\frac{\mu _{r}^{2}}{\Lambda^{2}}\right)}
\left[ 1 - \frac{\beta_{1}}{\beta_{0}^{2}}
\frac{\ln\left(\ln \left(\frac{\mu _{r}^{2}}{\Lambda^{2}}\right) \right)}
{\ln \left(\frac{\mu _{r}^{2}}{\Lambda^{2}}\right)}
\right] \, ,
\end{equation}
is improved to
satisfy renormalization-group invariance at the 2-loop
le\-vel~\cite{morretal}. The constants $\beta_{0}$ and $\beta_{1}$
are the familiar perturbative coefficients of the QCD beta-function,
\begin{equation}
\label{beta}
\beta_0=11-\frac{2}{3}\,n_f\, ;\hspace{24pt}
\beta_1=102-\frac{38}{3}\,n_f \, ,
\end{equation}
and the constant $d$ is given by\footnote{Strictly speaking, the constant
$d$ is known only to $1$-loop accuracy. In Ref.~\cite{morretal}, only
the ultraviolet divergent part of the $2$-loop correction to the
instanton density has been computed.}
\begin{equation}
\label{d}
d=\frac{C_1}{2}\,{\rm e}^{-3\,C_2+n_f\,C_3}\, ,
\end{equation}
with $C_{1}=0.466$, $C_{2}=1.54$, and $C_{3}=0.153$, in the $\overline{\rm
MS}$-scheme.
In our case, we should of course take $n_{f}=1$ in Eqs.~(\ref{beta}) and
(\ref{d}).
Before analytic continuation, the amplitude
${\mathcal A}_{\mu \,\mu ^{\prime}}$
entering Eq.~(\ref{ampl_inst}) takes the following form in Euclidean space,
\vfill\eject
\begin{eqnarray}
\label{ampi}
\lefteqn{ {\mathcal A}^{a}_{\mu \,\mu ^{\prime}} = -\ii\,e_{q}\,
\lim_{p^{2}\to 0}p^{2}\,
{\rm tr}\left( \sigma^{a}\,A_{\mu ^{\prime}}^{(I)}(p)\right)\times }
\\
\nonumber
&&
\chi_{R}^{\dagger}(k_{2})\,
\left[
\lim_{k_{2}^{2}\to 0}\,(\ii k_{2})\,\kappa (-k_{2})\,
{{\mathcal V}_\mu ^{(t)}}(q,-k_{1})\right .
\\
\mbox{}&&+
\left .
{{\mathcal V}_\mu ^{(u)}}(q,-k_{2})\,\lim_{k_{1}^{2}\to 0}\,
\overline{\phi}(-k_{1})\,(-\ii\,\overline{k}_{1})
\right]
\chi_{L}(k_{1})\, ,
\nonumber
\end{eqnarray}
with contributions ${{\mathcal V}_\mu ^{(t,u)}}$ from the diagrams
on the left and right in Fig.~\ref{f3}, respectively,
\begin{eqnarray}
{{\mathcal V}_\mu ^{(t)}}(q,-k_{1})&\equiv&
\int d^{4}x\,{\rm e}^{-\ii\,q\cdot x} \,
\left[
\overline{\phi}(x)\,\overline{\sigma}_\mu \,
\lim_{k_{1}^{2}\to 0}\,S^{(I)}\,(x,-k_{1})\,(-\ii\,\overline{k}_{1})\,
\right] ,
\label{tinst}\\
{{\mathcal V}_\mu ^{(u)}}(q,-k_{2})&\equiv&
\int d^{4}x\,{\rm e}^{-\ii\,q\cdot x} \,
\left[
\lim_{k_{2}^{2}\to 0}\,(\ii k_{2})\,\overline{S}^{(I)}\,(-k_{2},x)\,
\sigma_\mu \,\kappa(x)\right]\, ,
\label{uinst}
\end{eqnarray}
and generic notation for various Fourier transforms involved,
\begin{equation}
\label{fourier}
f(\ldots,k,\ldots)=\int d^4 x\,{\rm e}^{-\ii\,k\cdot x}\,
f(\ldots,x,\ldots)\, .
\end{equation}
The LSZ-amputation of the classical instanton
gauge field $A^{(I)}_{\mu ^{\prime}}$
in Eq.~(\ref{ampi}) and the quark zero modes $\kappa$ and $\overline{\phi}$
in Eqs.~(\ref{tinst}) and (\ref{uinst}), respectively,
is straightforward~\cite{r},
\begin{eqnarray}
\lim_{p^{2}\to 0}p^{2}\,
{\rm tr}\left( \sigma^{a}\,A_{\mu ^{\prime}}^{(I)}(p)\right) &=&
\frac{2\,\pi^{2}}{g_{s}}\,\rho^{2}\,
{\rm tr}\left[ \sigma^{a}\,U\,\left[
p_{\mu ^{\prime}}-\sigma_{\mu ^{\prime}}\,\overline{p}\right]\,U^{\dagger}
\right]\, ,
\label{lszgluon}\\
\lim_{k_{2}^{2}\to 0}\,
(\ii k_{2})^{\alpha\dot\alpha}\,\kappa^{i}_{\dot\alpha}(-k_{2})&=&
2\,\pi\,\rho^{3/2}\,U^{i}_{\ \beta}\,\epsilon^{\beta\alpha}\, ,
\label{lszkappa}\\
\lim_{k_{1}^{2}\to 0}\,\overline{\phi}_{j}^{\dot\gamma}(-k_{1})\,
(-\ii\,\overline{k}_{1})_{\dot\gamma\delta}
&=&2\,\pi\,\rho^{3/2}\,\epsilon_{\beta\delta}\,
\left( U^{\dagger}\right)^{\beta}_{\ j}\, .
\label{lszbarphi}
\end{eqnarray}
On the other hand, the LSZ-amputation of the quark propagators
$S^{(I)}$ and $\overline{S}^{(I)}$ in
Eqs.~(\ref{tinst}) and (\ref{uinst}), respectively,
is quite non-trivial and has important physical consequences, as we
shall see below. We give here only the final result and refer the interested
reader to Appendix A where the details of the calculation can be
found:
\vfill\eject
\begin{eqnarray}
\label{onshellsi}
\lefteqn{ \lim_{k_1^2\to 0}\,
S^{(I)}\,(x,-k_1)\,(-\ii\,
\overline{k_1}) =}
\\[0.8ex]
\nonumber
&&
\frac{- 1}{\sqrt{\Pi_x}}\,
{\rm e}^{\ii\,k_1\cdot x}\,
\left[
1 +
\frac{1}{2}\,\frac{\rho^2}{x^2}\,
\frac{\left[ U\,x\,\overline{k_1}\,U^\dagger
\right]}{k_1\cdot x}\,
\left( 1- {\rm e}^{-\ii\,k_1\cdot x}\right)
\right]
\, ,
\\[1.6ex]
\label{onshellsibar}
\lefteqn{
\lim_{k_2^2\to 0}\,
\left( \ii\,k_2\right)\,
\overline{S}^{(I)}\,(-k_2,x)
=}
\\[0.8ex]
\nonumber
&&
\frac{- 1}{\sqrt{\Pi_x}}\,
{\rm e}^{\ii\,k_2\cdot x}\,
\left[
1 +
\frac{1}{2}\,\frac{\rho^2}{x^2}\,
\frac{\left[ U\,k_2\,\overline{x}\,U^\dagger
\right]}{k_2\cdot x}\,
\left( 1- {\rm e}^{-\ii\,k_2\cdot x}\right)
\right]
\, .
\end{eqnarray}
It should be noted that the first terms in
Eqs.~(\ref{onshellsi}), (\ref{onshellsibar}), corresponding to the
$1$ in square brackets, were argued to be present on general grounds
already in Ref.~\cite{bbb}. The remaining terms, however, have
not been given in the literature. As we shall see below,
they play a very important r{\^o}le in ensuring electromagnetic
gauge invariance.
The Fourier transforms entering Eqs.~(\ref{tinst}) and (\ref{uinst}),
respectively,
can now be done with the help of Eqs.~(\ref{onshellsi}) and
(\ref{onshellsibar}). The result is (see Appendix B),
\begin{eqnarray}
\label{tvertex}
\lefteqn{
{\mathcal V}^{(t)}_{\mu \ j\,\alpha}\,(q,-k_{1})
=
2\,\pi\,\ii\,\rho^{3/2}\,
\left( U^{\dagger}\right)^\gamma_{\ j}\,
\left\{
\frac{1}{2}\,
\frac{\left[ \epsilon\,k_{1}\,\overline{\sigma}_\mu \right]_{\gamma\alpha}}
{q\cdot k_{1}}\, f\left( \rho\,\sqrt{q^{2}} \right) \right .}
\\[1.6ex]
\nonumber
&&
\left .
+ \left[
\frac{\left[ \epsilon\,\left(q-k_{1}\right)
\,\overline{\sigma}_\mu \right]_{\gamma\alpha}}
{(q-k_{1})^2}
-\frac{1}{2}\,
\frac{\left[ \epsilon\,k_{1}\,\overline{\sigma}_\mu \right]_{\gamma\alpha}}
{q\cdot k_{1}}
\right]\,f\left(\rho\,\sqrt{\left( q-k_{1}\right)^2}\right)
\right\}\, ,
\nonumber
\\[2.4ex]
\label{uvertex}
\lefteqn{
{\mathcal V}^{(u)\ \alpha\, i}_{\mu \ }\,(q,-k_{2}) =
2\,\pi\,\ii\,\rho^{3/2}\,
U^i_{\ \gamma}\,
\left\{
\frac{1}{2}\,
\frac{\left[ \sigma_\mu \,
\overline{k_{2}}\,\epsilon\right]^{\alpha\gamma}}
{q\cdot k_{2}}\, f\left( \rho\,\sqrt{q^{2}}\right) \right .}
\\[1.6ex]
\nonumber
&&
\left .
+ \left[
\frac{\left[ \sigma_\mu \,
\left(\overline{q}-\overline{k_{2}}\right)\,\epsilon
\right]^{\alpha\gamma}}
{(q-k_{2})^2}
-\frac{1}{2}\,
\frac{\left[ \sigma_\mu \,
\overline{k_{2}}\,\epsilon\right]^{\alpha\gamma}}
{q\cdot k_{2}}
\right]\,f\left(\rho\,\sqrt{\left( q-k_{2}\right)^2}\right)
\right\}\, ,
\nonumber
\end{eqnarray}
with the shorthand (``form factor''),
\begin{equation}
\label{form}
f(\omega )\equiv \omega\,K_{1}(\omega ),
\end{equation}
in terms of the Bessel-K function, implying the normalization,
\begin{equation}
\label{formnorm}
f(0)=1\, .
\end{equation}
The next step is to insert Eqs.~(\ref{tvertex}), (\ref{uvertex}),
and (\ref{lszgluon}-\ref{lszbarphi}) into Eq.~(\ref{ampi}) and to
perform the integration over the colour orientation according to
Eq.~(\ref{ampl_inst}) by means of the relation,
\begin{eqnarray}
\label{colorint}
\lefteqn{
\int dU\,U^k_{\ \beta^\prime}\,(U^\dagger )^{\gamma^\prime}_{\ l}\,
U^i_{\ \tau}\,(U^\dagger )^\gamma_{\ j} =}
\\
\nonumber
&&\frac{1}{6}\,\left[
\delta_\tau^{\ \gamma^\prime}\,\delta^i_{\ l}\
\delta_{\beta^\prime}^{\ \gamma}\,\delta^k_{\ j}
+
\delta_\tau^{\ \gamma}\,\delta^i_{\ j}\
\delta_{\beta^\prime}^{\ \gamma^\prime}\,\delta^k_{\ l}
+
\epsilon_{\tau\,\beta^\prime}\,\epsilon^{i\,k}\
\epsilon^{\gamma^\prime\,\gamma}\,\epsilon_{l\,j}
\right] \, .
\end{eqnarray}
After analytic continuation to Minkowski space we find for the
scattering amplitude, Eq.~(\ref{ampl_inst}),
\begin{eqnarray}
\label{ampi2}
&& {\mathcal T}_{\mu \,\mu ^{\prime}}^a\,
\left( \gamma^\ast + {\rm g}\rightarrow
\overline{{\rm q}_L} + {\rm q}_R \right)
= -\ii\,\frac{4}{3}\,\pi^4\,\frac{e_q}{g_s}\,\sigma^a \int\limits_0^\infty
d\rho\,d(\rho ,\mu _{r} )\,\times
\\[0.8ex]
&&
\chi_R^\dagger (k_2)
\left[ \left(
\sigma_{\mu ^{\prime}}\overline{p}-p\overline{\sigma}_{\mu ^\prime}\right)
V(q,k_1;\rho )\overline{\sigma}_\mu
-\sigma_\mu \overline{V}(q,k_2;\rho )
\left(
\sigma_{\mu ^\prime}\overline{p} - p\overline{\sigma}_{\mu ^\prime}
\right)
\right]
\chi_L(k_1) ,
\nonumber
\end{eqnarray}
with the four-vector $V_{\lambda}$,
\begin{eqnarray}
\nonumber
V_\lambda (q,k;\rho ) &\equiv&
\left[
\frac{\left( q-k\right)_\lambda}{-(q-k)^2}
+\frac{k_{\lambda}}{2 q\cdot k}
\right]\rho\sqrt{-\left( q-k\right)^2}\,
K_1\left(\rho\sqrt{-\left( q-k\right)^2}\right)
\\[0.8ex]
\label{V}
\mbox{}&&
-
\frac{k_{\lambda}}{2 q\cdot k}\rho\sqrt{-q^{2}}\,
K_1\left(\rho\sqrt{-q^{2}}\right) .
\end{eqnarray}
At this stage of our instanton calculation,
the gauge-invariance con\-st\-raints, Eqs.~(\ref{gaugeinv}),
can easily be checked. While the relation
${\mathcal T}_{\mu \,\mu ^{\prime}}^a\,p^{\mu ^\prime}=0$ holds trivially,
the electromagnetic (e.m.) current conservation
$q^\mu \,{\mathcal T}_{\mu \,\mu ^{\prime}}^a=0$ follows again
from the relations (\ref{com}) of the $\sigma$-matrices, the Weyl-equations
(\ref{weyleq}) and the on-shell conditions $k_1^2=k_2^2=0$.
Electromagnetic current conservation provides also for a non-trivial
check of our result for the amputated quark propagators, which differs
somewhat from the result quoted in Ref.~\cite{bbb}:
If we keep only the first terms in
Eqs.~(\ref{onshellsi}), (\ref{onshellsibar}), corresponding to the
$1$ in square brackets, e.m. current conservation would only hold
for a restricted set of momenta in phase space, namely for
$(q-k_1)^2=(q-k_2)^2$.
Furthermore, we note one of the main results of this paper:
The integration over the instanton size $\rho$ in Eq.~(\ref{ampi2})
is {\it finite}. In particular, the {\it good infrared} behavior
(large $\rho$) of the integrand is due to the exponential decrease
of the Bessel-K function for large $\rho$ in Eq.~(\ref{V}). Its origin,
in turn, can be traced back to the ``feed-through'' of the factor
$1/\sqrt{\Pi_x}$, by which the amputated (current) quark propagators
(\ref{onshellsi}) and (\ref{onshellsibar}) in the $I$-background
essentially differ from the respective amputated
free propagators. If the current-quark propagators
in Eqs.~(\ref{tinst}) and (\ref{uinst}) are naively approximated by the
free ones (c.\,f. Eqs.~(\ref{si}), (\ref{sibar})),
\begin{equation}
S^{(0)}(x,y)=\frac{x-y}{2\pi^2(x-y)^4}\,;\hspace{24pt}
\overline{S}^{(0)}(x,y)=\frac{\overline{x}-\overline{y}}{2\pi^2(x-y)^4}\, ,
\end{equation}
the result is both gauge variant and contains an IR-divergent
piece in the $\rho$ integration.
We have thus demonstrated explicitly and to our
knowledge for the first time that the typical hard scales ($Q^{2},\ldots$)
in deep-inelastic scattering provide a dynamical IR cutoff for the
instanton size (at least in leading semi-classical approximation).
Now we are ready to perform the final integration over the instanton
size $\rho$ by inserting the instanton density, Eq.~(\ref{density}), into
Eq.~(\ref{ampi2}). The result is:
\begin{eqnarray}
\label{ampifin}
&& {\mathcal T}_{\mu \,\mu ^{\prime}}^a\,
\left( \gamma^\ast + {\rm g}\rightarrow
\overline{{\rm q}_L} + {\rm q}_R \right)
=
\\[2.4ex]
\nonumber
&& -\ii\frac{\sqrt{2}}{3}d\pi^{3}e_q
\left( \frac{2\,\pi}{\alpha_s(\mu _{r} )}\right)^{13/2}
{\exp}\left[-\frac{2\,\pi}{\alpha_s(\mu _{r} )}\right]\,
2^{\,b}\Gamma \left(\frac{b+1}{2}\right)
\Gamma \left(\frac{b+3}{2}\right)
\sigma^a\,\times
\\[2.4ex]
&&
\chi_R^\dagger (k_2)
\left[ \left(
\sigma_{\mu ^{\prime}}\overline{p}-p\overline{\sigma}_{\mu ^\prime}\right)
v(q,k_1;\mu _{r} )\overline{\sigma}_\mu
-\sigma_\mu \overline{v}(q,k_2;\mu _{r} )
\left(
\sigma_{\mu ^\prime}\overline{p} - p\overline{\sigma}_{\mu ^\prime}
\right)
\right]
\chi_L(k_1) ,
\nonumber
\end{eqnarray}
with the four-vector $v_{\lambda}$,
\begin{eqnarray}
\label{v}
&&v_\lambda (q,k;\mu _{r} ) \equiv
\\[1.6ex]
\nonumber
&&
\frac{1}{\mu _{r}}\,\left\{
\left[
\frac{\left( q-k\right)_\lambda}{-(q-k)^2}
+\frac{k_{\lambda}}{2 q\cdot k}
\right] \left(
\frac{\mu _{r}}{\sqrt{-\left( q-k\right)^2}}\right)^{b+1}
-
\frac{k_{\lambda}}{2 q\cdot k}
\left( \frac{\mu _{r}}{\sqrt{-q^{2}}}\right)^{b+1}
\right\} .
\end{eqnarray}
In Eqs.~(\ref{ampifin}) and (\ref{v}), the variable $b$ is a shorthand for the
effective power of $\rho\mu _{r}$ in the instanton density,
Eq.~(\ref{density}),
\begin{equation}
\label{beff}
b\equiv
\beta_0+\frac{\alpha_s(\mu _{r} )}{4\,\pi}\, \left( \beta_1 -
12\,\beta_0 \right) \, .
\end{equation}
The next steps consist in contracting the amplitude, Eq.~(\ref{ampifin}),
with the gluon polarization vector $\epsilon_{g}^{\mu ^{\prime}}(p)$ and
taking the modulus squared of this amplitude according to Eq.~(\ref{wmunu_n}).
After applying Eq.~(\ref{compl}),
the remaining spinor traces can be evaluated, in
principle, by repeated use of Eq.~(\ref{com}). For the actual calculation,
however, we used FORM and, for an independent check, the HIP package
for MAPLE. The final result for the relevant projections
(c.f. Eqs.~(\ref{diffcross}), (\ref{projF2}), and (\ref{projFL}))
of the contribution of the $I$-induced process (\ref{instproc})
to the differential gluon structure tensor is found to be
\begin{eqnarray}
\nonumber
\lefteqn{
-g_{\mu \nu}\, \frac{d{\mathcal W}^{\mu \nu\,{(2)}}_{g}}{dt}
\, \left( \gamma^\ast + {\rm g}\rightarrow
\overline{{\rm q}_L} + {\rm q}_R \right)
=}
\\[1.6ex]
\label{gmunu}
&&\frac{e_q^2}{16}\,
{\mathcal N}^{2}\,
\left( \frac{2\,\pi}{\alpha_s(\mu _{r} )}\right)^{13}\,
{\exp}\left[-\frac{4\,\pi}{\alpha_s(\mu _{r} )}\right]\,
\left( \frac{\mu _{r}^{2}}{Q^{2}}\right)^{b}\,\times
\\[1.6ex]
\nonumber
&& \frac{1-x}{Q^{2}}\,
\left[
\left(\frac{Q^{2}}{-t}\right)^{b+1} +
\left(\frac{Q^{2}}{-u}\right)^{b+1} +
2\,t\,u\,
\frac{
\left( \left(\frac{Q^{2}}{-t}\right)^{\frac{b+1}{2}}
-1 \right)
\left( \left(\frac{Q^{2}}{-u}\right)^{\frac{b+1}{2}}
-1 \right)
}
{(t+Q^{2})(u+Q^{2})}
\right] \, ,
\\[2.4ex]
\nonumber
\lefteqn{
\frac{p_\mu \,p_{\nu}}{p\cdot q} \,
\frac{d{\mathcal W}^{\mu \nu\,{(2)}}_{g}}{dt}
\, \left( \gamma^\ast + {\rm g}\rightarrow
\overline{{\rm q}_L} + {\rm q}_R \right)
=}
\\[1.6ex]
\label{pmupnu}
&&\frac{e_q^2}{16}
{\mathcal N}^{2}\,
\left( \frac{2\,\pi}{\alpha_s(\mu _{r} )}\right)^{13}\,
{\exp}\left[-\frac{4\,\pi}{\alpha_s(\mu _{r} )}\right]
\,
\left( \frac{\mu _{r}^{2}}{Q^{2}}\right)^{b}\,\times
\\[1.6ex]
\nonumber
&&
\frac{(1-x)^{2}}{Q^{2}}\,
t\,u\,
\left[
\frac{\left(\frac{Q^{2}}{-t}\right)^{\frac{b+1}{2}}
+\frac{u}{Q^{2}}\frac{x}{1-x}}{t+Q^{2}}
-\frac{\left(\frac{Q^{2}}{-u}\right)^{\frac{b+1}{2}}
+\frac{t}{Q^{2}}\frac{x}{1-x}}{u+Q^{2}}
\right]^{2} \, .
\end{eqnarray}
Here we have introduced the shorthand
\begin{equation}
{\mathcal N} \equiv \sqrt{\frac{2}{3}}\,\pi^{2}\,d\
2^{\,b}\, \Gamma \left(\frac{b+1}{2}\right)\,
\Gamma \left(\frac{b+3}{2}\right)\, .
\end{equation}
In Eqs.~(\ref{gmunu}) and (\ref{pmupnu}), the $t\leftrightarrow u$ symmetry
is manifest.
\begin{figure}
\begin{center}
\epsfig{file=dsigdz.eps,width=9cm}\vfill
\hspace{8pt}
\epsfig{file=fixangle.eps,width=8cm}
\caption[dum]{\label{f4}
Differential cross-sections, $d\sigma /d\cos\theta\ [\mbox{nb}]$,
of the $I$-induced {\it chirality-violating} process (\ref{instproc})
and the perturbative {\it chirality-conserving} process
(\ref{pertproc}), both for
fixed c.m. scattering angle, $\theta=90^{\,\circ}$,
$\Lambda=0.234$ GeV, $e_{q}=2/3$, and $\mu _{r}=Q$.
Top: For fixed $x=0.25$, as function of $Q$ [GeV].
Bottom:
For fixed $Q=10$ GeV, as function of $x$.}
\end{center}
\end{figure}
Upon inserting Eqs.~(\ref{gmunu}), (\ref{pmupnu}) into
Eqs.~(\ref{diffcross}), (\ref{projF2}), and (\ref{projFL}), we see
that the contribution of the $I$-induced process
(\ref{instproc}) to the differential cross-section $d\sigma /dt$
and the differential gluon structure functions,
$d{\mathcal F}^{(2)}_{2\,g}/dt, d{\mathcal F}^{(2)}_{L\,{g}}/dt$,
is well-behaved as long as we avoid the (collinear) singularities for
$t,u\rightarrow 0$. This is illustrated in Fig.~\ref{f4},
where we compare the differential cross-sections, $d\sigma /\cos\theta$,
of both the $I$-induced process, Eq.~(\ref{instproc}), and the perturbative
process, Eq.~(\ref{pertproc}), where
\begin{equation}
\label{kt}
t=-\frac{Q^{2}}{2\,x}\,
\left( 1- \cos \theta
\right)
\, .
\end{equation}
We note that the renormalization-scale dependence of the $I$-induced
cross-section in Fig.~\ref{f4} is very small, due to the
re\-nor\-ma\-li\-za\-tion-group im\-proved density (\ref{density}).
Let us address at this point the important question concerning the
range of validity of the present calculation. Specifically, let us
examine the constraints emerging from the requirement of the
dilute instanton gas approximation following Refs.~\cite{cdg,ag,as,svz}.
Along these lines one finds that instantons with size
\begin{equation}
\label{rhoc}
\rho > \rho_{c}\simeq 1/(500\ {\rm MeV})
\end{equation}
are ill-defined semi-classically~\cite{svz},
corresponding to a breakdown of the dilute gas approximation.
On the other hand, using the form of our $\rho$ integral in
Eqs.~(\ref{ampi2}), (\ref{V}), we may determine the average
instanton size $\langle \rho\rangle$ contributing for a given
virtuality
\begin{equation}
\label{virt}
{\mathcal Q}=\min \left( Q,
\sqrt{-t}=\frac{Q}{\sqrt{x}}\sin\frac{\theta}{2},
\sqrt{-u}=\frac{Q}{\sqrt{x}}\cos\frac{\theta}{2}
\right)\, ,
\end{equation}
according to
\begin{equation}
\label{avrho}
\langle \rho\rangle \equiv
\frac{\int\limits_{0}^{\infty}d\rho\,
\rho\,\rho^{b+1}\,K_{1}(\rho\,{\mathcal Q})}
{\int\limits_{0}^{\infty}d\rho\, \rho^{b+1}\,K_{1}(\rho\,{\mathcal Q})}
\simeq
\frac{b+3/2}{\mathcal Q} \, .
\end{equation}
Hence, with Eq.~(\ref{rhoc}) and Eq.~(\ref{beff}), we find that
the virtuality $\mathcal Q$ should obey
\begin{equation}
\label{qmin}
{\mathcal Q}\ (>)> (5-6)\ {\rm GeV}\, .
\end{equation}
In particular, our results in Fig.~\ref{f4} (top) for
the $I$-induced differential
cross-section, $d\sigma /d\cos\theta$, at $\theta=90^{\,\circ}$ and $x=0.25$,
should be taken seriously only for $Q\ (= {\mathcal Q})> (5-6)$ GeV
(since here $\sqrt{-t}=\sqrt{-u}=Q/\sqrt{2x}>Q$).
Thus, like in the perturbative case, {\it fixed-angle scattering
processes at high $Q^2$ are reliably calculable in (instanton) perturbation
theory} (at least in leading semi-classical approximation).
Next, we note that the contributions (\ref{gmunu}), (\ref{pmupnu}) of
the $I$-induced exclusive process (\ref{instproc})
to the differential gluon structure functions are much more singular
($\propto (-t, -u)^{-(b+1)}$) for $t\to 0,u\to 0$ than the perturbative
ones ($\propto (-t, -u)^{-1} $). This leads to a much stronger
scheme dependence~\cite{mrs} than in the perturbative case.
Let us have a closer look at this feature.
We regularize the collinear divergence of the $t$ integral
along the same lines as in
perturbation theory, i.e. we restrict the integration to the interval
$\{ -Q^{2}/x+\mu _{c}^{2},-\mu _{c}^{2}\}$. On account of
Eqs.~(\ref{projF2}), (\ref{projFL}), we then obtain for
the hard contributions of the $I$-induced exclusive process
(\ref{instproc}) to the gluon structure functions,
\begin{eqnarray}
\nonumber
{\mathcal F}_{2\,g}^{(\overline{{\rm q}_L}{\rm q}_R)}\,(x,Q^2;\mu _{c}^{2})
& =&\frac{e_q^2}{8}\,
{\mathcal N}^{2}\,
\left( \frac{2\,\pi}{\alpha_s(\mu _{r} )}\right)^{13}\,
{\exp}\left[-\frac{4\,\pi}{\alpha_s(\mu _{r} )}\right]
\,\left( \frac{\mu _{r}^{2}}{\mu _{c}^{2}}\right)^{b}
\\[1.6ex]
\label{F2inst}
\mbox{}&& \times
\frac{x\,(1-x)}{b}\,
\left[1+{\mathcal O}\left( \frac{\mu _{c}^{2}}{Q^{2}}\right)\right]
\, ,
\\[2.4ex]
\nonumber
{\mathcal F}_{L\,g}^{(\overline{{\rm q}_L}{\rm q}_R)}\,(x,Q^2;\mu _{c}^{2})
& =&\frac{e_q^2}{2}\,
{\mathcal N}^{2}\,
\left( \frac{2\,\pi}{\alpha_s(\mu _{r} )}\right)^{13}\,
{\exp}\left[-\frac{4\,\pi}{\alpha_s(\mu _{r} )}\right]
\,\left( \frac{\mu _{r}^{2}}{\mu _{c}^{2}}\right)^{b}\,
\frac{\mu _{c}^{2}}{Q^{2}}\,
\\[1.6ex]
\label{FLinst}
\mbox{}&& \times \,
\frac{x\,(1-x)^{2}}{b-1}\,
\left[1+{\mathcal O}\left(\frac{\mu _{c}^{2}}{Q^{2}}\right)\right]
.
\end{eqnarray}
In the Bjorken limit, $Q^{2}/\mu _{c}^{2}\to \infty$,
but with $\mu _{c}^{2}/\mu _{r}^{2}$
fixed, we find from Eqs.~(\ref{F2inst}) and (\ref{FLinst}),
respectively,
\begin{eqnarray}
\label{F2instbj}
\lefteqn{
\lim_{Q^{2}\to \infty}\,
{\mathcal F}_{2\,g}^{(\overline{{\rm q}_L}{\rm q}_R)}(x,Q^2;\mu _{c}^{2})
=}
\\[1.6ex]
\nonumber
&&
\frac{e_q^2}{8}\,
{\mathcal N}^{2}\,
\left( \frac{2\,\pi}{\alpha_s(\mu _{r} )}\right)^{13}\,
{\exp}\left[-\frac{4\,\pi}{\alpha_s(\mu _{r} )}\right]
\,\left( \frac{\mu _{r}^{2}}{\mu _{c}^{2}}\right)^{b}\,
\frac{x\,(1-x)}{b}
\, ,
\\[2.4ex]
\label{FLinstbj}
\lefteqn{
\lim_{Q^{2}\to \infty}\,
{\mathcal F}_{L\,g}^{(\overline{{\rm q}_L}{\rm q}_R)}(x,Q^2;\mu _{c}^{2})
\equiv }
\\[1.6ex]
\nonumber
&&\lim_{Q^{2}\to \infty}\,\left[
{\mathcal F}_{2\,g}^{(\overline{{\rm q}_L}{\rm q}_R)}
(x,Q^2;\mu _{c}^{2})
-2\,x\,{\mathcal F}_{1\,g}^{(\overline{{\rm q}_L}{\rm q}_R)}
(x,Q^2;\mu _{c}^{2})\right]
= 0\, .
\end{eqnarray}
Hence, in this limit, the considered $I$-induced process gives a
{\it scaling} contribution to ${\mathcal F}_{2\,g}$ and
the analogue of the Callan-Gross relation,
${\mathcal F}_{2\,g}=2\,x\,{\mathcal F}_{1\,g}$, holds.
In particular, this means that the {\it same} parton distribution
can absorb the infrared sensitivity of both structure functions,
${\mathcal F}_{2\,g}^{(\overline{{\rm q}_L}{\rm q}_R)}$
and ${\mathcal F}_{1\,g}^{(\overline{{\rm q}_L}{\rm q}_R)}$.
This is one of the prerequisites of factorization~\cite{pert}.
\section{Conclusions and Outlook}
In this paper, we studied QCD-instanton induced processes in
deep-inelastic lepton-hadron scattering.
The purpose of the present work was to shed further light on the important
questions around the IR-behaviour associated with the instanton size.
In order to eliminate unessential technical complications, we have reduced
the realistic task of evaluating
the $I$-induced cross-sections of the chirality violating multi-particle
processes (illustrated in Fig.~\ref{f1})
\begin{equation}
\gamma^{\ast}+{\rm g}\Rightarrow
\overline{{\rm u}_{L}}+{\rm u}_{R}\,
+ \overline{{\rm d}_{L}}+{\rm d}_{R}\,
+\overline{{\rm s}_{L}}+{\rm s}_{R}\,
+n_g\, {\rm g},
\end{equation}
to the detailed study of the {\it simplest} one,
without additional gluons and with just one massless flavour ($n_f=1$),
\begin{equation}
\gamma^{\ast}(q)+{\rm g}(p)\Rightarrow
\overline{{\rm q}_{L}}(k_{1})+{\rm
q}_{R}(k_{2})\, .
\end{equation}
We have explicitly calculated the corresponding fixed angle cross-section
and the contributions to the gluon structure functions
within standard instanton perturbation theory in leading
semi-classical approximation (Sect.~3). To this approximation,
{\it fixed-angle scattering processes at high $Q^2$ are reliably
calculable}. In the Bjorken limit, the considered $I$-induced process gives a
{\it scaling} contribution to ${\mathcal F}_{2\,g}$ and
the analogue of the Callan-Gross relation,
${\mathcal F}_{2\,g}=2\,x\,{\mathcal F}_{1\,g}$, holds.
All along we focused our main attention on the IR behaviour associated with
the instanton size. Gauge invariance was kept manifest along the
calculation.
As a central result of this paper and unlike Ref.~\cite{bb}, we found
{\it no} IR divergencies associated with the integration over the
instanton size $\rho$, which can even be perfomed analytically. We have
explicitly demonstrated that the typical hard scale ${\mathcal Q}$
in deep-inelastic scattering provides a {\it dynamical} infrared cutoff for
the instanton size, $\rho \mbox{\,\raisebox{.3ex {\mathcal O}(1/{\mathcal Q})$.
Thus, deep-inelastic scattering may indeed be viewed as a
distinguished process for studying manifestations of QCD-instantons.
In Ref.~\cite{bb}, the $I$-induced contribution to deep-inelastic scattering of
a virtual gluon from a real one~\cite{bbgg},
$g^\ast g \Rightarrow g^\ast g$,
served as a simplified r\^{o}le model for the splitting into a IR-finite
contribution ($\rho \mbox{\,\raisebox{.3ex 1/Q$) and an IR-divergent term (large $\rho$).
As speculated by one of the
authors~\cite{b}, the occurence of the IR-divergent term could well have
been due to the lacking gauge invariance of this model, associated with the
off-shellness of one of the initial gluons. In fact, one may enforce
gauge invariance by replacing the instanton gauge field
$A^{(I)}_\mu (x)$ describing the virtual gluon by the familiar
gauge-invariant operator
\begin{equation}
G^{(I)}_\mu (x)=e^\alpha\, [x+\infty,x]\,
G^{(I)}_{\mu \,\alpha}(x)\,[x,x+\infty],
\end{equation}
where $G^{(I)}_{\mu \,\alpha}(x)$ is the instanton field-strength,
and
\begin{equation}
[x,x+\infty]=P \exp\left\{ \ii\,g_{s} \int^\infty_0 d\lambda\,\,
e\cdot A^{(I)}(x+\lambda e)
\right\}\, ,
\end{equation}
is a gauge factor ordered along the lightlike line in the direction
$e_\mu =(q_\mu +x p_\mu )/(2p\cdot q)$. In this case the IR-divergent term is,
indeed, absent~\cite{b}.
Since our present calculation is manifestly gauge-invariant, the absence of
an IR divergent term fits well in line with these arguments.
A further main purpose of the present calculation was to provide a clean and
explicit discussion of most of the crucial steps involved in our subsequent
task~\cite{mrs} to calculate the {\it dominant} $I$-induced contributions
coming from final states with a large number of gluons (and three massless
flavours, say).
Let us close with some comments on the generalization
to the more realistic case with $n_g$ gluons in the final state, which is
entirely straightforward~\cite{mrs,q96}. Instead of Eq.~(\ref{ampi2}),
the corresponding amplitude involves, in leading semi-classical approximation,
the additional factors from the $n_g$ gluons (c.f. Eq.~(\ref{lszgluon})),
\begin{eqnarray}
\label{ampi2g}
\lefteqn{
{\mathcal T}_\mu ^{a\,a_1\ldots a_{n_g}}\,\left( \gamma^\ast + {\rm
g}\rightarrow \overline{{\rm q}_L} + {\rm q}_R +n_g\,{\rm g}\right)
= }
\\[1.6ex]
\nonumber
&&
\ii\,e_q\,4\,\pi^2\,\left(
\frac{\pi^3}{\alpha_s}\right)^{\frac{n_g+1}{2}}\,
\int dU
\int\limits_0^\infty d\rho\,d(\rho ,\mu _{r} )\,\rho^{2\,n_g}
\\[1.6ex]
\nonumber
&&\times \,
{\rm tr}\left[ \sigma^{a}\,U\,\left[
\epsilon_g(p)\cdot p-\epsilon_g(p)\,\overline{p}
\right]\,U^{\dagger} \right]\,
\prod_{i=1}^{n_g}
{\rm tr}\left[ \sigma^{a_i}\,U\,\left[
\epsilon_g(p_i)\,\overline{p_i}-\epsilon_g(p_i)\cdot p_i
\right]\,U^{\dagger} \right]
\\[1.6ex]
\nonumber
&&\times
\left\{
\left[
U \chi_R^\dagger (k_2 ) \epsilon
\right]
\left[\epsilon V(q,k_1;\rho)\overline{\sigma}_\mu
\chi_L(k_1)\, U^\dagger
\right]
\right .
\\[1.6ex]
\nonumber
\mbox{}&&
-
\left .
\left[ U \chi_R^\dagger (k_2)
\sigma_\mu \overline{V}(q,k_2;\rho)\epsilon
\right]
\left[
\epsilon\, \chi_L(k_1)\, U^\dagger
\right]
\right\} ,
\end{eqnarray}
where the four-vector $V_\lambda$ is again given by Eq.~(\ref{V}).
Besides the enhancement by a factor of $(\pi^3/\alpha_s)^{1/2}$, each
additional gluon gives rise to a factor of $\rho^2$ under
the $I$-size integral. The IR-finiteness of this integral is, however,
not altered by the
presence of the additional overall factor of $\rho^{2\,n_g}$, on account of
the exponential cutoff $\propto \exp[ -\rho {\mathcal Q}]$ from the Bessel-K
function in the ``form-factors'' contained in $V_\lambda(q,k;\rho)$,
Eq.~(\ref{V}). We also note, that the amplitude (\ref{ampi2g}) satisfies
e.m. gauge invariance.
In analogy to electro-weak
$(B+L)$-violation~\cite{m}, one expects~\cite{bb,rs} the sum of
the final-state gluon contributions to exponentiate, such that
the total $I$-induced $\gamma^\ast$g cross-section takes the form
(at large $Q^2$),
\begin{eqnarray}
\label{exp}
\sigma^{{ (I)}}_{\gamma^\ast { g}}(x,Q^2) &\equiv&
\sum_{n_{ g}}\sigma^{{ (I)}}_{\gamma^\ast { g}\,n_{ g}}(x,Q^2)
\\[1.6ex]
\nonumber
&\sim& \int_x^1 dx^\prime\int\limits^{Q^2\frac{{x^\prime}}{x}}
\frac{dQ^{\prime 2}}{Q^{\prime 2}}\,\ldots\,
\frac{1}{Q^{\prime 2}}\,
\exp\left[-\frac{4\pi}{\alpha_s(Q^\prime)}\,F(x^\prime )\right]\, ,
\end{eqnarray}
where the so-called ``holy-grail function''~\cite{m}
$F({x^\prime} )$ (normalized to F(1)=1) is expected to decrease towards
smaller ${x^\prime}$, which implies a dramatic growth of
$\sigma^{{ (I)}}_{\gamma^\ast { g}}(x,Q^2)$ for decreasing $x$.
\section*{Appendix A}
Here we want to derive Eqs.~(\ref{onshellsi}) and
(\ref{onshellsibar}) for the LSZ-amputated quark propagators. Let us
first consider the Fourier transform of the
quark propagator (\ref{si}) which we write as
\begin{equation}
S^{(I)}(x,-k)=\frac{1}{\sqrt{\Pi_x}}
\sum_{i=1}^3 s^{(i)} (x,-k) \, ,
\label{sumsi}
\end{equation}
where
\begin{eqnarray}
\label{s1}
\lefteqn{
s^{(1)}(x,-k)
= }
\\[1.6ex]
\nonumber
&&
2\left[ x- (-\ii\partial_k)\right]\,
(-\ii\overline{\partial}_k)\,(-\ii\partial_k)\,
\int \frac{d^4y}{(2\pi )^2} \frac{{\rm e}^{\ii k\cdot y}}
{\left( (x-y)^2\right)^2 \left( y^2+\rho^2\right)^{1/2}
\left( y^2\right)^{1/2}} \, ,
\\[2.4ex]
\label{s2}
\lefteqn{
s^{(2)}(x,-k)
=}
\\[1.6ex]
\nonumber
&&
2\,\frac{\rho^2}{x^2}\left[ x- (-\ii\partial_k)\right]
U x (-\ii\overline{\partial}_k)\, U^\dagger
\int \frac{d^4y}{(2\pi )^2} \frac{{\rm e}^{\ii k\cdot y}}
{\left( (x-y)^2\right)^2 \left( y^2+\rho^2\right)^{1/2}
\left( y^2\right)^{1/2}} \, ,
\\[2.4ex]
\label{s3}
\lefteqn{
s^{(3)}(x,-k)
=}
\\[1.6ex]
\nonumber
&&
\frac{\rho^2}{x^2}
\sigma_\mu U x\,
\left[ \overline{x}- (-\ii\overline{\partial}_k)\right]
\sigma_\mu (-\ii\overline{\partial}_k)\, U^\dagger
\int \frac{d^4y}{(2\pi )^2} \frac{{\rm e}^{\ii k\cdot y}}
{\left( x-y\right)^2 \left( y^2+\rho^2\right)^{3/2}
\left( y^2\right)^{1/2}} \, .
\end{eqnarray}
Our strategy to analyze the $k^2\to 0$ limit of Eqs.~(\ref{s1}-\ref{s3})
starts by partially evaluating the master integral,
\begin{equation}
I\,(-k;x,\rho ;\alpha , \beta , \gamma )
\equiv
\int \frac{d^4y}{(2\pi )^2} \frac{{\rm e}^{\ii k\cdot y}}
{\left( (x-y)^2\right)^\alpha \left( y^2+\rho^2\right)^\beta
\left( y^2\right)^\gamma} \, ,
\label{masterint}
\end{equation}
by means of the
Feynman parametrization (see e.g. \cite{yn}),
\begin{eqnarray}
\label{feynman}
\lefteqn{
\frac{1}{A^\alpha B^\beta C^\gamma}
=}
\\[1.6ex]
\nonumber
&& \frac{\Gamma (\alpha +\beta +\gamma ) }
{\Gamma (\alpha )\,\Gamma (\beta )\, \Gamma (\gamma )}\,
\int_0^1 da\,a \,
\int_0^1 db\,
\frac{(ab)^{\alpha -1} (a(1-b))^{\beta -1}\,(1-\alpha)^{\gamma
-1}} {
\left[ ab\,A +a\,(1-b)\,B + (1-a)\,C\right]^{\alpha +\beta +\gamma}}
\, .
\end{eqnarray}
With the help of Eq.~(\ref{feynman}),
it is possible to show that Eq.~(\ref{masterint}) can be expressed as
\begin{eqnarray}
\label{masterintfin}
\lefteqn{
I\,(-k;x,\rho ;\alpha , \beta , \gamma )
=}
\\[1.6ex]
\nonumber
&& \frac{2^{1-(\alpha +\beta +\gamma )}}{ \Gamma (\alpha )\,
\Gamma (\beta )\, \Gamma (\gamma )}\,
\int_0^1 da\,a^{\alpha +\beta -1}\,(1-a)^{\gamma -1}
\int_0^1 db\,b^{\alpha -1}\,(1-b)^{\beta -1}\,
{\rm e}^{\ii\,k\cdot x\,ab}
\\[1.6ex]
\nonumber
&&
\times \,\left(
\frac{\sqrt{x^2 ab(1-ab)+\rho^2 a(1-b)}}{\sqrt{k^2}}
\right)^{2-(\alpha +\beta +\gamma )}
\\[1.6ex]
\nonumber
&&
\times \,K_{2-(\alpha +\beta +\gamma )}
\left(
\sqrt{k^2}\sqrt{x^2 ab(1-ab)+\rho^2\, a(1-b)}
\right) .
\end{eqnarray}
Next we insert Eq.~(\ref{masterintfin}) into
Eqs.~(\ref{s1})-(\ref{s3}), perform the various derivatives, and expand
the integrand with respect to $k^2\to 0$. Finally, the remaining Feynman
parameter integrations are done. After this procedure we find:
\vfill\eject
\begin{eqnarray}
\lim_{k^{2}\to 0}{s^{(1)}}
(x,-k)(-\ii\,\overline{k}) &=&
-{\rm e}^{\ii\,k\cdot x}\, ,
\\[1.6ex]
\lim_{k^{2}\to 0}{s^{(2)}}
(x,-k)(-\ii\,\overline{k}) &=&- \frac{1}{2}\frac{\rho^2}{x^2}
\frac{\left[ U x \overline{k} U^\dagger
\right]}{k\cdot x}
\left[ {\rm e}^{\ii\,k\cdot x} -
\frac{\ii}{k\cdot x}
\left(1-{\rm e}^{\ii\,k\cdot x}\right)\right]
,
\\[1.6ex]
\lim_{k^{2}\to 0}{s^{(3)}}
(x,-k)(-\ii\,\overline{k}) &=& \frac{1}{2}\frac{\rho^2}{x^2}
\frac{\left[ U x\overline{k} U^\dagger
\right]}{k\cdot x}
\left[ 1 -
\frac{\ii}{k\cdot x}
\left(1-{\rm e}^{\ii\,k\cdot x}\right)\right]
.
\end{eqnarray}
Thus, on account of Eq.~(\ref{sumsi}), the on-shell
residuum of the quark propagator (\ref{si}) is given by
Eq.~(\ref{onshellsi}). A similar reasoning leads to
Eq.~(\ref{onshellsibar}) for the residuum of the quark propagator
(\ref{sibar}).
\section*{Appendix B}
Our task is to derive Eqs.~(\ref{tvertex}), (\ref{uvertex}), corresponding
to the $\gamma^{\ast}$- quark vertex ${\mathcal V}_\mu ^{(t,u)}$ in the
leading-order $I$-induced amplitude. We will concentrate on
the derivation of Eq.~(\ref{tvertex}), since the derivation of
Eq.~(\ref{uvertex}) is completely analogous.
Let us recall the definition of ${\mathcal V}_\mu ^{(t)}$, but now with
indices written explicitly,
\begin{eqnarray}
{\mathcal V}^{(t)}_{\mu \ m \lambda}\,(q,-k) =
\int d^4x\,{\rm e}^{-\ii\,q\cdot x}\
\overline{\phi}^{\dot\alpha}_l (x)\,
\overline{\sigma}_{\mu \,\dot\alpha\alpha}\,
\lim_{k^2\to 0}\,
{{S}^{(I)\,\alpha\dot\beta\ l}}_m\,(x,-k)\,
\left(-\ii\,\overline{k}_{\dot\beta\lambda}\right)
.
\label{phivertex1}
\end{eqnarray}
Inserting Eqs.~(\ref{phibar}) and (\ref{onshellsi})
into Eq.~(\ref{phivertex1}) we obtain for the vertex,
\begin{eqnarray}
{\mathcal V}^{(t)}_{\mu \ m \lambda}\,(q,-k) &=&
-\frac{\rho^{3/2}}{\pi}\,
\int d^4x\,{\rm e}^{-\ii\,( q-k)\cdot x}\,
\frac{1}{\left( x^2+\rho^2\right)^2}
\label{phivertex2}
\\
&\times &
\epsilon_{\gamma\delta}\,
\left[ x\,\overline{\sigma}_\mu \right]^\delta_{\ \lambda}\,
\left[
\left( U^\dagger\right)^\gamma_{\ m}
+\frac{1}{2}\,\frac{\rho^2}{x^2}\,
\frac{\left[ x\,\overline{k}\,U^\dagger\right]^\gamma_{\ m}}
{k\cdot x}\,
\left( 1-{\rm e}^{-\ii\,k\cdot x}\right)
\right]\, .
\nonumber
\end{eqnarray}
The matrix structure in Eq.~(\ref{phivertex2}) can be simplified
using
\begin{equation}
\epsilon_{\gamma\delta}\,
\left[ x\,\overline{\sigma}_\mu \right]^\delta_{\ \lambda}\,
\left[ x\,\overline{k}\,U^\dagger\right]^\gamma_{\ m}
= x^2\,\epsilon_{\gamma\delta}\,\left( U^\dagger\right)^\gamma_{\ m}\,
\left[ k\,\overline{\sigma}_\mu \right]^\delta_{\ \lambda}\, ,
\end{equation}
which follows from the transposition rules of the $\sigma$-matrices.
Thus Eq.~(\ref{phivertex2}) can be rewritten as
\begin{eqnarray}
{\mathcal V}^{(t)}_{\mu \ m \lambda}\,(q,-k) &=&
-\frac{\rho^{3/2}}{\pi}\,
\int d^4x\,{\rm e}^{-\ii\,( q-k)\cdot x}\,
\frac{1}{\left( x^2+\rho^2\right)^2}
\label{phivertex3}
\\
&\times &
\epsilon_{\gamma\delta}\,\left( U^\dagger\right)^\gamma_{\ m}\,
\left[
\left[ x\,\overline{\sigma}_\mu \right]^\delta_{\ \lambda}
+\frac{1}{2}\,\rho^2\,
\left[ k\,\overline{\sigma}_\mu \right]^\delta_{\ \lambda}\,
\frac{1}{k\cdot x}\,
\left( 1-{\rm e}^{-\ii\,k\cdot x}\right)
\right]\, .
\nonumber
\end{eqnarray}
The remaining $d^4x$ integration in Eq.~(\ref{phivertex3}) can be
done with the help of the following formulae ($k^2=0$ is always understood),
\begin{eqnarray}
\label{int1}
\lefteqn{
\int d^4x\,{\rm e}^{-\ii\,(q-k)\cdot x}\,
\frac{x}{\left( x^2+\rho^2\right)^2}
=}
\\[1.6ex]
\nonumber
&&
-2\,\pi^2\,\ii\,\frac{q-k}{\left( q-k\right)^2}\,
\rho\,\sqrt{\left( q-k\right)^2}\,
K_1\left(\rho\,\sqrt{\left( q-k\right)^2}\right)\, ,
\\[2.4ex]
\label{int2}
\lefteqn{
\int d^4x\,{\rm e}^{-\ii\,(q-k)\cdot x}\,
\frac{1}{\left( x^2+\rho^2\right)^2\,\left( k\cdot x\right)}
= }
\\[1.6ex]
\nonumber
&&
2\,\pi^2\,\ii\,
\frac{1}{q\cdot k}\,\frac{1}{\rho^2}\,
\rho\,\sqrt{\left( q-k\right)^2}\,
K_1\left(\rho\,\sqrt{\left( q-k\right)^2}\right)\, ,
\\[2.4ex]
\label{int3}
\lefteqn{
\int d^4x\,{\rm e}^{-\ii\,q\cdot x}\,
\frac{1}{\left( x^2+\rho^2\right)^2\,\left( k\cdot x\right)}
= }
\\[1.6ex]
\nonumber
&&
2\,\pi^2\,\ii\,
\frac{1}{q\cdot k}\,\frac{1}{\rho^2}\,
\rho\,\sqrt{q^2}\,
K_1\left(\rho\,\sqrt{q^2}\right)\, .
\end{eqnarray}
By means of these basic integrals, we obtain finally for the
vertex,
\begin{eqnarray}
\lefteqn{
{\mathcal V}^{(t)}_{\mu \ m \lambda}\,(q,-k)
=
2\,\pi\,\ii\,\rho^{3/2}\,
\epsilon_{\gamma\delta}\,\left( U^\dagger\right)^\gamma_{\ m} }
\label{phivertex4}
\\
&&\times
\Biggl\{
\frac{\left[ \left(q-k\right)
\,\overline{\sigma}_\mu \right]^\delta_{\ \lambda}}
{(q-k)^2}\,
\rho\,\sqrt{\left( q-k\right)^2}\,
K_1\left(\rho\,\sqrt{\left( q-k\right)^2}\right)
\nonumber
\\
\mbox{}&&-
\frac{1}{2}\,
\frac{\left[ k\,\overline{\sigma}_\mu \right]^\delta_{\ \lambda}}
{q\cdot k}\, \left[
\rho\,\sqrt{\left( q-k\right)^2}\,
K_1\left(\rho\,\sqrt{\left( q-k\right)^2}\right)
-\rho\,\sqrt{q^2}\,
K_1\left(\rho\,\sqrt{q^2}\right)
\right]
\Biggr\}\, ,
\nonumber
\end{eqnarray}
in accordance with Eq.~(\ref{tvertex}).
\vspace{10pt}
\section*{Acknowledgements}
We would like to acknowledge helpful discussions with
V. Braun and V. Rubakov.
\vspace{10pt}
| proofpile-arXiv_065-483 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section*{Introduction}
There has been recently an important activity in the study of {$N=2$} supersymmetric
hierarchies (KP \cite{popo1,aratyn,dasb1,ghosh,dasp}, generalizations of KdV
\cite{bokris,ikrim}, Two Bosons \cite{dasb2}, NLS \cite{kriso,krisoto,dasb3},
etc..). The most usual tools
in this field are the algebra of
$N=1$ pseudo-differential operators and Gelfand-Dickey type Poisson brackets
\cite{geldik}.
Although these systems have {$N=2$} supersymmetry, only for very few of them
with very low number of fields is a formulation in extended superspace known.
It is the purpose of this paper to partially fill this gap. The formalism which
we shall present here partly originates from the article \cite{delma}.
It turns out that in order to construct the Lax operators of
{$N=2$} supersymmetric hierarchies,
one should not use the whole algebra of {$N=2$} pseudo-differential operators, but
rather the subalgebra of pseudo-differential operators preserving chirality.
These operators were first considered in \cite{popo3}.
They will be defined in section \ref{main}, where we also study the KP Lax
equations and the two associated Hamiltonian structures. It turns out that the
first (linear) bracket
is associated with a non-antisymmetric $r$ matrix
\cite{semenov}. Because of that, the second
(quadratic) bracket is not of pure Gelfand-Dickey type. The main result of this
paper is that we find two
possibilities for this quadratic bracket. In fact, we show that there
is an invertible map in the KP phase space which sends one of the
quadratic Poisson structure into the other. However, this map does
not preserve the Hamiltonians.
In section \ref{reduc}, we study the possible reductions of
the KP hierarchy by looking for Poisson subspaces in the phase space. These
are different depending on the quadratic bracket which is used.
Among these reductions, there are two different hierarchies with the {$N=2$}
classical super-${\cal W}_n$ algebra \cite{lupope} as a hamiltonian structure.
In particular, two of the three known {$N=2$} supersymmetric extensions
of the KdV hierarchy \cite{Mathieu1} are found.
They correspond to $a=-2$ and $a=4$ in
the classification of Mathieu. These and some other examples are described in section \ref{examples}. Notice that from the known cases with a low number of fields \cite{Mathieu1,math2,popo2,yung1,Ivanov1,yung2},
one expects for any $n$ three hierarchies with
super-${\cal W}_n$ as a hamiltonian structure. So our construction does not exhaust the possible cases.
We also found two hierarchies which Poisson structure
is the classical ``small" $N=4$ superconformal algebra. In one case the
evolution equations are $N=4$ supersymmetric, while in the other they
are only {$N=2$} supersymmetric.
Finally, in section \ref{n1susy} we give the relation of our formulation with the usual formulation of the {$N=2$} supersymmetric KP Lax equations in $N=1$ superspace \cite{inami,dasb1,dasp}.
\setcounter{equation}{0}
\section{N=2 KP hierarchy \label{main}}
\paragraph{{$N=2$} supersymmetry}
We shall consider an {$N=2$} superspace with space coordinate $x$ and two
Grassmann coordinates $\theta$, $\bar\theta$. We shall use the
notation ${\underline x}$ for the triple of coordinates
$(x,\theta,\bar\theta)$. The supersymmetric covariant derivatives
are defined by
\begin{equation}
\partial\equiv{\partial\over\partial x},\,\,D={\partial\over\partial\theta}
+\bar\theta\partial,\,\,\bar D={\partial\over\partial\bar\theta}
+\theta\partial, D^2=\bar D^2=0,\,\,\{ D,\bar D\}=\partial
\label{n2alg}\end{equation}
Beside ordinary superfields $H({\underline x})$ depending
arbitrarily on Grassmann coordinates, one can also define chiral
superfields $\varphi({\underline x})$ satisfying
$D\varphi =0$ and antichiral superfields $\bar\varphi({\underline x})$
satisfying $\bar D\bar\varphi =0$.
We define the integration over the {$N=2$} superspace to be
\begin{equation}
\int d^3{\underline x}\, H(x,\theta,\bar\theta)= \int dx\bar DDH(x,\theta,\bar\theta)
\vert_{\theta=\bar\theta=0}.
\end{equation}
The elements of the associative algebra of {$N=2$} pseudo-differential operators ($\Psi$DOs) are the operators
\begin{equation}
P = \sum_{i <M} ( a_{i} +b_i[D,\bar D]+\alpha_{i} D + \beta_{i} \overline{D} )\partial^{i}
\label{pdo}\end{equation}
where $a_{i}$, $b_{i}$ and $\alpha_{i}$, $\beta_{i}$ are respectively even and odd {$N=2$} superfields.
However, this algebra is not very manageable.
In particular, the set of strictly pseudo-differential operators ($M=0$ in\reff{pdo}) is not
a proper subalgebra, but only a Lie subalgebra.
Also, there are too many fields in these operators. We expect the phase space of the
{$N=2$} KdV hierarchies to consist of the supercurrents of the {$N=2$}
${\cal W}_n$ algebras. In extended superspace, these supercurrents are bosonic superfields,
and there is one such superfield for a given integer dimension.
But in \reff{pdo}, each power of $\partial$ corresponds to four superfields, two even ones
of integer dimension and two odd ones of half-integer dimension. It is thus clear that one
has to restrict suitably the form of the {$N=2$} operators. It turns out
that a possible
restriction is to define the set $\cal C$ of
pseudo-differential operators $L$ preserving chirality of the form
\begin{footnote}
{Operators of this type were first considered in \cite{popo3}}
\end{footnote}
\begin{equation}
L=D{\cal L}\bar D,\,\,\,\,\,\,{\cal L}= \sum_{i <M} u_{i}\partial^{i}
\label{cpdo}\end{equation}
The coefficient functions $u_i$ are bosonic {$N=2$} superfields. These operators satisfy
$DL=L\bar D=0$.
The product of two chiral operators is again a
chiral operator. The explicit product rule is easily worked out
\begin{equation}
LL'= D \left(
{\cal L}\partial {\cal L'} +(D.{\cal L})(\bar{D}.{\cal L'}) \right) \bar{D},
\end{equation}
where we have used the notation
\begin{equation}
(D.{\cal L})=\sum_{i <M}(Du_{i})\partial^{i}.
\end{equation}
Notice that $I=D \partial^{-1}\bar{D}$ is the unit of the algebra $\cal C$.
We could have used as well the algebra $\bar{\cal C}$ of $\Psi$DOs
satisfying $\bar D\bar L=\bar L D=0$. Notice that the product of an element in ${\cal C}\,\,$ by
an element in $\bar{\cal C}$ vanishes. In fact ${\cal C}\,\,$ and $\bar{\cal C}$
are related by transposition, $L^t=-\bar D{\cal L}^tD\in \bar{\cal C}$.
Although the transposition leads from ${\cal C}$ to $\bar{\cal C}$,
there exists an anti-involution which acts inside ${\cal C}$. It is
given by
\begin{equation}
\tau(L)=DL^t\partial^{-1}\bar
D,\,\,\,\tau(L_{1}L_{2})=\tau(L_{2})\tau(L_{1}).
\end{equation}
Notice that it does not make sense in the algebra $\cal C$ to multiply a
$\Psi$DO by a function. However, it is possible to multiply on the left by a chiral
function $\phi$, $D\phi=0$
\begin{equation}
\phi L= D\phi{\cal L}\bar D={\lambda}(\phi) L,\,\,\,{\lambda}(\phi)\equiv
D\phi\partial^{-1}\bar D,
\end{equation}
and on the right by an antichiral function $\bar\phi$, $\bar D\bar\phi=0$
\begin{equation}
L\bar\phi = D{\cal L}\bar\phi\bar D=L{\bar\lambda}(\bar\phi),\,\,\,
{\bar\lambda}(\bar\phi)\equiv D\partial^{-1}\bar\phi\bar D.
\end{equation}
We define the residue of the pseudo-differential operator $L$ by
$\mbox{\rm res} L=u_{-1}$ \cite{Mathieu1}.
The residue of a commutator is a total derivative,
$\mbox{\rm res}[L,L']=D\bar\omega+\bar D\omega$. The trace of $L$ is the integral of the
residue
\begin{equation}
\mbox{\rm Tr}L=\int d^3{\underline x}\,\mbox{\rm res}L,\,\,\,\,
\mbox{\rm Tr}[L,L']=0.
\end{equation}
$\cal C$ can be divided into two proper subalgebras
${\cal C} = {\cal C}_{+} \oplus {\cal C}_{-}$, where $L$ is in ${\cal C}_{+}$ if $\cal L$ is a differential operator and $L$ is in ${\cal C}_{-}$ if $\cal L$ is a
strictly pseudo-differential operator ($M=0$ in \reff{cpdo}). We shall note
\begin{equation}
L=L_++L_-, \,\,\,L_+=D{\cal L}_+\bar D\in {\cal C}_+,\,\,\,
L_-=D{\cal L}_-\bar D\in{\cal C}_-.
\end{equation}
Here an important difference with the usual bosonic and $N=1$ cases occurs. For
any two $\Psi$DOs $L$ and $L'$ in $\cal C$ one has $\mbox{\rm Tr}(L_{-}L'_{-})= \int d^3{\underline x}\,\,\mbox{\rm res}(L)\,\,\mbox{\rm res}(L') \neq 0$. While ${\cal C}_{+}$ is an isotropic subalgebra, ${\cal C}_{-}$
is not. One important consequence of this fact is that if one defines the
endomorphism $R$ of $\cal C$ by $R(L)={1\over 2}(L_+-L_-)$, then $R$ is a non-antisymmetric classical $r$ matrix,
\begin{equation}
\mbox{\rm Tr} (R(L)L'+LR(L'))=-\int d^3{\underline x}\,\mbox{\rm res} L\,\mbox{\rm res} L'.
\end{equation}
Notice that a non-antisymmetric $r$ matrix in the context of bosonic KP Lax
equations first appeared in \cite{kuper}.
\paragraph{KP equations}
Let us now write the evolution equations of the {$N=2$} supersymmetric KP hierarchy.
We consider operators $L=D{\cal L}\bar D$ in $\cal C$ of the form
\begin{equation}
{\cal L}=\partial^{n-1}+\sum_{i=1}^{\infty}V_i\partial^{n-i-1}.
\label{KPop}\end{equation}
$L$ has a unique $n$th root in $\cal C$ of the form
\begin{equation}
L^{1\over n}=D(1+\sum_{i=1}^{\infty}W_i\partial^{-i})\bar D,
\end{equation}
and we are led to consider the commuting flows (see \cite{popo3})
\begin{equation}
{\partial\over\partial t_k}L=[(L^{k\over n})_+,L]=[R(L^{k\over n}),L].
\label{KPeq}\end{equation}
There are symmetries of these equations which may be described as
follows. Let us first introduce a chiral, Grassmann even superfield $\varphi$ which
satisfies
\begin{equation}
{\partial\over\partial t_k}\varphi=(L^{k\over n})_+.\varphi
\label{fieq}\end{equation}
where the right-hand side is the chiral field obtained by acting with
the differential operator $(L^{k\over n})_+$ on the field $\varphi$.
Then the transformed operator
\begin{equation} s(L)= \lambda(\varphi^{-1})L\lambda(\varphi)
\label{simil}\end{equation}
satisfies an evolution equation of the same form \reff{KPeq} as that
of $L$.
We may also consider an antichiral, Grassmann odd superfield
$\bar\chi$ which satisfies
\begin{equation}
{\partial\over\partial t_k}\bar\chi=-(L^{k\over n})^t_+.\bar\chi
\label{chieq}\end{equation}
Then the transformed operator
\begin{equation} \sigma(L)=(-1)^n\lambda((D\bar\chi)^{-1})\tau(L)\lambda(D\bar\chi)
\label{chitra}\end{equation}
satisfies an evolution equation of the same form \reff{KPeq} as that
of $L$, with the direction of time reversed.
\paragraph{Poisson brackets}
The Lax equations \reff{KPeq} are bi-hamiltonian with respect to two compatible Poisson brackets
which we now exhibit.
Let $X$ be some $\Psi$DO in $\cal C$ with coefficients independent of
the phase space fields $\{V_i\}$, then define the linear functional
$l_{X}(L) = \mbox{\rm Tr}(LX)$. The generalization of the first Gelfand-Dickey bracket
is obvious and reads
\begin{equation}
\{ l_{X},l_{Y} \}_{(1)} (L) = \mbox{\rm Tr} \left( L[X_{+},Y_{+}]-L[X_{-},Y_{-}] \right).
\label{pb1}\end{equation}
This is nothing but the linear bracket associated with the matrix $R$.
Now we turn to the construction of the second bracket. It will turn out
more complicated than the standard Gelfand-Dickey bracket because of the non-antisymmetry of the $r$ matrix. An analogous situation in the bosonic case is studied in \cite{Oevel}. We finally found two different
possibilities.
In order to write them down, we need to be able to separate the residue of a $\Psi$DO in
$\cal C$ into a chiral and an antichiral part. For an arbitrary
superfield $H({\underline x})$, we define
\begin{equation}
H=\Phi[H]+\bar\Phi[H],\,\,\, D\Phi[H]=0,\,\,\, \bar D\bar\Phi[H]=0.
\end{equation}
This is not a local operation in $\cal C$. An explicit form may be
chosen as
\begin{equation}
\Phi[H]=D\bar D\int d^{3}{\underline x}'\Delta({\underline
x}-{\underline x}')H({\underline x}'),\,\,
\bar \Phi[H]=\bar DD\int d^{3}{\underline x}'\Delta({\underline
x}-{\underline x}')H({\underline x}'),
\label{chipro}\end{equation}
where $\Delta$ is the distribution
\begin{eqnarray}
&\Delta({\underline x}-{\underline x}')=
(\theta-\theta')(\bar\theta-\bar\theta')\epsilon(x-x'),&\label{distri}\\
&\partial\epsilon(x-x')=\delta(x-x'),\,\,\, \epsilon(x-x')=-\epsilon(x'-x).
\nonumber\end{eqnarray}
In the following, we shall use the short-hand notations
$\Phi[\,\mbox{\rm res}[L,X]]=\Phi_X$, $\bar\Phi[\,\mbox{\rm res}[L,X]]=\bar\Phi_X$.
In general, $\Phi_{X}$ will not satisfy the same boundary conditions as the
phase space fields do. However, we noted earlier that in the case of a commutator,
the residue is a total derivative,
$\,\mbox{\rm res} [L,X]=D\bar\omega+\bar D\omega$.
Here $\omega$ and $\bar\omega$ are differential polynomials in the fields. Then
one easily shows that $\Phi_{X}=D\bar\omega+\alpha$,
$\bar\Phi_{X}=\bar D\omega-\alpha$,
where $\alpha$ is a constant reflecting the arbitrariness in the
definition of $\Phi$, $\bar\Phi$. Up to this constant, $\Phi_{X}$ will
respect the boundary conditions.
We are now in a position to write the two possibilities for the second bracket
as
\begin{footnote}
{The Poisson brackets (\ref{pb2p},\ref{pb2m}) may be put in the
general form introduced in \cite{Maillet}
\begin{equation}
\{ l_{X},l_{Y} \}_{(2)}^{a,b} (L) =\mbox{\rm Tr} \left( LXa(LY)+XLb(LY)-LXc(YL)-XLd(YL)
\right)\end{equation}
However, the price to pay is that $a$, $b$, $c$, $d$ are non-local endomorphisms of $\cal C$. As an example, for the first quadratic bracket one finds
\begin{equation}
a(X)={1\over 2}(X_++\lambda(\Phi[\,\mbox{\rm res} X]))-{1\over 2}
(X_--\lambda(\Phi[\,\mbox{\rm res} X])),\,\,\, b(X)=\bar\lambda(\bar\Phi[\,\mbox{\rm res} X]).
\end{equation}
One easily checks in particular that $a$ is a non-local antisymmetric $r$ matrix.}
\end{footnote}
\begin{equation}
\{ l_{X},l_{Y} \}_{(2)}^a (L) =\mbox{\rm Tr} \left( LX(LY)_{+}-XL(YL)_{+}+
\Phi_Y LX+XL\bar\Phi_Y\right),
\label{pb2p}\end{equation}
and
\begin{equation}
\{ l_{X},l_{Y} \}_{(2)}^b (L) =\mbox{\rm Tr} \left( LX(LY)_{+}-XL(YL)_{+}+
\Phi_Y XL+LX\bar\Phi_Y\right).
\label{pb2m}\end{equation}
These expressions do not depend on the arbitrary constant $\alpha$.
Checking the antisymmetry
of the Poisson brackets and the Jacobi identity can be done with
a little effort. As usual,
the first bracket is a linearization of the two quadratic ones, that is to say
\begin{equation}
\{ l_{X},l_{Y} \}_{(2)}^{a,b} (L+zD\partial^{-1}\bar D)
=\{ l_{X},l_{Y} \}_{(2)}^{a,b} (L)
+z\{ l_{X},l_{Y} \}_{(1)} (L),
\end{equation}
and the linear bracket is compatible with each of the two quadratic brackets.
Introducing
the hamiltonians
${\cal H}_{k} = {n\over k}\mbox{\rm Tr}(L^{k\over n})$, the KP evolution equations \reff{KPeq}
may be written as
\begin{equation}
\partial_{t_k} \left( l_{X}(L) \right)
= \{ l_{X},{\cal H}_{k+n} \}_{(1)} (L)
= \{ l_{X},{\cal H}_{k} \}_{(2)}^{a,b} (L)
\end{equation}
\paragraph{Poisson maps}
Before turning to the study of the reductions of the KP hierarchies,
let us exhibit some relations between the two quadratic brackets. We
will use the invertible map in $\cal C$
\begin{equation}
p(L)=\partial^{-1}\tau(L)=D\partial^{-1}L^t\partial^{-1}\bar D.
\label{poimap}\end{equation}
Then a straightforward calculation leads to
\begin{equation}
\{ l_{X}\circ p,l_{Y}\circ p\}_{(2)}^a=-\{ l_{X},l_{Y}\}_{(2)}^b\circ
p,
\end{equation}
which shows that \reff{pb2p} and \reff{pb2m} are equivalent Poisson
brackets. However there is no relation between the hamiltonians
$\mbox{\rm Tr}(L^{k\over n})$ and $\mbox{\rm Tr}(p(L)^{k\over n-1})$.
There is another relation between the two brackets, which involves the
chiral superfield $\varphi$ satisfying the evolution equation
\reff{fieq}. Let us introduce the linear functional
$l_t=\int d^3{\underline x}(t\varphi)$, where $t({\underline x})$ is a Grassmann
even superfield. We consider an enlarged phase space including $\varphi$,
and extend the Poisson bracket \reff{pb2p} to this phase space by
\begin{equation}
\{ l_t,l_{Y} \}_{(2)}^a (L,\varphi)=\int d^3{\underline x} t((LY)_+.\varphi
+\Phi_Y\varphi),\,\,\,\{ l_{t},l_{t'} \}_{(2)}^a=0.
\end{equation}
Then one finds
\begin{equation}
\{ l_{X}\circ s,l_{Y}\circ s\}_{(2)}^a=\{ l_{X},l_{Y}\}_{(2)}^b\circ s,
\end{equation}
where the transformation $s$ has been defined in \reff{simil}. Notice
that the hamiltonians are invariant functions for the transformation
$s$, $\mbox{\rm Tr}(L^{k\over n})=\mbox{\rm Tr}(s(L)^{k\over n})$.
A last relation uses the antichiral superfield $\bar\chi$ satisfying
the evolution \reff{chieq}. Let us introduce the linear functional
$l_{\bar t}=\int d^3{\underline x}(\bar t\bar\chi)$,
where $\bar t({\underline x})$ is a Grassmann
odd superfield. We consider an enlarged phase space including $\bar\chi$,
and extend the Poisson bracket \reff{pb2p} to this phase space by
\begin{eqnarray}
&\{ l_{\bar t},l_{Y} \}_{(2)}^a (L,\bar\chi)=\int d^3{\underline x}
{\bar t}(-(LY)^t_+.\bar\chi
+\Phi_Y\bar\chi),&\\ &
\{ l_{\bar t_1},l_{\bar t_1} \}_{(2)}^a=-2\int d^{3}{\underline
x}\bar t_1\bar\chi\bar\Phi[\bar t_2\bar\chi],&
\end{eqnarray}
where $\Phi$, $\bar\Phi$ are defined in equations (\ref{chipro},\ref{distri}).
Notice that this is a non-local Poisson bracket. One finds
\begin{equation}
\{ l_{X}\circ\sigma,l_{Y}\circ\sigma\}_{(2)}^a=-\{
l_{X},l_{Y}\}_{(2)}^b\circ\sigma,
\end{equation}
where the transformation $\sigma$ has been defined in \reff{chitra}.
\setcounter{equation}{0}
\section{Reductions of the KP hierarchy \label{reduc}}
In order to obtain consistent reductions of the KP hierarchy, we need to find Poisson
submanifolds of the KP phase space. Considering first the
quadratic bracket \reff{pb2p}, we rewrite it as
\begin{eqnarray}
&\{ l_{X},l_{Y} \}_{(2)}^a (L) =\mbox{\rm Tr} X\xi_{l_Y}^a,&\nonumber\\
&\xi_{l_Y}^a=(LY)_{+}L-L(YL)_{+}+\Phi_Y L+L\bar\Phi_Y.&
\label{hvf}\end{eqnarray}
$\xi_{l_Y}^a$ is the hamiltonian vector field associated with the function $l_Y$. One easily checks that if L has the form \reff{KPop}, then for any $Y$,
$\xi_{l_Y}^a$ has the form $D(\sum_{i<n-1}\xi_i\partial^i)\bar D$.
It is obvious from \reff{hvf} that for any $Y$, if $L$ is in ${\cal C}_+$,
then $\xi_{l_Y}^a$ is also in ${\cal C}_+$. This means that the constraint
\begin{equation}L=L_+
\label{kdv}\end{equation}
defines a Poisson submanifold. The hierarchies obtained in this way are
the {$N=2$} supersymmetric KdV hierarchies studied by Inami and Kanno
\cite{inami}, and the Lax operators \reff{kdv} already appeared in
\cite{popo3}. The lowest order cases will be presented in the next section.
Another possible reduction is to take $L$ of the form
\begin{equation} L=L_++D\,\varphi\partial^{-1}\bar\varphi\bar D,\,\,\,\,\,
D\varphi=\bar D\bar\varphi=0.
\label{nls}\end{equation}
where $\varphi$ and $\bar\varphi$ are Grassmann even or odd chiral superfields.
With $L$ of the form \reff{nls} and $Y$ arbitrary, one finds
\begin{equation}
(\xi_{l_Y}^a)_-=D((LY)_+.\varphi+\Phi_Y\varphi)\partial^{-1}\bar\varphi
+\varphi\partial^{-1}(-(YL)_+^t.\bar\varphi+\bar\Phi_Y\bar\varphi))\bar D,
\end{equation}
Noticing that $(LY)_+.\varphi$ is a chiral superfield and
$(YL)_+^t.\bar\varphi$ an antichiral superfield, it is easily checked that
$\xi_{l_Y}^a$ is indeed tangent to the submanifold defined by the constraints \reff{nls}.
It is possible to consider an enlarged phase space which coordinates are the fields in $L$
and $\varphi$, $\bar\varphi$. Let us introduce the linear functionals
\begin{equation}l_t=\int d^3{\underline x}(\varphi t),\,\,
l_{\bar t}=\int d^3{\underline x}(\bar t\bar\varphi),
\end{equation}
where $ t$ and $\bar t$ are general superfields, of the same
Grassmann parity as $\varphi$ and $\bar\varphi$.
In this enlarged phase space, the second Poisson bracket, in the case when
$\varphi$ and $\bar\varphi$ are Grassmann even, is defined by \reff{pb2p}
and
\begin{eqnarray}
&\{ l_t,l_{Y} \}_{(2)}^a (L,\varphi,\bar\varphi)=\int d^3{\underline x} ((LY)_+.\varphi
+\Phi_Y\varphi)t, &\label{lfi}\\
&\{l_{\bar t},l_{Y} \}_{(2)}^a (L,\varphi,\bar\varphi)=\int d^3
{\underline x}\,\bar t(-(YL)_+^t.\bar\varphi
+\bar\Phi_Y\bar\varphi),&
\nonumber\end{eqnarray} and
\begin{eqnarray}
&\{ l_t,l_{\bar t} \}_{(2)}^a (L,\varphi,\bar\varphi)=
\int d^3{\underline x}\, ({ L}_+.\bar t)t,&\label{bose}\\&
\{ l_{t_1},l_{t_2} \}_{(2)}^a=0,\,\,\,\,
\{ l_{\bar t_1},l_{\bar t_2} \}_{(2)}^a=0.&
\nonumber\end{eqnarray}
In the case when
$\varphi$ and $\bar\varphi$ are Grassmann odd,
the last two lines should be modified to
\begin{eqnarray}
&\{ l_t,l_{\bar t} \}_{(2)}^a (L,\varphi,\bar\varphi)=
\int d^3{\underline x} (({ L}_+.\bar t)t-2\varphi t\Phi[\bar t\bar\varphi]),
\label{fermi}&\\&
\{ l_{t_1},l_{t_2} \}_{(2)}^a=2\int d^3{\underline x}\,\varphi t_1
\Phi[\varphi t_2],\,\,\,\,
\{ l_{\bar t_1},l_{\bar t_2} \}_{(2)}^a=-2\int d^3{\underline x}
\,\bar t_1\bar\varphi
\bar\Phi[\bar t_2\bar\varphi],&
\nonumber\end{eqnarray}
where the applications $\Phi$ and $\bar\Phi$ have been defined in
\reff{chipro}.
The lowest order case is $L=D(1+\varphi\partial^{-1}\bar\varphi)\bar D$. Then
if $\varphi$ and $\bar\varphi$ are odd, the equation
${d\over dt}L=[L^2_+,L]$ is the {$N=2$} supersymmetric extension of the NLS
equation \cite{roelo}. The next-to-lowest order case is $L=D(\partial +H+\varphi\partial^{-1}\bar\varphi)\bar D$.
If $\varphi$ and $\bar\varphi$ are even, the hamiltonian structure \reff{pb2p} reduces in
this case to the classical version of the ``small'' $N=4$ superconformal algebra. Although
the Poisson algebra contains $4$ supersymmetry
generators, the evolution equations \reff{KPeq} have only {$N=2$} supersymmetry.
This case was first obtained by another method which will be
given, as part of a detailed study, in \cite{dgi}.
We now turn to the second quadratic bracket \reff{pb2m}. We rewrite it as
\begin{eqnarray}
&\{ l_{X},l_{Y} \}_{(2)}^b (L) =\mbox{\rm Tr} X\xi_{l_Y}^b,&\nonumber\\
&\xi_{l_Y}^b=(LY)_{+}L-L(YL)_{+}+L{\lambda}(\Phi_Y)+{\bar\lambda}(\bar\Phi_Y)L.&
\label{hvfm}\end{eqnarray}
It is easily seen that neither the condition \reff{kdv}, nor the more complicated condition
\reff{nls} are admissible reductions in this case. The easiest way to
find Poisson subspaces for the bracket \reff{pb2m} is to apply the
map \reff{poimap} to the Poisson subspaces of the first quadratic
bracket. From \reff{kdv}, we are then lead to the restriction:
\begin{equation}
L=L_++D\bar D\partial^{-1}H\partial^{-1}D\bar D
\label{a4}\end{equation}
With $L$ of the form \reff{a4} and $Y$ arbitrary, one finds
\begin{equation}
(\xi_{l_Y}^b)_-=D\bar D\partial^{-1}((LY)_+.H-(YL)_+^t.H+\,\mbox{\rm res}[L,Y]H)
\partial^{-1}D\bar D,
\end{equation}
which directly shows that condition \reff{a4} defines a Poisson submanifold for the
Poisson bracket
\reff{pb2m}. It turns out that \reff{a4} also defines a Poisson submanifold for
the linear Poisson
bracket \reff{pb1}. To show this we rewrite the linear bracket as
\begin{equation}\{ l_{X},l_{Y} \}_{(1)} (L) =\mbox{\rm Tr} X\eta_{l_Y},\,\,\,
\eta_{l_Y}=[L,Y]_+-[L,Y_+]+{\lambda}(\Phi_Y)+{\bar\lambda}(\bar\Phi_Y).
\end{equation}
With $L$ of the form \reff{a4} and $Y$ arbitrary, one finds
\begin{equation}
(\eta_{l_Y})_-=D\bar D\partial^{-1}((Y_+-Y_+^t).H+\,\mbox{\rm res}[L,Y])\partial^{-1}D\bar D.
\end{equation}
Thus the reduced hierarchies defined by condition \reff{a4} are
bi-hamiltonian. The lowest order cases will be studied in the next section.
Notice that the transformation \reff{simil} maps the systems satisfying the condition \reff{nls} with Grassmann even fields $\varphi$ and $\bar\varphi$
into systems satisfying condition \reff{a4} with
\begin{equation}
H= \varphi\bar\varphi+\varphi^{-1}L_+.\varphi.
\end{equation}
Analogously, the transformation \reff{chitra} maps the systems satisfying the condition \reff{nls} with Grassmann odd fields $\varphi$ and $\bar\varphi$
into systems satisfying condition \reff{a4} with
\begin{equation}
H= (-1)^n\left(\bar\varphi\varphi+(D\bar\varphi)^{-1}D(L_+^t.\bar\varphi)\right).
\end{equation}
Such transformations may be found in \cite{kriso,bokris}.
Finally we may consider the image of the Poisson subspace defined by
\reff{nls} under the map $p$. One finds the condition
\begin{equation}
L=L_++D\bar D\partial^{-1}(H+\bar\varphi\partial^{-1}\varphi)\partial^{-1}D\bar D.
\label{n4}\end{equation}
The lowest order case is when $L_{+}=D\bar D$. The hamiltonian structure \reff{pb2m}
reduces in
this case to the classical version of the ``small'' $N=4$ superconformal algebra.
The equation ${d\over dt}L=[(L^{3})_{+}, L]$ becomes, after suitable
redefinitions, the $N=4$ supersymmetric extension of the KdV equation
derived in \cite{delivan} and written in {$N=2$} superspace in
\cite{dik}.
One can again consider an enlarged phase space which coordinates are
the fields in $L$ and $\varphi$, $\bar\varphi$. The second quadratic bracket
in this phase space is easily obtained from the first one by applying the map
$p$ to the first quadratic bracket. $p$ acts as the identity on $\varphi$ and
$\bar\varphi$. As a consequence the Poisson brackets \reff{bose} and \reff{fermi} keep the same form, whereas \reff{lfi} should be modified to
\begin{eqnarray}
&\{ l_t,l_{Y} \}_{(2)}^b (L,\varphi,\bar\varphi)=\int d^3{\underline x} (\,\mbox{\rm res}\left(\tau((YL)_+)\lambda(\varphi)\right)
+\Phi_Y\varphi)t, &\\
&\{l_{\bar t},l_{Y} \}_{(2)}^b (L,\varphi,\bar\varphi)=\int d^3
{\underline x}\,\bar
t(-\,\mbox{\rm res}\left(\bar\lambda(\bar\varphi)\partial^{-1}\tau((LY)_+)\partial\right)
+\bar\Phi_Y\bar\varphi).&
\nonumber\end{eqnarray}
\setcounter{equation}{0}
\section{Examples and comparison with other works \label{examples}}
This paragraph is devoted to the presentation of the simplest integrable equations obtained using our formalism.
Considering first the condition \reff{kdv}, the simplest example is the lax operator $L = D (\partial + W) \bar{D}$. Then the
evolution equation
\begin{equation}
{d\over dt}L=[L^{3\over 2}_+,L],
\end{equation}
leads to the equation
\begin{equation}
8\partial_{t} W = 2W_{xxx}+6\left( (DW)(\overline{D}W) \right)_{x}-
\left( W^3 \right)_{x},
\end{equation}
which coincide after the redefinition $W=2i\Phi$ with the $a=-2$ {$N=2$}
extension of the KdV equation in the classification of
Mathieu \cite{Mathieu1,math2}. The Lax operator given in \cite{Mathieu1} may be
obtained from $L$ in the following way. Let us consider the operator
\begin{equation}
L_{-2} = L+L^t=\partial^2+W[D,\bar D]+(DW)\bar D-(\bar D W)D.
\label{eqam2}\end{equation}
$L$ is in $\cal C$ and $L^t$ is in $\bar{\cal C}$. If we remember that the product of an element in $\cal C$ and an element in $\bar{\cal C}$ always vanishes, we immediately get that a square root of $L_2$ with highest derivative term equal to $\partial$ is $(L_{-2})^{1\over 2}=L^{1\over 2}-(L^{1\over 2})^t$. From this we deduce the relation $(L_{-2})^{3\over 2}=L^{3\over 2}
-(L^{3\over 2})^t$. As a consequence $L_{-2}$ satisfies the evolution equation
\begin{equation}{d\over dt}L_{-2}=[L^{3\over 2}_+,L]+([L^{3\over 2}_+,L])^t
=[(L_{-2})^{3\over 2},L_{-2}],\end{equation}
which is thus an equivalent Lax representation for equation \reff{eqam2}.
As the next example, we consider the Lax operator
\begin{equation}
L=D(\partial^2+V\partial+W)\bar D
\end{equation}
Then the evolution equation ${d\over dt}L=[L^{2/3}_{+},L]$ should coincide, after suitable redefinitions, with one of the three {$N=2$} supersymmetric extensions of the Boussinesq equations derived in \cite{Ivanov1}. Indeed
one can check that the Lax operator they give for the $\alpha = -1/2$
equation may be written as
$L^{(1)}=L+\bar D\partial^2 D$. Then one easily obtains
$(L^{(1)})^{2\over 3}=L^{2\over 3}+\bar D\partial D$, and the evolution equation for $L^{(1)}$ is easily deduced from that of $L$
\begin{equation}
{d\over dt}L^{(1)}={d\over dt}L=[(L^{(1)})^{2\over 3},L^{(1)}].
\end{equation}
Turning now to condition \reff{a4}, the lowest order case corresponds to
the Lax operator
$L=D\bar D+D\bar D\partial^{-1}W\partial^{-1}D\bar D$. Then the
equation ${d\over dt}L=[(L^{3})_{+}, L]$ becomes, after suitable
redefinitions, the {$N=2$} supersymmetric extension of the KdV equation
with parameter $a=4$,
\begin{equation}
\partial_{t}W = W_{xxx} + \frac{3}{2}\left( [ D ,\overline{D} ] W^2 \right)_{x} - 3\left( (DW)(\overline{D}W) \right)_{x} + (W^3)_{x}
\end{equation}
Notice that, all integer powers of $L$ define
conserved charges in this case (an alternative Lax operator with the
same property was derived in \cite{krisoto}).
The last example that we shall study is the Lax operator
\begin{equation}
L = D \left( \partial + V \right) \overline{D} + D\overline{D}\partial^{-1}W\partial^{-1}D\overline{D}.
\end{equation}
Then the equation
\begin{equation}
\partial_{2} L = [L_{+},L]
\end{equation}
explicitely reads
\begin{eqnarray}
\partial_{2} V &=& 2 W_{x}\\
\partial_{2} W &=& [ D, \overline{D} ]W_{x} +VW_{x} +(DV)( \overline{D}W)+( \overline{D}V)(DW).
\end{eqnarray}
This equation is identical, up to a rescaling of time, to the {$N=2$} supersymmetric extension
of the Boussinesq equation with parameter $\alpha = -2$
derived in \cite{Ivanov1}.
\setcounter{equation}{0}
\section{From $N=2$ to $N=1$ superspace \label{n1susy}}
$N=2$ extensions of the KP and KdV hierarchies
have been studied in several articles
\cite{inami,ghosh,dasb1,dasp} using an $N=1$ superspace formalism. In this section we wish to relate the KP
hierarchies that we described in section \ref{main} to those given in
the litterature. The first step will be to relate our $N=2$ algebra ${\cal C}$
of pseudo-differential operators to the $N=1$ algebra of pseudo-differential operators.
An operator $L=D{\cal L}\bar D$ in ${\cal C}$ should be considered as acting on a chiral
object $\Psi$, $D\Psi$=0, and this action writes
\begin{equation}
L .\Psi=D{\cal L}\bar D .\Psi={\cal L}\partial .\Psi+(D .{\cal L})\bar D .\Psi.\label{R1}
\end{equation}
We shall use the following combinations of the chiral derivatives
\begin{equation}
D_1=D+\bar D,\,\, D_2=-D+\bar D,\,\, D_1^2=-D_2^2=\partial,\,\,
\{ D_1,D_2\}=0.\end{equation}
Then the action of $L$ on $\Psi$ is
\begin{equation}
L .\Psi=({\cal L}\partial+(D .{\cal L})D_1) .\Psi.
\end{equation}
We then choose to associate to the $N=2$ pseudo-differential operator $L$
the $N=1$ pseudo-differential operator $\underline L$ given by
\begin{equation}
{\underline L}={\cal L}\vert_{\theta_2=0}\partial+
(D .{\cal L})\vert_{\theta_2=0}D_1.
\end{equation}
It is easily checked that this correspondence respects the product,
$\underline{LL'}=\underline{L}\,\,\underline{L'}$. It also has the property
\begin{equation}
\underline{L_+}={\underline L}_{>0}.
\end{equation}
That is to say that the image of an $N=2$ differential operator is a strictly differential
$N=1$ operator, without the non-derivative term. Notice also the useful relations
\begin{eqnarray}
& \,\mbox{\rm res} (\underline{L})=(D. {\,\mbox{\rm res}}(L))\vert_{\theta_2=0},\,\, &\\
&\mbox{\rm Tr}(L)=\mbox{\rm Tr}(\underline{L})\equiv
\int d^2{\underline x} {\,\mbox{\rm res}}(\underline{L}),\,\,\,\int
d^2{\underline x}\equiv\int dxd\theta_1&
\end{eqnarray}
where the residue of the operator $\underline L$ is the coefficient of
$D_1^{-1}\equiv D_1\partial^{-1}$. From now on, all expressions will be written in $N=1$
superspace, and we drop the index of $D_1$ and $\theta_1$. The KP hierarchy described in
section \ref{main} may be described in $N=1$ superspace as follows. We consider an operator
$\underline L$ of the form
\begin{equation}
\underline{L}=D^{2n}+\sum_{p=1}^\infty w_pD^{2n-p-1}
\end{equation}
and consider evolution equations
\begin{equation}
{\partial\over\partial t_k}\underline{L}=[\underline{L}^{k\over n}_{>0},L]
\label{R9}\end{equation}
This is nothing but the non-standard supersymmetric KP hierarchy described in \cite{ghosh,dasb1}. The evolution equations (\ref{R9}) admit
the conserved quantities $H_p=\mbox{\rm Tr}(\underline{L}^{p\over n})$, and they are bi-hamiltonian.
The first Poisson bracket is easily deduced from its $N=2$
counterpart (\ref{pb1}). With
$l_{\underline{X}}=\mbox{\rm Tr}(\underline{L}\,\,\underline{X})$, we have
\begin{equation}
\{ l_{\underline{X}},l_{\underline{Y}}
\}_1=\mbox{\rm Tr} L([\underline{X}_{>0},\underline{Y}_{>0}]-
[\underline{X}_{\leq 0},\underline{Y}_{\leq 0}])
\end{equation}
As in the $N=2$ formalism, this is a standard bracket associated with a non-antisymmetric
$r$ matrix. As a consequence, the two quadratic brackets
deduced from (\ref{pb2p}) and \reff{pb2m} are
quite complicated. They involve the quantity $\psi_{\underline{X}}$ defined up to a constant by
$D\psi_{\underline{X}}= {\,\mbox{\rm res}}[{\underline{L}}\, ,{\underline{X}}]$. The first one is
\begin{eqnarray}
&\{ l_{\underline{X}},l_{\underline{Y}}
\}_2^a(L)=\mbox{\rm Tr}(\underline{L}\,\,\underline{X}(\underline{L}\,\,\underline{Y})_+
-\underline{X}\,\,\underline{L}(\underline{Y}\,\,\underline{L})_+)
+\int d^2{\underline x} (-\psi_{\underline{Y}}\, {\,\mbox{\rm res}}[{\underline{L}},{\underline{X}}]\nonumber&\\&
+{\,\mbox{\rm res}}[{\underline{L}}\, ,{\underline{Y}}]\,
{\,\mbox{\rm res}}(\underline{X}\,\,\underline{L}\,D^{-1})
- {\,\mbox{\rm res}}[{\underline{L}}\, ,{\underline{X}}]\,
{\,\mbox{\rm res}}(\underline{Y}\,\,\underline{L}\,D^{-1})
).&
\end{eqnarray}
The Poisson bracket \reff{pb2m} becomes
\begin{eqnarray}
&\{ l_{\underline{X}},l_{\underline{Y}}
\}_2^b(L)=\mbox{\rm Tr}(\underline{L}\,\,\underline{X}(\underline{L}\,\,\underline{Y})_+
-\underline{X}\,\,\underline{L}(\underline{Y}\,\,\underline{L})_+)
+\int d^2{\underline x} (\psi_{\underline{Y}}\, {\,\mbox{\rm res}}[{\underline{L}},{\underline{X}}] \nonumber&\\&
+{\,\mbox{\rm res}}[{\underline{L}}\, ,{\underline{Y}}]\,
{\,\mbox{\rm res}}(\underline{L}\,\,\underline{X}\,D^{-1})
- {\,\mbox{\rm res}}[{\underline{L}}\, ,{\underline{X}}]\,
{\,\mbox{\rm res}}(\underline{L}\,\,\underline{Y}\,D^{-1})
),&
\end{eqnarray}
and already appeared in \cite{dasp}.
It is not a difficult task to obtain the $N=1$ restrictions which correspond to the {$N=2$} conditions (\ref{kdv},\ref{nls},\ref{a4},\ref{n4}). Some of the lax
operators obtained in this way are already known, in particular those satisfying \reff{kdv} from \cite{inami} and the lowest order operator coming from
\reff{nls} with odd $\varphi$ and $\bar\varphi$, which is the super-NLS Lax operator obtained in \cite{dasb1}.
\setcounter{equation}{0}
\section{Conclusion}
An easy generalization of the hierarchies presented in this article would be to consider multi-components KP hierarchies, that is to say replace the fields
$\varphi$ and $\bar\varphi$ in \reff{nls} and \reff{n4} by a set of $n+m$ fields
$\varphi_i$ and $\bar\varphi_i$, $n$ of them being Grassmann even and the other
$m$ being Grassmann odd.
For the lowest order case of equation \reff{nls}, such a generalization has been considered in \cite{bokris}. The Lax representation that we propose for such hierarchies has the advantage that one does not need to modify the definition of the residue. For the next to lowest order case of equation \reff{nls}, and
the lowest order case of equation \reff{n4}, it should be possible to obtain
in this way hierarchies based on $\cal W$-superalgebras with an arbitrary number of supersymmetry charges.
Little is known about the matrix Lax formulation of the hierarchies presented here. In the case of operators satisfying
condition \reff{kdv},
such a matrix Lax formulation was constructed in $N=1$ superspace
by Inami and Kanno \cite{inami,ina1}. It involves the loop superalgebra based on $sl(n\vert n)$. What we know about the matrix Lax formulation in {$N=2$} superspace for hierarchies based on Lax operators satisfying conditions \reff{kdv} or \reff{nls} will be reported elsewhere. Notice that we obtained the form \reff{kdv} of the scalar Lax operators from a matrix Lax representation, and only later became aware of reference \cite{popo3} where these operators also appear.
| proofpile-arXiv_065-484 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Motivations}
The CLEO Collaboration has very recently presented measurements of the
branching ratios for $B\to\rho\ell\bar\nu$ and $B\to\pi\ell\bar\nu$
decays ($\ell{=}e,\mu$)~\cite{cleo} and has announced its intention to
measure the corresponding differential decay rates. These various
measurements represent an excellent opportunity to determine the
poorly known CKM matrix element $|V_{ub}|$. Such determinations require
understanding the non-perturbative, strong-interaction corrections to
the elementary $b-u-W$ coupling contained in the matrix elements of
the weak currents $V^\mu{=}\bar u\gamma^\mu b$ and $A^\mu{=}\bar
u\gamma^\mu\gamma^5 b$ between $B$ and $\pi$ or $\rho$ meson states.
It is to calculate these matrix elements that we resort to the
lattice.
$heavy\to light$ quark decays, such as the ones that concern us here,
are also interesting because they enable one to test heavy-quark
symmetry (HQS). For these decays, HQS is weaker than for $heavy\to
heavy$ quark decays: it only applies in a limited region around the
zero recoil point $q^2{=}q^2_{max}$, where $q$ is the four-momentum
transferred to the leptons, and imposes no normalization condition on
the relevant form factors at $q^2_{max}$. Nevertheless, because both
the mass and the spin of the heavy quark can be varied in lattice
calculations, the deviations from the heavy-quark limit due to finite
heavy-quark mass and spin effects can be measured.
\section{Limitations}
Current day lattice calculations, with lattice spacings on the order
of $3\mbox{ GeV}^{-1}$, do not permit one to simulate $b\to light$ quark
decays over their full kinematical range. The problem is that the
energies and momenta of the particles involved, whose orders of
magnitude are set by the $b$ quark mass ($m_b\simeq 5\mbox{ GeV}$), are large
on the scale of the cutoff in much of phase space.
To limit these energies in relativsitc
lattice quark calculations, one performs the simulation with
heavy-quark mass values $m_Q$ around that of the charm ($m_c\simeq
1.5\mbox{ GeV}$), where discretization errors remain under control. Then one
extrapolates the results up to $m_b$ by fitting heavy-quark scaling
relations (HQSR) with power corrections to the lattice results
(see \sec{hqex}). Another approach is to work with discretized
versions of effective theories such as Non-Relativistic QCD (NRQCD) or
Heavy-Quark Effective Theory (HQET) in which the mass of the heavy
quark is factored out of the dynamics. All approaches, however,
are constrained to relatively small momentum transfers because of the
limited applicability of HQS and because of momentum-dependent
discretization errors. So one can only reconstruct the $q^2$
dependence of the relevant form factors in a limited region around
$q^2_{max}$ and one is left with the problem of extrapolating these
results to smaller $q^2$.
$heavy\to light$ quark decays are difficult in any theoretical
approach. Indeed, they require understanding the underlying QCD dynamics
over a large range of momentum transfers from $q^2_{max}{=}26.4\mbox{ GeV}^2
(20.3\mbox{ GeV}^2)$ for semileptonic $B\to\pi$ ($B\to\rho$) decays, where
the final state hadron is at rest in the frame of the $B$ meson, to
$q^2{=}0$ where it recoils very strongly.
\section{$\bar B^0\to\rho^+\ell^-\bar\nu$ and a
Model-Independent Determination of $|V_{ub}|$
\protect\footnote[1]
{The results presented here were obtained on a $24^3\times 48$ lattice
at $\beta=6.2$ from 60 quenched
configurations, using an $\ord{a}$-improved SW action for the quarks.
(Please see \protect\cite{JoF96} for details.)}
}
\label{sec:btorho}
One solution to the problem of the limited kinematical range of the
lattice results is to ignore the problem or rather rely on the
ingenuity of experimental groups to provide measurements of partial
rates in the region where lattice results are available. Combined with
lattice results for $B\to\rho\ell\bar\nu$ decays, such experimental
measurements will enable a model-independent determination of
$|V_{ub}|$~\cite{btorho}. Rates should be sufficient since our
lattice results span a range of $q^2$ from $\sim 14.4\mbox{ GeV}^2$ to
$q^2_{max}$ over which the partially integrated is
$4.6\er{4}{3}|V_{ub}|^2 ps^{-1}$. This represents approximatively
$1/3$ of the total rate obtained from light-cone sumrules (LCSR) in
\re{PBa96}, whose results at large $q^2$ agree well with ours.
\subsection{Form Factors and Heavy-Quark Extrapolation}
\label{hqex}
To describe $\bar B^0\to\rho^+\ell^-\bar\nu$ decays, we must evaluate
the matrix element $\langle \rho^+(p',\eta)|V^\mu-A^\mu|\bar B^0(p)\rangle$,
traditionally decomposed in terms of four form factors $A_1$,
$A_2$, $A$ and $V$ which are functions of $q^2$, where $q{=}p-p'$. We
calculate this matrix element for four
values of the heavy-quark mass around that
of the charm. Then, to obtain $A_1$ at the scale of the
$B$ meson we fit, to the lattice results, the HQSR
\begin{eqnarray}
A_1(\omega,M)\alpha_s(M)^{2/\beta_o}\sqrt{M}&= c(\omega)\big(1+
\frac{d(\omega)}{M}\nonumber\\
&+\ord{\Lambda^2/M^2}\big)
,\label{eq:hqsr}
\end{eqnarray}
where $M$ is the mass of the decaying meson, $\beta_0{=}11-2n_f/3$ and
$\Lambda$ is an energy characteristic of the light degrees of freedom.
This scaling relation holds for $\omega{=}(M^2+m^2-q^2)/2Mm$ close to
1 and the fit parameters $c$ and $d$ are independent of $M$. Here
$m$ is the mass of the final state meson. Once $c$ and $d$ are fixed,
it is trivial to obtain $A_1(M{=}m_B)$ at the corresponding value of
$\omega$. Furthermore, $d$ determines the size of corrections to the
heavy-quark limit. Repeating this procedure for many values of
$\omega$, one obtains the $q^2$ dependence of the desired form factor
around $q^2_{max}$. The resulting $A_1(q^2,m_B)$ is plotted together
with the LCSR result of \re{PBa96} and the lattice results of
APE~\cite{ape} and ELC~\cite{elc}. Agreement amongst these four
calculations is excellent as it for $A_2$ and $V$~\cite{PBa96,JoF96},
which are obtained in an entirely analogous way.
\begin{figure}[tb]
\setlength{\epsfxsize}{60mm}\epsfbox[93 290 460 550]{a1.eps}
\unit=0.8\hsize
\point 0.90 -0.00 {\large{$q^2(\mbox{GeV}^2)$}}
\point 0.30 0.55 {\Large{$A_1(q^2)$}}
\caption{$A_1$ vs. $q^2$ from UKQCD (crosses),
APE\protect\cite{ape} (right-hand dot), ELC\protect\cite{elc}
(left-hand dot) and LCSR\protect\cite{PBa96} (curve). (Adapted from
\protect \re{PBa96}.)}
\label{fig:comp}
\end{figure}
\subsection{Rates}
Having determined $A_1$, $A_2$ and $V$, we can compute
$(1/|V_{ub}|^2)$$d\Gamma/dq^2$. Our results are plotted in \fig{fig:rate}
(squares). In the region of $q^2$ accessed, we can
legitimately expand, around $q^2_{max}$, the helicity amplitudes that
appear in the rate. Thus we fit, to the lattice points,
the parametrization
\begin{eqnarray}
\frac{10^{12}}{|V_{ub}|^2}
\frac{d\Gamma}{dq^2}&\simeq&\frac{G^2_F}{192\pi^3m_B^3}
q^2\lambda(q^2)^{1/2} \nonumber\\
&\times& a^2\left(1+b(q^2-q^2_{max})\right)
\label{eq:rate}
\end{eqnarray}
where $\lambda(q^2){=}(M^2+m^2-q^2)-
4M^2m^2$. We find $a{=}4.6\er{4}{3}\pm
0.6 \mbox{ GeV}$ and $b{=}(-8\er{4}{6}) 10^{-2} \mbox{ GeV}^{-2}$ where the second
error on $a$ is systematic, all other errors being statistical. With
$a$ and $b$ determined, the only unknown in \eq{eq:rate} is
$|V_{ub}|$. Therefore, a fit of the parametrization of \eq{eq:rate} to
an experimental measurement of the differential decay rate around
$q^2_{max}$ determines $|V_{ub}|$. In this determination, $a$ plays
the role of ${\cal F}(1)$ in the extraction of $|V_{cb}|$ from
semileptonic $B\to D$ or $D^*$ decays \cite{MNe94} and $b$ the role
of ${\cal F}'(1)$. The difference,
here, is that $a$ is not determined by HQS up to small radiative and
power corrections. It is a genuinely non-perturbative quantity.
Another way of determining $|V_{ub}|$ from the lattice results is to
compare partially integrated rates from $q^2\ge 14\mbox{ GeV}^2$ to $q^2_{max}$
given by \eq{eq:rate} to the corresponding experimental measurements.
Both these methods yield $|V_{ub}|$ with approximatively 10\% statistical
and 12\% theoretical uncertainties.
\begin{figure}[tb]
\setlength{\epsfxsize}{60mm}\epsfbox[30 100 500 530]{vub.ps}
\caption{The data points are our lattice results and the solid
curve, the fit to \protect\eq{eq:rate}.}
\label{fig:rate}
\end{figure}
\subsection{A Test of HQS}
In \fig{fig:sem_over_rad} we compare semileptonic $B\to\rho$ form factors
with those governing the short distance contribution to radiative
$B\to K^*\gamma$ decays for which the relevant hadronic matrix element
is $\langle K^*(p',\eta)|\bar
s\sigma^{\mu\nu}q^\nu b_R|B(p)\rangle$, with $q{=}p-p'$. This
matrix element is traditionally decomposed in terms of three form
factors, $T_1$, $T_2$ and $T_3$. The comparison is made for three
initial meson masses: $M{=}m_D$, $M{=}m_B$ and $M\to\infty$.
For identical final-state vector
mesons (in
\fig{fig:sem_over_rad} all light-quarks involved have the same mass,
slightly larger than that of the strange), HQS predicts
\begin{equation}
V(q^2)=2T_1(q^2),\quad A_1=2iT_2(q^2)
\ ,\label{eq:sor}
\end{equation}
for $q^2$ around $q^2_{max}$ or, equivalently,
$\omega$ close to 1.
\begin{figure}[tb]
\setlength{\epsfxsize}{30mm}\epsfbox[170 290 285 690]{sem_over_rad.ps}
\caption{Ratios $V/2T_1$ and $A_1/2iT_2$ for 5 values of $\omega$ and three
initial meson masses. The solid lines are the HQS predictions.}
\label{fig:sem_over_rad}
\end{figure}
While $V/2T_1$ displays large
$1/M$ corrections at the $D$ and even $B$ meson scale, $A_1/2iT_2$
exhibits no such corrections even at the $D$ scale.
Both ratios, however, converge to 1 in the heavy-quark limit
which gives us confidence that we control the heavy-quark-mass dependence
of the various form factors. Furthermore,
these ratios can help constrain the possible $q^2$ dependences
of the various form factors around $q^2_{max}$ at $M{=}m_B$.
\section{$\bar B^0\to\pi^+\ell^-\bar\nu$ and Dispersive Constraints}
A second solution to the problem of the limited kinematical reach of
lattice simulations of $heavy\to light$ quark decays is
to combine lattice results for the relevant form factors around
$q^2_{max}$ with dispersive bound techniques to obtain improved,
model-independent bounds for the form factors for all $q^2$~\cite{btopi}.
For the
case of $\bar B^0\to\pi^+\ell^-\bar\nu$ decays, whose hadronic matrix
element, $\langle \pi^+(p')|V^\mu|\bar B^0(p)\rangle$, is traditionally decomposed
in terms of two form factors $f^+(q^2)$ and $f^0(q^2)$, one can use the
kinematical constraint, $f^+(0)${=}$f^0(0)$, to further constrain the
bounds.
\subsection{Dispersive Bounds}
The subject of dispersive bounds in semileptonic decays
has a long history going back to S. Okubo {\it et al.}~who
applied them to semileptonic $K\to\pi$ decays~\cite{SOkS71}.
C. Bourrely {\it et al.}~first combined these techniques with
QCD and applied them to semileptonic $D\to K$ decays \cite{CBoMR81}.
Very recently,
C.G. Boyd {\it et al.}~applied them to $B\to\pi\ell\bar\nu$
decays \cite{CBoGL95}.
The starting point for $B\to\pi\ell\bar\nu$ decays is the polarization
function
\begin{eqnarray}
\Pi^{\mu\nu}(q){=}i\int d^4x\ e^{iq\cdot x} \langle 0|T\left(V^\mu(x)
V^{\nu\dagger}(0)\right)|0\rangle\nonumber\\
{=}(q^\mu q^\nu-g^{\mu\nu}q^2)\,\Pi_T(q^2)+q^\mu q^\nu \,\Pi_L(q^2)
\ ,\label{twopoint}
\end{eqnarray}
where $\Pi_{T(L)}$ corresponds to the propagation of a
$J^P{=}1^-\,(0^+)$ particle. The corresponding spectral functions,
$\mbox{Im}\,\Pi_{T,L}$, are sums of positive contributions coming from
intermediate $B^*$ ($J^P{=}1^-$), $B\pi$ ($J^P{=}0^+\mbox{ and }1^-$),
$\ldots$ states and are thus upper
bounds on the $B\pi$ contributions.
Combining, for instance, the bound from $\mbox{Im}\,\Pi_L$ with the
dispersion relation ($Q^2{=}-q^2$)
\begin{eqnarray}
\chi_L(Q^2)&=&\frac{\partial}{\partial Q^2} (Q^2\Pi_L(Q^2))\nonumber\\
&=&\frac{1}{\pi}\int_0^\infty
dt\frac{t\,\mbox{Im}\,\Pi_L(t)}{\left(t+Q^2\right)^2}
\ ,\label{eq:disprelS}
\end{eqnarray}
one finds
\begin{eqnarray}
\chi_L(Q^2)\ge\frac{1}{\pi}\int_{t^+}^\infty
dt\,k(t,Q^2)|f^0(t)|^2
\ ,\label{eq:chilbnd}
\end{eqnarray}
where $t_{\pm}{=}(m_B\pm m_\pi)^2$ and $k(t,Q^2)$ is a kinematical
factor. Now, since $\chi_L(Q^2)$ can be calculated analytically in
QCD for $Q^2$ far enough below the resonance region (i.e. $-Q^2\ll
m_b^2$), \eq{eq:chilbnd} gives an upper bound on the weighted integral
of the magnitude squared of the form factor $f^0$ along the $B\pi$
cut. To translate this bound into a bound on $f^0$ in the region of
physical $B\to\pi\ell\bar\nu$ decays is a problem in complex
analysis (please see \re{btopi} for details).
A similar constraint can be obtained from $\Pi_T$ for $f^+$. There, however,
one has to confront the additional difficulty that $f^+$ is not analytic
below the $B\pi$ threshold because of the $B^*$ pole.
The beauty of the methods of \re{CBoMR81} is that they enable one to
incorporate information about the form factors, such as their
values at various kinematical points, to constrain the bounds.
For the case at hand, however, these methods must be generalized in two
non-trivial ways. In constructing these generalizations, one must
keep in mind that the bounds: 1) form inseparable pairs; 2) do not
indicate the probability that the form factor
will take on any particular value within them.
\subsection{Imposing the Kinematical Constraint}
The first problem is that \eq{eq:chilbnd} and the equivalent
constraint for $f^+$ yield independent bounds on the form
factors which do not satisfy the kinematical constraint
$f^+(0){=}f^0(0)$. The bounds on $f^+$ require $f^+(0)$ to lie within
an interval of values $I_+$ and those on $f^0$, within an interval
$I_0$. Together with these bounds, however, the kinematical constraint
requires $f^+(0){=}f^0(0)$ to lie somewhere within $I_+\cap I_0$.
Thus, we seek bounds on the form factors which are consistent
with this new constraint.
A natural definition is to require these new bounds to be the envelope
of the set of pairs of bounds obtained by allowing $f^+(0)$ and
$f^0(0)$ to take all possible values within the interval $I_+\cap
I_0$. In \re{btopi}, I show how this envelope can be constructed
efficiently and that the additional constraint can only improve the
bounds on the form factors for all $q^2$. Also, as a by product, one
obtains a formalism which enables one to constrain bounds on a
form factor with the knowledge that it must lie within an interval of
values at one or more values of $q^2$.
\subsection{Taking Errors into Account}
As they stand, the methods of \re{CBoMR81} can only accommodate exact
values of the form factors at given kinematical points and contain no
provisions for taking errors on these values into account. Of course,
the results given by the lattice do carry error bars. More precisely,
the lattice provides a probability distribution for the value of the
form factors at various kinematical points. What must be done, then,
is to translate this distribution into some sort of probability
statement on the bounds.
The conservative solution is to consider the probability that complete
pairs of bounds lie within a given finite interval at each value of
$q^2$. Then, using this new probability, one can define upper and
lower $p\%$ bounds at each $q^2$ as the upper and lower boundaries of
the interval that contains the central $p\%$ of this
probability.\footnote{The density of pairs of bounds increases
toward the center
of the ``distribution'' as long as the distribution of the lattice results
does.} These bounds indicate that there is at least a $p\%$ probability that
the form factors lie within them at each $q^2$.
\subsection{Lattice-Constrained Bounds}
To constrain the bounds on $f^+$ and $f^0$, I use the lattice
results of the UKQCD Collaboration
\cite{DBuetal95}, to which I add a large range of systematic
errors to ensure that the bounds obtained are conservative. Because of
these systematic errors, the probability distribution of the lattice
results is not known. I make the simplifying and rather conservative
assumption that the results are uncorrelated and gaussian distributed.
I construct the required probability by generating 4000 pairs of
bounds from a Monte-Carlo on the distribution of the lattice results.
My results for the bounds on the form factors are shown in
\fig{fig:lcsr}. I have plotted the two form factors back-to-back to
show the effect of the kinematical constraint. Without it, the bounds
on $f^+$ would be looser, especially around $q^2{=}0$, where phase
space is large. Since $f^+$ determines the rate, the kinematical
constraint and the bounds on $f^0$ are important.
Also shown in \fig{fig:lcsr} is the LCSR result of \re{VBeBKR95} which
has two components: for $q^2$ below $15\mbox{ GeV}^2$, the $q^2$ dependence
of $f^+$ is determined directly from the sumrule; for larger $q^2$,
pole dominance is assumed with a residue determined from the same
correlator. Agreement with the bounds is excellent. In \re{btopi}, the
bounds are compared with the predictions of more authors as well as
with direct fits of various parametrizations to the lattice results.
Though certain predictions are strongly disfavored, the lattice results
and bounds will
have to improve before a firm conclusion can be drawn as to the
precise $q^2$ dependence of the form factors.
\begin{figure}[tb]
\setlength{\epsfxsize}{64mm}\epsfbox[80 300 470 545]{lcsr.ps}
\unit=0.8\hsize
\point 0.95 -0.07 {\large{$q^2(\mbox{GeV}^2)$}}
\point 0.20 0.55 {\Large{$f^0(|q^2|)$}}
\point 0.770 0.55 {\Large{$f^+(q^2)$}}
\caption{$f^0(|q^2|)$ and $f^+(q^2)$ versus $q^2$.
The data points are the lattice results of UKQCD\protect\cite{DBuetal95}
with added systematic errors.
The pairs of fine curves are, from the outermost
to the innermost, the 95\%, 70\% and 30\% bounds. The shaded curve is
the LCSR result of \protect\re{VBeBKR95}.}
\label{fig:lcsr}
\end{figure}
The bounds on $f^+$ also enable one to constrain the $B^*B\pi$
coupling $g_{B^*B\pi}$ which determines the residue of the $B^*$ pole
contribution to $f^+$.
The constraints obtained are poor because $f^+$ is weakly bounded at
large $q^2$. Fitting the lattice results for $f^0$ and $f^+$ to a
parametrization which assumes $B^*$ pole dominance for $f^+$ and which
is consistent with HQS and the kinematical constraint gives the more
precise result $g_{B^{*+}B^o\pi^+}=28\pm 4$.\footnote{The result of
this fit is entirely compatible with our bounds
on $f^+$ and $f^0$.}
However, because this
result is model-dependent, it should be taken with care.
\subsection{Bounds on the rate and $|V_{ub}|$}
As was done for the form factors, one can define the probability of
finding a complete pair of bounds on the rate in a given interval and
from that probability determine confidence level (CL) intervals for the
rate. The resulting bounds are summarized in \tab{tab:rate}. They
were obtained by appropriately integrating the
4000 bounds generated for $f^+(q^2)$,
taking the skewness of the resulting ``distribution''
of bounds on the rate into account.
\begin{table}[tb]
\caption{Bounds on rate in units of $|V_{ub}|^2\,ps^{-1}$ and on $f^+(0)$.
\label{tab:rate}}
\begin{tabular}{ccc}
\hline
$\Gamma\left(\bar B^0\to\pi^+\ell^-\bar\nu\right)$ & $f^+(0)$ & CL\\
\hline
$2.4\to 28$ & $-0.26\to 0.92$ & 95\% \\
$2.8\to 24$ & $-0.18\to 0.85$ & 90\% \\
$3.6\to 17$ & $0.00\to 0.68$ & 70\% \\
$4.4\to 13$ & $0.10\to 0.57$ & 50\% \\
$4.8\to 10$ & $0.18\to 0.49$ & 30\% \\
\hline
\end{tabular}
\end{table}
The CL
bounds obtained can be used, in conjunction with the branching ratio
measurement of CLEO \cite{cleo}, to determine $|V_{ub}|$. One finds
\begin{equation}
|V_{ub}|10^4\sqrt{\tau_{B^0}/1.56\,ps}=(34\div 49)\pm 8\pm 6
\ ,\label{eq:vub}
\end{equation}
where the range given in parentheses is that obtained from the
30\% CL bounds on the rate and represents the most probable
range of values for $|V_{ub}|$. The first set of errors is obtained
from the 70\% CL bounds and the second is obtained by
combining all experimental uncertainties in quadrature and applying them to
the average value of $|V_{ub}|$ given by the 30\% CL results.
This determination of $|V_{ub}|$ has a theoretical error of approximately
37\%. Though non-negligible, this error is quite reasonable given
that the bounds on the rate are completely model-independent
and are obtained from lattice data which lie in a limited kinematical domain
and include a conservative range of systematic errors.
\section{Conclusion and Outlook}
Because HQS applies to $heavy\to light$ quark decays in a rather
limited way, it is not possible to determine the full $q^2$ dependence
of the relevant form factors from the lattice alone. The flip side of
the coin is that the model-independent information provided by lattice
calculations about these decays, though limited, is still very
important, because the relevant matrix elements are not anchored at zero
recoil by HQS, up to small radiative and power corrections, as they
are in $heavy\to heavy$ quark decays.
I have presented two approaches by which the information provided
by the lattice on exclusive semileptonic $b\to u$ decays can
be used to extract $|V_{ub}|$. Both approaches
will benefit from forthcoming, improved
lattice results. The lattice-constrained bounds would also benefit
enormously from an increase in the range of accessible $q^2$.
Finally, the techniques developed in \re{btopi} to construct lattice-improved
bounds for $B\to\pi\ell\bar\nu$ decays
are in principle applicable to limited results obtained by
non-lattice means and to other processes such as $B\to\rho\ell\bar\nu$
and $B\to K^*\gamma$ decays.
| proofpile-arXiv_065-485 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
The galactic microquasar GRS1915+105 is one of the most
intriguing objects in astrophysics today. This object shows hard
X-ray, radio, and infrared (IR) flares with variability on timescales
from minutes to months (Mirabel and Rodriguez, 1996). In addition,
VLA monitoring has revealed collimated ejection events exhibiting
superluminal motion interpreted as synchrotron emission from jets with
relativistic bulk motions (Mirabel and Rodriguez, 1994). Recently,
Sams {\it et al.} (1996) have reported observations of extended
near-IR ($2.2 \mu$m) emission near the beginning of a hard X-ray flare
(Sazonov and Sunyaev, 1995). While the position angle of the near-IR
extensions is consistent with the radio jets, the nature of the
emission is not well-determined. Detailed multi-wavelength studies of
the flaring of GRS1915+105 imply the presence of other gas and/or dust
in the region surrounding the compact object, in addition to the jets
(Mirabel and Rodriguez, 1996). Thus, possible sources of the extended
IR flux include reprocessing of the hard X-ray flare on an ejected gas
or dust disk, a wind outflow, or the radio jets, or synchrotron
emission from the jets themselves (Sams {\it et al.}, 1996).
\section{Observations}
We observed GRS1915+105 with the COB infrared imager on the
Kitt Peak National Observatory\footnote{KPNO is operated by AURA Inc.
under contract to the National Science Foundation.} 2.1-m telescope on
October 16 and 17, 1995, roughly 3 months after Sams {\it et al.}
(1996). We used the K-band ($2.2 \mu$m) filter, with a
0.2-arcsec/pixel plate scale. On October 16, we took four 30-second
exposures through light cirrus, with seeing of $\sim 0.7$-arcsec
full-width half-maximum (FWHM). On October 17, we again took four
30-second exposures, this time under photometric conditions, but with
$\sim 1.1$-arcsec FWHM seeing. During each sequence of exposures, we
moved the telescope around the position of GRS1915+105 in order to
avoid defects in the array. We subtracted the sky background and
flatfielded each exposure, and then combined each set of four
exposures. Figure 1 shows a contour map of the resulting image for
the October 16 data. Photometry of the combined images gives K-band
magnitudes of $K=13.51 \pm 0.05$ on October 16 and $K=13.43 \pm 0.03$
on October 17 (see also Eikenberry and Fazio, 1995).
\section{Analysis}
\subsection{Limits on point-like emission}
In our analysis, we first concentrate on possible emission
from a feature similar to that seen by Sams {\it et al.} (1996).
While the separation between the stellar counterpart of GRS1915+105
and the extension of Sams {\it et al.} (1996) is less than 1/2 of our
image FWHM, if the feature's brightness ($K=13.9$) remained constant
it would be the source of 60\% of the photons in our images, and
simple PSF-subtraction should reveal its presence. From each of the
background-subtracted and flatfielded (but uncombined) 30-second
exposures, we extract a $5 \times 5$-arcsec region centered on
GRS1915+105. For our PSF, we extract an identical region centered on
a star near GRS1915+105 with a very similar K-band flux (Star A in
Figure 1). For each exposure, we scale the PSF and subtract it from
the images of GRS1915+105. We see no evidence for extended structure
in any of the PSF-subtracted exposures. We then combine the four
PSF-subtracted exposures for each night. Again, we find no evidence
for extended structure in the combined PSF-subtracted images (see
Figure 2).
We then apply a simple sliding-cell source-detection algorithm
to the PSF-subtracted images. In this approach, we take a model PSF
(a 2-D gaussian fit to Star A) and center it on a pixel in the image.
We multiply the image pixel values by the corresponding PSF values at
their location, and sum to obtain the total of the products. We then
perform the same operation on the error image for the PSF-subtracted
image (including the uncertainties in the PSF), summing the error
products in quadrature. The ratio of the image sum to the error sum
then gives the statistical significance of any point source at the
image pixel location. Applying this algorithm to the PSF-subtracted
images from both nights, we find no source with a statistical
significance $>1 \sigma$ at any location.
In order to estimate the sensitivity of our observations and
analysis techniques, we perform a simple Monte Carlo simulation.
First, we model the GRS1915+105 region as 2 point sources separated by
0.3-arcsec - the star plus a southern jet - as seen by Sams {\it et
al.} (1996). We ignore the northern jet due to its much lower flux.
Second, (using the model PSF for both point sources), we select a
relative normalization of the model PSFs for the two point sources,
scale and add them with the appropriate positional offsets, and then
rescale the sum to give the same number of counts as in the real
GRS1915+105 image. Next, we add normally-distributed random numbers
(having standard deviations determined from the quadrature-summed
uncertainties of both the GRS1915+105 and Star A (PSF) images) to each
pixel of the simulated image. Finally, we take the image of Star A,
scale it, and subtract it from the simulated image, exactly as with
the actual GRS1915+105 images.
In Figure 3, we present a typical simulated result of the PSF
subtraction for an extended component with $K=13.9$. If the extended
emission seen by Sams {\it et al.} (1996) had remained unchanged, we
would have unambiguously found it in our data on both nights. In
order to place an upper limit on any point-like flux at this position,
we then decrease the flux of the extended emission in the model and
repeat the simulation process. We set our upper limit on the flux of
the extended component at the point where, for 100 Monte Carlo
simulations as described above, we detect the extended component in
the PSF-subtracted image at the 95\% confidence level using the
sliding-cell source-detection algorithm. For October 16, the limit is
$K>16.4$, while for October 17 (when the seeing was poorer), the limit
is $K>15.8$.
\subsection{Limits on extended emission}
The limit on point-like emission is useful in confirming that
the Sams {\it et al.} (1996) feature was transient, as expected for
emission from the radio-emitting jets. However, since this feature is
associated with the radio-emitting jets of GRS1915+105, it will
exhibit superluminal motion, and may also expand at high velocities.
Thus, we have performed further analyses searching for possible
non-point-like emission from the jet.
We perform this search using, once again, the sliding-cell
algorithm described above. However, instead of using Star A as a PSF,
we use a broadened PSF for the jet emissions. Applying the algorithm
to the stellar-PSF-subtracted images, we find no evidence of extended
emission using trial jet-PSFs with FWHM values of 0.8, 1.0, 1.25, and
1.5 arcsec. Given that the Sams {\it et al.} feature was point-like
with their $\sim 0.2$ arcsec resolution, even if the feature expanded
at $0.5 c$ (much faster than the limt for the radio-emitting jets), it
could have expanded only to a FWHM of 1.25 arcsec, given the 12.5 kpc
distance to GRS1915+105 (Rodriguez {\it et al.}, 1995). Thus, we
conclude that their is no evidence for infrared jet emissions in our
data.
As with the point-like emission, we perform a Monte Carlo
simulation to estimate the sensitivity of our observations and
analysis techniques. We now assume that the jet has moved 1.2 arcsec
farther from GRS1915+105 - an apparent velocity of $1.0c$, which is
less than is observed in the radio jets (Mirabel and Rodriguez, 1994).
We also assume a worst-case jet FWHM of 1.5 arcsec for the
source-detection algorithm - an expansion velocity $>0.5c$, which is
much greater than observed in the radio jets. This approach gives an
upper limit of $K>17.7$ at the 95\% confidence level for any infrared
jet emission.
\section{Discussion}
Given that the hard X-ray flaring activity which began in late
June or early July 1995 (Sazonov and Sunyaev, 1995) continued through
the time of these observations (Harmon {\it et al.}, 1995), the drop
of a factor $>10$ in the $2.2 \mu$m flux at the observed location of
Sams {\it et al.} (1996) places strong constraints on several of the
proposed explanations for the extended near-IR emission. In
particular, hypotheses involving the reprocessing of the X-ray
emission on stellar winds, ejected dust or gas disks, or other
steady-state or slow-moving structures do not appear to explain such
behavior. This, in addition to the appearance of the IR features
oppositely oriented about GRS1915+105, their position angle match with
the radio jets, and the similarities of the North/South flux asymmetry
to that in the radio (Sams et al., 1996), seems to confirm the
identification of the features seen by Sams {\it et al.} as infrared
jets.
If the features are indeed due to infrared jets, then by the
time of our observations, the southern (bright) jet would have moved
more than 1 arcsec from GRS1915+105, and we have an upper limit of
$K>17.7$ in this region. If the near-IR flux arises from reprocessing
of the X-ray emission on the jet, the reprocessing efficiency may have
dropped by this factor $>33$ due to the increased distance between the
X-ray source and the jet and/or changes in the X-ray opacity of the
jets. Alternatively, if the IR flux arises from synchrotron processes
in the jet, then we can place an upper limit on the radiative lifetime
of the IR-emitting particles, using the time separation between our
observations and those of Sams {\it et al.} (1996) and the fact that
our upper limit is a factor 33 lower in flux than the Sams {\it et
al.} feature. Thus, we find that the $1/e$ radiative lifetime of the
IR-emitting electroncs is $\tau <26$ days. For synchrotron emission,
the relativistic electrons producing the IR emission have shorter
radiative lifetimes than the radio-emitting electrons by a factor of
$\sqrt{\nu_{IR} / \nu_{radio}} \sim 10^2$, independent of the magnetic
field strength. Since the radio-emitting jets have lifetimes
significantly less than 1 year, we find that our limits are compatible
with the hypothesis that the Sams {\it et al.} feature arises from
synchrotron processes in the radio-emitting jets.
\section{Conclusions}
We have presented near-infrared K ($2.2 \mu$m) band
observations of the galactic microquasar GRS1915+105 on October 16 and
17, 1995 with a 0.2 arcsec/pixel plate scale under good seeing
conditions. Using PSF subtraction of the stellar image of
GRS1915+105, we find no evidence of near-infrared emissions as seen by
Sams {\it et al.} (1996) in July, 1995. Simple modelling shows that
we would have detected any such extended emission at the 95\%
confidence level down to a limit of $K>16.4$, as compared to the
$K=13.9$ jet observed by Sams {\it et al.} (1996). The fact that the
IR flux at this location dropped by a factor $>10$ during a time when
the hard X-ray flux increased seems to rule out reprocessing of the
hard X-ray emission on slow-moving or steady-state structures near the
compact object as a viable explanation for the extended IR emission,
and confirms the hypothesis that the extended IR emission arises from
the radio-emitting jets.
If the features are indeed due to infrared jets, then by the
time of our observations, the southern (bright) jet would have moved
more than 1 arcsec from GRS1915+105, and we have an upper limit of
$K>17.7$ in this region, a factor of $>33$ drop in the IR flux. This
allows us to place an upper limit on the radiative lifetime of the
feature of $\tau <26$ days. These limits are consistent with the
hypothesis that the Sams {\it et al.} feature was due to synchrotron
processes in the radio-emitting jets of GRS1915+105.
\acknowledgements
We would like to thank I.F. Mirabel for bringing the near-IR
jets in GRS1915+105 to our attention, M. Merrill for assisting with
the COB observations, and the anonymous referee for his/her helpful
comments. S. Eikenberry is supported by a NASA Graduate Student
Researchers Program fellowship through Ames Research Center.
| proofpile-arXiv_065-486 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
This is the fourth paper of a series where the results of the EFAR project
are presented. In Wegner et al. (1996, hereafter Paper I) the galaxy and
cluster sample was described, together with the related selection
functions. Wegner et al. (1997, hereafter Paper II) reports the analysis of the
spectroscopic data. Saglia et al. (1997, hereafter Paper III) derives the
photometric parameters of the galaxies. In this paper we describe the fitting
technique used to derive these last quantities.
A large number of papers have been dedicated to galaxy photometry.
The reader should refer to the
{\it Third Reference Catalogue of Bright Galaxies} (RC3, de
Vaucouleurs et al. 1991) for a complete review of
the subject. By way of introduction we give here only a short summary of the
methods and tests adopted and performed in the past to derive the photometric
parameters of galaxies.
Using photoelectric measurements, photometric parameters
have been derived by fitting curves of growth. The RC3 values are computed
by choosing the optimal curve between a set of 15 (for $T=-5$ to
$T=10$, see Buta et al. 1995), one for each type $T$ of galaxies.
Photoelectric data are practically free from sky subtraction errors
($<0.5$\%), but can suffer from contamination by foreground objects.
Typically, 5-10 data points are available per galaxy, with apertures
which do not exceed 100 arcsec and do not always bracket the
half-luminosity diameter. Burstein et al. (1987) (who fit the
$R^{1/4}$ curve of growth to derive the photometric parameters of a
set of ellipticals) discuss the systematic effects associated with
these procedures. The total magnitudes $M_{TOT}$ and
effective radii $R_e$ derived are biased depending on the set of data
fitted. The errors in both quantities are strongly correlated, so that
$\Delta \log R_e -0.3 \Delta \langle SB_e\rangle\approx$ constant,
where $\langle SB_e \rangle = M_{TOT}+5\log R_e+2.5\log (2\pi)$ is the
average surface brightness inside $R_e$.
This constraint
(Michard 1979, Kormendy \& Djorgovski 1989 and references therein)
stems from the fact that the product $R{\langle I \rangle}^{0.8}$
varies only by $\pm$5\% for {\it all} reasonable growth curves (from
$R^{1/4}$ to exponential laws) in a radius range $0.5R_e\le R\le 1.5
R_e$ (see Figure 1 of Saglia, Bender \& Dressler 1993). Here $\langle
I \rangle$ is the average surface brightness inside $R$. If the
galaxies considered are large ($R_e>10$ arcsec), no seeing corrections
are needed (see Saglia et al. 1993).
Until the use of CCD detectors, differential luminosity profiles of galaxies
were obtained largely from photographic plates. The procedure required
to calibrate the nonlinear response of the plates and to
digitize them is very involved. As a consequence, it was possible to
derive accurate luminosity profiles or two-dimensional photometry
only for a small number of galaxies
(see, for example, de Vaucouleurs \& Capaccioli 1979).
Using this sort of data, Thomsen \& Frandsen (1983) derived $R_e$ and
$M_{TOT}$ for a set of brightest elliptical galaxies in clusters at
redshifts $<0.15$. They fit a two-dimensional $R^{1/4}$ law convolved
with the appropriate point-spread function and briefly investigated
the systematic effects of sampling (pixel size), signal-to-noise
ratio, and shape of the profile on the derived photometric
quantities. Lauberts \& Valentijn (1989) digitized and calibrated the
blue and red plates of the ESO Quick Schmidt survey to derive
the photometric parameters of a large set of southern galaxies. Here
the total
magnitudes are not corrected for extrapolation to infinity, but are
defined as the integrated magnitude at the faintest measured surface
brightness (beyond the 25 B mag arcsec$^{-2}$
isophote) for which the luminosity profile is monotonically decreasing.
In addition, the catalogue gives the parameters derived by
fitting a ``generalized de Vaucouleurs law''
($I=I_0\exp(-(r/\alpha)^N)$;
compare to Eq. \ref{r1n}) to the surface brightness profiles.
The last 15 years have seen the increased use of CCDs for
photometry. CCDs are linear over a large dynamic range, can be
flatfielded to better than 1\% and allow one to eliminate possible
foreground objects during the analysis of the data. Large samples of
CCD luminosity profiles for early-type galaxies have been collected by
Djorgovski (1985), Lauer (1985), Bender, D\"obereiner \& M\"ollenhoff
(1988), Peletier et al. (1990), Lucey et al. (1991), J\o rgensen,
Franx \& Kj\ae rgaard (1995). Using CCDs one can derive photometric
parameters by fitting a curve-of-growth to the integrated surface
brightness profile. One major concern of CCD photometry is sky
subtraction. If the CCD field is not large enough compared to the
half-luminosity radii of the galaxies, then the sky value determined
from the frame may be systematically overestimated (due to
contamination of the sky regions by galaxy light), leading to
systematically underestimated $R_e$ and $M_{TOT}$. This problem might
however be solved with the construction of very large chips or
mosaics of CCDs (see MacGillivray et al. 1993, Metzger, Luppino \&
Miyazaki 1995).
Among the most recent studies of galaxy photometry is the Medium Deep
Survey performed with the Hubble Space Telescope. Casertano et
al. (1995) analyse 112 random fields observed with the Hubble Space
Telescope Wide Field Camera prior to refurbishment to study the
properties of faint galaxies. They construct an algorithm which fits
the two-dimensional matrix of data points to perform a disk/bulge
classification. The $R^{1/4}$ and exponential components are
convolved with the point spread function (psf) of the HST and Monte
Carlo simulations are
performed to test the results. Disk-bulge decomposition is attempted
only for a few cases (see Windhorst et al. 1994), because the data are
in general limited by the relatively low signal-to-noise and by the spatial
resolution.
In order to derive total magnitudes, galaxy photometry involves
extrapolation of curves of growth to infinity, and therefore relies on
fits to the galaxy luminosity profiles. Recently, Caon, Capaccioli \&
D'Onofrio (1993, hereafter CCO) and D'Onofrio, Capaccioli \& Caon
(1994) focused on the use of the $R^{1/4}$ law to fit the photometry
of ellipticals.
CCO find a correlation with the galaxy size and argue that if an
$R^{1/n}$ law (Sersic 1968, see Eq. \ref{r1n}) is used to fit the
luminosity profiles, then smaller galaxies ($\log R_e[{\rm kpc}]<0.5$) are best
fitted with exponents $1<n<4$, while larger ones ($\log R_e[{\rm
kpc}]>0.5$) have $n>4$. Half-light radii and total magnitudes derived
using these results may differ strongly from those using $R^{1/4}$
extrapolations. Finally, Graham et al. (1996) find that the extended
shallow luminosity profiles of BCG are best fit by $R^{1/n}$ profiles
with $n>4$.
To summarize,
the EFAR collaboration has collected photoelectric and CCD photometry
for 736 galaxies (see Colless et al. 1993 and Paper III), 31\% of
which appear to be spirals or barred objects. The remaining
69\% can be subdivided in cD-like (8\%), pure E (12 \%) and mixed E/S0
(49\%); the precise meaning of these classifications is explained in
detail in \S 3.4 of Paper III.
We derived circularly averaged luminosity profiles for all of
the objects. Isophote shape analysis can only be reliably performed for the
subset of our objects which are large and bright enough, and will be
discussed in
a future paper. Since 96\% of the EFAR galaxies have ellipticities
smaller than 0.4, the use of circularized profiles does not introduce
systematic errors on the photometric parameters derived (see \S
\ref{decomposition}) and has the advantage of giving robust results
for even the smaller, fainter objects in our sample.
The galaxies show a large variety of profile
shapes. Typically, each object has been observed several
times, using a range of telescopes, CCD detectors, and exposure
times, under different atmospheric and seeing conditions, with
different sky surface brightnesses.
Deriving homogeneous photometric parameters from the large EFAR data
set has required the construction of a sophisticated algorithm to (i)
optimally combine the multiple photoelectric and CCD data of each
object, (ii) fit the resulting luminosity profiles with a model
flexible enough to describe the observed variety of profiles, (iii)
classify the galaxies morphologically and (iv) produce reliable
magnitudes and half-luminosity radii.
This paper describes our method as applied in Paper III. It explores
the sources of random and systematic errors by means of Monte Carlo
simulations, and develops a scheme to quantify the precision of the
derived parameters objectively. The fitting algorithm searches for the
best combination of the seeing-convolved, sky-corrected $R^{1/4}$ and
exponential laws. This approach fulfils the requirement (ii) above: it
produces convenient fits to the extended components of cD luminosity
profiles, it models the profile range observed in E/S0 galaxies (from
galaxies with flat cores to clearly disk-dominated S0s), and it
reproduces the surface brightness profiles of spirals. Moreover, for
the E/S0s and spirals, this approach determines the parameters of
their bulge and disk components, to assist classification (requirement
(iii)). Finally, this approach minimizes extrapolation (requirement
(iv)) which is the main source of uncertainty involved in the
determination of magnitudes and half-luminosity radii.
Would it be possible to reach the same goals with another choice of
fitting functions? We demonstrate here (\S \ref{profiles}) that the
$R^{1/n}$ profiles quoted above can be seen as a ``subset'' of the
$R^{1/4}$ plus exponential models and therefore might not meet
requirement (ii). In addition, for $n>4$ they require large
extrapolations and therefore might fail to meet requirement (iv). What
is the physical interpretation of the two components of our fitting
function? There are cases (the above cited cD galaxies and the
galaxies with cores) where our two-component approach provides a good
fitting function, but the ``disk-bulge'' decomposition is not
physical. However, we argue that the systematic deviations from a
simple $R^{1/4}$ law observed in the luminosity profiles of our
early-type galaxies are the signature of a disk. We will investigate
this question further in a future paper, where the isophote shape
analysis of the largest and brightest galaxies in the sample will be
presented. Would it be worth improving the present scheme by, for
example, allowing for a {\it third} component (a second $R^{1/4}$ or
exponential) to be fit? This could produce better fits to barred
galaxies or to galaxies with cores and extended shallow
profiles. However, it is not clear that the systematic errors related
to extrapolation and sky subtraction could be reduced. Summarizing,
the solution adopted here fulfils our requirements (i)-(iv).
This paper is organized as follows. \S \ref{fitting} describes the
three-step fitting technique. This involves the algorithm for the
combination of multiple profiles of the same object (\S
\ref{combination}), our two-component fitting technique with the
additional option of sky fitting (\S \ref{diskbulge}), and the
objective quality assessment of the derived parameters (\S
\ref{quality}). \S \ref{montecarlo} presents the results of the Monte
Carlo simulations performed to test the fitting
procedure and assess the precision of the derived photometric
parameters. We explore a large region of the parameter space ($R_{eB},
h$, D/B, $\Gamma$, see \S \ref{diskbulge} for a definition of the
parameters) and test the performance of the fitting algorithm (\S
\ref{parameter}). In \S \ref{sky} we investigate the systematic
effects introduced by possible errors on sky subtraction and test the
algorithm to correct for this effect (see \S \ref{diskbulge}). The
influence of the limited radial extent of the profiles (\S
\ref{extension}), of the signal-to-noise ratio (\S \ref{snratio}), and
of seeing and pixellation (\S \ref{seeing}) are also investigated. The
profile combination algorithm is tested in \S \ref{testcombination}.
In \S \ref{decomposition} we assess the effectiveness of using the
fitting algorithm to derive the parameters of bulge and disk
components of a simulated galaxy. A number of different
profiles are considered in \S \ref{profiles} to test their systematic
effect on the photometric parameters. We show that the $R^{1/n}$
profiles can be reproduced by a sequence of $R^{1/4}$ plus exponential
profiles, with small systematic differences ($<0.2$ mag arcsec$^{-2}$)
over the radial range $R_e/20<R<5R_e$ (see discussion above).
In \S \ref{discussion} we discuss
how to estimate the precision of the derived photometric parameters.
In \S \ref{conclusions} we summarize our results in terms of the
expected uncertainties on the derived photometric parameters.
\section{The fitting procedure}
\label{fitting}
The algorithm devised to fit the luminosity profiles of EFAR galaxies
(see Paper III) involves three connected steps, (i) the combination of
multiple profiles, (ii) the two-component fitting, and (iii) the
quality estimate of the results. In the first step, the multiple CCD
luminosity profiles available for each object are combined taking into account
differences in sensitivity or exposure time, and sky subtraction
errors. A set of multiplicative and additive constants is determined
($k_i$, $\Delta_i$), which describe respectively the relative scaling
due to sensitivity and exposure time and the relative difference in
sky subtraction errors. The absolute value of the scaling is the
absolute photometric calibration of the images. This is accomplished
as described in Paper III, making use of the photoelectric aperture
magnitudes and absolute CCD calibrations. The absolute value of the
sky correction can be fixed either to zero or to a percentage of the
mean sky, or passed to the second step to be determined as a result of the
fitting scheme.
The second step fits these combined profiles. The backbone of the
fitting algorithm is the sum of the seeing-convolved $R^{1/4}$ and the
exponential laws. We have discussed the advantages of this choice in
the Introduction. This combination produces a variety of luminosity
profiles which can fit a large number of realistic profiles to high
accuracy. The photometric parameters derived from this approach do not
require large extrapolations, if the available profiles extend to at
least $4R_e$. When galaxies with disk and bulge components (E/S0s and
spirals) are seen at moderate inclination angles (as it is the case
for the EFAR sample, where 96\% of galaxies have ellipticities less
than 0.4, see Paper III), then the algorithm is also able, to some
extent, to determine the parameters of the two components. In Paper
III this information is used, together with the visual inspection of
the images and, sometimes, the spectroscopic data, to classify each
EFAR object as E, E/S0, or spiral. While we believe that in these
cases the two components of the fits are indicative of the presence
of two physical components, additional investigation is certainly required to
test this conclusion. This will involve the
isophote shapes analysis (Scorza \& Bender 1995), the fitting of the
two-dimensional photometry (Byun \& Freeman 1995, de Jong 1996), the
colors and metallicities (Bender \& Paquet 1995), and the kinematics
(Bender, Saglia \& Gerhard 1994) of the objects. We intend to address
some of these issues in future papers for a selection of large and
bright EFAR galaxies.
The third step assigns quality parameters to the derived photometric
parameters. Several factors determine how accurate these parameters can
be expected to be. \S \ref{montecarlo} explores in detail the effects of
sky subtraction errors, radial extent, signal-to-noise ratio,
seeing and sampling, and goodness of fit. A global quality parameter
based on these results quantifies the precision of the final results.
\subsection{Profile combination}
\label{combination}
The first step of the fitting algorithm is to combine the multiple
profiles available for each galaxy. Fitting each profile separately,
and averaging the
results produces severely biased
results if the fitted profiles differ in their signal-to-noise ratio,
seeing and sampling, radial extent, and sky subtraction errors. Only
a simultaneous fit can minimize the biasing effects of
these factors (see \S \ref{testcombination}).
Apart from the very central regions of galaxies, where seeing and
pixel size effects can be important, the profiles of the same object
taken with different telescopes and instruments differ by
a normalization (or multiplicative constant) only and an additive
constant. The first takes into account differences in the
efficiency and transparency, while the second adjusts for the relative
errors in the sky subtraction. Let $I_i(R)$, $i=1$ to $n$ denote the
$n$ available
profiles in counts per arcsec$^2$ at a distance $R$ from the center,
and consider the profile $I_{max}(R)$ as the one having the maximum radial
extent. In general the radial grids on which the profiles $I_i(R)$
have been measured will not be the same, but it will always be
possible to (spline) interpolate the values of $I_{max}(R)$ on each of
the grid points of the other profiles $I_i(R)$. The normalization $k_i$ of
the
profiles $I_i(R)$ relative to the profile $I_{max}(R)$ and the quantity
$\Delta_i$ (related to $\Delta_i/k_i$ the correction to the sky value
of the profile $I_i(R)$) are the
multiplicative and additive constants to be sought, so that:
\begin{equation}
\label{corrected}
I_i'(R)=k_iI_i(R)-\Delta_i.
\end{equation}
The $k_i$ and $\Delta_i$ constants are determined by minimizing the
$\chi^2$-like functions (see the related discussion for
Eq. \ref{chitot}):
\begin{equation}
\label{chid}
\chi^2_i=\sum_{R>R_c} w_i(R)\left(I_{max}(R)-k_i I_i(R)+\Delta_i\right)^2.
\end{equation}
The inner cutoff radius $R_c$ is 6 arcsec or half of the maximum
extent of the profile, if this is less than 6 arcsec. This cutoff
minimizes the influence of seeing, while retaining a
reasonable number of points in the sums. Here
$w_i(R)=1/\sigma_i(R)^2$ are the relative weights of the data points,
which are related to the expected errors for the profile
$I_i$:
\begin{equation}
\label{weightw}
\sigma_i(R)=\frac {\sqrt{G_i I_i(R)+G_i\hbox{Sky}_i+RON_i^2/S_i^2}}
{\sqrt{2\pi R/S_i}},
\end{equation}
where $S_i$, $G_i$ and $RON_i$ are the scale (in arcsec/pixel), the gain and
the readout noise of the CCD used to obtain the profile $I_i$ (see
Table 2 of Paper III). The denominator of Eq. \ref{weightw} assumes
that all of the pixels in the annulus at radius $R\neq 0$ have been averaged
to get $I(R)$ and therefore underestimates the
errors if some pixels have been masked to delete background or
foreground objects superimposed on the program galaxies. If $R=0$ (i.e., the
central pixel) the following equation is used:
\begin{equation}
\label{weightw0}
\sigma_i(R=0)={\sqrt{G_i I_i(R=0)+G_i\hbox{Sky}_i+RON_i^2/S_i^2}}.
\end{equation}
The weight in this fit monotonically increases with radius.
The errors $\sigma_{\mu_i}$ on the surface brightness magnitudes
$\mu_i=-2.5 \log I_i$ are related to Eqs. \ref{weightw} and
\ref{weightw0} through:
\begin{equation}
\label{sigmai}
\sigma_{\mu_i}=\frac{2.5 \sigma_i(R)\log e}{I_i(R)},
\end{equation}
By requiring $\partial \chi^2_i/ \partial k_i=0$ and $\partial
\chi^2_i/ \partial \Delta_i=0$ we solve the linear system
for $k_i$ and $\Delta_i$.
At this stage the relative sky corrections are known for all of the
profiles except the most extended one. This last correction
$\Delta_{max}$ can either be computed as part of the fitting program
(see Eqs. \ref{chikmax} and \ref{chidmax}), or fixed to a given value.
In \S \ref{montecarlo} the strategy of setting the mean
percentage sky errors (for a given galaxy) to zero will be tested
extensively against the above. For this case one requires:
\begin{equation}
\label{meanzero}
\frac{\Delta_{max}}{\hbox{Sky}_{max}}+\sum_i \frac{\Delta_i}{k_i \hbox{Sky}_i}=0.
\end{equation}
In general, Eq. \ref{meanzero} is not a good choice and gives rise to
systematic errors (see Fig. \ref{figskyerror}), however it is
preferred when the sky fitting solution (Eq. \ref{chidmax}) requires
excessively large extrapolations. Forty percent
of the fits presented in Paper III
are performed using Eq. \ref{meanzero}.
Note that for both Eq. \ref{meanzero} and \ref{chidmax} described
below, the
value of $\Delta_{max}$ is determined iteratively,
by minimizing Eq. \ref{chid}, having
redefined $I_{max}(R)$ as $I'_{max}(R)$, where
$I'_{max}(R)=I_{max}(R)-\Delta_{max}$, and repeating the procedure
until it convergences. Four or five iterations are needed to
reach a precision $<10^{-5}$ when Eq. \ref{meanzero} is used.
Convergence is reached while performing the non-linear fitting of \S
\ref{diskbulge}, when using Eq. \ref{chidmax}. Sky corrections, as computed
in Paper III, are less than 1 \% for 80\% of the cases examined.
The absolute scaling, $k_{max}$, of the $I_{max}(R)$ profile represents the
photometric calibration of the profiles. This is performed as described
in Paper III using the photoelectric aperture magnitudes and CCD zero-points.
In the following we set $k_{max}=1$.
\subsection{$R^{1/4}+$exponential law fitting}
\label{diskbulge}
The surface brightness profiles of each galaxy are
modeled simultaneously by assuming that they can be represented by
the sum of a de Vaucouleurs law (the ``bulge'' component indicated by B)
and an exponential component (the ``disk'' component indicated by D):
\begin{equation}
\label{fittingfun}
f(R,R_{eB},h,D/B,\Gamma,S)_{B+D}=f_B+f_D,
\end{equation}
where $R_{eB}$ is the half-luminosity radius of the bulge component,
$h$ the exponential scale length of the disk component, $D/B$ the disk
to bulge ratio, $\Gamma$ the FWHM of the seeing profile, and $S$ the
pixel size. Both laws are seeing-convolved as described by Saglia et
al. (1993) and take into account the effects of finite pixel
size. Definitions and numerical details can be found in the Appendix.
The results presented in Paper III show
that Eq. \ref{fittingfun} gives fits with respectably small
residuals. The differences in surface brightness $\Delta
\mu=\mu-\mu_{fit}$ are typically less than 0.05 mag arcsec$^{-2}$, while those
between the integrated aperture magnitudes are a factor two
smaller. However our formal values of reduced $\chi^2$ (see discussion
below) indicate that very few galaxies (less than 10\%) have
luminosity profiles that are fit well by the model disk and bulge. Over 90\%
of the fits have reduced $\chi^2$ larger than 2. In this sense
Eq. \ref{fittingfun} is not a statistically good representation of the
galaxy profiles.
A hybrid non-linear least squares algorithm is used to find the
$R_{eB}$, $h$, D/B and the vector of seeing values which gives the
best representation $f_{B+D}(R)$ of the profiles $I_i(R)$, taking into
account the sky corrections $\Delta_i/k_i$. The algorithm uses the
Levenberg-Marquardt search (Press et al. 1986), repeated several times
starting from randomly scattered initial values of the parameters. The
search is repeated using the Simplex algorithm (Press et
al. 1986). The best of the two solutions found is finally chosen.
This approach minimizes the biasing influence of the possible presence
of several nearly-equivalent minima of Eq. \ref{chitot}, a problem
present especially when low disk-to-bulge ratios are considered (see
discussion in \S \ref{parameter}).
All of the
profiles $I_i(R)$ available for a given galaxy are fitted simultaneously
determining the appropriate value of the
seeing $\Gamma_i$, for each single profile $i$.
The minimization is performed on the function:
\begin{equation}
\label{chitot}
\chi^2_{totB+D}=\sum_i\left(\sum_{R,\lambda_i f_{B+D}>-\Delta_i/k_i}
T_>^2+\sum_{R,\lambda_i f_{B+D}<-\Delta_i/k_i} T_<^2\right),
\end{equation}
where:
\begin{equation}
\label{termmaj}
T_>=-2.5\log \left[\frac{\lambda_i f_{B+D}(R,R_{eB},h,D/B,\Gamma_i,S_i)+\Delta_i/k_i}{I_i(R)}\right]\frac{p_i}{\sigma_{\mu_i}},
\end{equation}
and:
\begin{equation}
\label{termmin}
T_<=-2.5 \log \left[\frac{\lambda_i f_{B+D}(R,R_{eB},h,D/B,\Gamma_i,S_i)}
{I_i(R)-\Delta_i/k_i}\right]\frac{p_i}{\sigma_{\mu_i}}
\end{equation}
The penalty function $p_i$ is introduced to avoid unphysical solutions
and increases $\chi^2_{totB+D}$ to very large values when $D/B<0$ or
when the values of of $R_{eB}$ or $h$ become too large ($>300''$) or
too small ($<1''$). The use of the
$T_>$ and $T_<$ terms ensures that the arguments of the logarithm are
always positive. The sky correction is usually applied to the fitting
function (see Eq. \ref{termmaj}). However, data points where
$\lambda_i f_{B+D}+ \Delta_i/k_i<0$ (this may happen when a negative
sky correction $\Delta_i/k_i$ is applied) are included using
Eq. \ref{termmin}, which applies the sky correction to the data
points. Note also that Eq. \ref{chitot} is the weighted sum of the
squared {\it magnitude} residuals. This is to be preferred to the
weighted sum of the squared linear residuals, which is dominated by the data
points of the central parts of the galaxies.
The model normalization relative to the profile $I_i(R)$, $\lambda_i$,
is determined by requiring
$\partial \chi^2_{\lambda_i}/ \partial \lambda_i=0$, where:
\begin{equation}
\label{chikmax}
\chi^2_{\lambda_i}=\sum_R w_i(R)\left(I_i(R)-\lambda_i f_{B+D}(R)-
\Delta_i/k_i\right)^2.
\end{equation}
Note that the ratios $\lambda_{max}/\lambda_i$ can in principle differ
from the constants $k_i$, because of (residual) seeing effects (see,
e.g., $R_c$ in Eq. \ref{chid}) and systematic differences between
model and fitted profiles. In fact, the differences are smaller than
8\% in 85\% of the fits performed with more than one profile (see
Paper III). When a bulge-only or a two-component model is used, the
total magnitude of the fitted galaxy, in units of the $I_{max}(R)$
profile, is computed as $M_{TOT}=-2.5 \log (L_B+L_D)$, where
$L_B=\lambda_{max} R_{eB}^2$ (see Eq. \ref{bulge}, with this
normalization one has $I_{eB}=\lambda_{max}/(7.22\pi)$) is the
luminosity of the bulge and $L_D=(D/B)L_B$ is the luminosity of the
disk. When a disk-only model is used, then $M_{TOT}=-2.5 \log L_D$,
where $L_D=\lambda_{max}h^2$ (see Eq. \ref{disk}, with this
normalization one has $I_0=\lambda_{max}/(2\pi)$).
Note again that the photometric
calibration of these magnitudes $M_{TOT}$ to apparent magnitudes $m_{T}$ is
performed in Paper III using photoelectric aperture magnitudes and CCD
zero-points.
The sky correction to the profile $I_{max}$ can be set to a given
value (zero for no sky correction, using Eq. \ref{meanzero} for zero
mean percentage sky correction). Alternately, a fitted sky
correction $\Delta_{max}$ can be determined by additionally requiring
$\partial \chi^2_{\lambda_{max}}/ \partial \Delta_{max}=0$, where:
\begin{equation}
\label{chidmax}
\chi^2_{\lambda_{max}}=\sum_R w_{max}(R)\left(I_{max}(R)-\lambda_{max}f_{B+D}(R)-
\Delta_{max}\right)^2.
\end{equation}
If the resulting $\Delta_{max}$ produces
$\lambda_{max}f_{B+D}+\Delta_{max}<0$ at any $R$,
Eq. \ref{termmin} is used to compute
the corresponding contribution to Eq. \ref{chitot}. When using
Eq. \ref{chidmax}, the constants $k_i$ and $\Delta_i$ are computed
again using $I'_{max}(R)=I_{max}(R)-\Delta_{max}$ (see the profile
combination iterative algorithm in \S \ref{combination}). The
Monte Carlo simulations of \S \ref{montecarlo} show that
Eq. \ref{chidmax} gives an unbiased estimate of the sky corrections
when the $f_{B+D}$ is a good model of the fitted profiles.
Eq. \ref{meanzero} is to be preferred when large extrapolations are
obtained; 60\% of the
fits presented in Paper III are performed using
Eq. \ref{chidmax}.
One might use the equivalent of Eq.
\ref{chidmax} for the profiles $I_i(R)$ to compute the corrections
$\Delta_i$ directly from the fit, without having to go through Eq. \ref{chid}.
This would automatically take into account the seeing differences
of the profiles. However, tests show that this approach does not produce
the correct relative sky corrections between the profiles, if the fitting
function does not describe the fitted profiles well.
Finally, one might try deriving $\lambda_i$ and $\Delta_i$ by minimizing
Eq. \ref{chitot} for these two additional parameters. The adopted
solution, however,
speeds up the CPU intensive, non-linear minimum search, since
$\lambda_i$ and $\Delta_i$ are computed analytically.
The fit is repeated using a pure de Vaucouleurs law (D/B=0) and a pure
exponential law (B/D=0). In analogy with Eq. \ref{chitot}, two other
$\chi^2_{tot}$ are considered for these fits, $\chi^2_{totB}$ and
$\chi^2_{totD}$. A (conservative) $3\sigma$ significance test (see discussion
after Eq. \ref{chido}) is performed to decide whether
the addition of the second component improves the fit significantly.
The bulge-only fit is taken if:
\begin{equation}
\label{chibo}
\frac{\chi^2_{totB}}{\chi^2_{totB+D}}-1<3\sqrt{\frac{2}{N^{free}_{B+D}}}.
\end{equation}
The disk-only fit is taken if:
\begin{equation}
\label{chido}
\frac{\chi^2_{totD}}{\chi^2_{totB+D}}-1<3\sqrt{\frac{2}{N^{free}_{B+D}}}.
\end{equation}
The number of degrees of freedom of the $R^{1/4}$ plus exponential law
fit is $N^{free}_{B+D}=N_{data}-N_{sky}-3-2N_{prof}$, where $N_{data}$ is
the number of data points involved in the sum of Eq. \ref{chitot},
$N_{sky}=1$ if the sky fitting is activated, zero otherwise, and
$3+2N_{prof}$ are the number of parameters fitted ($R_{eB},h,D/B$,
$M_{TOT}$, $N_{prof}$ seeing values and $N_{prof}-1$ normalization constants
$\lambda_i$, where $N_{prof}$ is the number of fitted profiles). If
the errors $\sigma_{\mu_i}$ are gaussian, $\chi^2_{totB+D}$
follows a $\chi^2$ distribution of $N^{free}_{B+D}$ degrees of freedom.
If the bulge plus disk model is a good representation of the data,
then the $\chi^2_{totB+D}\approx N^{free}_{B+D}$ in the mean, with an
expected dispersion $\sqrt{2N^{free}_{B+D}}$. In this case
Eqs. \ref{chibo} and \ref{chido} are a 3$\sigma$ significance test on
the conservative side, meaning that one-component models are
preferred, if two-component models do not improve the fit by more than
$3\sigma$. In fact, Paper III shows that only 10\% of the
fits are statistically ``good'' ($\chi^2_{totB+D}\approx
N^{free}_{B+D}$). The median reduced $\chi^2$,
$\hat\chi^2=\chi^2_{totB+D}/ N^{free}_{B+D}$, is $\approx 6$, indicating
the existence of statistically significant systematic deviations from
the simple two-component models of Eq. \ref{fittingfun}. Fortunately,
the tests performed in \S \ref{montecarlo} show that reliable
photometric parameters can be obtained even in these cases. Note that
fits based on the $R^{1/n}$ profiles {\it do not} give better results:
Graham et al. (1996) obtain reduced $\chi^2\approx 10$ for their
sample of brightest cluster galaxies. Eqs. \ref{chibo}
and Eq. \ref{chido} as applied in Paper III select a bulge-only fit in
14 \% of the cases, and a disk-only fit in less than 1\%. In the 85\%
of the cases when both components are used, the median value of the
significant test is 16$\sigma$, with significance larger than
5$\sigma$ in 90\% of the cases. In the following
sections and plots we shall indicate the reduced $\chi^2$ with $\chi^2$.
Total magnitudes of galaxies are extrapolated values. In order to
quantify the effect of the extrapolation, we also derive the percentage
contribution to $M_{TOT}=-2.5 \log (L_B+L_D)$ due to the extrapolated
light beyond the radius $R_{max}$ of the last data point. In 80\%
of the galaxies examined in Paper III this extrapolation is less than 10\%.
The half-luminosity radius $R_e$ (and the $D_n$ diameter, see Paper III) of
the best-fitting function is computed using Eqs. \ref{bulgegrowth} and
\ref{diskgrowth}, so that seeing effects are taken into account.
Finally, the contamination of the sky due to galaxy light is estimated
by computing the mean surface brightness in the annulus with radii
$R_i^{max}$ and $2R_i^{max}$, where $R_i^{max}$ is the radius of the
last data point of the profile $i$. Galaxy light contamination is less
than 0.5 \% of the sky in 80\% of the cases studied in Paper III.
Using the appropriate seeing-convolved tables (see \S \ref{diskbulge}),
the fitting algorithm can also be used to fit a $f_\infty$ $\Psi=12$ model
(see description in Saglia et al. 1993 and the appendix here) plus
exponential, or a smoothed $R^{1/4}$ law plus exponential. These
additional fitting models are useful to study the effects of the
central concentration and radial extent of galaxies (see \S \ref{seeing}).
\subsection{Quality parameters}
\label{quality}
The third step in the fitting procedure assigns quality estimates to
the derived photometric parameters. Several factors determine their
expected accuracy. (i) Low signal-to-noise
images provide fits with large random errors. (ii) Images
of small galaxies observed under poor seeing conditions and/or with
inadequate sampling (a detector with large pixel size) give systematically
biased fits. (iii) Images of large galaxies taken with a small detector
give profiles with too little radial extent and fits involving
large, uncertain extrapolations. (iv) Sky subtraction errors bias the
faint end of the luminosity profiles and therefore the fitted
parameters. Finally, (v) bad fits to the luminosity profiles provide biased
quantities. The effects of these possible sources of errors
are estimated by means of Monte Carlo simulations in \S \ref{montecarlo}.
Based on these results, one can assign the quality estimates
$Q_{max}$, $Q_\Gamma$, $Q_{S/N}$, $Q_{\hbox{Sky}}$, $Q_{\delta \hbox{Sky}}$, $Q_{E}$,
$Q_{\chi^2}$ according to the rules listed in Table \ref{tabquality},
where increasing values of the quality estimates correspond to
decreasing expected precision of the photometric parameters derived
from the fits. The global quality parameter $Q$:
\begin{equation}
\label{qtot}
Q=Max(Q_{max},Q_{\Gamma},Q_{S/N},Q_{\hbox{Sky}},Q_{\delta \hbox{Sky}},Q_E,Q_{\chi^2}),
\end{equation}
assumes values $1,2,3$, corresponding to expected precisions on total
magnitudes $\Delta M_{TOT}
\approx 0.05,0.15,0.4$, on the logarithm of the half-luminosity radius
$\Delta \log R_e \approx 0.04, 0.1,0.3$ and on the combined quantity
$FP=\log R_e-0.3\langle SB_e \rangle$ $\Delta FP
\approx 0.005, 0.01,0.03$ (see \S \ref{discussion}, Fig. \ref{figqualdis}).
Paper III shows that 16\% of EFAR galaxies have
$Q=1$, 73\% have $Q=2$ and 11\% $Q=3$.
Note that $M_{TOT}$ and, therefore, $FP$ are subject to the additional
uncertainty due to the photometric zero-point. In Paper III we extensively
discuss this source of error and find that it is smaller than 0.03 mag
per object, for all of the cases (86\%) where a photoelectric or a CCD
calibration has been collected.
\begin{table}[ph]
\caption[*]{The definition of the quality parameters}
\label{tabquality}
\begin{tabular}{cccccccc}
&&&&&&&\\
\tableline
\tableline
$R_{max}/R_e^f$ & $Q_{max}$ & $R_e^f/\Gamma^f$ & $Q_\Gamma$ & $S/N$ & $Q_{S/N}$ &
Extrap & $Q_E$ \\
\tableline
$\le 1$ & 3 & $\le 2$ & 2 & $\le 300$ & 2 & $\ge0.3$ & 3 \\
$>1$,$\le2$ & 2 & $>2$ & 1 & $>300$ & 1 & $<0.3$ & 1 \\
$>2$ & 1 & & & & & & \\ \hline
&&&&&&&\\
\tableline
\tableline
$\chi^2$ & $Q_{\chi^2}$ & $\mu_{\hbox{Sky}}-\langle SB_e^f \rangle$ & $Q_{\hbox{Sky}}$ &
$|\delta \hbox{Sky/Sky}|$ & $Q_{\delta \hbox{Sky}}$ & & \\
\tableline
&&&&&&&\\
$\ge25$ & 3 & $\le0.75$ & 2 & $>0.03$ & 3 & &\\
$\ge12.5$, $<25$ & 2 & $>0.75$ & 1 & $>0.01$,$<0.03$ & 2 & &\\
$<12.5$ & 1 & & & $<0.01$ & 1 & &\\
\tableline
\end{tabular}
\end{table}
\section{Monte Carlo Simulations}
\label{montecarlo}
The fitting procedure described in the previous section has been
extensively tested on simulated profiles with the goals of
checking the minimization algorithm and quantifying the effects of the
errors described in \S \ref{quality}. Luminosity
profiles of models with known parameters have been fitted, to compare
input and output values. In all of the following figures the output
parameters of the fit are indicated with the superscript $f$ for
``fit'' (for example, $\Gamma^f$).
As a first step (\S \ref{parameter}-\ref{testcombination}, Figures
\ref{figresdb}-\ref{figcombination}), we ignore possible systematic
differences between test profiles and fitting functions (such as the
ones possibly present when fitting real galaxies, see discussion in
Paper III) and generate a number of $R^{1/4}$ plus exponential model
profiles of specified $R_{eB}$, $h$, $D/B$ ratio, seeing $\Gamma$ and
total magnitude, using the seeing-convolved tables described in the
Appendix. A constant can be added (subtracted) to simulate an
underestimated (overestimated) sky subtraction. Given the pixel size,
the sky value, the gain and readout noise, appropriate gaussian noise
is added to the model profile following Eqs. \ref{weightw} and
\ref{weightw0}. The maximum
extent of the profiles can be specified to simulate the finite size of
the CCD. The profile is truncated at the radius where noise (or the
sky subtraction error) generates negative counts for the model.
The signal-to-noise ratios computed in the
following refer to the total
number of counts in the model profile out to this radius. The
parameter space explored in all of the simulations discussed in \S
\ref{parameter}-\ref{seeing} is displayed in Figure \ref{figparspace}
and covers the region where the EFAR galaxies are
expected to reside (see Paper III). Different symbols identify the
models (see caption of Fig. \ref{figparspace}). As a second step (\S
\ref{decomposition}-\ref{profiles}, Figures
\ref{figbulgedisk}-\ref{figr1npar}), we explore the influence of
systematic differences between test profiles and fitting functions.
In \S \ref{decomposition} we show that fitting circularized profiles
of moderately flattened galaxies (as the ones observed in Paper III)
allows good determinations of the photometric parameters and also of
the bulge and disk components. In \S \ref{profiles} we fit the
$R^{1/n}$ profiles, achieving two results. First, we quantify the
influence of the quoted systematic effects on the fitted photometric
parameters. Second, we suggest that the possible correlation between
galaxy sizes and exponent $n$ (see discussion in the
Introduction) reflects the presence of a disk component
in early-type galaxies.
\S \ref{discussion} summarizes the results by calibrating the quality
parameter $Q$ of Eq. \ref{qtot}.
\subsection{The parameter space}
\label{parameter}
In this section we discuss the results obtained by fitting the models
indicated by the crosses in Figure \ref{figparspace}. For clarity the
parameters are also given in Table \ref{tabpara}.
No sky subtraction
errors are introduced and the sky correction algorithm is not used. The
detailed analysis of the possible sources of systematic errors
discussed in \S \ref{sky}-\ref{snratio} is performed on the same
sample of models. More extreme values of the
parameters are used when testing the effects of seeing and resolution
(\S \ref{seeing}). The profiles tested in this section extend out to
$4 R_e$, have a pixel size of 0.4 arcsec and normalization of $10^7$
counts, with $G_i=RON_i=1$ (see Eq. \ref{weightw}),
corresponding to $S/N\approx 1000$.
\begin{table}[hp]
\centering
\caption[*]{The parameters of the models indicated by the crosses of
Figure \ref{figparspace} (see \S \ref{parameter}). A model for each
combination of parameters in the two blocks separately has been generated.
$D/B=\infty$ indicates exponential models ($B/D=0$). }
\label{tabpara}
\begin{tabular}{ll}
&\\
\tableline
\tableline
Parameter & Values \\
\tableline
&\\
$R_{eB}('')$ & 4, 8, 12, 16, 20, 32\\
$h('')$ & 4, 8, 12, 16, 20, 32\\
$D/B$ & 0, 0.1, ..., 1, 1.2, 1.6, 2, 3.2, 5, $\infty$\\
$\Gamma('')$ & 1.5, 2.5\\
Sky/pixel & 1000 \\
&\\
\tableline
&\\
$R_{eB}('')$ & 2, 3\\
$h('')$ & 3, 6\\
$D/B$ & 0, 0.2, 0.4, 0.8, 1.6, $\infty$\\
$\Gamma('')$ & 1.5, 2.5\\
Sky/pixel & 500 \\
&\\
\tableline
\end{tabular}
\end{table}
Figure \ref{figresdb} shows the precision of the reconstructed
parameters. Total
magnitudes are derived with a typical accuracy of 0.01 mag, $R_e$ and
$\Gamma$ to 3\%, $R_{eB}$ and $h$ to $\approx 8$\%, $D/B$ to $\approx
10$\%. The errors $\Delta M_{TOT}=M_{TOT}-M_{TOT}^f$ and $\Delta R_e
=\log R_e/R_e^f$ are highly
correlated, with insignificant differences from the relation $\Delta FP =
\Delta R_e-0.3(\Delta M_{TOT}+5\Delta R_e)= \Delta R_e-0.3\Delta
\langle SB_e \rangle$. Galaxies with faint ($D/B<0.3$) and shallow
($h/R_{eB}>2$) disks show the largest deviations. This partly reflects
a residual (minimal) inability of the fitting program to converge to
the real minimum $\chi^2$ (there are 3 points with $\chi^2>10$), but stems
also from the degeneracy of the Bulge plus Disk fitting. Figure
\ref{figbadfit} (a) and (b) show the example of a model with $D/B=0.1$
and $h/R_{eB}=5$ where a very good fit is obtained ($\chi^2=1.3$) yet
there is a 0.05 mag error on $M_{TOT}$ and the disk solution is
significantly different from the input model. Note that the largest deviations
$\Delta M_{TOT}$ and $\Delta R_e$ are associated with the largest
extrapolations ($\approx 20$\%). In the following sections we shall
see that extrapolation is the main source of uncertainty, when
determining total magnitudes and half-luminosity radii. The
uncertainties $\Delta R_{eB}$ on the bulge scale length are smallest
with bright bulges, while those on the disk scale length $\Delta h$ are
smallest with bright disks. The algorithm to opt for one-component
best-fits (Eqs. \ref{chibo}-\ref{chido}) identifies successfully all of
the one-component models tested (bulges plotted at $\log D/B =-1.1$
and $\log h/R_{eB}=-1.1$, disks plotted at $\log D/B =1.1$ and $\log
h/R_{eB}=1.1$ in Figure \ref{figresdb}). For only two models (with
$D/B=0.1$ and large $h/R_{eB}$) is the bulge-only fit preferred (using
the 3$\sigma$ test) to the
two-component fit (circled points in Figure \ref{figresdb}).
Figure \ref{figquatexp} shows the results obtained by fitting a pure
bulge or a pure disk. As before, no sky subtraction error is introduced
and the sky correction algorithm is not activated. Neglecting one of the
two components strongly
biases the derived total magnitudes and half-luminosity radii. In the
case of the $R^{1/4}$ fits, already test models with values of D/B as small as
$\approx 0.2$ give fitted
magnitudes wrong by 0.2 mag, and $R_e$ by more than 30\%. The
systematic differences correlate with the amount of extrapolation
involved, and large extrapolations yield strongly overestimated
magnitudes and half-luminosity radii. However, the resulting
correlated errors $|\Delta FP|$ are almost always smaller than 0.03. In the
case of pure exponential fits, the derived total magnitudes and
half-luminosity radii are always smaller than the true values, since
very little extrapolation ($<1$\%) is involved. Consequently, a
positive, correlated error $\Delta FP$ ($\approx +0.03$) is
obtained. Finally, note that pure bulge fits are {\it bad} fits of the
surface brightness profiles ($\chi^2>10$), but may appear to give
acceptable fits of the integrated magnitude profiles. (One can easily
show that the differences in integrated magnitudes are the weighted
mean of the differences in surface brightness magnitudes). Figure
\ref{figbadfit} (c) and (d) shows such an example for an $R^{1/4}$ fit
to a model with $D/B=0.8$ and $h/R_{eB}=1$. The residuals in the
integrated magnitude profile are always smaller than 0.07 mag, but a
$\chi^2=181$ is derived, with $\Delta M_{TOT}=0.32$ and
$R_e^f/R_e=1.65$. These considerations suggest that magnitudes and
half-luminosity radii derived by fitting the $R^{1/4}$ curve of growth
to integrated magnitude profiles (Burstein et al. 1987, Lucey et
al. 1991, J\o rgensen et al. 1995, Graham 1996) may be subject to systematic
biases, as indeed Burstein et al. (1987) warn in their Appendix.
This might be important for the sample of
J\o rgensen et al. (1995), where substantial disks are detected in a large
fraction of the galaxies by means of an isophote shape analysis. It is
certainly very important for
the sample of cD galaxies studied by Graham (1996, see discussion in
Paper III).
These objects have luminosity profiles which differ strongly from an $R^{1/4}$
law.
Finally, note that the {\it systematic} errors shown in Figure
\ref{figquatexp} (and in the figures of the following sections) cannot
be simply estimated by considering the shape of the $\chi^2$ function
near the minimum. Figure \ref{figcontour} shows the $1,2,3,5\sigma$
contours of constant $\chi^2$ for an $R^{1/4}$ fit to a
$h/R_{eB}=0.5$, $D/B=0.1$ $R^{1/4}$ plus exponential model. The
reduced $\chi^2$
(8.47 at the minimum) has been normalized to 1, so that $1\sigma$
corresponds to a (normalized) $\chi^2=1+\sqrt{2/N_{free}}=1.11$. The
errors, estimated at the $5\sigma$ contour, underestimate the
differences between the fit and the model by a factor 2. This results from
the extrapolation involved and can be as large as one order of magnitude
for models with larger D/B ratios.
\begin{figure}
\plotone{f1.eps}
\caption[f1.eps]{The parameter space of the $R^{1/4}$ plus exponential
profile of the Monte Carlo simulations discussed in Figure
\ref{figresdb}-\ref{figgamma}. Models of
Figs. \ref{figresdb}-\ref{figskyfix}: crosses (see also Table 2). Models of
Fig. \ref{figextension}: skeletal triangles. Models of
Fig. \ref{figflux}: open triangles. Models of
Figs. \ref{figseeing}-\ref{figgamma}: open squares. Models of
Fig. \ref{figcombination}: open pentagons. Models of Fig.
\ref{figbulgedisk}: open hexagons. The small dots show the
position of the EFAR galaxies as determined in Paper 3. The parameters
of bulge only models are shown with $h=0$. The parameters of disk only
models are shown at $R_{eB}=0$. See discussion in \S \ref{montecarlo}.}
\label{figparspace}
\end{figure}
\begin{figure}
\plotone{f2.eps}
\caption[f2.eps]{The reconstructed parameter space for the models
indicated
by the crosses in Fig. \ref{figparspace}. No sky error is present.
The quantities plotted on the y-axis are defined as $\Delta
M_{TOT}=M_{TOT}-M_{TOT}^f$, $\Delta R_e=\log R_e/R_e^f$, $\Delta FP =
\Delta R_e-0.3(\Delta M_{TOT}+5\Delta R_e)=\Delta R_e-0.3\Delta
\langle SB_e \rangle$, $\Delta (D/B)=\log [(D/B)/(D/B)^f$], $\Delta
R_{eB}=\log R_{eB}/R_{eB}^f$, $\Delta h = \log h/h^f$, $\Delta \Gamma
= \log \Gamma/\Gamma^f$. On the x-axis, the first three boxes show the
input parameters of the models in the logarithm units ($\log D/B$,
$\log h/R_{eB}$, $\log R_e/\Gamma$). The last three boxes show the
differences in magnitudes between the assumed sky value and the
average effective surface brightness of the models ($\mu_{\hbox{\rm
Sky}}-\langle SB_e \rangle$), the logarithm of the reduced $\chi^2$,
and the fraction of light extrapolated beyond $R_{max}$ used in the
determination of
$M^f_{TOT}$. Models with $D/B=0$ (pure $R^{1/4}$ laws) are plotted at
$\log D/B=-1.1$ and $\log h/R_{eB}=-1.1$. Models with $B/D=0$ (pure
exponential laws) are plotted at $\log D/B=1.1$ and $\log
h/R_{eB}=1.1$. Models with $D/B\neq 0$ which have been fitted with one
component are circled. See \S \ref{parameter} for a discussion of
the results.}
\label{figresdb}
\end{figure}
\begin{figure}
\plotone{f3.eps}
\caption[f3.eps]{(a) A circular disk plus bulge model with $D/B=0.1$
and $h/R_{eB}=5$
(crosses). The dotted curves show the luminosity profiles
$\mu(R)=-2.5\log I(R)$ of the bulge and the disk components, the dashed
curve the fitted disk component. (b) The differences $\Delta \mu$ (in
mag arcsec$^{-2}$, open squares) between
model surface brightness and the fitted one (dotted curve) (see \S
\ref{parameter}).
(c) The $R^{1/4}$ fit (solid curve) to the surface brightness
magnitude profile of a circular disk plus bulge model with $D/B=0.8$
and $h/R_{eB}=1$ (crosses, one point in every
four). (d) The differences $\Delta mag$ between the $R^{1/4}$
integrated
magnitudes and the fitted ones (solid curve, see \S \ref{parameter}). Note
that $|\Delta mag| <0.07$ even if large deviations $\Delta \mu$ are present.
(e) The fit to the circularized profile of a flattened bulge plus an
inclined disk
model (see \S. \ref{decomposition}). The luminosity profile of the model
(crosses, one point in every four; the bulge
and the disk components, with the listed parameters, are the full
curves) is best fitted by an $R^{1/4}$ plus exponential law (dashed
curves) with parameters $R_{eB}^f=16.34$ arcsec, $h^f=13.93$ arcsec,
$(D/B)^f=0.13$, $R_e=17.51$ arcsec.
(f) The residuals $\Delta \mu$ of the fit (open squares, one point in every
four) and the differences
between the growth curves $\Delta mag$ (full curve).}
\label{figbadfit}
\end{figure}
\begin{figure}
\plotone{f4.eps}
\caption[f4.eps]{The effects of fitting disk plus bulge test
profiles by either
a single bulge (first three rows of plots) or a single disk (last three rows
of plots) model.
The test models are indicated by the crosses of
Fig. \ref{figparspace}. $\Delta M_{TOT}$, $\Delta R_e$, and $\Delta FP$
are defined as in Fig. \ref{figresdb}, x-axis as in Fig.
\ref{figresdb}. See \S \ref{parameter} for the a discussion of the results.
Note the change of scale on the ordinate axis with respect to Figure
\ref{figresdb}.}
\label{figquatexp}
\end{figure}
\begin{figure}
\plotone{f5.eps}
\caption[f5.eps]{Illustration of the underestimation of the errors.
The contours of constant $\chi^2$ near the minimum
of an $R^{1/4}$ fit to a $h/R_{eB}=0.5$, $D/B=0.1$ disk plus bulge model.
The cross shows the best-fit solution, the circle near the upper left corner
gives the real parameters
of the model. The errors estimated at
the $5\sigma$ contour underestimate the differences between the model
and the fit by a factor 2 (see
\S \ref{parameter}).}
\label{figcontour}
\end{figure}
\subsection{Sky subtraction errors}
\label{sky}
Sky subtraction errors can induce severe systematic errors on the
derived photometric parameters of galaxies. Figure \ref{figskyerror}
shows the parameters derived from the $R^{1/4}$ plus exponential
models examined in the previous section, where now the sky has been
overestimated or underestimated by $\pm 1$\%. The sky correction
algorithm is not activated.
The biases become increasingly large as the sky brightness approaches
the effective surface brightness of the models. As expected,
underestimating the sky (a negative sky error) produces
total magnitudes that are too bright and
half-luminosity radii that are too large relative to the true ones.
The size of
the bias correlates with the extrapolation needed to derive
$M_{TOT}^f$. The opposite happens when the sky is overestimated, but
the amplitude of the bias is smaller, because there is no extrapolation.
The correlated error $\Delta FP$ remains small ($\approx
0.05$), except for the cases where large extrapolations are
involved. The D/B ratio is ill determined, with better precision for
models with extended disks ($h/R_{eB}>2.5$). The scale length of the
bulge is better determined for low values of $D/B$ (dominant bulge),
the scale length of the disk component is better determined for large
values of $D/B$ (dominant disk). The parameter least affected is the
value $\Gamma$ of the seeing, which is determined in the inner, bright
parts of the models, where sky subtraction errors are unimportant.
Bulge-only or disk-only models appear to be fit best
by two-component models (crosses and triangular crosses in Figure
\ref{figskyerror}).
Finally, note that reasonably good fits ($\chi^2<10$)
to the surface brightness profiles are always obtained, in spite of
the large errors on the reconstructed parameters.
The biases discussed above can be fully corrected when the sky-fitting
algorithm of Eq. \ref{chidmax} is applied. Figure \ref{figskyfix} shows
the reconstructed parameters of the models considered in \S \ref{parameter},
where sky subtraction errors of 0, $\pm1$\%, $\pm3$\% have been introduced.
For most of the models examined, the errors on the derived quantities are no
more than a factor 2 larger than those shown in Fig. \ref{figresdb}. The sky
corrections are computed to better than 0.5\% precision.
Larger errors $\Delta M_{TOT}$ and $\Delta R_e$ are obtained for models with
relatively weak ($D/B<0.3$) and extended disks ($h/R_{eB}>2.5$), where the
degeneracy
discussed in \S \ref{parameter} is complicated by the sky subtraction
correction. These cases give reasonably good fits ($\chi^2<10$), but are identified by
the large extrapolation ($>0.3$) involved. Models with concentrated disks
($h/R_{eB}<0.2$) can also be difficult to reconstruct, when $h/\Gamma\approx
1$. For some of these problematic fits, one-component solutions are preferred
by Eqs. \ref{chibo}-\ref{chido} (circles in Eq. \ref{figskyfix}).
\begin{figure}
\plotone{f6.eps}
\caption[f6.eps]{The biases introduced by a $\pm 1$\% sky subtraction
error.
Quantities plotted as in Fig. \ref{figresdb}. Models with $D/B=0$ which
have been fitted with two components are shown as crosses.
Models with $B/D=0$ which have
been fitted with two components are shown as triangular crosses.
Note the change of scale on the ordinate axis with respect to
Figure \ref{figresdb}. See discussion in \S \ref{sky}.}
\label{figskyerror}
\end{figure}
\begin{figure}
\plotone{f7.eps}
\caption[f7.eps]{The effects of the sky fitting algorithm.
The parameters of the models of
Fig. \ref{figskyerror} with the sky subtraction errors of 0, $\pm1$\%,
$\pm3$\%, are reconstructed using the sky fitting algorithm. Quantities
and symbols plotted as in Figures \ref{figresdb} and \ref{figskyerror}.
In addition, the difference
$\Delta$dSky=dSky/Sky-dSky$^f/$Sky on the sky correction is plotted.
Note the change of scale on the ordinate axis with respect to Figure
\ref{figresdb}. See discussion in \S \ref{sky}.}
\label{figskyfix}
\end{figure}
A common problem of CCD galaxy photometry is
the relatively small field of view, particularly with the older smaller CCDs.
If the size (projected on the sky)
of the CCD is not large enough compared to the half-luminosity radius of
the imaged galaxy, then the sky as determined on the same frame
will be contaminated by galaxy light and biased to values larger
than the true one. Total
magnitudes and half-luminosity radii can therefore be biased to smaller
values, the effect being more important for intrinsically large galaxies,
which tend to have low effective surface brightnesses.
The mean surface brightness in the annulus with radii
$R_i^{max}$ and $2R_i^{max}$ (see \S \ref{diskbulge}) predicted by the fit
allows us to estimate the size of the contamination.
\subsection{Radial extent}
\label{extension}
Photoelectric photometry of large, nearby galaxies rarely goes beyond
1 or 2 $R_e$ (Burstein et al. 1987) and the same applies for the surface
photometry obtained with smallish CCDs. The typical profiles obtained
in Paper III extend to a least 4 $R_e$, but a small fraction of them are
less deep, reaching 1 or 2 $R_e$ only. Here we investigate the effect of
the radial extent of the profiles, keeping the normalization of the
profiles fixed ($10^7$ counts, $S/N\approx 10^3$).
Sky subtraction errors of $0$,
$\pm3$\% are introduced and the sky fitting is activated. Figure
\ref{figextension} shows the cumulative distributions of the errors on the
derived photometric
parameters as derived from the simulations, for a range of $R_{max}$ values.
When $R_{max}=R_e$, rather
large errors are possible (0.3 mag in the total
magnitude, $>30$\% in $R_e^f$).
The main source of error is again the large extrapolation
involved when $R_{max}\approx R_e$, coupled with the sky correction
which becomes unreliable for these short radial extents. As soon
as $R_{max}\ge3R_e$ the errors reduce to the ones discussed in
\S \ref{parameter}. The same kind of trend is observed for the
parameters of the two component ($\Delta (D/B)$, $\Delta R_{eB}$,
$\Delta h$). The seeing values are less affected, as they are
sensitive to the central parts of the profiles only. Finally, note that
in all cases very good fits are obtained ($\chi^2\approx 1$).
\begin{figure}
\plotone{f8.eps}
\caption[f8.eps]{The effect of the radial extent of the profiles
on the precision
of the derived parameters. The cumulative distributions of the errors on the
derived photometric
parameters as derived from the simulations are shown for a range of $R_{max}$
values (full lines: $R_{max}=R_e$, dotted lines: $R_{max}=2R_e$, dashed lines
$R_{max}=3R_e$, long-dashed lines: $R_{max}=4R_e$). Good reconstructions are
obtained when $R_{max}/R_e>2$ (see \S \ref{extension}).}
\label{figextension}
\end{figure}
\subsection{Signal-to-noise ratio}
\label{snratio}
For most of the galaxies discussed in Paper III, multiple profiles are
available with integrated signal-to-noise ratios $S/N> 300$, the
normalization used in the previous sections. But for some of the luminosity
profiles a smaller number of total counts has been collected (see Figure
\ref{figparspace}). Here we
investigate how the signal-to-noise ratio of the profiles affects the
outcome of the fits. As
before, the subset of models of \S \ref{sky} is used with
$R_{max}\le4 R_e$ (see comment at the beginning of
\S \ref{montecarlo}). Sky subtraction errors of $0$, $\pm3$\% are
introduced and the sky fitting is activated. Figure \ref{figflux}
shows how the errors on the derived parameters increase when the
signal-to-noise ratio is reduced. For fluxes as low as about $10^5$
($S/N\approx 10^2$) all of the derived photometric parameters become
uncertain (0.2 mag in the total
magnitudes, 20\% variations in the derived $R_e$, large spread $\Delta (D/B)$,
$\Delta R_{eB}$, $\Delta h$,
$\Delta \Gamma$), as large extrapolations and uncertain sky
corrections are applied. In all cases very good fits are obtained
($\chi^2\approx 1$).
\begin{figure}
\plotone{f9.eps}
\caption[f9.eps]{The effect of the signal-to-noise ratio of the
profiles on the
precision of the derived parameters. Good reconstructions are obtained
when $S/N>300$ (see \S \ref{snratio}).
Note the change of scale on the ordinate axis with respect to Figure
\ref{figresdb}.}
\label{figflux}
\end{figure}
\subsection{Seeing and sampling effects}
\label{seeing}
Some of the galaxies considered in Paper III are rather small, with
$R_e<4''$. Here we investigate the effects of seeing and pixel
sampling, when $R_e\approx\Gamma\approx$pixel size. Figure
\ref{figseeing} shows that reliable parameters can be derived down to
$R_e\approx\Gamma$, with pixel sizes 0.4-0.8 arcsec, with only a
small increase of the scatter for $R_e<2\Gamma$.
A small systematic effect is caused by the choice of the psf. Saglia et al.
(1993) demonstrate that a good approximation of the psfs observed during
the runs described in Paper III is given by the $\gamma$ psf
with $\gamma=1.5-1.7$. We adopt $\gamma=1.6$ for the fits. Here
we test the effect of having $\gamma=1.5$ or 1.7 with a pixel size of
0.8 arcsec. Figure \ref{figgamma} shows that if $\gamma=1.5$ is the true
psf of the observations, then the half-luminosity radius,
the total luminosity, the scale length of the bulge will be slightly
overestimated, and the disk to bulge ratio slightly underestimated.
A small systematic trend is observed in the correlated errors $\Delta FP$.
The scale length of the disk component is less affected. The sky
corrections are also biased, but do not strongly affect the photometric
parameters, because of the high average surface brightness of the small
$R_e$ models. Seeing values suffer a very small, but systematic effect.
The opposite trends are observed if the true $\gamma$ is 1.7.
In all cases very good fits are obtained
($\chi^2\approx 1$). The systematic differences
become unimportant for $R_e>2\Gamma$.
Finally, the seeing values derived can be systematically biased, if
the central concentration of the fitted galaxies does not match the
one of the $R^{1/4}$ plus exponential models. We investigate this
effect by fitting the $\Psi=12$ plus exponential or the smoothed
$R^{1/4}$ plus exponential models discussed in
\S \ref{diskbulge}. We find that in the first case the seeing
value is underestimated which compensates for the higher concentration of
the $\Psi=12$ component. The shallow radial decline of the luminosity
profile in the outer parts introduces systematic biases in the
reconstructed parameters, similar to those discussed for the $R^{1/n}$
profiles, for large values of $n$ (see \S \ref{profiles}). The
half-luminosity radii and total magnitudes derived are underestimated
by 20\% and 0.2 mag respectively, when a $\Psi=12$ model
with no exponential component is fitted. The biases are reduced when
models with an exponential component are constructed. In the case of
the smoothed $R^{1/4}$ law, the seeing value is overestimated to fit
the lower concentration
of the smoothed $R^{1/4}$ component. No biases are introduced on the
other reconstructed parameters.
\begin{figure}
\plotone{f10.eps}
\caption[f10.eps]{The effect of seeing and pixel sampling of the
profiles on the
precision of the derived parameters. Different symbols indicate different
pixel sizes (small dot 0.4 arcsec, triangles 0.6 arcsec, squares 0.8 arcsec).
Note the expanded ordinate scale with respect to Figures
\ref{figquatexp}-\ref{figflux}. See discussion in \S \ref{seeing}.}
\label{figseeing}
\end{figure}
\begin{figure}
\plotone{f11.eps}
\caption[f11.eps]{The effect of the choice of the psf on the
precision of the derived parameters. Open triangles for $\gamma=1.5$, dots for
$\gamma=1.6$ and open squares for $\gamma=1.7$. Fits performed with the
$\gamma=1.6$ psf overestimate (underestimate) magnitudes and half-luminosity
radii of models constructed with $\gamma=1.5$ ($\gamma=1.7$; see \S
\ref{seeing}). Note the expanded ordinate scale with respect to Figures
\ref{figquatexp}-\ref{figflux}.}
\label{figgamma}
\end{figure}
\subsection{Tests of profile combination}
\label{testcombination}
In order to test the combination algorithm described in
\ref{combination}, four profiles with different $\Gamma$, pixel sizes,
normalizations, gain, readout noise, and sky subtraction errors (see
Table \ref{tabcombination}; these parameters match the typical
values of the profiles of Paper III) are generated for the set of
models identified
by the open pentagons of Figure \ref{figparspace}. Figure
\ref{figcombination} shows the result of the test. The abscissa plots the
residuals $\Delta$ of the parameters derived using the fitting
procedure with profile combination. $\Delta$ dSky and $\Delta
\Gamma$ are averaged over the four obtained values. The ordinate plots
the {\it mean} of the residuals of the parameters derived by fitting
each single independently as crosses, and the residuals of each fit
as dots. The profile combination algorithm
obtains better precision on all of the parameters with the exception of
$\Gamma$, where the maximum deviation is in any case smaller than 8\%.
\begin{figure}
\plotone{f12.eps}
\caption[f12.eps]{The profile combination algorithm and the
precision of the derived parameters. The x-axis plots the
residuals $\Delta$ of the parameters derived using the fitting
procedure with profile combination. $\Delta$ dSky and $\Delta
\Gamma$ are averaged over the four obtained values. The y-axis plots
the {\it mean} of the residuals of the parameters derived by fitting
each single independently as crosses, and the residuals of each fit as dots
(see discussion in \S \ref{testcombination}).}
\label{figcombination}
\end{figure}
\clearpage
\begin{table}[ph]
\centering
\caption[*]{The parameters of the multiple profiles test (see \S
\ref{testcombination}).}
\label{tabcombination}
\begin{tabular}{lrrrrr}
&&&&\\
\tableline
\tableline
Profile & 1 & 2 & 3 & 4 \\
\tableline
&&&&\\
Pixel size $('')$ & 0.4 & 0.606 & 0.862 & 0.792 \\
Sky per pixel & 300 & 350 & 250 & 1500 \\
$\delta$Sky/Sky & $+1$\% & $-0.5$\% & $+1.5$\% & $+0.5$\% \\
$R_{max}/R_e$ & 4 & 3 & 4.5 & 2.5 \\
Normalization & $10^7$ & $5\times 10^6$ & $10^7$ & $5\times 10^6$ \\
Gain & 1 & 3 & 1 & 2 \\
Ron & 1 & 4 & 1 & 5 \\
$\Gamma ('')$ & 2 & 2.1 & 1.5 & 2.4 \\
\tableline
\end{tabular}
\end{table}
\subsection{``Bulge'' and ``Disk'' components}
\label{decomposition}
The discussion of the previous sections shows that for a large fraction of the
parameter space, i.e. when deep enough profiles are available, with large
enough objects, not only can
the global photometric parameters $R_e$ and $M_{TOT}$ be reconstructed
with high accuracy, but also the parameters of the $R^{1/4}$ and the
exponential components. Here we investigate further if reliable
``bulge'' and ``disk'' parameters can be derived, when the profiles
analysed are constructed from the superposition of these two components.
With this purpose, we constructed a number of two-dimensional frames
(filled triangles in Figure \ref{figparspace})
as the sum of a flattened $R^{1/4}$ bulge and an exponential disk of
given inclination. The bulge (disk) frames follow
an exact $R^{1/4}$ (exponential) law with $R_{eB}=12\sqrt{b/a}$ arcsec
($h=10\sqrt{\cos(i)}$ arcsec) along the minor axis. Three flattenings
of the bulge ($b/a=1,0.7,0.4$), four inclinations for the disk
($i=0^\circ,30^\circ,60^\circ,80^\circ$, where $i=0^\circ$ is
face-on and $i=90^\circ$ edge-on) and five values of the disk to bulge
ratio ($D/B=0,0.5,1,2,\infty$) are considered. The resulting models
are normalized to $10^7$ counts. The pixel size is 0.6 arcsec.
The circularly averaged luminosity profiles are derived following the
same procedure adopted for the observed galaxies (see Paper III) and extend
out to $\approx 4-6R_e$. A 1\% sky error is introduced and
the sky fitting procedure is activated. Note that the maximum
flattening of the EFAR galaxies is $b/a=0.5$, with 96\% of the
galaxies having $b/a>0.6$ (see Paper III). This corresponds to (pure) disk
inclinations $i\le 60^\circ$.
Figure \ref{figbulgedisk} shows the reconstructed parameters as a
function of the inclination angle of the disk, for the different
flattenings of the bulge, using the sky fitting procedure. The
horizontal bars show models with $D/B=0.5$.
The plot at the bottom right shows the scale lengths of the flattened
bulge (filled symbols) or of the inclined disk as a function of the
flattening angle (open symbols, $i=arccos(b/a)$) or of the inclination
angle, normalized to the $b/a=1$ or $i=0^\circ$ values. When $D/B$ is
low ($\le 0.5$) the errors are very small for {\it every} inclination angle.
For larger values of $D/B$, reliable
photometric parameters are obtained for $i<60^\circ$, but as soon as
the disk is nearly edge-on, total magnitudes and half-luminosity radii
are overestimated (by 0.1 mag and 20\% respectively). The integrated
circularized profiles, in fact, converge more slowly than the ones
following the isophotes. The correlated errors $\Delta FP$ always remain
very small. Similarly, the parameters of the two
components are reconstructed well for $i<60^\circ$, but badly underestimate
the disk when it is nearly edge-on. However, a decent
fit is obtained, by increasing the half-luminosity radius of the bulge
component (see Fig. \ref{figbadfit} (e) and (f)).
The sky correction is returned to better than
0.5\% for $i<80^\circ$. The systematic effects
connected to the flattening of the bulge are small for the range of
ellipticities considered here ($b/a\ge0.6$).
These results indicate two potential problems, (i) galaxies
may be misclassified due to the presence of an edge-on disk component
not being recognized, or (ii) the photometric parameters may be
systematically overestimated. However, these problems
do not apply to the EFAR sample, where $b/a>0.5$ always
and $b/a\ge 0.6$ for 96\% of the galaxies. Therefore, galaxies with bright
edge-on disks are only a very small fraction. Galaxies with faint edge-on
disks, which may not show large averaged flattenings, have low $D/B$
ratios and therefore are not affected by problem (ii). In a future
paper we will address the question whether in these cases the isophote
shape analysis might detect these faints disks and improve on point (i).
Finally, the two-dimensional frames described here have been used to calibrate
the estimator of the galaxy light contamination described in \S
\ref{diskbulge}. We measured the sky in the same way as for the real
frames of Paper III,
by considering some small areas around the simulated galaxies. We find that
the predicted galaxy light contamination overestimates the measured sky
excess by at least a factor two, and therefore can be used as a rather robust
upper limit to the galaxy light contamination.
\begin{figure}
\plotone{f13.eps}
\caption[f13.eps]{The reconstructed parameters of the bulge plus disk
models as a function of the inclination $i$ of the disk. Different symbols
indicate different flattenings of the bulge. The horizontal bars show
models with $D/B=0.5$. The plot at the bottom right
shows the scale lengths of the flattened bulge (open symbols) or of the
inclined disk (filled symbols)
as a function of the flattening angle ($i=arccos(b/a)$) or of the
inclination angle, normalized to the $b/a=1$ or $i=0^\circ$ values.
Good reconstructions of the parameters are obtained when the
inclination is less than $60^\circ$ (see \S \ref{decomposition}).}
\label{figbulgedisk}
\end{figure}
\subsection{$R^{1/n}$ luminosity profiles}
\label{profiles}
The tests described above show that our fitting algorithm is able to
reconstruct the parameters of a sum of an
$R^{1/4}$ plus an exponential law accurately. In these cases sky subtraction
errors can also be corrected efficiently. Even so, we do find in
Paper III that luminosity profiles of real early-type
galaxies show systematic differences from $R^{1/4}$ plus exponential
profiles, yielding to a median reduced $\chi^2$ of 6. Here we quantify
the systematic effects that would be produced in this case, by
studying the case of the $R^{1/n}$ profiles.
CCO fitted the luminosity profiles of 52 early-type
galaxies using the $R^{1/n}$ law introduced by Sersic (1968):
\begin{equation}
\label{r1n}
I(R)=I_e^n 10^{-b_n\left[\left(\frac{R}{R_e^n}\right )^{1/n}-1\right]},
\end{equation}
where $b_n\approx 0.868n-0.142$, $R_e^n$ is the half-luminosity
radius, and $I_e^n$ the surface brightness at $R_e^n$.
The total luminosity is $L_T=K_nI_e^n{R_e^n}^2$, where $\log
K_n\approx 0.03[\log(n)]^2+0.441\log(n) +1.079$. Eq. \ref{r1n} reduces
to Eq. \ref{bulge} for $n=4$ and to Eq. \ref{disk} for $n=1$.
For large values of $n$, Eq. \ref{r1n} describes a luminosity profile which
is very peaked near the center and has a very shallow
decline in the outer parts. Ciotti (1991) computes the curve of
growth related to Eq. \ref{r1n} analytically for integer values of
$n$ and finds that while already $\approx 13$\% of the total light is
included inside $R<0.05 R_e^n$, only 80\% of the total light is
included inside $6R_e^n$ for $n=10$.
We fitted Eq. \ref{r1n},
modified to have a core at $R<0.05 R_e^n$, to an $R^{1/4}$ plus
exponential model for
$n=0.5$ to $n=15$ out to $6R_e^n$. Fig. \ref{figr1nfit} shows the
results of the fit for a selection of models. With the
exception of the $n=0.5$ model, all of the $R^{1/n}$ profiles can be
described by a combination of an $R^{1/4}$ and an exponential
component, with residuals less than 0.2 mag arcsec$^{-2}$ for $R\le 4R_e$.
For $n<4$ the
residuals increase to 0.4 mag arcsec$^{-2}$ at $R>5R_e^n$, where the fits are
increasingly brighter than the $R^{1/n}$ profiles. For large
values of $n$ the residuals reach -0.4 mag arcsec$^{-2}$ at $R>5
R_e^n$, where the
fits are increasingly fainter than the $R^{1/n}$ profiles. The
relation between $n$ and the parameters of the decomposition is shown
in Fig. \ref{figr1npar}. Models with $1<n<4$ are fitted using a
decreasing amount of the exponential component, with a scale length
comparable to the one of the $R^{1/4}$ component. Models with $n>4$
are fitted with an increasing amount of the exponential component,
with increasingly large scale length. Half-luminosity radii are
progressively underestimated, being $\approx 60$\% of the true values at
$n=15$. Correspondingly, total magnitudes are also underestimated, by
0.25 magnitudes at $n=15$.
A possible problem can emerge for large values of $n$, if
the sky fitting algorithm is activated. The dotted curves in
Fig. \ref{figr1npar} show that if the sky subtraction algorithm is
activated (Eq. \ref{chidmax}), then larger systematic effects are
produced. Note that the computed sky correction (dotted curve of
Fig. \ref{figr1npar}) is $\approx 0$\ for $n \approx 1$ or $n\approx
4$ only. For
$n>4$ the correction is used to reduce the systematic negative
differences in the outer parts of the profiles. A comparison between
the fitted sky corrections and the upper limits on the possible galaxy
light contamination (see \S \ref{diskbulge} and
\ref{decomposition}) gives an important consistency check. In the
case shown in Fig. \ref{figr1npar} the fitted sky corrections are
twice as large as the upper limits on the galaxy light
contamination. In a real case this, together with the rather large values of
$\chi^2$, would hint at an uncertain fitted sky correction.
The fact that the $R^{1/n}$ sequence can be approximated by a
subsample of $R^{1/4}$ plus exponential models suggests a possible
reinterpretation of CCO's results: the variety of profile shapes of
early-type galaxies is caused by the presence of a disk component.
Moreover, the use of the $R^{1/n}$ profiles to determine the
photometric parameters of galaxies of large $n$ is dangerous, since
the extrapolation involved is large and the fitted
profiles barely reach 2 or 3$R_e^n$, as derived from the fit.
This problem is much smaller using the $R^{1/4}$ plus exponential
approach.
\begin{figure}
\plotone{f14.eps}
\caption[f14.eps]{The fits to the $R^{1/n}$ law. Two plots are drawn
for each value of $n$ (given in the top right corner). In the top plot the
crosses (one point in every seven) show the luminosity profiles
$\mu(R)=-2.5 \log I(R)$ of the
$R^{1/n}$ law as a function of $R/R_e$. The dotted and dashed curves
show the best-fitting $R^{1/4}$ and exponential laws respectively. In
the bottom plot the residuals (full curves) in mag arcsec$^{-2}$ from the fits
to the $R^{1/n}$ law are shown. The dashed curve shows the residuals
(in mag) from the curves of growth (see discussion in \S \ref{profiles}).}
\label{figr1nfit}
\end{figure}
\begin{figure}
\plotone{f15.eps}
\caption[f15.eps]{The relation between $n$ and the parameters of the
decomposition (see \S \ref{profiles}).
The full curves refer to the results obtained with no sky subtraction errors.
The dotted curves show the results obtained when the sky fitting algorithm
is activated.}
\label{figr1npar}
\end{figure}
\subsection{Discussion}
\label{discussion}
We conclude our tests by discussing the quality parameters defined in Table
\ref{tabquality} and their use to estimate the size of the systematic
errors present.
The definitions given in Table \ref{tabquality} have been derived
after inspection of Figures \ref{figresdb} to \ref{figr1npar}, with
the desired goal of identifying three classes of precision, $\Delta M_{TOT}\le
0.05$, $\Delta M_{TOT}\le 0.15$, $\Delta M_{TOT}>0.15$.
The parameters $Q_E$, $Q_{max}$,
$Q_{\chi^2}$, $Q_{S/N}$, and $Q_\Gamma$ are directly
related to the simulations. Their low values imply that
the fits involve a small extrapolation, extend to large enough radii,
give low surface brightness residuals with a large enough
signal-to-noise
ratio and good spatial resolution. The definitions of $Q_{\hbox{Sky}}$
and $Q_{\delta \hbox{Sky}}$ deal with the accuracy of the sky
subtraction, taking into
account that high surface brightness galaxies suffer less from this
problem, and that large sky corrections indicate a lower quality of
the data. Low values of $Q$ (see Eq. \ref{qtot}) imply low values of all
quality parameters.
Figure \ref{figqualdis} shows the cumulative distributions of the errors
$\Delta M_{TOT}$, $\Delta R_e$ and $\Delta FP$ derived from
all the performed disk plus bulge fits with sky correction algorithm
activated, as a function of the different quality parameters. The two
most important parameters regulating the precision of the photometric
parameters are the level of extrapolation and the goodness of the fit, followed
by the sky subtraction errors. A low $Q_E$ fixes the maximum possible
overestimate of the parameters. A low $Q_{\chi^2}$ with a low $Q_E$
constrains the underestimate and the reliability of the sky correction.
The ranges of the errors match the desired goal of identify three classes
of precisions.
Finally, it is sobering to note that the constraints
needed to achieve $Q=1$, high precision total magnitudes and $R_e$ are
rather stringent. Only 16\% of EFAR galaxies have $Q=1$.
Most of the existing {\it published} values of $M_{TOT}$ and $R_e$
of galaxies are far below this precision, because of the restricted
radial range probed by photoelectric measurements or small CCD chips,
because of sky subtraction errors, and also by the use of
the pure $R^{1/4}$ curve of growth fitting (see Figure \ref{figquatexp}).
The related observational problems
can be somewhat reduced with the use of large CCDs (see Introduction),
but the {\it a priori} limiting factor of galaxy
photometry, the extrapolation, will always remain with us at a certain level.
On the other hand, the errors on $M_{TOT}$ and $R_e$ are strongly
correlated, so that the quantity $\log R_e-0.3\langle SB_e \rangle$ is
always well determined. This fact allows the
accurate distance determinations achieved using the Fundamental Plane
correlations despite the systematic errors in the photometric quantities.
\begin{figure}
\plotone{f16.eps}
\caption[f16.eps]{The precision of the reconstructed total
magnitudes $M_{TOT}$,
the half-luminosity radii $R_e$ and the combined quantity $FP=\log R_e - 0.3
\langle SB_e\rangle $.
The cumulative distributions of the errors
$\Delta M_{TOT}$, $\Delta R_e$ and $\Delta FP$ derived from
all the performed disk plus bulge fits with sky correction algorithm
activated are shown as a function of the different quality parameters
defined in \S \ref{quality}. The full lines plot the distributions when
the parameters have value of 1, the dotted ones when the value is 2,
the dashed ones when the value is 3.
The distributions derived by selecting on the global
quality parameter $Q$ match the precision ranges identified in \S
\ref{discussion}.}
\label{figqualdis}
\end{figure}
\section{Conclusions}
\label{conclusions}
We constructed an algorithm to fit the circularized profiles of the
(early-type) galaxies of the EFAR project, using a sum of a
seeing-convolved $R^{1/4}$ and an exponential law. This choice allows
us to fit the large variety of profiles exhibited by the
EFAR galaxies homogeneously. The procedure provides
for an optimal combination of multiple profiles. A sky fitting option
has been developed. A conservative upper limit to the sky
contamination due to the light
of the outer parts of the galaxies is estimated.
From the tests described in previous sections we draw the following
conclusions:
1) The reconstruction algorithm applied to simulated $R^{1/4}$ plus
exponential profiles shows that random errors are negligible if the
total signal-to-noise ratio of the profiles exceeds $300$. Systematic
errors due to the radial extent of the profiles are minimal if
$R_{max}/R_e> 2$. Systematic errors due to sky subtraction are
significant (easily larger than 0.2 mag in the total magnitude)
when the sky surface brightness is of the order of the
average effective surface brightness of the galaxy. They can be
reliably corrected for as long as the fitted profiles show small
systematic deviations ($\chi^2<12.5$).
2) Strong systematic biases (errors larger than 0.2 mag in the total
magnitudes) are present when a simple $R^{1/4}$ or exponential
model is used to fit test profiles with disk to bulge ratios as low as 0.2.
3) The use of the shape of the (normalized) $\chi^2$ function badly
underestimates the (systematic) errors on the photometric parameters.
4) Systematic biases emerge when test profiles are derived for systems with
significant disk components seen nearly edge-on, or when the fitted luminosity
profile declines more slowly than an $R^{1/4}$ law. The parameters
of bulge plus disk systems can be determined to better than $\approx 20$\%
if the disk is not very inclined ($i<60^\circ$).
5) The sequence of $R^{1/n}$ profiles, recently used to fit the
profiles of elliptical galaxies by Caon et al. (1993), is equivalent to
a subset of $R^{1/4}$ and exponential profiles, with appropriate
scale lengths and disk-to-bulge ratios, with moderate systematic biases
for $n\le 8$ and residuals less than 0.2 mag arcsec$^{-2}$ for $R\le
4R_e$. This suggests that the variety
of luminosity profiles shown by early-type galaxies is due to the
frequent presence of a weak disk component.
6) A set of quality parameters has been defined to control the
precision of the estimated photometric parameters. They take into
account the amount of extrapolation involved to derive the total
magnitudes, the size of the sky correction, the average surface
brightness of the galaxy relative to the sky, the radial extent of
the profile, its signal-to-noise ratio, the seeing value and the
reduced $\chi^2$ of the fit. These are combined into a single quality
parameter $Q$ which correlates with the expected precision of the
fits. Errors in total magnitudes $M_{TOT}$ less than 0.05 mag and in
half-luminosity radii $R_e$ less than 10\% are expected if $Q=1$, and
less than 0.15 mag and 25\% if $Q=2$.
89\% of the EFAR galaxies have
fits with $Q=1$ or $Q=2$. The errors on the combined Fundamental Plane
quantity $FP=\log R_e -0.3\langle SB_e\rangle$, where $\langle SB_e
\rangle$ is the average effective surface brightness, are smaller than
0.03 even if $Q=3$. Thus systematic errors on $M_{TOT}$ and $R_e$
only marginally affect the distance estimates which involve $FP$.
\acknowledgments {RPS acknowledges the support by DFG grants SFB 318
and SFB 375. GW is grateful to the SERC and Wadham College for a
year's stay in Oxford, and to the Alexander von Humboldt-Stiftung for
making possible a visit to the Ruhr-Universit\"at in Bochum. MMC
acknowledges the support of a Lindemann Fellowship, a DIST Collaborative
Research Grant and an Australian Academy of Science/Royal Society
Exchange Program Fellowship. This work was partially supported by NSF
Grant AST90-16930 to DB, AST90-17048 and AST93-47714 to GW,
AST90-20864 to RKM, and NASA grant NAG5-2816 to EB. The entire
collaboration benefitted from NATO Collaborative Research Grant 900159
and from the hospitality and monetary support of Dartmouth College,
Oxford University, the University of Durham and Arizona State
University. Support was also received from PPARC visitors grants to
Oxford and Durham Universities and a PPARC rolling grant:
``Extragalactic Astronomy and Cosmology in Durham 1994-98''.}
| proofpile-arXiv_065-487 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{ Introduction}
The well-studied quasicrystals fall essentially into two families,
the Al-transition metal family exemplified by i(AlPdMn)
and the Frank-Kasper family \cite{FK1,FK2,shoe57,shoe87} exemplified by
i(AlCuLi). Decagonal quasicrystals are known experimentally only in the first
family, except for one claim \cite{chinese}.
Recently, three dimensional model systems have been found which,
in simulations, freeze into a quasicrystal phase with
Frank-Kasper local structure:
a monatomic system with a special potential
freezes into a dodecagonal quasicrystal \cite{dzug}, and
a system of large (L) and small (S) atoms with Lennard-Jones
interactions freezes into an icosahedral phase \cite{rotcool},
in fact a simplified version of the Henley-Elser structure model for
i(AlCuLi) and i(AlZnMg) \cite{henels}.
Since Frank-Kasper systems are dominated by $sp$ conduction electrons,
it is believed that simple pair potentials (as used in these
simulations)
can offer a good approximation to the structure energy \cite{haf}.
The L and S atoms used by Roth et al.\ \citeyear{rotcool} had single-well
potentials where the ideal distances
are non-additive $(r_{LL} \approx r_{LS} \approx 1.15 r_{SS})$.
This paper reports on a new {\it decagonal} Frank-Kasper phase
which froze from the liquid in molecular dynamics simulations
of a model system using practically the same potentials as
Roth et al.\ \citeyear{rotcool} (see Sec.~\ref{simulations}).
No realization of this structure in nature has been established
(except conceivably in d(FeNb), as discussed in Sec.~\ref{comparison}).
However, in the absence of a {\it realistic} model system, a toy model
system with an equilibrium quasicrystal phase can be quite useful.
(To date, microscopically derived potentials for particular alloy systems have
not been used in simulations of freezing from the melt \cite{mihII}, but
only for comparisons of trial structures at zero temperature, in
the Al-transition metal case (Phillips, Deng, Carlsson and Murray, 1991; Zou
and Carlsson, 1994; Widom, Phillips, Zou and Carlsson, 1995; Widom and
Phillips, 1993; Phillips and Widom, 1994; Mihalcovi{\v c}, Zhu, Henley and
Oxborrow, 1996; Mihalcovi{\v c} et al.\ 1996b)
\nocite {carlsson1,carlsson2,carlsson3,wid92,phil,mihI,mihII}).
Of course, there is some fundamental interest in the existence
of any microscopic model of interacting atoms
which has a quasicrystal equilibrium state.
Furthermore, within such a model
one can study the phonon/phason coupling,
the energy changes and barriers corresponding to tile flips,
the phason elastic constants with their temperature dependence,
and can derive from microscopics
an effective Hamiltonian in terms of tile-tile interactions (Mihalcovi{\v c} et
al.\ 1996a).
In two dimensions, the ``binary tiling'' case \cite{lan86,wid87}
has served this purpose,
but it has quite unrealistic bond radii, in contrast to
our model potentials
(which were in fact tailored to favor the i(AlCuLi) structure \cite{rotstab}).
The bulk of our paper
presents an idealized decoration model which was constructed by
abstracting the patterns observed in our simulations.
The model has several variants, corresponding to closely related
atomic structures, yet described by different tilings
(See Sec.~\ref{dsdm}).
Thus it exemplifies a system where an effective tile-tile Hamiltonian
may select a less disordered subensemble of a random tiling ensemble
\cite{jeong}.
We have constructed the acceptance domains of a quasiperiodic
version of our model (Sec.~\ref{environments}).
Finally, by applying information from the ideal model,
we adjusted the potentials
so as to obtain a better-ordered structure after quenching (Sec.~\ref{XXX}).
In the concluding section,
we have discussed how our model is related to others, and how
realistically it extends beyond our small system size.
\section { Initial Simulations} \label{simulations}
Our decagonal phase was first seen in a slow-cooling simulation of
a binary liquid. (Further details about this simulation and other structures
observed, as well as the motivation for our initial choice of parameters for
the potential, can be found (Roth et al.\ 1995). \nocite{rotcool}
The interaction is described by Lennard-Jones potentials
\begin{equation}
v_{\alpha \beta}(r) = 4\epsilon_{\alpha \beta}
\left(\left(\frac{\sigma_{\alpha \beta}}{r}\right)^{12} -
\left(\frac{\sigma_{\alpha \beta}}{r}\right)^{6}\right)
\label{eq-LJ}
\end{equation}
The bond parameters $\sigma_{\alpha \beta}$ are $\sigma_{SS} = 1.05 $,
$\sigma_{SL} = 1.23 $, and $\sigma_{LL} = 1.21 $. These and all other
simulation parameters are given in reduced or Lennard-Jones units, and are
indicated by a star. The corresponding bond lengths are $2^{1/6}
\sigma_{\alpha\beta}$.
In overall outline, these distances are appropriate for a
typical tetrahedrally-close-packed structure,
in which the small atoms have icosahedral coordination
shells and the large atoms have Frank-Kasper coordination
shells with larger coordination number.
(Frank-Kasper structures are by definition
``tetrahedrally close-packed (tcp)'',
meaning that neighboring atoms always form tetrahedra
(Samson, 1968; Samson, 1969; Shoemaker et al.\ 1957; Frank and Kasper, 1958;
Frank and Kasper, 1959)\nocite {sam1,sam2,shoe57,FK1,FK2}.
This is not strictly true in our model -- or that of Henley and Elser
(1986) -- since there is a small number of non-tcp environments.)
The coupling constants were set to $\epsilon_{SS} = \epsilon_{LL} =
0.656$ for same species and to $\epsilon_{SL} = 1.312$ for different
species, in order to prevent phase separations into monatomic domains.
To save computation time we cut off the potential
smoothly at $r_{c} = 2.5 \sigma_{SL}$.
For the simulation we have modified the standard molecular dynamics method
to allow us to control pressure and temperature as described
by Evans \& Morriss (1983). \nocite{eva}
A constant cooling rate may be introduced by the method described by
Lan\c{c}on \& Billard (1990). \nocite{lanbil}
The equations of motion are integrated in a fourth-order
Gear-predictor-corrector-algorithm (see for example Allen \& Tildesley
(1987)).
The time increment $\delta t^{*}$ is
adjusted after testing for numerical stability. We
find that $\delta t^{*} = 0.00462$ is an appropriate value.
For simplicity the masses of small and large atoms are set to unity.
Our simulation box is a cube containing 128 small (S) and 40 large (L)
atoms. The initial positions are set with a random number generator,
and then relaxed. The generated liquid is equilibrated at an initial
temperature of about $T^{*} = 1.0$. At high cooling rates we observe
a transition form a supercooled liquid to a glass at about $T^* = 0.5$.
If the cooling rate is lowered to $T^{*}/t^{*} = 1.74\times 10^{-5}$
the supercooled liquid transforms sharply to an ordered solid at about $T^* =
0.6$, which analysis reveals is a quasicrystal (sometimes icosahedral and
sometimes decagonal). The solid may still contain defects,
but the order can be improved by annealing. The annealed structures are
quenched down to $T^* = 0$ for structure analysis. Subsequent annealing runs
with $5 \times 10^{5}$ steps at constant temperatures of $T^* = 0.5 $ and 0.4
show that
the nucleated structures are stable in the sense that they do not change
except for defect annealing. More details about the cooling simulations can be
found by Roth et al.\ \citeyear{rotcool}.
The new decagonal structure was observed at cooling rates of $T^{*}/t^{*} =
1.74\times 10^{-6}$ and $1.74\times 10^{-5}$.
The tenfold axis is aligned with the $\langle 1 1 0 \rangle$
diagonal of the cubic simulation cell; the structure is stacked periodically,
with a stacking vector parallel to this direction.
Fig.~\ref{fig-proj} shows a projection of the simulation box onto a
plane normal to this axis; the atom positions are highly ordered and
the approximant tenfold symmetry within this plane is obvious.
Because of the constraint of fitting into the fixed, periodic simulation cell,
the structure is distorted somewhat (in that the $x/y$ ratio within the
layers is squeezed relative to decagonal symmetry).
By examining slices transverse to the decagonal axis (not shown),
we could distinguish three different types of
layers transverse to the tenfold axis, which we label ``A'',
``B'', and ``X''. Let $c$ be the lattice constant in the stacking
direction; the simulation box extends over two periods ($2c$) before
encountering the same atomic layer.
Most of the atoms in our structure are in A and B layers,
which are separated by $\sim 0.8$ units.
The A layer (at height $0$) consists mostly of pentagons linked by
corners; the B layer (at $c/2$) also consists of pentagons, but these are
bigger and share edges instead.
The basic motifs in the A/B layers are therefore pentagonal antiprisms.
Both the A and B layers contain a mix of L and S atoms.
Between the A and B layers at $c/4$ and $3c/4$ are the much sparser X and $\rm
X'$ layers.
These contain only S atoms centering the pentagons in the A/B layers,
thus forming chains of interpenetrating icosahedra along the $c$ direction.
The X and $\rm X'$ layers are identical in projection on the
quasiperiodic plane.
The complete stacking sequence is $\rm AXBX'$.
In order to discuss the structure in greater detail, it is convenient
to present an idealized model; we shall do this in the
next section.
The centers of the columns of icosahedra are shown linked by
added lines in Fig.~\ref{fig-proj}.
Observe that the structures we found in simulations are not strictly
periodic in the stacking direction:
the axis centering a column of icosahedra may jump in position
from one layer to the next.
We also observed icosahedral structures
under the same simulation conditions.
Consequently, both phases might coexist in single (larger) sample.
Such a coexistence is well-known
experimentally in the Al-transition metal structure family,
most commonly among metastable, rapidly-quenched quasicrystals
but also for stable phase i(AlPdMn) and d(AlPdMn)
\cite{alpdmn}.
The simultaneous stability (or near stability)
of both structures suggests that the local order is very similar.
Indeed, we shall show that our decagonal structure model is
quite closely related to the well-known `Henley-Elser'
\cite{henels} model.
\section { Decoration Models}
\label{dsdm}
The most convenient framework for making sense of our structures,
without any bias about the kind of long-range order,
is a decoration of a tiling with atoms.
The same decoration scheme, with a finite number of site types and
positional parameters,
can model a random structure as a decoration
of a random tiling, or
an ideal quasiperiodic structure as
a decoration of a quasiperiodic tiling.
However, in attempting to describe an idealized structure model,
we suffer from an embarrassment of riches.
That is, there are several related structure models describing
this kind of order.
The models differ by (i) breaking a local symmetry in the position
of an atom (which introduces matching-rule-like arrows)
(ii) by allowing additional kinds of tiles
(iii) by constraining the packing of tiles (this
may create larger tiles as composites of smaller ones).
Structure models are conveniently presented using idealized positions,
represented as integer combinations of some finite set of basis vectors,
but obviously the real atoms undergo displacements from these positions.
Two structural models are called
``topologically equivalent'' (Mihalcovi{\v c} et al.\ 1996a) \nocite{mihI}
when they converge to the same
structure after equilibration at low temperature
(or at zero temperature by relaxation under the pair potentials.
Although apparently different, the models are physically
indistinguishable.
Thus the concept of topological equivalence is crucial in
comparing different structure models and in organizing the
above-mentioned family of related structure models.
We will begin (Sec. \ref{basictiles})
with a simple simple tiling of three tiles --
the fat and skinny Penrose rhombus plus distorted hexagon ``Q'' --
from which the other tilings can be derived.
This is not itself a good tiling model, because the packing is
poor in the skinny rhombus, but it functions as a ``zero-order''
approximation for constructing better models.
Therefore, we go on to consider two different modifications
which correct this:
(i) the triangle-rectangle tiling (Sec.~\ref{trirect}).
and (ii) the ``two-level tiling'' (Sec. \ref{twolevel})
The relation of each modified model to the Penrose tiling decoration
is an example of a
``differentiation'' in the terminology of Mihalkovi\v{c} et al.\
(1996b); \nocite{mihII}
this means that a single environment in one model corresponds to
several subclasses in the second model, which have somewhat different
atomic decorations.
\subsection{ Basic Tiles}\label{basictiles}\label{basic}
Next we present the simplest possible version of the model,
phrased as a decoration of the Penrose rhombi without matching rules.
\subsubsection{ Penrose Rhombi}\label{penrose}
Consider the decoration of the fat and skinny Penrose rhombi,
as shown in Fig.~\ref{fig-basic}(a,b).
This structure is not the best packed, but it is certainly
one of the simplest.
Each large atom is at a position
dividing the long diagonal of a fat rhombus in the ratio
$\tau^{-1}:\tau^{-2}$ where $\tau$ is the golden mean $(1+\sqrt 5)/2$.
Notice that there is a complete symmetry
between the A and B layers. Furthermore, the decorating
atoms sit on the midpoint of each tile edge, so the decoration
does not enforce the Penrose matching rules: it may be applied to
any random tiling of these rhombi.
The centering atoms of the pentagons are called $\alpha$. They are located in
the X layer at the corners of the rhombi.
In the A layer,
the mid-edge atoms are called $\beta$
and the rhombus interior atoms $\lambda$.
In the B layer,
the equivalent mid-edge atoms are called $\gamma$
and the rhombus interior atoms are $\mu$.
There are two alternative criteria for assigning the species: either
according to interatomic distances in the model, or according to
coordination number \cite{henels};
the two criteria are obviously correlated and (for all
models presented in this paper) they lead to the same assignments.
We assign $\beta/\gamma$ mid-edge sites to be S atom, since
they have coordination 12 and tend to be squeezed by the
$\alpha$ atoms at either end of the edge.
The $\alpha$ sites are also assigned S atoms, since
they have coordination 12 and are squeezed along the $z$ axis
where they form chains.
(The interlayer spacing is less than an interatomic spacing, since
the other atoms don't form vertical chains.)
Finally, the $\lambda/\mu$ sites, which all gave coordination 16,
are taken to be L atoms.
\subsubsection{ Q Tile }
In the Penrose decoration, described so far,
the A and B layers are equivalent by a $10_5$ screw symmetry,
and correspondingly $\beta$ is equivalent to $\gamma$ and
$\lambda$ is equivalent to $\mu$.
However, in the decorations we present later, the A and B layers will be
inequivalent. In those more complicated structures, a single Greek
label refers to several subclasses of atoms having similar
but not identical local environments. (Usually their coordination
shells are topologically identical, but there is some variation in
which neighbors are L and S atoms.)
The Penrose tiling structure forces a ``short'' linkage between two vertices
when they are related by the short diagonal of the skinny rhombus.
Here ``linkage'' denotes a connection between the centers of
neighboring motifs, which form the vertices of a rigid network.
(In this case, the motif is the chain of centered pentagons,
and the network is the tiling.)
The $\alpha$ atoms projecting to those two vertices
have {\it non}-icosahedral coordination shells.
(There are similar to the awkward coordination shells occurring
in the oblate rhombohedron \cite{henels}).
We believe these are unfavorable energetically.
(Indeed, nothing resembling a skinny rhombus is found in Fig.~1.)
Therefore, we shall make an additional hypothesis:
{\it true chains of icosahedra are energetically favorable and
so their frequency should be maximized}.
To increase the number of chains of icosahedra, we must somehow
eliminate the short linkages.
This is possible whenever two skinny tiles and a fat tile
form a distorted hexagon; they
can be joined in a composite tile Q
as shown in Fig.~\ref{fig-basic}(c).
The only change in atomic positions in forming the Q tile is
along the axis over the central vertex, where the
the $\alpha$ atoms forming the chain
are replaced by $\lambda'$ atoms
(in A layers, if the adjacent atoms along the tile edges are in a B layer,
and otherwise vice versa).
Note there are half as many new
atoms $\lambda'$ as there were of old $\alpha$ atoms;
they should be L atoms because of this larger spacing
(or alternatively, because they have coordination 15).
We will call this operation a ``chain-shift''.
Note that the Q tile is considered
to have a symmetry between the top apex and bottom apex
(the light dashed lines in Fig.~\ref{fig-basic}(c)
only illustrate the topological equivalence to the
apparently asymmetric Penrose tile decoration.)
The Q decoration should have
the full symmetry of the Q tile's exterior;
after the chain-shift, this require only small adjustments of
position and species.
The $\lambda$ site from the fat tile is renamed $\lambda'$
since it has coordination 15 after the chain-shift and is symmetry-equivalent
to the $\lambda'$ site from the internal vertex.
The non-Frank-Kasper $\gamma$ atom on the edge shared by skinny rhombi
is converted into a $\mu$ L atom (upper part of Fig.~\ref{fig-basic}(c))
just like the $\mu$ site from the fat rhombus.
Finally, the other two non-Frank-Kasper $\gamma$ sites on internal edges
are renamed ``$\delta$''
and must be shifted slightly so as to lie
symmetrically on the bisector of the long Q diagonal
(since their neighbors are symmetric around that bisector.)
They divide the bisector line almost into thirds
(we used $0.312:0.376:0.312$ in the initial configuration
for simulations in Sec.~\ref{secopt}).
Since the $\delta$ atoms have coordination 14, and are somewhat
squeezed with their neighbors along the bisector line,
they have been designated S atoms.
(However, note that
coordination $14$ was found to be
borderline in the closely related icosahedral structure:
analogous sites occurring on the short axis of the
rhombic dodecahedron tile (see below)
are occupied by small atoms in i(AlCuLi) but by large atoms in
i(AlMgZn)\cite{henels}).
To maximize the number of proper centered-pentagon chains,
then, the original tiling should be arranged
with skinny rhombi grouped in pairs wherever possible.
This is essentially the same mathematical problem
as that of packing equal disks on Penrose tile vertices
\cite{hen86}; those the disks correspond to the
pentagon-chain motif in the present model.
\subsubsection{ Icosahedral Tiling Relationship} \label{icorel}
It is interesting to note the close relationship of this tiling
to the three-dimensional ``Henley-Elser'' decoration of the
Ammann rhombohedra by L and S atoms as a model of Frank-Kasper
icosahedral quasicrystals such as $i$(AlCuLi) and $i$(AlZnMg).
Let us orient rhombohedra so that one edge lies in the vertical direction
as shown in Fig.~\ref{fig-PROR}, but shear them slightly so that
the vertical edge has length $c$ and the other five kinds of
edge have vertical component $c/2$ and horizontal components
of length $a$ along one of the five basis vectors.
Projecting the prolate and oblate rhombohedra onto the plane
perpendicular to the selected axis generates respectively the
fat and skinny rhombus of the (2D) Penrose tiling.
Thus, every tiling of the rhombi can be extended
vertically as a stacking of rhombohedra with period one edge length
along the special axis.
The layer of this stacking is a puckered slab dual to a grid-plane,
analogous to one of the ``tracks''
in a two-dimensional Penrose tiling \cite{dunkat}.
Note that, in contrast to a generic rhombohedral tiling,
{\it every} rhombohedron in this stacking has edges
in the vertical direction.
In fact, it turns out that our decagonal decoration of Penrose tiles
is ``topologically equivalent'' to the icosahedral decoration
of rhombohedra by Henley \& Elser (1986) \nocite{henels}
\footnote {A similar, but quite imperfect,
relationship can be made between the
icosahedral $i$(AlMn) (Elser \& Henley 1985; Mihalcovi{\v c} 1996b)
\nocite{elshen,mihII}
and the decagonal model based on the Al$_{13}$Fe$_4$ structure
\cite{hendec}}.
To see this, cut the rhombohedra along horizontal planes such
as those shown dashed in Fig. 2(a,b)
-- all the atoms in the PR and OR lie in or near these
rather densely packed layers.
Then reassemble them into prisms with horizontal rhombic faces.
The atom positions form a decoration of the prisms,
shown in Fig.~\ref{fig-3D}(c,d),
which is clearly equivalent to that in Fig.~\ref{fig-basic}(a,b).
(The ideal positions by Henley \& Elser (1986), \nocite{henels}
projected onto the plane,
are {\it exactly} those used here in the tiling;
the vertical positions of the $\lambda$ atoms, however,
must be adjusted by $\sim 0.026 c$.)
The third tile of the Henley-Elser model, the
rhombic dodecahedron (RD), is a little bit more complicated.
(see Fig.~\ref{fig-RD}). Before it can be stacked,
two skew triangular prisms at opposite sides of the RD, forming together one
PR, must be removed.
If this truncated RD is cut into layers and reassembled,
it produces an elongated hexagonal prism which
is decorated exactly as the Q tile in Fig.~\ref{fig-basic}(c).
The structure obtained by transforming the Henley-Elser structure
not only has the atoms in exactly the same positions (after
the small vertical shifts) as our decagonal model,
but also has exactly the same assignment of S and L atoms.
That is not very surprising, since the species were assigned
according to the local environments in both models.
\subsection{ Triangle-Rectangle Tiling}
\label{trirect}
Returning to the 2D tiling,
there are now two possible schemes to handling the
skinny rhombi everywhere in the tiling.
In one scheme, we imagine first that {\it all} skinny tiles are
paired and absorbed into Q tiles: a generic tiling of this sort is
the ``fR/Q (fat rhombus and Q) tiling''.
Then the Q tile can be divided into
a rectangle (R) and two isosceles triangles (T), with new
edges of length $b\equiv (1+\tau^{-2})^{1/2} a$,
while each rhombus
can be divided into two isosceles triangles of the same kind.
The two objects can form random tilings of the plane,
most of which cannot be regrouped into rhombi \cite{ox}.
The decoration introduced in this paper is compatible with an
arbitrary triangle-rectangle (T/R) tiling.
Note that, in this variant of the structure model,
each tile comes in an A and a B flavor,
denoting the level at which its mid-edge atoms sit;
tiles adjoining each other by a $b$ edge have opposite flavors.
The triangle and rectangle have long been known as the building blocks for the
so-called pentagonal Frank-Kasper phases (see for example
Samson (1968); Samson (1969)). \nocite{sam1,sam2}
The 12-fold symmetric quasicrystals, which are typical Frank-Kasper phases,
are built from a triangle and square
on which the atomic arrangements are similar, but not identical,
to those in the T and R tiles in our model.
Furthermore, a T/R description like ours was used by
He et al.\ \citeyear{chinese}
to describe
the $\rm Fe_{52}Nb_{48}$ quasicrystal, the only real
material claimed to be a Frank-Kasper decagonal.
(See Sec.~\ref{comparison};
Nissen \& Beeli (1993)
\nocite{beeli}
also used a rhombus-triangle-rectangle tiling
to analyze TEM images of a Fe-Nb decagonal quasicrystal.)
However, the decagonal T/R structure has been systematically
studied as an abstract (random) tiling only recently
\cite {ox,cock}.
We shall now consider energy energy differences between different
T/R tilings which might cause additional ordering.
It would seem unfavorable energetically for two rectangles to
adjoin along an $a$ edge,
since the $\delta$ and $\beta/\gamma$ atoms along the bisecting axis
are tight within each tile and would be squeezed against each other;
let us add this to the constraints defining our tiling ensemble.
Then Cockayne's quasiperiodic T/R packing\cite{cock}
satisfies the constraint just mentioned; it satisfied a
second constraint, in that rectangles
don't adjoin along $b$ edges, either.
With the latter constraint, every rectangle must
be capped with triangles forming a complete Q tile, and then all other
triangles must be paired along their $b$ edges forming complete fat rhombi:
thus the structure is more economically described as an
fR/Q tiling with the constraint that Q's may not lie side-to-side.
(We presume the ensemble of such packings is a random tiling, but this
is not proven.)
Every fR/Q structure, of course, can be considered as a fat and skinny (fR/sR)
rhombus tiling produced by decomposing the Q tile.
It has been pointed out \cite{mihox,ox}
that an fR/Q tiling gives the
optimal disk packing
(i.e. maximal density of $\alpha$-chains, in our models)
of all Penrose tiling, since
{\it all} the skinny rhombi are grouped into pairs, unlike the two-level tiling
described below.
Possibly the additional constraint on the T/R tiling
is justified by the energetic cost of adjoining along the $b$ edge --
the $\lambda'$ atoms inside one R press against
the $\delta$ atoms (in the same layer) inside the other R --
but we have not identified any compelling argument and hence
do not propose the fR/Q tiling as the optimal geometry.
\subsection{Two-Level Tiling} \label{twolevel}
The decagonal structure observed after cooling from the
melt (Sec.~I) had the additional property
that its A layer is inequivalent to its B layer.
We shall now turn to a structure model which adopts this
as a postulate.
Chain-shifts occur only at vertices where all the edges are decorated with
B layer atoms. Naturally, the B atoms in turn shift away from the midpoint
of the edge towards the partially vacated columns. (This was observed
in the simulation, in the fact that the B layer pentagons were bigger
than A layer pentagons.)
This has the effect of the double-arrow matching rule in the Penrose tiling.
An arbitrary tiling obeying only the double-arrow matching rules
(Tang and Jari\'c, 1990) is equivalent to a random tiling
of three composite supertiles Q, K, and S\footnote{
The three tiles are called ``Q, K, S'' after the corresponding
three local environments found in the Penrose tiling where double-arrow
edges can meet. The same tiles are called ``h,c,s'' respectively
by Li \& Kuo (1991).
\nocite{li2}
}
which are obtained by merging Penrose tiles along the double-arrow edges.
Since the atoms decorating the outer edges of the Q, K, and S tiles
remain symmetric about the middle of the edge,
they do {\it not} enforce the (single-arrow) Penrose matching rules
on these edges: all random tilings of the plane by Q, K, S tiles
are a priori permissible.
The random tiling of these tiles has been called the
``two-level'' tiling (2LT)\cite{henART}; these tiles were independently
proposed by Li \& Kuo (1991). \nocite{li2}
Fig.~\ref{fig-2LTdec} shows a fragment of the 2LT, including one of each tile.
As a special case, the Penrose tiling,
modified by combining rhombi along their double-arrow edges,
becomes a simple quasiperiodic Q, K, S tiling.
The ``Q'' tile decoration was described already;
the star tile S is also straightforward, being a composite of five fat rhombi.
Note that
no chain-shift is necessary at the center of the S tile
since there is no skinny tile adjoining;
thus this site forms an exception, around which the A/B symmetry breaking
(between large and small pentagons) is opposite to the pattern around all other
$\alpha$-chains.
Otherwise, all the edge sharing pentagons are in layer A and the
corner sharing pentagons are in layer B.
The ``K'' tile is a grouping of a skinny rhombus
with three fat rhombi; there is a chain-shift on the central vertex, which
becomes known as a $\kappa$ site.
This is the second option for absorbing the extra
single skinny rhombi, instead of pairing all of them in Q tiles.
However, the atomic arrangements in the K are still somewhat awkward.
The four $\gamma$ atoms surrounding the internal vertex
can only relax to form a sort of pentagon with a missing corner,
meaning that the $\kappa$ coordination shell
is not a Frank-Kasper polyhedron but a distorted octahedron.
We could have arrived directly at the two-level tiling
from analysis of Fig.~\ref{fig-proj}, if we
accepted as a postulate that in the linkage between
two chain-motifs, the edge-sharing pentagons are always in the B layer.
We quickly arrive at a geometry where the
angles between linkages are all multiples of $2\pi/5$.
The smallest tiles enclosed by such edges are the Q, K and S
tiles of the ``two-level'' tiling (Fig. \ref{fig-2LTdec}).
The details of this derivation are in Appendix A.
Reviewing Sec.~\ref{dsdm},
we can identify a hierarchy of successively more ordered tiling geometries.
The Penrose tiling is the most ordered;
it can be broken up into
the ``two-level'' tiling of Q, K, S tiles.
The K and S tiles can be broken up further into fat and skinny rhombi;
finally the fat rhombi can be broken into triangles, and the
Q tiles broken into triangles and rectangles.
\section{ Quasiperiodic Models and Stoichiometry}
\label{environments}
Even when a quasicrystal model has been formulated as a tile decoration,
it is often illuminating to represent it
as a quasiperiodic structure obtained by a cut
through a higher-dimensional space,
since
(i) the space-group and diffraction properties are most concretely
understood in this fashion
(ii) different site types are represented as domains of the
acceptance domains in the ``perp'' space, thus
the relations between different site types are
visible in the spatial relation of the domains.
(iii) The number density (or frequency) of a particular species or site type
is proportional to the area of its portion of the acceptance domain,
which is convenient for computing a definite stoichiometry.
This description is helpful even for atomic models based on
random tilings. These, when ensemble-averaged, produce structures of
partially-occupied sites. Mapped into perp space, such
distributions can usually be {\it approximated} as
quasiperiodic acceptance domains convolved with a Gaussian
in perp space\cite{henART}.
The first step is to find a quasiperiodic tiling.
This is trivial for the two-level (Q/K/S) tiling, but highly nontrivial
for the T/R (or for the fR/Q) tiling; in that
case, the tiling can only be constructed by an elaborate deflation,
and the corresponding acceptance domains have fractal boundaries
\cite{cock}.
Therefore, we will limit ourselves
to presenting the acceptance domains based on the Q/K/S-tiling decoration,
where the tiles are
arranged in the perfect Penrose tiling.
(A Penrose tiling is turned into a two-level (Q/K/S) tiling simply by
erasing the double-arrow edges).
Also in this section, we tabulate the contents in atoms of each tile.
This approach to stoichiometry is a more powerful than the approach using
the acceptance domain, since it can be applied even to random tilings.\\[1ex]
\subsubsection{ Idealized Structure and Acceptance Domain}
In order to construct the acceptance domain,
we first make a crude idealization
of the Penrose-rhombus decoration which we presented first
(Fig.~\ref{fig-basic}(a,b)).
All $\beta$ atoms lie on single-arrow edges
and all $\gamma$ atoms lie on double-arrow edges.
Let us shift the $\beta$ atoms and the
$\gamma$ atoms in the direction of Penrose's arrows,
such that each divides its edge
in the ratio $\tau^{-1}:\tau^{-2}$.
(This ratio is technically convenient in allowing
the ideal coordinates to be written as integer combinations
of the five basis vectors.
They are actually most simply treated using a
deflated Penrose tiling basis with lattice constant $\tau^{-1} a$.)
In going to the Q/K/S tiles for in the quasiperiodic structure,
we retain these same positions but change the label and
the species as described in Sec.~\ref{twolevel}; note in particular
that the displacement properly places the $\delta$ atoms on the
bisecting symmetry axis of the Q tile.
The B layer forms Penrose's packing of regular pentagons, one
centered on every vertex of the 2LT.
Each ``internal'' vertex (where double-arrows meet) is also surrounded
by a pentagon, but these are irregular and have many short distances.
The acceptance domains are shown in Fig.~\ref{fig-acc}.
Note there is a separate domain for layers A, B, and X of the real structure;
in addition, as always, the vertices project into five flat two-dimensional
layers in three-dimensional perp-space, so we must show separate
shapes for each perp-space layer.
Finally we have shaded the domain with dark and light shading to distinguish
L and S atom occupancy.
Although this structure has unreasonable distances, it is
``topologically equivalent'' (Mihalcovi{\v c} et al.\ 1996a)\nocite{mihI} to
the
(reasonable) version presented in figure \ref{qksdecl}.
We could alternatively compute the acceptance domains for the
decoration in figure \ref{qksdecl},
with the atoms moved to mid-edge positions (except for those in the Q
tile).
Relative to Fig.~\ref{fig-acc},
the $\beta$, $\gamma$, and $\delta$ subdomains
each would get broken up into five pieces,
and shifted in the parallel space direction.
Since the mid-edge atom positions are also
special crystallographic positions, several of the shifted subdomains
will reunite there and form a new piece of acceptance domain
which is centered on a mid-edge site in the 5-dimensional hypercubic lattice.
We forgo to present these domains since this second
picture is less compact and not as appealing as the first one.
The space group of this quasiperiodic decoration is $P10 /m m m$. In other
variant decorations in which the
A/B symmetry is not broken, we would
get an additional system of mirror planes perpendicular to
the 10-fold axis and this changes the space group to $ P10_{5} /m m c$.
The point
group of the motif with the highest symmetry is $5 /m m$
\cite{hendec}.
\subsubsection{ Stoichiometry}
The most general approach to stoichiometry is to count the number of atoms
associated with each tile.
Table I summarizes the contents of each tile.
The combination Q+S has contents 29L+14S while 2K (which covers
the same area as Q+S) has contents 28L+14S.
This suggests that the K tile is too loosely packed;
indeed, the average coordination number in K is also a bit low.
Table II gives the coordination numbers and the net stoichiometry for each
decoration.
In the case of the 2LT (Q/K/S) decoration, this agrees with
the stoichiometry which can be read off from Fig.~\ref{fig-acc}.
It can be seen that the stoichiometry is close to $L S_2$ in both cases.
(The pure Penrose tiling decoration, an unrealistic model,
has $L_{0.198} S_{0.802}$.)
That seems to be considerably fewer L atoms than in the T(AlZnMg)
icosahedral approximant \cite{berg}
or Samson's AlMg-type structures, but it is comparable
to i(Al$_6$CuMg$_3$) in which
only $30\%$ of the atoms are L (Mg) \cite{sam1,sam2}.
\section{ Further Simulations}
\label{XXX}
In section III, we worked out the geometric properties of the idealized model,
in several closely related variants based on different tiling rules.
We derived the coordination configuration for all sites:
all atoms have a reasonable distance from their
neighbours and most of them are tetrahedrally close packed.
The idealized model, in any of the variants presented in
the last two sections, is well-packed
and (nearly) tetrahedrally close-packed around every atom.
Hence it is expected to be (locally) stable against disordering
by thermal fluctuations.
To check this stability, we ran additional simulations
starting with a configuration from the ideal model.
A major motivation of these simulations was to locate the
region of parameter space in which the quasicrystal phase may be
thermodynamically stable; hence, we attempted to adapt the
potentials to the structure model, as explained in Subsec.~\ref{secopt}.
As a result of the optimization there
we can predict the ideal c/a ratio
and the density of the structure, as well as
the ratios of bond radii for LL, LS and SS pairs which seem
most conducive for (decagonal) quasicrystal-forming.
We used not only Lennard-Jones (LJ) potentials as in eq.~(\ref{eq-LJ})
but also the Dzugutov potential, with independent parameters for
the three ways of pairing species $\alpha$ and $\beta$.
Both pair potentials have a single attractive well with a minimum at radius
$2^{1/6} \sigma_{\alpha\beta}$,
and having depth $\epsilon_{\alpha\beta}$;
here $\sigma_{\alpha\beta}$ and $\epsilon_{\alpha\beta}$
are called the ``bond parameter'' and ``interaction parameter''.
The Dzugutov potential \cite{dzug}
has an additional repulsive maximum at a radius $\sim 1.6$
times that of the minimum,
and a height $\sim 0.5 \epsilon _{\alpha\beta}$;
this is designed to disfavor the square arrangements found in
fcc, bcc or hcp structures.
\subsection{ Optimization of Parameters}
\label{secopt}
To find the optimal potential parameters, we minimized the potential energy
at $T=0$ while varying these parameters. For this purpose we used a fixed
ideal structure model with the ideal atom position and without relaxation.
The sample size was a $p/q~=~3/2$ approximant\footnote{
the basis vectors are $(2 q, 0), (p-q, p), (-p, q), (-p, -q), (p-q, -p)$, and
the periods are\\ $2(2 p-q) \tau+2(3 q-p)$ in $x$-direction and $4
\sin(\pi/10)(p \tau+q)$ in $y$-direction.} of a perfect Penrose tiling
with 10 periods of XAXB layers and ideal size $l_x \cdot l_y \cdot l_z = 18.95
\cdot 16.09 \cdot 22.93$, containing 3270 small and 1420 large atoms. Samples
of different size do not yield different results within the accuracy we
achieve.
In a three-dimensional structure with two species of atoms L and S,
bond and interaction parameters
$\sigma_{\alpha\beta}$ and $\epsilon_{\alpha\beta}$ must each be determined
for pairs $\alpha\beta$ equal to $LL$, $LS$, and $SS$.
Of course, one of the $\epsilon_{\alpha\beta}$'s
can be eliminated in principle,
in the choice of the energy unit.
We chose the sample size/shape (which was kept fixed in the $xy$ plane
of the layers) so as to
the Penrose tile edge length at exactly $a=2$
(the nearest-neighbour distance is roughly $a/2$.)
Thus all three $\sigma_{\alpha\beta}$'s are nontrivial parameters.
Results obtained in optimizing potentials for binary icosahedral Frank-Kasper
structures (Roth et al.\ 1990)\nocite{rotstab} with similar local structure
and potentials, show that
$\sigma_{\alpha\beta}$ are largely independent of
$\epsilon_{\alpha\beta}$, for $\epsilon_{\alpha\beta}$ in the range of $0.5 <
\epsilon_{\alpha\beta} < 2.0$.
Therefore we set $\epsilon_{LS} \equiv 1$ and $\epsilon_{LL} = \epsilon_{LS}
\equiv 1/2$. Thus, we actually varied only the three
$\sigma_{\alpha\beta}$'s.
Before we can carry out the optimization itself we have to determine the
lattice constant $c$ in the stacking direction.
(The A/B layer spacing $c/2$ should also be roughly a lattice constant.)
This is done in a poor man's minimization by
scanning the interval $0.8 < c < 4.0$ and looking for the minimum of the
potential energy $E(c)$ in the following way:
having fixed $a$ (and consequently $l_x, l_y$)
we calculate the
partial radial distribution functions $g_{\alpha\beta}(r)$
for the ideal model using a given layer distance $c$.
Then, without annealing or relaxing, we compute the partial potential energy
\begin{equation}
E_{\alpha\beta}(\sigma_{\alpha\beta}, c) = \int_{0}^{r_{c}} v_{\alpha\beta}
\left(\frac{r}{\sigma_{\alpha\beta}}\right)\, g_{\alpha\beta}(r, c)\, 4\pi
r^{2}\, dr
\end{equation}
dependent on the cut-off radius $r_{c}$ and the pair potential
$v_{\alpha\beta}$ and minimize it as a function of
$\sigma_{\alpha\beta}$.
Using this procedure we find that the total potential
energy $E = E_{LL} + E_{LS} + E_{SS}$ is minimzed if each
$E_{\alpha\beta}$ is optimized separately.
The optimal $\sigma_{\alpha\beta}$ for Lennard-Jones potentials are
$\sigma_{SS} = 1.029$, $\sigma_{SL} = 1.137$, and $\sigma_{LL} = 1.189$.
For the potentials used by Dzugutov we get $\sigma_{SS} = 1.108$,
$\sigma_{SL} = 1.034$, and $\sigma_{LL} = 0.929$.
The result reflects the fact that the Dzugutov potentials are very
short ranged and that they have a maximum repulsive for second nearest
neighbours, whereas even in the cut-off and smoothed version of the
Lennard-Jones potential interactions with the second or
third set of neighbours are still attractive.
We also found that the optimal A/B interlayer distance for Lennard-Jones
and for Dzugutov potentials is about $c/2= 1.04\pm 0.01$
The uncertainty quoted here
reflects the slight variation of the spacing depending on
the choice of $\epsilon_{\alpha\beta}$ and $r_{c}$.
\subsection{ Results}
The constant temperature and pressure molecular dynamics
simulation method described in Sec.~\ref{simulations} has been applied to
study the
thermodynamic stability of the ideal structure. To simulate non-cubic
structures
it has been extended to allow independent changes of the box lengths $l_{k}$.
Thus during the MD simulations the box size could vary in contrast to the
optimization, were the box was fixed.
The simulation sample was the same as the one used in Sec.~\ref{secopt}.
With Dzugutov's potentials the melting temperature at $P^{*} = 0.01 $ is
$T^{*}_{\rm m} = 1.23$. (In fact, the structure does not melt; it
vaporizes.)
Of course, in an infinite box the ratio $l_x/l_y = 2 \sin 36^\circ \sim 1.1756$
is fixed by pentagonal symmetry. In a finite system, however,
$l_x/l_y$ should deviate
slightly from the ideal ratio due to the ``phonon-phason'' coupling
(Lubensky, 1988, and references therein)
since the periodic boundary conditions force a nonzero
phason strain.
We found the equilibrium value $l_y/l_x = 1.177$ in the simulation,
independent of the temperature and very close to the ideal value.
The A and B layers are mirror planes so they should still be flat in the
relaxed structure; on the other hand, the X layers should pucker slightly
after relaxation, with the $\rm X'$ layer puckered in the opposite direction.
The simulation, however, show that the layers remain essentially flat.
We have also recorded the atomic mobility by monitoring the mean square
displacement of the atoms. It turns out that even close to the melting point
($T = 0.9 T^{*}_{\rm m}$)
long-range diffusion is {\em not} observable at all,
and only very few atoms are seen to jump to a new position.
Those atoms which do jump are all in the skinny rhombus part of the K-tile.
Similar results have been obtained for icosahedral and dodecagonal
structures \cite{rothdiff} where oblate rhombohedra and skinny
rhombic prisms resp.\ play the same role as the flat rhombus in the K-tile.
Most of the jumps in the K tile take place around the $\alpha$ atom at the
top of the tile (see Fig. \ref{qksdecl}) and around the interior vertex.
Possibly the jumping in this environment indicates that our model is erroneous
there:
perhaps a different arrangement or choice species should occur
in the atoms surrounding the interior vertex of the K tile, or perhaps
the correct model is a T/R tiling which has no K tiles
at all.
It is also possible that the correct structure model should be thought
of simply as a random tiling of Penrose rhombi
(Sec. \ref{penrose});
then ``phason fluctuations'' are realized by tile flips
that entail switching which vertex of a skinny rhombus undergoes a
``chain-shift''.
Our conclusion is that the structure is very stable, and that the
choice of the atoms positions is reasonable. The jumping atoms do not affect
the stability since they are separated by large distances.
\section{ Discussion}
\subsection{ Summary}
Now we are back at the beginning: we started with a structure found by
computer simulations, and recognized that the structure could be described
by a tiling model. We derived quite a number of tilings envolving fR,
sR, T, R, Q, K, and S tiles, all of which are compatible with a
decoration with atoms that include the original model. Although the
fR/Q and T/R models seem to be superior to the Q/K/S tiling and the
variants involving sR tiles, there may be properties that favour the
latter. We tested one variant (Q/K/S) of the tiling models again with
computer simulation to derive structure and potential parameters, and
to find out if the geometically reasonable model is also reasonable with a
simple interaction model. Since the Q/K/S tiling is somehow
cumbersome, but on the other hand includes most of the properties of
the other tilings, its stability assures us that the other tiling
types will also be stable.
Up to now we have only treated each tiling for itself and described
its properties. In this final section we would like to address the
interrelationship of the tiling types, and if there is a hierarchy
of tilings. This includes the A/B layer symmetry breaking
which occurs in some of the models. In addition to computer
simulations it will be interesting to compare the tiling models to
experiment and to other structure models.
Furthermore we want to ask if and how it is possible to change
a certain tiling which means reshuffling the tiles, introducing
defects like random stacking and atomic jumps. These moves may also
lead to the a transformation of one tiling type into another.
Although such a
process is not suitable for MD simulations, it may be
studied in Monte Carlo simulations with properly chosen interactions.
\subsection{ Hierarchy of Tiling Descriptions}
Our approach is an example of a general approach which may be fruitful
in understanding the relation of interatomic interactions to long-range
order in quasicrystals. An ensemble of tilings is proposed;
via a decoration rule, these correspond one-to-one to an ensemble
of low-energy atomic structures. However, the energy
is slightly different for each of these structures; these energy
differences may be considered as a ``tiling Hamiltonian''
${\cal H}_{tile}$
which is
purely a function of the tile arrangement. Often the tiling
Hamiltonian is a sum
of one-tile and neighbor-tile pair interactions (Mihalcovi{\v c} et al.\
1996a,1996b)
\nocite{mihI,mihII};
alternatively, Jeong and Steinhardt (1994) proposed a
family of tiling Hamiltonians with a (favorable) energy $-V$ for each
occurrence of a special local pattern, which they called the ``cluster''.
The ground states of ${\cal H}_{tile}$
are a subset of the original ensemble;
most often this constrained sub-ensemble can be described by a
tiling of larger (``super'') tiles with their own packing rules.
In the ``cluster'' Hamiltonians studied by Jeong and Steinhardt (1994),
the degree of long-range order is enhanced in the supertiling --
indeed for one special ``cluster''
one just obtains the quasiperiodic Penrose tiling --
and even a random tiling of supertiles has greatly reduced phason fluctuations.
Evidently, if
${\cal H}_{tile}$
can be divided into
a sum of successively weaker parts, there may be
a corresponding hierarchy of successively larger supertiles.
Furthermore, as the temperature is raised, the constraints
due to the weakest terms of
${\cal H}_{tile}$
are broken, so the size of the relevant
tiles may decrease with temperature. (The associated changes in
diffraction pattern were discussed by Lan\c{c}on, Billard, Burkov \& de
Boissieu (1994)).
\nocite{burkov}
In Sec.~\ref{dsdm} , we discovered just such a hierarchy of similar
structures, some more constrained than others, as follows: the fR/Q tiling is
a special case of either the fR/sR/Q (fat and skinny rhombus and Q) tiling, or
of the T/R tiling.
The 2-level (Q/K/S) tiling is also a special case of the fR/sR/Q, which is
a modification of the fR/sR tiling (Subsec.~\ref{basic}).
We postulated (at the start of Sec.~\ref{dsdm}) that
the dominant term in our tiling-Hamiltonian
favors the chain-motif, consisting of $\alpha$ atoms surrounded by
alternating pentagons. (Incidentally, this is a ``cluster''
in the sense of Jeong and Steinhardt (1994).)
Then the sub-ensemble which minimizes that term is the T/R
random tiling, which (see below) is also consistent with
a structure model proposed by He et al.\ \citeyear{chinese}.
Due to the limitations of our simulation, and because we did not
perform relaxation at $T=0$ to extract the parameters of
${\cal H}_{tile}$,
we could not prove that the chain-motif (and hence T/R tiling)
is favored, nor could we
measure the smaller energy differences
that would decide which subensemble of T/R tilings
most faithfully models the structure.
We could only conjecture (in Sec.~\ref{trirect})
that there should be an additional term
in ${\cal H}_{tile}$,
disfavoring Q tiles adjoining by $a$ edges.
Granting that, it followed that
the best tiling to represent our structures is something
between the T/R tiling and the fR/Q tiling
of Cockayne (1994).
\nocite{cock}
\subsection{ Issues of Symmetry Breaking in our
Simulation}
There remains a serious doubt that the apparent decagonal structure
we observed in our simulations might be only an artifact of
the geometric restrictions imposed by the small simulation box:
the correct phase in the thermodynamic limit
might be an icosahedral random tiling.
Indeed, Sec.~\ref{icorel},
showed that our decagonal structure is essentially
a special layered packing of the icosahedral tiles;
rhombohedral tiling might
find a layered arrangement in a quench, if that happens
to fit well with the periodic boundary conditions.
Furthermore, out of seven quenches with
the same potentials (Roth et al.\ 1995), \nocite{rotcool}
five quenches led to icosahedral structures and two to decagonal ones.
Even if we grant that the true structure {\it is} decagonal,
there is a second question whether we may be over-interpreting accidental
effects of the finite-size box used in our initial simulations:
does an A/B symmetry-breaking occur in the thermodynamic limit?
This is the fundamental difference between the 2LT (Q/K/S) model
and the T/R tiling model.
Apart from the observation in our initial simulation, there is
no convincing reason for the 2LT (Q/K/S) model, since the K tile
seems to be poorly packed.
There is no other numerical evidence that the A/B symmetry-breaking
really occurs in an extended structure.
It is quite possible that the shape and size of each layer
in our simiulation box happened to allow
only a tiling which lacks the $180^\circ$ symmetry
axis (out of the plane) which would make A and B edges equivalent.\footnote{
A simple matching rule has been given that favours the
Q/K/S tiling over T/R and fR/Q: if you consider the pentagonal antiprisms
around the $\alpha$ atoms, then there should always be two different atoms on
the opposite sides of a diameter (true in the Q/K/S case). In the T/R tiling
there are only two vertex environments that fullfil this condition, in the
fR/Q tiling, there is only one. But with this vertex environments alone it is
not
posssible to produce a fR/Q or T/R tiling. Thus the Q/K/S tiling is
favoured. The rule also works if you have sR tiles: In an sR/fR and an sR/fR/Q
tiling it either forces the assembly of Q/K/S tiles or induces chain shifts.
}
To pursue this further, we should consider whether there
is any physical cause for such a symmetry breaking.
Rather than start with the Q/K/S tiles, it may be better to start
by imagining a fR/sR/Q tiling and asking
what might drive the symmetry breaking,
which from this viewpoint is a displacive instability
of the $\gamma$ atoms, perhaps coupled to some chain-shifts.
(After all, on every skinny rhombus such a symmetry-breaking occurs
locally; the question is whether the pattern of displacements
can be propagated from one skinny to the next.)
It is plausible that, once A/B symmetry breaking is granted,
it will be advantageous to group the rhombi into S and K tiles.
Both of these questions might be addressed (at zero temperature) by
relaxing a variety of decagonal and icosahedral structures under
a variety of potentials, to see which is more stable energetically,
in the spirit of calculations done on some Al-transition metal
quasicrystals (Phillips and Widom, 1994; Mihalcovi{\v c} et al.\ 1996b)
\nocite{phil,mihII}.
\subsection{ Comparison to Other Structure Models}
\label{comparison}
The Fe-Nb structure model outlined by He et al.\ \citeyear{chinese}
was not formulated in terms of a tiling of any kind, nor as
a deterministic rule for packing the plane with motifs;
instead, it is largely a scheme for analyzing high-resolution images.
Although there is a possibility that they misinterpreted images of
a periodic approximant of the quasicrystal,
or a microcrystalline mixture of approximants,
it is still interesting and economical to decribe it using
quasicrystal tiling element.
Their model is based on the same motifs (pentagon-chains) as ours
and the lines in figure 3(b) of He et al.\ \citeyear{chinese},
which indeed
outline triangle and rectangle tiles, are the same as the $a$ linkages
in our model.
Since the sample they imaged
was rather small and defective (much like out first simulation),
we can only be tentative
which variant of our structure model it should be identified with.
In principle a T/R tiling could be a
fR/Q tiling with the $b$ edges drawn in.
However, the image of He et al.\ \citeyear{chinese}
includes pairs of rectangles adjoining
by a $b$ edge, which is a defect from the fR/Q viewpoint;
note also it clearly does {\it not} have a A/B layer ordering.
Thus it is most plausibly idealized as a random T/R tiling;
The icosahedral relationship described in Sec.~\ref{icorel}
allows the decagonal structure to grow on the icosahedral
one. Congruent icosahedral/decagonal grain boundaries should therefore be
possible. The decagonal phase could be viewed as an approximant to the
icosahedral phase, and thin decagonal bands in the icosahedral phase may be
regarded as stacking defects. Furthermore,
since the icosahedral and the
decagonal phases are similar in composition,
phase transformations between them should also be
possible.
Thus our Lennard-Jones system might serve as a toy system for
investigating the behavior $i$-AlPdMn.
(However, we must re-emphasize that
the atomic arrangement in an Al-transition metal
quasicrystal certainly differs from the Frank-Kasper quasicrystal
described here.)
We turn briefly to another approach to constructing structure models --
that based on atomic clusters (Elser and Henley, 1985; Henley and Elser, 1986;
Mihalcovi{\v c} et al.\ 1996a)\nocite{elshen,henels,mihI},
not to be confused with the ``clusters'' of Jeong and Steinhardt (1994)!
Now, Al-Mn type structure models have a common motif of
$\rm Al_6 Mn_4$ tetrahedra \cite{kreiner};
by joining several of these along their faces larger clusters
are formed which are also observed in these structures.
On the other hand, Frank-Kasper models have a common motif of
``truncated tetrahedra'' surrounding every L atom with coordination 16;
indeed, this motif is frequent in our structures.
(see Fig.~\ref{friauf}; the full coordination shell, with 4 additional
atoms, is the ``Friauf polyhedron''.)
These are combined into larger clusters \cite{sam1,sam2} in exactly the same
fashion.
Thus, if an Al-Mn model can be represented as packing of $\rm Al_6 Mn_4$
tetrahedra (plus atoms needed to fill the interstices), then by
replacing each of these by a truncated tetrahedron we produce a
(hypothetical) Frank-Kasper model.
For example, the ``Mackay Icosahedron'' \cite{elshen} is
a combination of 20 tetrahedra which maps to the
Bergman polyhedron (Bergman et al.\ 1957)\nocite{berg}.
It is well known that that crystal phases $\alpha$-AlMnSi and
$R$-AlCuLi are bcc packings of the respective clusters,
and this suggested that the same cluster-cluster networks could
describe the related icosahedral quasicrystals i-AlMnSi and i-AlCuLi
\cite{henCCT}.
A combination of 5 tetrahedra produces a pentagonal-bipyramid motif
which is the basis for crystalline approximants
[such as $\rm Al_{13}Fe_4$ and $\rm T_3(AlMnZn)$]
and for conjectured structure models of {\it decagonal} Al-transition metal
quasicrystals \cite{hendec}.
The corresponding combination in a Frank-Kasper structure
[consisting of 5 truncated tetrahedra arranged around
a central axis]
is the ``VF'' cluster, a common motif in large-unit-cell Frank-Kasper
alloys of simple metals \cite{sam1,sam2}.
If we assume that such clusters are linked as in $\rm Al_{13}Fe_4$,
we produce a new hypothetical Frank-Kasper decagonal model
{\it different} from the one presented in the present paper.
(The ``VF'' motif is rarer in our models -- it occurs only
when five fat rhombi meet to form a star.)
In the new structure model, the centers of neighboring clusters
are separated by $\tau a$ in the horizontal planes
and by $\pm c/2$ vertically. We have not observed or investigated
such a model in simulations.
\subsection{ Beyond Two Dimensions}
To describe real structures, we must go beyond
static, perfectly stacked structures.
The real structure has 2 extra dimensions:
time, and the periodic direction $z$.
Whether its ultimate state is random-tiling or a locked quasiperiodic
tiling, a tiling can improve its order under annealing
only by reshuffling of its tiles.
For example, recoognizing the fundamental
reshuffling in the case of the binary tiling
permitted an accelerated Monte Carlo move (Widom et al.\ 1987)\nocite{wid87}).
Note that, although the tile rearrangement apparently
involves a large volume, it is common that
the reshuffling requires the motion of only a single atom \cite{wid92}.
The problem of reshuffling has an obvious relation
to that of stacking disorder: a realistic model
must handle configurations in which the structure does not
repeat precisely along the fivefold axis.
Indeed, the structure as quenched
was not strictly periodic in the stacking direction.
Of course, such nonperiodicity may be considered a defect, and
perhaps blamed on incomplete equilibration; however there are
two senses in which it is necessary even in equilibrium.
First, reaching equilibrium usually demands some tile reshufflings.
But in a 3D atomic structure based on a 2D tiling, every
tile reshuffling requires rearranging an entire column of the atomic structure.
Obviously this is easiest done one bit of the column at a time; the
intermediate state is one with stacking randomness.
Second, in the random-tiling explanation of the thermodynamic
stability of quasicrystals, the contribution of tile-reshuffling
entropy to the free energy is decisive; this entropy can be
extensive only in an ensemble with stacking disorder \cite{henART}.
Recently, Ritsch, Nissen, and Beeli \citeyear{ritsch} have argued that
common features
in high-resolution transmission electron microscope images of d(AlCoNi)
are evidence of stacking randomness in this decagonal.\\[1ex]
\subsubsection{ Reshuffling}\label{reshuffling}
The reshufflings in the case of the two-level tiling
are simple and are shown in Fig.~\ref{fig-old2}.
We can trade off $\rm Q+S \leftrightarrow 2 K$
(Fig.~\ref{fig-old2}(a))
or $\rm Q+K \leftrightarrow K+Q$ (Fig.~\ref{fig-old2}(b)),
This reshuffling
is the simplest fundamental move which can be used to visit
from one state to another.
Local reshufflings of the T/R tiling are not
possible; instead it is necessary to form a ``defect tile''
such as a skinny rhombus that is normally absent in that tiling.
A defect tile
can move through the tiling in what is called a ``zipper move''
\cite {ox} since it leaves behind a rearranged structure.
One must check what reshufflings imply for
the atoms. In particular, the number of each species ought to be conserved;
otherwise the reshuffling is either blocked,
or it creates substitutional defects,
or it only takes place in a structure which is already
substitutionally disordered in equilibrium.
In a rhombus tiling, the phason strain
(which can be regarded as the deviation
of the projection plane embedded in higher dimensional space from its ideal
position)
determines the number density of each kind of rhombus; since the rhombus
content is conserved under reshuffling, so is the atomc content.
The same thing is true in our T/R tiling,
but not in the two-level (Q/K/S) tiling, because of the
possibility of trading off $\rm Q+S \leftrightarrow 2 K$.
(When divided into Penrose rhombi, both combinations have
the same contents).
Indeed, our 2LT (Q/K/S) atomic structure model has serious problems with
the reshufflings:
the $\rm Q+S \leftrightarrow 2 K$ reshuffling
changes the content
of the tiles from 29 L+ 14 S to 28 L +14 S, so one large atom disappears or
has to become an interstitial. The $\rm Q+K \leftrightarrow K+Q$ reshuffling
conserves the net atomic content of the tiles;
however, the atomic rearrangement in which atoms jump the
least distance would put atoms on the wrong site for their species.
Simulations have shown that such substitutions are not
acceptable; even a few of them destroy the stability of the structure.
We have to admit that no better moves have been found up to now.
In a T/R tiling, the question of rearrangements is quite different
since the only possible update move is a ``zipper'', is an
entire chain of tiles which closes on itself \cite{ox},
just as in triangle-square tilings \cite{oxhen}.
Thus, the intermediate state of a rearrangement involves not
only stacking defects, but also defects within each layer
(which would appear as special tiles other than T or R).
\subsubsection{ Stacking Randomness}
In a nonperiodic stacking, we may presume the
tilings describing adjacent layers are similar.
Thus the spatial sequence of layers is much like the
temporal sequence of a 2D tiling undergoing a series of reshufflings
\cite{henART}.
Each violation of periodicity presumably costs energy;
most likely, one or two ways of doing so are less costly than any others,
so it is a reasonable approximation to postulate as
a {\it constraint } on the stacked tiling ensemble that layers can
be related spatially only in such ways (point stacking defects).
In the case of the Q/K/S tiling model, the obvious stacking defects are
flips $\rm Q+S \leftrightarrow 2 K$ and $\rm Q+K \leftrightarrow K+Q$
from one layer to the next. Since the positions of
the atoms are similar in both states, these defects should be not too costly in
energy.
An interesting experiment would be
to create a stacking of layers from
{\it completely different} 2D tilings, and
see what structure it relaxes to.
For the {\it dodecagonal} Frank-Kasper structure,
in the monatomic case \cite{rothdiff},
the atoms rearrange -- while moving only short distances -- until
all the layers are {\it identical} and defect free; on the other hand,
in the binary case of the same dodecagonal structure
(L and S atoms, similar to the present paper)
the structure cannot be healed with just
short-distance moves; instead substitutional defects are introduced.
\section*{ Acknowledgments}
J.R. would like to acknowledge a post doctoral fellowship from the DFG and the
hospitality at Cornell University, Ithaca, where most of this work has been
carried out. C.L.H. was supported by the U.S. D.O.E. grant DE-FG02ER89-45404.
| proofpile-arXiv_065-488 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
\label{sec:intro}
After ten years of research in the field of high-$T_c$
superconductors\cite{BedMue} (HTSC), many of their properties have not
yet been understood. In particular, the symmetry of the
superconducting gap\cite{Lyo,Dyn,Sch} is still controversial. Usually,
one assumes that the superconducting condensate in the HTSC can be
described by an order parameter $\Delta_{\mbf k}$, which depends only on
the quasi momentum ${\mbf k}$, but not on band index $n$. Retardation
effects are also often neglected, i.e.\ the gap is assumed to be
independent of the frequency $\omega$.
A wide range of experimental techniques can be employed to investigate
the properties of the gap function. Among these, Raman scattering has
played an important role.\cite{DevEin,KraCar} The dependence of the
Raman response on the directions of polarization of the incident and
scattered light yields several independent spectra which provide a
considerable number of constraints on the assumed ${\mbf k}$-dependence
of the gap function $\Delta_{\mbf k}$. However, Raman scattering is not
sensitive to the phase of the gap.
The Raman spectra at temperatures below $T_c$ shows, in most HTSC, a
clear gap-like structure which lies in the energy range of the optical
phonons at the ${\it \Gamma}$ point. These phonons have been
identified for most HTSC,\cite{ThoCar} and the subtraction of the
corresponding structures from the spectra has become a standard
procedure to isolate electronic structures containing gap
information. Electronic Raman scattering spectra are now available for
many high-$T_c$ materials and, since they exhibit similar general
features, most of these data are considered to be reliable. In this
paper, we attempt to interpret these spectra from a theoretical point
of view based on the full 3D one-electron band structure. We pay
attention to both, line shapes and {\it absolute} scattering
efficiencies.
The theory of electronic Raman scattering in superconductors was
pioneered by Abrikosov and coworkers in two important
papers.\cite{AbrFal,AbrGen} In the first, they developed a theory for
the scattering efficiency of {\it isotropic} Fermi liquids under the
assumption that the attractive interaction between quasiparticles can
be neglected. In the second paper, they extended this approach to
anisotropic systems, introduced the effective mass vertex concept, and
included Coulomb screening. The current form of the theory, developed
mainly by Klein {\it et al.},\cite{KleDie} takes into account the
attractive pairing interaction and emphasizes the role of gauge
invariance as well as the polarization dependence for anisotropic
gaps. In order to compare the theoretical predictions with the
experiment, we evaluate them numerically in a quantitative manner
(including {\it absolute} scattering efficiencies!) and compare them
to the experimental findings.
Several calculations of the electronic Raman scattering efficiency of
HTSC have already been published. Some of them use highly simplified
2D band structures and a decomposition of the Raman vertex
$\gamma_{\mbf k}$ in Fermi surface (FS) harmonics\cite{Allen} or
Brillouin zone (BZ) harmonics, as well as FS integrations instead of
the required BZ integrations.\cite{DevEin,DevRepl,DevEinPRB} The
results of these calculations depend very strongly on the number of
expansion coefficients used for $\gamma_{\mbf k}$ and their relative
values. Another approach\cite{KraMaz} involves the use of band
structures calculated in the framework of the local density
approximation\cite{KohSha} (LDA) using the LMTO
method.\cite{AndJep,AndLie} Within the approximations of the LDA, this
Raman vertex is exact, i.e.\ the only errors made in such a
calculation arise from limitations of the LDA method itself and from
the discretization of the Brillouin zone or Fermi surface. Some of
these calculations, however, suffer from the fact that only the
imaginary part of the Tsuneto function\cite{Tsu} has been used, and
that only 2D integrations were performed.\cite{KraCar}
The present approach\cite{CarStr} is based on the full 3D LDA-LMTO
band structure. It uses a BZ integration, screening effects are
included, and both the real and imaginary part of the Tsuneto function
are used as required by the theory. Electronic Raman spectra are
calculated for ${\rm YBa}_2{\rm Cu}_3{\rm O}_7$ (Y-123) and ${\rm
YBa}_2{\rm Cu}_4{\rm O}_8$ (Y-124). The orthorhombicity of the
cuprates is also taken into account in the Raman vertex since we use
as starting point the band structure of the {\it orthorhombic}
materials.
The cuprates under consideration are not only of interest because of
their superconducting, but also of their strange normal-conducting
properties. Usual metals should show peaks in their Raman spectra at
their plasma frequencies corresponding to Raman shifts of a few
${\rm eV}} \def\meV{{\rm meV}$. The optimally doped cuprates, in contrast, show a very broad
electronic background (from $0$ to about $1\,{\rm eV}} \def\meV{{\rm meV}$ Raman shift), which
is almost independent of temperature and frequency. The spectra of the
underdoped HTSC, such as Y-124, show some temperature dependence at
low frequencies ($\hbar\omega\ll kT$). It is possible to explain these
peculiarities, together with other properties, by assuming a certain
form of the quasiparticle lifetime, as was done in the Marginal Fermi
Liquid theory.\cite{Var,VarII}
For the superconducting state, various forms for the gap function have
been proposed. That which has received most experimental support has
${d_{x^2-y^2}}$ symmetry, i.e., $B_{1g}$ symmetry in tetragonal HTSC. The
power of Raman scattering to confirm such gap function has been
questioned, because, among other difficulties to be discussed below,
it only probes the {\it absolute} value of the gap function, i.e.\ it
cannot distinguish between a ${d_{x^2-y^2}}$-like gap function (for
instance $\cos2\phi$), and a $\left|\cos2\phi\right|$ gap function,
which corresponds to anisotropic $s$ ($A_{1g}$) symmetry. However, it
was pointed out that addition of impurities can be used to effect
the distinction.\cite{DevImp}
This paper is organized as follows: in Sec.~\ref{sec:ldabands}, we
review the main properties of the band structures of the investigated
cuprates, as obtained by LDA-LMTO calculations.
Sec.~\ref{sec:gtheory} discusses the theory of electronic Raman
scattering in systems with an anisotropic band structure. We first
introduce the basic concept of Raman vertex, and then present an
expression relating the scattering efficiency to the Raman
susceptibility. In Sec.~\ref{sec:stheory} and~\ref{sec:ntheory} we
derive expressions for the Raman susceptibility in the superconducting
and normal conducting phases, and discuss some effects not directly
contained in the presented form of the theory. Section~\ref{sec:expts}
is concerned with aspects of the experiments which have to be taken
care of, especially with regard to the comparison with the
theory. Finally, in Sec.~\ref{sec:numerics} the results of our
numerical calculation are presented and compared to the experimental
results. The difference between calculations involving only FS
averaging, and those in which such averaging is performed over the
whole BZ, are discussed.
\section{LDA band structure}
\label{sec:ldabands}
The basis of our calculation is the LDA-LMTO band structure of the
HTSC under consideration.\cite{AndLie} For the sake of further
discussion, we shall describe briefly such band structure.
The Fermi surface of $\rm YBa_2Cu_3O_7$ (Y-123)\cite{AndLie} consists
of four sheets, an even and an odd $pd\sigma$-like plane band, a
$pd\sigma$-like chain sheet and a very small $pd\pi$-like chain
sheet. The latter is predicted by the full-potential LMTO calculations
as well as LAPW calculations.\cite{OKAnd} We use the atomic spheres
approximation (ASA) to the LMTO, which does not reproduce this rather
small feature. In the case of Y-\nobreak123 the three $pd\sigma$-like
conduction bands extend from $-1\,{\rm eV}} \def\meV{{\rm meV}$ to $2\,{\rm eV}} \def\meV{{\rm meV}$ relative to the
Fermi energy. They are embedded in a broad valence band, which ranges
from $-7\,{\rm eV}} \def\meV{{\rm meV}$ to $2\,{\rm eV}} \def\meV{{\rm meV}$ and consists of 36 bands (mainly Cu-$d$ and
O-$p$ orbitals). Below $-7\,{\rm eV}} \def\meV{{\rm meV}$, there is a gap of $4\,{\rm eV}} \def\meV{{\rm meV}$. Above
the conduction band, there is another gap of $0.5\,{\rm eV}} \def\meV{{\rm meV}$, above which
are the lowest fully unoccupied bands which consist mainly of $d$
orbitals of Y and Ba.
The band structure of $\rm YBa_2Cu_4O_8$ (Y-124) shows similar
features. There is an additional $pd\sigma$-like chain band, while the
$pd\pi$-like chain bands are predicted by both, full-potential LMTO and
LAPW to contain no holes, i.e.\ to be completely filled.
An interesting feature of the band structure of both Y-123 and Y-124
is an extended saddle point\cite{AndJep} on the $k_x$-axis near the
$X$ point. This extended saddle point corresponds to a van Hove
singularity at approximately $25\,\meV$ (Y-123) and $110\,\meV$
(Y-124), respectively, below the Fermi level. As will be shown, the
comparatively large density of states in this energy region and the
warped nature of the corresponding bands has an influence on the
calculated electronic Raman spectrum.
The band structure which we used in our numerical calculations was
evaluated for Y-123 on a mesh of $48\times48\times12$ points in the
first BZ, involving $4373$ irreducible points. The band structure of
Y-124 is less sensitive to the resolution of the grid (because the
extended saddle point lies deeper with respect to the Fermi
surface). It was thus sufficient to use a $24\times24\times12$ mesh
with $1099$ irreducible points. The calculations of the
self-consistent potential have been performed in the ASA. Therefore,
the $pd\pi$ chain band around the $S$ point, which should be partly
filled, is completely filled. Because of the small number of states
involved, we do not think this should affect significantly our
results.
As stressed above, our calculations are based on a band structure
obtained within the LDA. We are aware of the fact, that the mean free
path for transport in the direction of the $c$-axis is smaller than
the size of the unit cell, i.e.\ that a description by means of a band
structure $\epsilon_{n{\mbf k}}$ may be questionable
(Ioffe-Regel-limit). Nevertheless a nontrivial band structure in the
$c$-direction may simulate some of the $c$-direction confinement
effects and represent, after integration along $k_z$, a reasonable 2D
band structure.
\section{General theory}
\label{sec:gtheory}
Two approaches have been used to derive the cross section (called
scattering efficiency when referred to unit path length in the solid)
of electronic Raman scattering in superconductors with anisotropic
Fermi surfaces. The first uses Green's
functions,\cite{AbrFal,AbrGen,KleDie,Kos} and the second the kinetic
equation.\cite{Woe,DevEin} Both start with the simplification of the
Hamiltonian, using ${\mbf k}\cdot{\mbf p}$ theory, which relates the Raman
vertex $\gamma_{\mbf k}$ to the inverse effective mass
tensor.\cite{AbrGen} We first briefly review this procedure and,
subsequently, the derivation of the expression for the scattering
efficiency using the diagrammatic approach.
\subsection{The Raman vertex}
To derive an expression for the efficiency for electronic Raman
scattering, one has to replace the momentum ${\mbf p}$ in the Hamiltonian
by ${\mbf p}-(e/c){\mbf A}$. This yields two distinct perturbation terms:
$H_{AA}=(r_0/2){\mbf A}^2$, quadratic in the vector potential, and
$H_A=-(e/mc){\mbf A}\cdot{\mbf p}$, linear in $A$ (we use the transverse
gauge; $r_0=e^2/(m_0c^2)$ denotes the classical electron radius). The
relevant states in the theory are composed of the state of electrons
in the sample plus the state of the photon field. The initial state of
the photon field has $n_L$ laser photons with wave vector ${\mbf k}_L$
and polarization ${\mbf e}_L$ and $n_S=0$ scattered photons with wave
vector ${\mbf k}_S$ and polarization ${\mbf e}_S$. The final state has one
laser photon less but one scattered photon (Stokes scattering). The
vector field can thus be written as a superposition of an incoming and
a scattered plane wave, ${\mbf A}={\mbf A}_L+{\mbf A}_S$ with
\begin{equation}
{\mbf A}_S=A_S^+{\mbf e}_S^* e^{-i{\mbf k}_Sr}~,\qquad
{\mbf A}_L=A_L^-{\mbf e}_L e^{i{\mbf k}_Lr}~,
\end{equation}
where $A_S^+$ contains the creation operator for the scattered photon
and $A_L^-$ the annihilation operator for the laser photons (note that
these are not hermitian).
Since Raman scattering is a second order process in ${\mbf A}$, the term
$H_{AA}$ has to be treated in first order perturbation theory. It is
therefore nonresonant and includes only intraband scattering. The
matrix elements are given by
\begin{eqnarray}
M^{(1)}_{n_fn_i}({\mbf q},{\mbf k})
&=& {1\over2}r_0\braket{n_f{\mbf k}+{\mbf q}}{{\mbf A}^2}{n_i{\mbf k}}
\nonumber\\
&=& r_0 \expv{A_S^+A_L}{\mbf e}_S^*{\mbf e}_L\delta_{n_in_f}~,
\label{fstpb}
\end{eqnarray}
whereas $\expv{A_S^+A_L}$ denotes a matrix element involving the
initial and final state of the photon field and
${\mbf q}={\mbf k}_L-{\mbf k}_S$ is the momentum transfer from the photon field
to the sample. For other values of ${\mbf q}$, the matrix element
$(\ref{fstpb})$ vanishes.
The second term, $H_A$, produces resonances via second order
perturbation theory. It has the form
\begin{equation}
M^{(2)}_{n_fn_i}({\mbf q},{\mbf k}) = r_0 \expv{A_S^+A_L}
\sum_{n_m} \Gamma^{(2)}_{n_fn_i;n_m}({\mbf q},{\mbf k})
\label{sndpb}
\end{equation}
with the expression
\begin{eqnarray}
\Gamma^{(2)}_{n_fn_i;n_m}({\mbf q},{\mbf k}) & = & \label{second}\\
\noalign{\hskip0.2\baselineskip}%
&&\hspace{-2.5cm}
{\braket{n_f{\mbf k}+{\mbf q}}{{\mbf e}_S^*{\mbf p}}{n_m{\mbf k}+{\mbf k}_L}
\braket{n_m{\mbf k}+{\mbf k}_L}{{\mbf e}_L{\mbf p}}{n_i{\mbf k}}
\over\epsilon_{n_i{\mbf k}}-\epsilon_{n_m{\mbf k}+{\mbf k}_L}+\omega_L+i0}
+ \nonumber \\
&&\hspace{-2.5cm}
+ {\braket{n_f{\mbf k}+{\mbf q}}{{\mbf e}_L{\mbf p}}{n_m{\mbf k}-{\mbf k}_S}
\braket{n_m{\mbf k}-{\mbf k}_S}{{\mbf e}_S^*{\mbf p}}{n_i{\mbf k}}
\over\epsilon_{n_i{\mbf k}}-\epsilon_{n_m{\mbf k}-{\mbf k}_S}-\omega_S+i0}~.
\nonumber
\end{eqnarray}
Here, $\omega_L$ and $\omega_S$ are the frequency of the incoming and
scattered light, respectively. Note that the states in the sum above
are states of the sample only. We have used Bloch states with band and
crystal momentum indices. The wavevectors of light, ${\mbf k}_L$ and
${\mbf k}_S$, can usually be neglected in the matrix elements of
Eq.~$(\ref{second})$ because $v_F\ll c$. For the same reason,
$\epsilon_{n_m,{\mbf k}+{\mbf k}_L}\approx\epsilon_{n_m,{\mbf k}}$. Therefore, we
introduce the symbol $\Gamma^{(2)}_{n_fn_i;n_m}({\mbf k})$ to denote
expression~$(\ref{second})$ with the light wavevectors set equal to zero.
If we now add the contributions of both terms in Eqs.~(\ref{fstpb})
and~(\ref{sndpb}) and introduce second quantization, we are left with
the {\it effective Hamiltonian}
\begin{equation}
H_R = r_0\expv{A_S^+A_L}\,\tilde\rho_{\mbf q}
\label{effham}
\end{equation}
as perturbation Hamiltonian leading to Raman scattering. The effective
density operator $\tilde\rho_{\mbf q}$ can be expressed in the form
\begin{equation}
\tilde\rho_{\mbf q} = \sum_{n_f,n_i,{\mbf k}} \gamma_{n_fn_i}({\mbf k})\,
c^+_{n_f,{\mbf k}+{\mbf q}} c_{n_i,{\mbf k}}~.
\label{flucop}
\end{equation}
using fermionic creation and annihilation operators for Bloch electrons
as well as the nondiagonal Raman vertex
\begin{equation}
\gamma_{n_fn_i}({\mbf k}) = {\mbf e}_S^*{\mbf e}_L\delta_{n_fn_i} +
\sum_{n_m} \Gamma^{(2)}_{n_fn_i;n_m}({\mbf k})~.
\label{raver}
\end{equation}
If we are interested mainly in the low frequency region, say Raman
shifts below $50\,\meV$, no {\it real} interband transitions of
significant weight are possible. This can easily be seen from the band
structure (Fig.~2 of Ref.~\onlinecite{AndJep}). Therefore, we
introduce the (intraband) {\it Raman vertex}
$\gamma_n({\mbf k})=\gamma_{nn}({\mbf k})$.
We proceed by discussing a very important simplification of
$(\ref{second})$ (with $n_i=n_f=n$ and ${\mbf q}=0$), the {\it effective
mass approximation}. Four different cases will be discussed. First,
the virtual intraband transition with $n_m=n_i$. In this case, up to
first order in $v_F/c$, we have $\braket{n_m{\mbf k}}{{\mbf p}}{n_i{\mbf k}} =
\braket{n_i{\mbf k}}{{\mbf p}}{n_m{\mbf k}}$ (remember that $n_i=n_f$) and
$\epsilon_{n_i{\mbf k}}-\epsilon_{n_m{\mbf k}}=0$. Then, it can be seen that
the contributions of virtual {\it intra\/}band transitions relative to
the contribution of virtual {\it inter\/}band transitions to
intermediate states are of the order of the Raman shift over the laser
frequency, i.e.\ $\omega/\omega_L\ll1$, and can therefore be
neglected. The second case are the virtual {\it inter\/}band
transitions involving bands which are much farther away from the FS
than the light frequency. Then, because of
$\abs{\epsilon_{n_i}-\epsilon_{n_m}}\gg\omega_L$, the light
frequencies $\omega_L$ as well as $\omega_S$ can be neglected in
$(\ref{second})$. The third case also involves virtual interband
transitions, but for bands at about the laser frequency above the
Fermi surface. Here, the scattering is resonant, and the spectra are
expected to depend strongly on the laser wavelength. One can try to
avoid this situation by using different laser lines. So we assume that
in the third case $\omega_L$ and $\omega_S$ also can be
neglected. Finally, the forth case consists of virtual interband
transitions to neighboring bands with $\Delta\epsilon\ll\omega_L$. In
this case, neglecting $\omega_L$ and $\omega_S$ is more difficult to
justify. We do it nevertheless and reach the approximate conclusion
that we can neglect the light frequencies in Eq.~(\ref{second}) and
can restrict the sum in $(\ref{sndpb})$ to all $n_m\not=n_i$. Then,
Eq.~(\ref{raver}) becomes completely equivalent to the expression for
the inverse effective mass from ${\mbf k}\cdot{\mbf p}$ theory and we can
write
\begin{equation}
\gamma_n({\mbf k})={m\over\hbar^2}
\sum_{i,j} {\mbf e}^*_{S,i}
{\partial^2\epsilon_{n{\mbf k}}\over\partial k_i\partial k_j}
{\mbf e}_{L,j}
\end{equation}
i.e.\ the Raman vertex is equal to the inverse effective mass
contracted with the polarization vectors of the laser light and the
scattered light, respectively.
Therefore, using the term $H_R$ with the intraband Raman vertex
$\gamma_{n{\mbf k}}=\gamma_{nn}({\mbf k})$ in Eq.~(\ref{flucop}) as
perturbation to the Hamiltonian for ${\mbf A}=0$ and treating this in
first order perturbation theory is, under the mentioned
restrictions, equivalent to taking into consideration both
terms $H_A$ and $H_{AA}$.\cite{AbrGen}
According to the LMTO calculations, for Y-123 and Y-124 there are
bands above a band gap between approximately $2\,{\rm eV}} \def\meV{{\rm meV}$ and $2.5\,{\rm eV}} \def\meV{{\rm meV}$
above the Fermi energy. These bands can present a problem with respect
to the former discussion, because they are almost resonant for typical
laser wavelengths like $514.5\,{\rm nm}$. The same is true for the
conduction bands, which extend until $2\,{\rm eV}} \def\meV{{\rm meV}$ above the Fermi
surface. Note that due to the strong on-site interaction at the Cu-$d$
orbitals, correlation effects are expected to be important in the
electronic structure. It is possible that at energies of the order of
$1\,{\rm eV}$ or more above the Fermi surface the picture of the
Hubbard bands is a better description of the band structure and may
explain the weak dependence of the Raman spectra on the laser
frequency observed for laser frequencies in the visible range. The
band structure shows many bands at about the laser frequency below the
Fermi energy. These should yield resonant contributions to the Raman
efficiency.
Because the Raman vertex $\gamma_{\mbf k}$ is, in the given
approximation, the second derivative of the energy with respect to
${\mbf k}$, the $A_{2g}$ component for tetragonal crystals vanishes in
this version of the theory ($A_{2g}$ is the symmetry of an
antisymmetric tensor). If one considers once more the effects of a
nearby resonance, it can be easily seen that the Raman tensor does not
have to be symmetric. This stresses again the questionability of the
effective mass approach if the scattering is resonant.
\subsection{The scattering efficiency}
Using the effective mass approach, we arrived at the effective
Hamiltonian $(\ref{effham})$ with the effective mass determining the
Raman vertex. This effective Hamiltonian is linear in
${\mbf A}_L\cdot{\mbf A}_S$. The derivation of the scattering efficiency
using linear response theory is now a straightforward task.
The first step is finding a relation between the Raman efficiency and
a dynamical structure factor of the sample. Then, in a next step, the
fluctuation-dissipation theorem is used to connect the dynamical
structure factor to the imaginary part of a susceptibility, in our
case the {\it Raman susceptibility}.
To establish the relation to the dynamical structure factor, we add
the time evolution factor $e^{-i\omega t}$ to the effective
Hamiltonian $(\ref{effham})$ and use the golden rule to find the
transition rate from a state $i$ to a state $f$ of the sample. Then, we
sum over all final states $f$ of the sample and do a thermal averaging
over the initial states $i$. The transition rate from a state with
$n_L\equiv n_{{\mbf k}_L{\mbf e}_L}$ laser photons and no scattered photon
to a state with $n_L-1$ laser photons and $n_S\equiv
n_{{\mbf k}_S{\mbf e}_S}=1$ scattered photon at a temperature $T$ is given
by the expression
\begin{equation}
\Gamma^T({\mbf k}_L,{\mbf e}_L;{\mbf k}_S,{\mbf e}_S) =
{2\pi\over\hbar} r_0^2\cdot \absii{\expv{A_S^+A_L}}\cdot
\tilde S^T({\mbf q},\omega)
\end{equation}
(the superscript $T$ denotes temperature dependency) whereas
\begin{eqnarray}
\tilde S^T({\mbf q},\omega) &=& \nonumber\\
&&\hspace{-1cm}\sum_{i,f}{e^{-\beta E_i}\over\cal Z}
\absii{\braket{f}{\tilde\rho_{\mbf q}}{i}} \delta(E_f-E_i+\hbar\omega)
\end{eqnarray}
is a {\it generalized dynamical structure factor} (of the sample!).
The partition function is denoted by ${\cal Z}$, and $\beta$ is the
inverse temperature. Now, we sum over all final states in a certain
region $d\Omega\,d\omega_S$ of $k$-space around ${\mbf k}_S$ and
normalize to the incoming flux $\hbar cn_L$. This yields the
expression
\begin{equation}
{d^2\sigma\over d\Omega\,d\omega}({\mbf q},\omega) =
{\omega_S\over\omega_L} r_0^2 \tilde S^T({\mbf q},\omega)
\label{rameff}
\end{equation}
for the differential cross section $d^2\sigma/d\Omega d\omega$ for a
given Raman shift $\omega$ and a given momentum transfer ${\mbf q}$. This
differential cross section is proportional to the scattering
volume. When performing the calculation for a scattering volume equal
to unity, $\sigma$ becomes the commonly used {\it Raman scattering
efficiency}.
Finally, one can define a linear response function, the {\it Raman
susceptibility}
\begin{equation}
\chi_{\rm Raman}({\mbf q},t) = {i\over\hbar}
\mathop{\rm Tr}\{{\cal Z}^{-1}e^{-\beta H_0}
[\tilde\rho_{\mbf q}(t),\tilde\rho_{-{\mbf q}}(0)]\}
\label{RamSus}
\end{equation}
and its Fourier-transformed $\chi_{\rm Raman}({\mbf q},\omega)$.
To relate the imaginary part of this quantity to the structure
function $\tilde S^T({\mbf q},\omega)$, we use the
fluctuation-dissipation theorem. The result is
\begin{equation}
\tilde S^T({\mbf q},\omega) = -{1\over\pi}(1+n_\omega)
\mathop{\rm Im}\nolimits\chi_{\rm Raman}({\mbf q},\omega)
\label{fdt}
\end{equation}
with the Bose factor $n_\omega$.
Equations $(\ref{rameff})$ and $(\ref{fdt})$ relate the Raman
efficiency directly to the imaginary part of the Raman
susceptibility. This evaluation of the Raman susceptibility shall be
given separately, {\rm (i)} in Sec.~\ref{sec:stheory} for the
superconducting phase and Raman shifts of the order of the gap, and
{\rm (ii)} in Sec.~\ref{sec:ntheory} for large Raman shifts in the
superconducting phase and for the normal phase.
\section{Theory: Superconducting phase}
\label{sec:stheory}
As pointed out in Ref.~\onlinecite{KleDie}, the Raman susceptibility
due to pair-breaking and including screening is given by a
polarization-like bubble made of a renormalized Raman vertex
$\Lambda_{\mbf k}$, a Raman vertex $\gamma_{\mbf k}$, and in between two
Green's function lines for Bogoliubov quasiparticles
(Fig.~\ref{screening}a). The vertex renormalization includes
corrections for Cooper-pair-producing attractive interaction as well
as the repulsive Coulomb interaction, the Dyson equation for the
vertex $\Lambda_{\mbf k}$ in the limit ${\mbf q}\to0$ is given by Fig.~13 in
Ref.~\onlinecite{KleDie}.
To show more clearly the effect of screening, we write the equation
for the Raman susceptibility as given in Fig.~\ref{screening}b
and~\ref{screening}c. Figure~\ref{screening}b (with $a=\gamma_{\mbf k}$
and $b=\gamma_{\mbf k}$) shows the unscreened susceptibility
$\chi_{\gamma\gamma}$ given by a bare polarization bubble with two
Raman vertices $\gamma_{\mbf k}$ and the contraction of a BCS-like ladder
sum with two Raman vertices. Therefore, $\chi_{\gamma\gamma}$ includes
the attractive Cooper-pair-producing interaction. We include Coulomb
screening by virtue of a RPA-like sum given in
Fig.~\ref{screening}c. The effect of screening on the electronic Raman
scattering can now easily be seen.\cite{AbrGen} If we denote by
$\chi_{ab}$ a bubble, renormalized by pairing interaction, with
vertices $a$ and $b$ at the ends as in Fig.~\ref{screening}b, the
RPA-chain can be easily summed up (see Fig.~\ref{screening}c) yielding
\begin{equation}
\chi_{\rm Raman}({\mbf q}\to0,\omega) = \chi_{\gamma\gamma}(\omega)
- {\chi_{\gamma 1}^2(\omega)\over\chi_{11}(\omega)}~,
\label{chir}
\end{equation}
where terms of order ${\mbf q}^2$ have been dropped. In
Eq.~$(\ref{chir})$ we have used the fact that
$V_{\mbf q}/(1-\chi_{11}V_{\mbf q})$ equals
$-\chi_{11}^{-1}(1-1/\varepsilon)$, and the factor $(1-1/\varepsilon)$
is $1+O({\mbf q}/q_{TF})^2$.
Without taking into account Coulomb interaction, the Green's functions
have a well-known massless pole (Goldstone mode) which is a
consequence of the breaking of gauge symmetry in the superconducting
phase.\cite{And} Coulomb interaction makes this pole acquire a finite
mass (which can be shown to correspond to the plasma frequency), so if
we correctly include Coulomb screening (not done in
Ref.~\onlinecite{BraCar}) we no longer have a Goldstone mode, but a
massive Anderson-Bogoliubov mode. This mode has the energy
$\hbar\omega_p$ ($\omega_p$ is the plasma frequency) at the ${\it\Gamma}$
point and is therefore negligible for the low energy behavior of the
Raman spectra.
The susceptibilities $\chi_{ab}$ in Fig.~\ref{screening}b are like a
ladder sum contracted with vertices $a_{\mbf k}$ and $b_{\mbf k}$ and can be
written as a sum
\begin{equation}
\chi_{ab}({\mbf q}{=}0, \omega) =
\sum_{\mbf k} a_{\mbf k} b_{\mbf k} \lambda_{\mbf k}(\omega)
\end{equation}
which involves the Tsuneto function\cite{Tsu}
$\lambda_{\mbf k}(\omega)$. For small values of ${\mbf q}$ (compared to the
inverse coherence length $\xi$ and the Fermi wave vector $k_F$), the
attractive interaction does not have to be taken into account in the
summation of the ladder, and the Tsuneto function is given simply by a
unmodified bubble and can be evaluated easily to be
\begin{eqnarray}
\lambda_{\mbf k}(\omega) &=& {\Delta^2_{\mbf k}\over E^2_{\mbf k}}
\tanh\left(E_{\mbf k}\over 2T\right)\times\nonumber\\
&&\times\left({1\over2E_{\mbf k}+\omega+i0}+{1\over2E_{\mbf k}-\omega-i0}\right)~.
\label{tsufct}
\end{eqnarray}
Equation~$(\ref{tsufct})$ involves the gap function $\Delta_{\mbf k}$
(which depends on the temperature) and the quasiparticle dispersion
relation $E_{\mbf k}^2=\xi_{\mbf k}^2+\Delta_{\mbf k}^2$ with
$\xi_{\mbf k}^2=(\epsilon_{\mbf k}-\epsilon_F)^2$. The constants $\hbar$ and
$k_B$ have been set equal to $1$. As already mentioned, vertex
corrections due to the pairing interaction are neglected. This
approximation is valid for $q\ll\xi^{-1},k_F$
(Ref.~\onlinecite{DevEinPRB}) and $\omega\ll\omega_p$, because the
Anderson-Bogoliubov pole at the plasma frequency need no longer be
included.
A first and very important fact in the expressions above is that they
contain only the absolute square of the gap function, i.e.\ Raman
scattering is {\it not phase sensitive}, and consequently cannot
distinguish between a strongly anisotropic $s$ gap
$\left|{d_{x^2-y^2}}\right|$ and a ${d_{x^2-y^2}}$ gap.
In the preceding calculation of the unscreened correlation functions
$\chi_{ab}$, we have neglected impurity scattering as well as
scattering between quasiparticles (collisionless regime). In isotropic
$s$-wave superconductors at $T=0$ and for Raman shifts
$\omega\ll2\Delta$, it is perfectly reasonable to neglect impurity
scattering, because in this regime pair breaking is not
possible.\cite{AndImp} Also, the scattering between quasiparticles can
be neglected because their density is very small for small
temperatures $T\ll T_c$. For $d$-wave superconductors this is no
longer true. The effect of impurities will be discussed in the next
subsection, whereas a discussion about scattering between
quasiparticles can be found in Sec.~\ref{sec:ntheory}.
The second term of $(\ref{chir})$, representing screening, vanishes if
the average of $\gamma_{\mbf k}\cdot\lambda_{\mbf k}$ does. The Tsuneto
function is fully symmetric, i.e.\ has $A_{1g}$ ($D_{4h}$ group) or
$A_g$ ($D_{2h}$) symmetry regardless of gap symmetry. As a
consequence, the screening term vanishes unless the Raman vertex has
the same symmetry as the crystal. In the tetragonal case,
$A_{1g}$-like vertices are screened, but $B_{1g}$- and $B_{2g}$-like
are not. This is different for orthorhombic HTSC of the YBCO-type. In
this case the Tsuneto function has $A_1$ symmetry, and the same is
true for the ${d_{x^2-y^2}}$-like component of the mass ($B_{1g}$ of
$D_{4h}$ group, $A_g$ of $D_{2h}$). Consequently, in these
orthorhombic crystals, the $B_{1g}$ component is also screened. This
discussion is also applicable to BISCO, but with interchanged roles of
$B_{1g}$ and $B_{2g}$ modes because of the different orientation of
the crystallographic unit cell with respect to the Cu-O bonds.
In tetragonal systems, the $B_{1g}$ component of the Raman vertex has
nodes at the same position as the gap function. This has severe
consequences for the low-energy part of the spectra.\cite{DevEinPRB}
In two dimensions, the existence of the nodes of the gap function in
the case of a ${d_{x^2-y^2}}$ gap results in a linear density of states at
low energies. If the vertex has a finite value in this region, the
imaginary part of the Raman susceptibility is also linear in the
frequency. If the vertex has a node, however, its magnitude squared
becomes quadratic with respect to the gap on the Fermi surface. This
causes two additional powers of the frequency to appear, the $B_{1g}$
component of the scattering efficiency is cubic at low
frequencies.\cite{DevEin} Two effects can alter this behavior: an
orthorhombic distortion and impurities.
In our calculations, we focus on a ${d_{x^2-y^2}}$-like gap function which
is only a function of the direction in $k$-space, but not of the
magnitude of ${\mbf k}$, since the values of the gap functions
sufficiently far from the Fermi surface do not affect the results. We
are using the same gap function for all bands involved.
\subsection{Effect of impurities}
\label{sec:stheory:imp}
In contrast to scattering at non-magnetic impurities in conventional
(isotropic) superconductors, the influence of impurity scattering
plays an important role for superconductors with anisotropic gaps and
its effect on the Raman spectrum is most pronounced for
superconductors which exhibit regions in $k$-space where the gap
almost or completely vanishes. It was shown\cite{BorHir,DevImp} that
in the case of $d$-wave pairing, impurity scattering can be described
by extending the nodal points on the 2D FS to small finite regions
with vanishing gap. This causes a nonvanishing density of states at
the Fermi energy. For anisotropic $s$-wave pairing the gap anisotropy
becomes smeared out leading to an increase of the minimum gap value
$\Delta_{\rm min}$. In the case of a $\abs{{d_{x^2-y^2}}}$ gap, this
minimum gap increases monotonically with the impurity concentration
$n_i$ for small values of $n_i$.
The renormalization of the gap function by the presence of impurities
causes an additional contribution, which is linear in the Raman shift
$\omega$ for small Raman shifts $\omega$, in the Raman
spectra.\cite{DevImp} This has consequences for the $B_{1g}$ spectrum
of a {\it tetragonal} crystal, which, according to the theory, has a
cubic $\omega$-dependence, because a linear frequency dependence is
added. As will be discussed in the next subsection, the
orthorhombicity of the YBCO compounds also causes a linear addition to
the cubic behavior of the $B_{1g}$ channel spectrum.
In the case of a $\abs{{d_{x^2-y^2}}}$-like, $A_g$ symmetry gap function
the impurity-induced minimal gap $\Delta_{\rm min}$ causes an
excitation-free region to show up in the electronic Raman spectrum
below a Raman shift of $2\Delta_{\rm min}$.
\subsection{Effect of orthorhombic distortion}
\label{sec:stheory:odist}
As already mentioned, orthorhombic distortions, i.e.\ deviations
from the tetragonal symmetry, have a different effect on Y-123 and on
Bi-2212. Consider the $B_{1g}\,(D_{4h})$ component of the inverse mass
tensor in a {\it tetragonal} high-$T_c$ superconductor with a
${d_{x^2-y^2}}$-like gap. The $B_{1g}\,(D_{4h})$ mass has its nodes in
directions diagonal to the axes of the copper planes; the same is true
for the gap function. As mentioned above, this results in the
$\omega^3$-dependence of the Raman efficiency for $B_{1g}\,(D_{4h})$
scattering, in contrast to the $\omega$-dependence predicted for
$A_{1g}$ and $B_{2g}$ scattering. Let us now consider the
orthorhombic distortion present in Y-123. The zeros of the
$B_{1g}\,(D_{4h})$ mass shift because there are no longer mirror
planes through the $(110)$ axes. For this reason, the low-energy part
of the spectrum acquires a linear component in addition to the
$\omega^3$ component of the $D_{4h}$ case.
In Bi-2212 the situation is different because the orthorhombic
crystallographic cell is rotated by $45^\circ$ with respect to the
$a$- and $b$-axes: the orthorhombic distortion preserves the mirror
planes $[a\pm b,c]$. Consequently, the $B_{1g}$ zeros stay at the
same position, the low-energy efficiency acquires no linear component.
\subsection{Effect of multilayers}
\label{sec:stheory:mlayer}
In systems with one layer of Cu-$\rm O_2$ planes per unit cell there
is only one sheet of Fermi surface and the mass fluctuations are
essentially intraband mass fluctuations, which are very sensitive to
the scattering polarizations. The scattering related to the average
mass is fully screened. The simplest $A_{1g}\,(D_{4h})$ scattering is
related to a Raman vertex of the form $\cos4\phi$ symmetry while
$B_{1g}\,(D_{4h})$ scattering is obtained for a $\cos2\phi$ vertex.
In multilayer systems, interband fluctuations between the various
sheets FS are also important. The lowest component of such
fluctuations corresponds to different {\it average} masses in each FS
sheet. Such fluctuations do not depend on the scattering polarizations
and lead to unscreened scattering of $A_g$ symmetry.
\subsection{Effect of sign change of $\gamma_{\mbf k}$
on the Fermi surface}
\label{sec:stheory:signch}
The behavior of the Raman vertex near the Fermi surface, especially
its sign, is crucial for the scattering efficiency and, in particular,
for the effect of screening. {\it Antiscreening}, i.e.\ an {\it
enhancement} of the scattering efficiency by screening, can occur if
the Raman vertex changes sign on the Fermi surface. This can be seen
by considering the screening part
\begin{equation}
\mathop{\rm Im}\nolimits{\chi_{\rm Scr}} = - \mathop{\rm Im}\nolimits{\chi_{\gamma1}^2\over\chi_{11}}
\label{scrpart}
\end{equation}
of the Raman susceptibility. A positive value of $\mathop{\rm Im}\nolimits{\chi_{\rm Scr}}$ enhances
the efficiency, i.e.\ corresponds to antiscreening.
To show how antiscreening arises, we first write the screening term
$\mathop{\rm Im}\nolimits\chi_{\rm scr}$ in terms of the real and imaginary parts
${\lambda'}\equiv\mathop{\rm Re}\nolimits\lambda$ and ${\lambda''}\equiv\mathop{\rm Im}\nolimits\lambda$ of the
Tsuneto function and the Raman vertex $\gamma$ as
\begin{equation}
\mathop{\rm Im}\nolimits{\chi_{\rm Scr}}={
{\expv{\gamma{\lambda'}}}^2\expv{{\lambda''}}-{\expv{\gamma{\lambda''}}}^2\expv{{\lambda''}}
-2\expv{\gamma{\lambda'}}\expv{\gamma{\lambda''}}\expv{{\lambda'}}\over
{\expv{{\lambda'}}}^2+{\expv{{\lambda''}}}^2}~.\label{screefo}
\end{equation}
The imaginary part of the Tsuneto function ${\lambda''}$ is a positive
$\delta$-function. Consequently, the quantity $\expv{{\lambda''}}$ is a
positive function of the Raman shift $\omega$. If $\gamma_{\mbf k}$
changes sign in a region around the Fermi surface, it is possible that
$\expv{\gamma{\lambda''}}$ changes sign as a function of $\omega$, i.e.\
has a zero. At the position of this zero, the second and the third
term in the numerator of $(\ref{screefo})$ vanish. The first term,
${\expv{\gamma{\lambda'}}}^2\expv{{\lambda''}}$, is positive and can become
dominant in Eq.~(\ref{screefo}). In this case antiscreening
results. In the Appendix~A will be shown that antiscreening is
particularly sensitive to the sign of the Raman vertex on parts of the
Fermi surface around the directions of the nodes of the gap function
$\Delta_{\mbf k}$.
\section{Theory: Normal phase}
\label{sec:ntheory}
In the normal phase, the exact mechanism which produces a finite Raman
intensity almost constant over a broad frequency and temperature
range, is not known. Therefore, we assume some scattering mechanism,
which implies a finite lifetime of the quasiparticles. Candidates for
this scattering are the quasiparticle-quasiparticle scattering in
Marginal Fermi Liquid theory\cite{Var} (MFL), impurity
scattering\cite{HirWoe} or scattering due to spin
fluctuations.\cite{QuiHir} A self energy with non-vanishing imaginary
part yields a susceptibility of the form
\begin{equation}
\chi_{ab}({\mbf q}{=}0, \omega)=\sum_k a_{\mbf k} b_{\mbf k}\nu_{\mbf k}(\omega)
\end{equation}
with the relaxation kernel (the function $f'$ is the derivative of the
Fermi function with respect to the energy)
\begin{equation}
\nu_{\mbf k}(\omega) = -f'(\xi_{\mbf k}) {i\Gamma_{\mbf k}\over\omega+i\Gamma_{\mbf k}}
\end{equation}
and its imaginary part
\begin{equation}
\mathop{\rm Im}\nolimits\nu_{\mbf k}(\omega) = -f'(\xi_{\mbf k})
{\omega\Gamma_{\mbf k}\over\omega^2+\Gamma_{\mbf k}^2}~.
\label{imnu}
\end{equation}
This can easily be seen by evaluating a bubble with two Greens function
lines for quasiparticles with an imaginary part $\Gamma_{\mbf k}$ of the
self energy.
Note that in the superconducting phase for Raman shifts larger than
$\sim\!\Delta$, the relaxation effects described by (\ref{imnu}) are
also of importance. The relevant relaxation kernel in this case is
\begin{equation}
\nu_{\mbf k}(\omega) = -f'(E_{\mbf k}) {\xi_{\mbf k}^2\over E_{\mbf k}^2}
{i\Gamma_{\mbf k}\over\omega+i\Gamma_{\mbf k}}~,
\label{drude}
\end{equation}
where $\xi_{\mbf k}^2=(\epsilon_{\mbf k}-\epsilon_F)^2$.
To describe the constant background in the Raman spectra in the normal
phase, one has to adopt the quasiparticle scattering rate of the MFL
theory\cite{Var,VarII}
\begin{equation}
\Gamma_{\mbf k}(\omega) \sim \max(\alpha T, \beta\omega)~.
\label{mfl}
\end{equation}
In order to evaluate the real part of $\nu_{\mbf k}$ using causality
arguments, and to prevent divergences, we introduce a high-frequency
cutoff $\omega_C$. Note that the nearly antiferromagnetic Fermi
liquid\cite{MMP,BarPin} (NAFL) and also the nested Fermi
liquid\cite{RuvVir} (NFL) yield a very similar quasiparticle
scattering rate. The former can also provide a mechanism, which
accounts for ${d_{x^2-y^2}}$ pairing. Similar results are obtained with
Luttinger liquid based results.\cite{CarStr}
Equation~(\ref{mfl}) yields a scattering continuum which is constant
for frequencies smaller than $\min(\alpha T/\beta,T)$ and for
frequencies larger than the temperature $T$, but with different
intensities. In the first case, $\Gamma_{\mbf k}$ is proportional to the
temperature, i.e.\ $\mathop{\rm Im}\nolimits\chi\sim\omega/T$. Multiplying by the Bose
factor $1+n_\omega\sim T/\omega$ a constant is found. In the second
case, $\Gamma_{\mbf k}\sim\omega$, and, consequently, $\mathop{\rm Im}\nolimits\chi={\rm
const}$. The Bose factor is also constant and one is left with a
constant Raman intensity. Note that in the first case, $\mathop{\rm Im}\nolimits\chi$
cancels the $\omega$- and $T$-dependence of the Bose factor. It has
been shown,\cite{DonKir,TZhou,VarII} that ${\rm Y}{\rm Ba}_2{\rm
Cu}_4{\rm O}_8$ does not exhibit this behavior. This has been
attributed to the breakdown of MFL theory for not optimally doped
cuprates.\cite{VarII} Actually, in this case the spectra are nearly
temperature independent {\it after} dividing them by the Bose
factor. We shall address this question once more at the end of this
section.
To discuss quasiparticle-quasiparticle (qp-qp) scattering, and its
influence on electronic Raman scattering, we start with the case of a
${d_{x^2-y^2}}$ gap. Suppose the nodes of this gap have a width $\delta_0$
in $k$-space on the Fermi surface due to impurity scattering. We use
the model of Eq.~(\ref{drude}) with a quasiparticle scattering rate
$\Gamma_{\mbf k}$ independent of ${\mbf k}$ and discuss first the case
$T=0$. Then it can be seen that the contribution of qp-qp scattering
to the imaginary part of the Raman susceptibility $(\ref{RamSus})$ for
low frequencies $\omega\ll\Delta_{\rm max}$ is proportional to the
Drude-like factor $\omega\Gamma/(\omega^2+\Gamma^2)$ (which is, for
small $\omega$ and low temperatures $T<\omega$, linear in $\omega$ if
$\Gamma=\hbox{const}$ (semiconductors) or
$\Gamma\sim\max(\omega^2,T^2)$ (FL), but constant as a function of
$\omega$ if $\Gamma\sim\max(\omega,T)$ (MFL). In the tetragonal case,
it is also proportional to the density of states at the Fermi surface
and in the case of $A_{1g}$ and $B_{2g}$ polarizations to the width
$\delta_0$, and in the case of $B_{1g}$ to the third power
$\delta_0^3$ of the width $\delta_0$. The discussion for BISCO is
analogous with the exception that $B_{1g}$ and $B_{2g}$ exchange their
role.
Finite, but small temperatures $T\ll\Delta_{\rm max}$ have the effect
of enlarging the widths $\delta_0$ linearly in temperature, i.e.\ the
temperature dependence of the contribution from qp-qp scattering is
proportional to ${\rm const}+T$. Note that for $T\appgeq0$, the Bose
factor changes the linear-in-$\omega$ dependence to a constant.
For the anisotropic $s$ gap of the form $\abs{{d_{x^2-y^2}}}$ which
acquires a finite minimum gap $\Delta_{\rm min}$ due to the presence
of impurities,\cite{DevImp} the situation is different. The frequency
dependence is also given by the factor
$\omega\Gamma/(\omega^2+\Gamma^2)$ in addition to the Bose factor. But
the temperature dependence is different. For temperatures
$T\ll\Delta_{\rm min}$ smaller than the minimal gap, the density of
quasiparticles is proportional to $\exp(-\Delta_{\rm min}/kT)$, i.e.\
the contribution of qp-qp scattering to the Raman efficiency is
exponentially small. At $kT\approx\Delta_{\rm min}$, this exponential
dependence on $T$ crosses over to a power law.
The background electronic Raman spectrum in the normal phase is almost
independent of temperature for nearly optimally doped high-$T_c$
compounds only. In the overdoped and underdoped case, the materials
seem to show Fermi liquid-like behavior concerning the quasiparticle
scattering rate $\Gamma_{\mbf k}$ (for small $\omega$).\cite{VarII,TZhou}
The temperature dependence of the scattering rate $\Gamma_{\mbf k}$, as
defined in $(\ref{drude})$, has been measured\cite{HacNem} for
optimally doped and overdoped Bi-2212, and, especially in the case of
the $B_{2g}\,(D_{4h})$ mode, the optimally doped sample shows
$\Gamma=\alpha T$, whereas for the overdoped sample
$\Gamma=\alpha'T^2+\Gamma_0$. Therefore, the overdoped sample shows
properties of a normal Fermi liquid which are predicted by theory to
have $\Gamma\sim\max(\omega^2,T^2)$. The $B_{1g}\,(D_{4h})$ mode
result for the optimally doped sample yields the puzzling
quasiparticle scattering rate $\Gamma={\rm const}$.
\section{Experimental spectra}
\label{sec:expts}
The experimental determination of {\it absolute} Raman scattering
intensities is plagued by a number of difficulties (a reason why
usually ``relative units'' are found in the literature). The first is
related to the presence of elastically scattered light in the spectra,
in particular when non-ideal sample surfaces are involved. Depending
on the quality of the spectrometer this leads to contributions
extending typically, for the parameters of the present work, up to
$50\,{\rm cm}^{-1}$ from the center of the laser line. These contributions
can be filtered out using a premonochromator or notch filters but, in
any case, Raman scattering measurements below $50\,{\rm cm}^{-1}$ remain
difficult. The measurements discussed here have been performed by
comparison with the known efficiency of silicon after correcting for
differences in the scattering volumes. The procedure leads to errors
of about 50\%.
We use for comparison with the calculation the experimental data of
Krantz {\it et al.}\cite{KraCar} in the case of Y-123, and Donovan
{\it et al.}\cite{DonKir} in the case of Y-124. Our
Figs.~\ref{Y123exp} and~\ref{Y124exp} are taken from these
publications. In the case of Fig.~\ref{Y123exp} we have corrected a
scale error in the abscissa found in Ref.~\onlinecite{KraCar}. In the
case of Fig.~\ref{Y124exp} we have calculated the $A_{1g}$ component
from the experimental results for the $(x'x')$ and $(xy)$
polarizations.
The classification of the measured spectra according to irreducible
representations of the symmetry group of the crystal is performed with
the use of the Raman tensor $\hat R$ which is related to the Raman
efficiency through the expression $I\sim{|{\mbf e}_L\hat R{\mbf e}_S|}^2$,
bilinear in the Raman tensor. In the calculations, the Raman tensor
does not appear explicitly, the inverse effective mass $\partial^2
E/(\partial k_i\partial k_j)$ playing its role. It is important to
note that the Raman efficiency as given by the theory
(Eqs.~(\ref{rameff}), (\ref{fdt}), and~(\ref{chir})) is bilinear in
the inverse effective mass of the Raman vertex (including the
screening part!), i.e. contains the same interferences as the approach
involving the Raman tensor. Note that the Tsuneto function $\lambda$
is fully symmetric. In the normal phase, the scattering kernel $\nu$
has been assumed to be the same for all scattering channels.
In most of the measurements of the Raman efficiency in orthorhombic
high-$T_c$ superconductors, an $A_{1g}$ component has been
given. Strictly, this irreducible representation does not exist in
$D_{2h}$ but only in $D_{4h}$. In orthorhombic crystals, the Raman
tensor contains two $A_g$ components which correspond to the $A_{1g}$
and $B_{1g}$ components of the tetragonal $D_{4h}$ case, and which are
not distinguishable in $D_{2h}$ because they transform in the same
way. Nevertheless, quantities can be constructed in the orthorhombic
case which correspond to the tetragonal $A_{1g}$ component.
One of these is $I^{(1)}=(I_{xx}+I_{yy})/2-I_{x'y'}$. Both, $I_{xx}$
and $I_{yy}$ contain $A_{1g}$ and $B_{1g}\,(D_{4h})$, and also an
interference term which cancels when $I_{xx}$ and $I_{yy}$ are added.
The $I_{x'y'}$ efficiency contains $B_{1g}$ and $A_{2g}\,(D_{4h})$. If
we assume that the antisymmetric component ($A_{2g}$ in $D_{4h}$) of
the Raman tensor $\hat R$ vanishes (i.e.\ $I_{xy}=I_{yx}$), $I_{x'y'}$
corresponds to tetragonal $B_{1g}$ and cancels the $B_{1g}$
contribution in $I_{xx}$ and $I_{yy}$. Provided that the $A_{2g}$
component of the Raman tensor vanishes, $I^{(1)}$ corresponds to the
$I_{A_{1g}}$ of the tetragonal case. Note that the antisymmetric
compoment $(R_{xy}-R_{yx})/2$ of the Raman tensor vanishes in the
effective mass vertex theory given in Sec.~\ref{sec:gtheory} because
of $\gamma_{xy}=\gamma_{yx}$ regardless of the symmetry of the
crystal, and also in the experiment in the case of tetragonal crystals
but not necessarily for orthorhombic crystals. The equality of
$I_{xy}$ and $I_{yx}$ in the calculation is an artifact of the theory.
A second possible construction for $A_{1g}$ is $I^{(2)}=I_{x'x'}-I_{xy}$.
The $I_{x'x'}$ efficiency contains $A_{1g}$ and $B_{2g}$
contributions. The interference term of these two contributions
vanishes in the tetragonal as well as the orthorhombic case.
Both, $B_{2g}\,(D_{4h})$ and $A_{2g}$ are contained in $I_{xy}$. But
if the $A_{2g}$ component of the Raman tensor vanishes, $I^{(2)}$ also
corresponds to the $I_{A_{1g}}$ of the tetragonal case. In one of the
experimental works\cite{KraCar} a different method to extract the
$A_{1g}$ component was used. Both of the expressions for $I^{(1)}$
and $I^{(2)}$ contain contributions of the $A_{2g}\,(D_{4h})$ Raman
tensor component. This component may be present in the experiment, but
not in the theory, a fact, that has to be kept in mind when comparing
the numerical results to the measurements. Note that the Raman
efficiencies in $(xy)$ and $(x'y')$ polarization configurations also
contain contributions from the antisymmetric part of the Raman tensor.
In view of these uncertainties in $A_{1g}$ we mainly focus in the next
section on the directly observable components of the Raman tensor.
We shall conclude this section by taking up again the question of the
validity of the effective mass approximation. In the experiment, this
can be checked in two ways. First, via the dependence of the spectra
on the laser frequency which should make it possible to distinguish
the contributions to the Raman efficiency resulting from resonant and
non-resonant transitions, respectively. The second way involves the
measurement of the $A_{2g}$ component of the mass. If the effective
mass approximation is valid, the Raman vertex should be symmetric
($\gamma_{xy}=\gamma_{yx}$), i.e.\ the $A_{2g}\,(D_{4h})$ component
should vanishes. A non-vanishing $A_{2g}$ component of the measured
scattering would cast doubts on the appropriateness of the effective
mass approximation.
\section{Numerical results and discussion}
\label{sec:numerics}
To carry out the numerical BZ and FS integrations, we employed a
tetrahedron approach.\cite{LehTau,JepAnd} The convergence of the
integrations was checked by using different meshes. In
Figs.~\ref{Y123} and~\ref{Y124}, the results of full BZ integrations
for Y-123 and Y-124, respectively, are plotted. The corresponding
spectra obtained through FS integrations can be seen in
Ref.~\onlinecite{CarStr}. The Bose factor has not been included, hence
the results apply to zero temperature. In both figures, the Raman
shift is given in units of the gap amplitude $\Delta_0$. Since the
calculated scattering efficiencies for BZ integrations, contrary to FS
integrations, are not only a function of the reduced frequency but
depend also weakly on the value of $\Delta_0$, we took for the
calculations $\Delta_0=220\,{\rm cm}^{-1}$. This value of $\Delta_0$ falls in
the range of $\Delta_0$'s determined by Raman scattering and other
methods. The delta-function peaks in the Tsuneto function have been
broadened phenomenologically by introducing a finite imaginary part
$\Gamma=0.3\Delta_0$ of the frequency variable $\omega$.
Figures~\ref{Y123} and~\ref{Y124} display spectra for each of the
polarization configurations $(yy)$, $(x'x')$, $(xx)$, $(x'y')$, and
$(xy)$, as well as the symmetry component $A_{1g}\,(D_{4h})$ (defined
by $I_{A_{1g}}=I_{x'x'}-I_{xy}$), the unscreened intensities, the
screening part $(\ref{scrpart})$, and the total intensities, equal to
the difference between unscreened and screening parts. Note that the
$(x'y')$ configuration corresponds to the $B_{1g}\,(D_{4h})$ component
because of the vanishing of the $A_{2g}$ component in the theory.
We discuss first the results for Y-123. The $A_{1g}$ component (in
the rest of this section we use tetragonal notation unless explicitly
stated) is subject to rather strong screening, however its unscreened
part is comparable to that of the $B_{1g}$ component. The relation
between the unscreened and the screened (total) spectral weight of the
$A_{1g}$ component is about three. Nevertheless, the shapes of the
unscreened and the screened parts are the same and, consequently, {\it
there is almost no shift in the peak position due to screening}
(contrary to the results of Ref.~\onlinecite{DevEin}). The peak is
located almost exactly at $2\Delta_0$. Note that there is no
antiscreening in the $A_{1g}$ component. The low-energy part of all
$A_{1g}$ spectra (screened and unscreened) is linear, as predicted by
the theory.
As already mentioned, the $(x'y')$ component (equal to the $B_{1g}$
component in the non-resonant case) is almost four times stronger than
its screened $A_{1g}$ counterpart. The screening is very small, its
nonvanishing being an effect of the distorted tetragonality of the
crystal. There is, in this case, a very small amount of antiscreening
in the region below $2\Delta_0$. As in the case of the $A_{1g}$
component, the $(x'y')$ component peaks at almost exactly the
$2\Delta_0$ frequency shift. The low-frequency part has an
$\alpha\omega+\beta\omega^3$ frequency dependence, the linear part
arising from the distorted tetragonality, i.e.\ the fact that the
$B_{1g}$ mass does not vanish at exactly the same position on the
Fermi surface as the gap function does.
The efficiency of the peak in the $(xy)$ configuration (equal to the
$B_{2g}$ component in the non-resonant case) is also four or five
times smaller than that of the $A_{1g}$ peak. The $(xy)$ peak is
located at about $1.3\Delta_0$, as expected from the fact that in the
neighborhood of the region where the gap is large, the $B_{2g}$ mass
vanishes. Consequently, the peak is not as sharp as in the former
cases and screening vanishes since these spectra correspond to a
nonsymmetric ($B_{1g}$) representation of the orthorhombic group
($D_{2h}$).
In the $A_{1g}$ and $(x'y')$ spectra there should be a small peak at
about $\omega=2\sqrt{\epsilon_{\rm vH}^2+\Delta_{\rm
max}^2}\approx3.9\Delta_0$ due to the van Hove singularity on the
$k_x$-axis near the $X$ point. The corresponding structure, however,
is very weak, and practically invisible in Fig.~\ref{Y123}. This is
not unexpected for a 3D calculation. These peaks appear strongly when
2D calculations are performed through BZ integrations.\cite{BraCar}
In general, the efficiencies in Y-124 (Fig.~\ref{Y124}) are about a
factor of three less than those for its Y-123 counterpart. Moreover,
the screening of the $A_{1g}$ component of Y-123 is much stronger than
that of Y-124. This may be, at least in part, due to the additional
chain band: The $(yy)$ component of Y-124 is less screened than the
$(yy)$ component of Y-123. At low frequencies, we correspondingly have
antiscreening even in $A_{1g}$, a fact which reveals a change of sign
of the effective mass on the Fermi surface (see
Sec.~\ref{sec:stheory:signch}). Due to this antiscreening, the peak
in the $A_{1g}$ spectrum is shifted from $2\Delta_0$ towards
approximately $1.6\Delta_0$. In contrast to the situation in Y-123,
the Y-124 spectra show clearly the influence of the van Hove
singularity on the spectra, as a small hump (vH) located near
$2\sqrt{\epsilon_{\rm vH}^2+\Delta_{\rm max}^2}\approx7\Delta_0$. In
the $A_{1g}$ spectrum this hump is almost screened out whereas in the
$(x'y')$ spectrum it appears slightly increased by the influence of
antiscreening.
To compare these predictions with the experiment let us first focus on
the peak positions. The experimental results for Y-123
(Fig.~\ref{Y123exp}, lower part) clearly show that the position of the
$(yy)$, $(x'x')$ and $(xx)$ peaks is at about $300\,{\rm cm}^{-1}$, whereas
the $(x'y')$ peak is located at $600\,{\rm cm}^{-1}$, i.e.\ at twice the
frequency of the former. This fact is in sharp contrast with the
calculated spectra and has been at the center of the controversy
concerning the topic at hand.\cite{KCComm,DevRepl} It has been
suggested by Devereaux {\it et al.}\cite{DevEin,DevRepl} that the
$B_{1g}$ component peaks at $2\Delta_0$, and the $A_{1g}$ component
becomes shifted down to almost $\Delta_0$ by the screening. This
interpretation contradicts our numerical results which clearly suggest
that the influence of screening on the position of the $A_{1g}$ mode
is usually smaller. The frequency renormalizations of phonons around
$T_c$ also seem to contradict the interpretation in
Refs.~\onlinecite{DevEin} and~\onlinecite{DevRepl}. It has been
shown\cite{FriTho} that lowering the temperature of the sample in the
superconducting phase causes the $A_{1g}$ $435\,{\rm cm}^{-1}$ phonon
(plane-oxygen, in-phase) to shift up in frequency and the $B_{1g}$
($D_{4h}$ notation) $340\,{\rm cm}^{-1}$ phonon (plane-oxygen, out-of-phase)
to shift down. This, in turn, implies an amplitude of the gap
$2\Delta_0$ between $300\,{\rm cm}^{-1}$ and $360\,{\rm cm}^{-1}$ and is consistent
with our interpretation of the electronic Raman spectra with the
$A_{1g}$ peak at $2\Delta_0$.
Note that the $(yy)$, $(x'x')$ and $(xx)$ spectra do {\it not} contain
contributions of the $A_{2g}\,(D_{4h})$ antisymmetric component of the
Raman tensor while the $(x'y')$ component does. So, the experimental
results may suggest that the shift of the position of the $(x'y')$
spectrum with respect to the peak position of the other spectra is
due to resonance effects. The $(xy)$ spectrum is also influenced by
the $A_{2g}$ component. It is difficult to determine its peak position
from Fig.~\ref{Y123}, but it seems to be located at the same position
as that of the $(yy)$, $(x'x')$ and $(xx)$ configurations. The
calculation predicts it to be located at about $1.3\Delta_0$, the
shift to $2\Delta_0$ can also be attributed to the existence of an
$A_{2g}$ component, like in the case of the $(x'y')$ configuration.
To compare the relative intensities of the spectra with different
polarizations, we refer to Table~\ref{peakh}, which lists them
together with the corresponding absolute intensities, both at the peak
position. The detailed results of our FS integration have already been
reported earlier.\cite{CarStr} We begin with Y-123 (upper panel in
Table~\ref{peakh}) and compare BZ integration results to the
experimental ones. With the possible exception of the $A_{1g}$
component (and the $(x'x')$ component, which is very similar to
$A_{1g}$), the agreement is rather good. The deviation of the $A_{1g}$
component may be attributed to screening, which is very sensitive
to sign changes and other details of the Raman vertex near the Fermi
surface (such as details of the band structure and especially the
exact position of the Fermi energy).
The second compound, Y-124 (lower panel in Table~\ref{peakh}), also
shows reasonable agreement between the results of the BZ integration
and the experiment. However, we also have problems with the $A_{1g}$
component, as we did for Y-123.
The measured absolute intensities agree particularly well with the
calculations in the case of Y-123. With the exception of $A_{1g}$, the
discrepancy between theory and experiment is only a factor of two,
which can easily be related to the difficulties in measuring absolute
scattering cross sections. In the case of Y-124, the discrepancy is a
bit larger, but a factor of four can still be considered good.
We should also keep in mind that resonances of $\omega_L$ or
$\omega_S$ with virtual interband transitions are expected to enhance
the simple effective mass Raman vertex, a fact which could also
explain why the measured scattering efficiencies are usually larger
than the calculated ones.
We close the discussion of the numerical results with a remark about
the Fermi surface integration. For Y-124, the results of the former
correspond rather closely to the results from the BZ integration. The
situation is different for Y-123. Here, the $(xx)$ peak height is
almost a factor four larger in the FS integration than in the BZ
integration. This is likely to result from the close proximity of the
van Hove singularity to the FS in the case of Y-123 ($25\,\meV$), as
compared to Y-124 ($110\,\meV$).
To verify the predictions related to the effect of orthorhombic
distortions as discussed in Sec.~\ref{sec:stheory:odist}, we performed
a fit of the function $\alpha\omega+\beta\omega^3$ to the
low-frequency part of the $B_{1g}$ data for Y-123 reported in
Ref.~\onlinecite{KraCar} and Ref.~\onlinecite{Hacetal} as well as for
Bi-2212 (taken from Ref.~\onlinecite{Staetal}) and to the results of
our numerical calculations for \hbox{Y-123}. The ratios of the cubic
vs.\ the linear part (at $\omega=300\,{\rm cm}^{-1}$) of the fit to
the low-frequency efficiency are given in Table~\ref{peakh}.
Both measurements for Y-123 agree in their large linear part, which
should be due mainly to the lack of exact tetragonality and the
presence of impurities. The results of the BZ integration show a
smaller linear part, because they do not take into account the
influence of impurities. Finally, the result for Bi-2212 is completely
different from the former results for Y-123. The linear part almost
vanishes, in agreement with the preceding discussion.
\section{Conclusions}
\label{sec:concl}
In spite of the striking ability to predict not only general features
of the observed spectra but also their peak intensities, our
calculations are not able to predict the relative positions of the
$A_{1g}$ and $B_{1g}$ peaks. According to Figs.~\ref{Y123}
and~\ref{Y124} the $A_{1g}$ spectrum should peak only slightly below
$2\Delta_0$ while $B_{1g}$ should peak at $2\Delta_0$. The
experimental data of Figs.~\ref{Y123exp} and~\ref{Y124exp}, however,
indicate that the $B_{1g}$ spectra peak nearly at twice the frequency
of $A_{1g}$. Since the observed $A_{1g}$ peak is considerably sharper
than that of $B_{1g}$, we may want to assign the $A_{1g}$ peak to
$2\Delta_0$. Our calculations show that it is impossible to reproduce
both peak frequencies with a simple gap of the form
$\Delta_0\cos2\phi$ where $\phi$ is the direction of the
${\mbf k}$-vector. A reasonable fit was obtained in
Ref.~\onlinecite{KraCar} with a two-dimensional FS which did not take
into account the chain component and assigned $d$- and $s$-like gaps
to the two bonding and antibonding sheets of the FS of the two planes
in an {\it ad hoc} way. Within the present 3-dimensional band
structure the FS cannot be broken up into bonding and antibonding
plane and chain components since such sheets are interconnected at
general points of $k$-space. It is nevertheless clear that there is no
reason why the gap function should be the same in the various sheets
for a given ${\mbf k}$-direction. Thus the remaining discrepancy in the
peak positions between theory and experiment could be due to a more
complicated $\Delta_{n{\mbf k}}$ than a simple $\Delta_0\cos2\phi$ used
here. Another possible source of this discrepancy is scattering
through additional excitations of a type not considered here (e.g.\
magnetic excitations) contributing to and broadening the $B_{1g}$
peak.
A BCS-like theory, which involves an attractive pairing potential as
well as the repulsive Coulomb potential and uses an anisotropic
${d_{x^2-y^2}}$-like gap function in connection with the effective mass
approximation used in the calculation of the absolute Raman scattering
efficiencies yields result which are in significant agreement with the
experimental spectra. One exception, the peak positions of the
$A_{1g}$ and the $B_{1g}$ components, remains unexplained. The theory
predicts them to be both located near $\omega=2\Delta_0$, but the
experiment shows the peak in $B_{1g}$ at almost twice the frequency of
the peak in $A_{1g}$. The weak $B_{2g}$ spectrum agrees in intensity
and peak position with calculations for a ${d_{x^2-y^2}}$-like gap. The
results of other experiments, involving the temperature dependence of
phonon frequencies,\cite{FriTho} suggest that the $A_{1g}$ peak
position corresponds to the gap amplitude $2\Delta_0$. The shifting of
the $B_{1g}$ peak towards higher frequencies may have an origin
different from the mass-fluctuation-modified charge-density
excitations described in the theoretical part of this paper but could
also be due to a multi-sheeted gap function, more complicated than the
simple ${d_{x^2-y^2}}$-like $\Delta_0\cos2\phi$ gap assumed in our
calculations. The initial variation of the $A_{1g}$ and $B_{1g}$
scattering efficiencies vs.\ $\omega$ are linear as expected for that
gap. The $B_{1g}$ symmetry becomes $A_g$ in the presence of the
orthorhombic distortion related to the chains. Consequently, the
scattering efficiency at low frequency is not proportional to
$\omega^3$ but should have a small linear component which is found
both in the calculated and the measured spectra. In the corresponding
spectrum of Bi-2212, with and orthorhombic distortion along $(x+y)$,
the $B_{1g}$ ($D_{4h}$) excitations also have a nonsymmetric $B_{1g}$
($D_{2h}$) orthorhombic character. Consequently, for small $\omega$ no
component linear in $\omega$ is found in the measured spectra.
We have performed our calculations using either BZ or FS integration.
In the case of Y-124 the spectra so obtained are very similar. For
Y-123 quantitative differences appear; they are probably related to
the presence of a van Hove singularity close to the FS. These
singularities appear as weak structures in the calculated spectra, as
expected for a 3D band structure.
\acknowledgments
We thank Jens Kircher for providing us with the LMTO band structures.
One of us (TS) also would like to thank his colleagues at the MPI for
numerous discussions on Raman scattering and superconductors. Thanks
are specially due to Igor Mazin for a critical reading of the
manuscript.
| proofpile-arXiv_065-489 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
Among many versions of the scalar-tensor theory of gravitation, the
prototype Brans-Dicke (BD) model [\cite{bd}] is unique for the
following four assumptions on the scalar field: (i) a nonminimal coupling of the simplest form; (ii) masslessness; (iii) no self-interaction; (iv) no direct matter coupling.
Although the model may not be fully realistic, it still seems to deserve further scrutiny as a testing ground of many aspects of wider class of the scalar-tensor theories. The model has been studied,
however, mainly as a classical theory. We attempt to take
quantum effects due to the matter coupling into account. It has
been argued that composition-independence that entails from the
assumption (iv) above would be violated as a quantum correction [\cite{yf1}]. We
came to realize, however, that the suspected contribution is canceled by other
terms arising from regularization;\footnote{More details on this analysis will be reported in another publication.} WEP is in fact a well-protected
and robust property of the
model beyond the classical level.
We discuss in this note another quantum effect which, as it turns out, has serious
cosmological consequences, beyond the extent of remedy expected by adjusting
the fundamental parameter of the model. For a possible way out we suggest to modify the assumption
(iv), making the scalar field almost ``invisible," thus reconciling with the absence of experimental evidences, still playing a cosmological role. We also discuss how the scalar field acquires a nonzero
self-mass due to the matter coupling, even having started with
the {\em classical}
assumption (ii), but likely in an entirely insignificant manner in practice.
We start with the basic Lagrangian
\begin{equation}
{\cal L}=\sqrt{-g}\left( \frac{1}{2} F(\phi) R
-\epsilon\frac{1}{2} g^{\mu\nu}\partial_{\mu}\phi \partial_{\nu}\phi
+L_{\rm m}\right),
\label{bd5_51}
\end{equation}
with
\begin{equation}
F(\phi)=\xi\phi^2.
\label{bd5_52}
\end{equation}
We use the unit system of $8\pi G=1.$\footnote{The units of
length, time and energy are $8.07\times 10^{-33}$cm, $2.71\times
10^{-43}$ sec, and $2.43\times 10^{18}$ GeV, respectively. Notice
also that the present age of the Universe is $\sim 10^{60}$.} The
scalar field $\phi$ and the constant $\xi$ are related to the original notation $\varphi$ and $\omega$, respectively, by
\begin{equation}
\varphi =\frac{1}{2}\xi\phi^2, \quad{\rm and }\quad \omega =\frac{1}{4\xi}.
\label{bd5_53}
\end{equation}
We also allow $\epsilon =\pm 1$, a minimum extension of the original model to avoid an immediate failure. As we see shortly, $\epsilon =-1$ does
not necessarily imply a ghost in the final result.
As the matter Lagrangian we choose, according to the assumption (iv),
\begin{equation}
L_{\rm m}= -\overline{\psi} \left( D\hspace{-.65em}/ +m_{0} \right)\psi,
\label{bd55_4}
\end{equation}
where $\psi$ stands for a simplified representative of the (spinor) matter
field.
In spite of (iv) $\phi$ couples in effect to the matter field {\em in the field equation.} This is inconvenient, however, when we try to apply the conventional technique of quantum field theory to the $\phi\,$-matter coupling. For this reason we apply a conformal transformation such that the nonminimal coupling is eliminated [\cite{dicke}]:
\begin{equation}
g_{\mu\nu}=\Omega^{-2}g_{*\mu\nu},\quad{\rm with}\quad \Omega =\left(\xi\phi^2\right)^{1/2}.
\label{bd55_5}
\end{equation}
Notice that we should have
\begin{equation}
\xi >0,
\label{bd55_5a}
\end{equation}
in order to ensure that the sign of the line element remains unchanged.
We in fact find that (\ref{bd5_51}) is now put into the form
\begin{equation}
{\cal L}=\sqrt{-g_{*}}\left( \frac{1}{2} R_{*} -\frac{1}{2} g^{\mu\nu}_{*}\partial_{\mu}\sigma\partial_{\nu}\sigma
+L_{*m} \right),
\label{bd3-16}
\end{equation}
expressed in terms of the new metric $g_{*\mu\nu}$ and the field $\psi_{*}=\Omega^{-3/2}\psi$. Also the {\em canonical} scalar field is now $\sigma$ as defined by
\begin{equation}
\phi(\sigma)=\phi_{0}\;e^{\beta\sigma},
\label{bd5-2}
\end{equation}
where
\begin{equation}
\beta =\frac{1}{\sqrt{6+\epsilon\xi^{-1}}}
=\sqrt{\frac{4\pi G}{3+2\epsilon\omega}}.
\label{bd5-2a}
\end{equation}
We emphasize that $\sigma$ is not a ghost if
\begin{equation}
\beta^2 >0,
\label{bd5-2b}
\end{equation}
even if $\epsilon =-1$ [\cite{fn}]; ``mixing" between the scalar field and (spinless part of) the metric field provides sufficient amount of positive contribution to overcome the negative kinetic part. The condition (\ref{bd5-2b}) is
equivalent to
\begin{equation}
\epsilon\xi^{-1} > -6,\quad {\rm or}\quad \epsilon\omega >-\frac{3}{2}.
\label{bd5-2c}
\end{equation}
We now have a direct $\sigma$-matter coupling expressed in terms of
the interaction Hamiltonian to which usual perturbation method is
readily applied. We point out, however, that EEP, hence WEP as well, remains intact because the deviation from geodesic arising from the transformation (\ref{bd55_5}) is given entirely in terms of $\sigma$ which is independent of any specific properties of individual particles; the fact that any motion is independent of the mass, for example, verified in one conformal frame obviously survives conformal transformations.
We call the conformal frames before and after the conformal
transformation J frame and E frame, respectively.\footnote{The name J
frame is used, following Cho's suggestion [\cite{cho}], after
P. Jordan [\cite{jd}] who was the first to discuss the nonminimal coupling. On the other hand, E frame is a reminder that this is a frame in which the standard theory of Einstein is formulated.}
\section{Quantum effect}
In E frame, in which we hereafter suppress the symbol $*$ for simplicity, we consider one-loop diagrams as shown in Fig. 1, due to the interaction
\begin{equation}
H'_{1}=m_{0}\xi^{-1/2}\phi^{-1}(\sigma)\overline{\psi}\psi,
\label{bd5-1}
\end{equation}
coming originally from the mass term in J frame.
As will be shown, $\sigma$ may keep moving with time, and so does the mass. But $\sigma$ moves so slowly compared with any of the microscopic time-scales, that the mass $m(t)$ {\em at each epoch} will be defined by
\begin{equation}
m(t)=m_{0}\xi^{-1/2}\phi^{-1}(\sigma(t)).
\label{bd5-3}
\end{equation}
Now consider the 1\hspace{.2em}-$\phi^{-1}$ diagram (a). Its contribution is given by the potential of $\sigma$;
\begin{equation}
V_{1}(\sigma)=im(\sigma)\int d^4 p \,{\rm Tr}\!
\left(\frac{1}{m(\sigma)+ip\hspace{-.4em}/}\right).
\label{bd5-4}
\end{equation}
Our consideration will be restricted to the Universe which is
sufficiently late to justify to ignore the effect of temperature and
spacetime curvature.\footnote{The temperature will be lower than $\sim
$TeV, for example, if $t\mbox{\raisebox{-.3em}{$\stackrel{>}{\sim}$}} 10^{-13}$sec, much earlier than the epoch of nucleosynthesis. Spacetime curvature will be important only for $t\mbox{\raisebox{-.3em}{$\stackrel{<}{\sim}$}} 10^{-28}$sec, corresponding to the temperature of $10^{11}$GeV.}
The integral in (\ref{bd5-4}) is quadratically divergent, but is expected to vanish if there is supersymmetry because the fermionic contribution given by (\ref{bd5-4}) is canceled by the same contribution from the bosonic partner. The cancellation would not be complete, however, if supersymmetry is broken at the mass scale
\begin{equation}
M_{\rm ssb}=rm,
\label{bd5-5}
\end{equation}
where the ratio $r$ of $M_{\rm ssb}$ to $m$, a representative of
ordinary particles taken roughly of the order of GeV, would be $\sim
10^3$--$10^4$, which we naturally choose to be a true constant.
The result may be given by
\begin{equation}
V_{1}(\sigma)=C_{1}r^2 m^4
=V_{1}(0)e^{-4\beta\sigma},
\label{bd5-6}
\end{equation}
where
\begin{equation}
V_{1}(0)=C_{1}r^2\left( m_{0}\xi^{-1/2}\phi_{0}^{-1} \right)^4,
\label{bd5-6a}
\end{equation}
with $C_{1}$ most likely of the order 1. Its sign, however, may not be known precisely because it depends on the details of supersymmetry breaking. If $C_{1}>0$, (\ref{bd5-6}) gives a positive exponential potential that would drive $\sigma$ toward infinity. We assume this to be the case.
\section{Cosmology}
We now consider the cosmological equations with a classical
potential $V=V_{1}$ which is the only potential \vspace{.3em}due to the assumption (iii):
\begin{eqnarray}
&&3H^2 =\frac{1}{2} \dot{\sigma}^2+V +\rho, \label{bd5-7}\vspace{.5em}\\
&&\ddot{\sigma}+3H\dot{\sigma}+V'(\sigma)=0,\vspace{.3em}\label{bd5-8}\\
&&\dot{\rho}+4H\rho =0,\label{bd5-9}
\end{eqnarray}
where $\rho$ is the matter density which we assume, for the moment, to
be relativistic. We also ignored possible terms representing the
coupling between $\sigma$ and $\rho$. This would be justified for our
purposes as long as we consider late epochs during which the coupling
is sufficiently weak [\cite{fn}].
With $V=V_{1}$, a set of analytic solutions of (\ref{bd5-7})--(\ref{bd5-9}) are obtained:
\begin{eqnarray}
a(t)&=& t^{1/2},\label{bd5-10}\\
\sigma(t)&=&\frac{1}{2\beta}\ln t,\label{bd5-11}\\
\rho(t)&=& \frac{3}{4}\left( 1- \frac{1}{4}\beta^{-2} \right)t^{-2},
\label{bd5-12}
\end{eqnarray}
with
\begin{equation}
V_{1}(0)=\frac{1}{16\beta^2}.
\label{bd5-12a}
\end{equation}
Notice that the condition $\rho >0$ is met if
\begin{equation}
\beta^{-2}<4,\quad {\rm or}\quad \epsilon\xi^{-1}<-2,\label{bd5-13}
\end{equation}
which, combined with (\ref{bd55_5a}), is satisfied only if $\epsilon =-1$. This is the reason why we decided to allow an ``apparent" ghost in J frame. Then (\ref{bd5-13}) translates into
\begin{equation}
\xi^{-1}>2,\quad{\rm or}\quad \omega > \frac{1}{2},
\label{bd5-13a}
\end{equation}
a much milder constraint than those derived from the observation.
With $\epsilon =-1$, however, (\ref{bd5-2c}) implies
\begin{equation}
\xi^{-1}< 6,\quad{\rm or}\quad \omega < \frac{3}{2},
\label{bd5-13b}
\end{equation}
which is ruled out immediately by the solar-system experiments, giving $\omega\mbox{\raisebox{-.3em}{$\stackrel{>}{\sim}$}} 10^3$ [\cite{will}]. We nevertheless continue our analysis as long as theoretical consistency is maintained.
From (\ref{bd5-11}) also follows
\begin{equation}
\phi(t) =\phi_{0}t^{1/2},
\label{bd5-15}
\end{equation}
where $\phi_{0}$ is determined by identifying (\ref{bd5-6a}) with
(\ref{bd5-12a}); $\phi_{0}^4=16 C_{1}r^2 m_{0}^4 \beta^2 \xi^{-2}$,
which, if used in (\ref{bd5-3}), yields
\begin{equation}
m(t)= \frac{1}{2} \left( C_{1}r^2 \beta^2 \right)^{-1/4}t^{-1/2}.
\label{bd5-16}
\end{equation}
It is interesting to notice that the behavior $m(t)\sim t^{-1/2}$ follows simply because the potential should be proportional to $m^{4}$, as shown in (\ref{bd5-6}), and it must decay like $t^{-2}$ because it is part of the energy density appearing on the right-hand side of the 00-component of the Einstein equation.
Obviously the assumption $r=\:$const is crucial in the above argument. We could obtain $m=\:$const if $r(t)\sim t^{-1}$, but with a highly unreasonable consequence that $r$ should be as large as $\sim 10^{63}$ at the Planck time. Also the dependence $V\sim m^4$ is common to any diagrams of many $\phi^{-1}$'s, as in (b) and (c) in Fig. 1. Including them results simply in affecting the overall size of the potential.
We arrived at (\ref{bd5-16}) in E frame in which the standard technique of quantum field theory can be applied. We should also notice, however, that we use some type of microscopic
clocks in most of the measurements. The time unit of atomic clocks, for example, is provided typically by the frequency $\sim m\alpha^4.$
It is also important to recognize that the cosmic time is usually assumed to be measured in the same time unit.
If the time unit $\tau(t)$ itself changes with time, the new time $\tilde{t}$ measured in this unit would be defined by
\begin{equation}
d\tilde{t}=\frac{dt}{\tau(t)}.
\label{bd5-20}
\end{equation}
In conformity with special relativity, the scale factor $a(t)$ in Robertson-Walker cosmology is transformed in the same manner:
\begin{equation}
\tilde{a}=\frac{a(t)}{\tau(t)}.
\label{bd4-3}
\end{equation}
These two relations can be combined to a conformal transformation
\begin{equation}
d\tilde{s}^2 =\tau^{-2}ds^2, \quad{\rm or}\quad
g_{\mu\nu}=\tau^2 \tilde{g}_{\mu\nu}.
\label{bd4-4}
\end{equation}
In the prototype BD model, there is no mechanism to make $\alpha$
time-dependent, hence $\tau\sim m^{-1}$. Combining this with $m\sim
\phi^{-1}$ as derived from (\ref{bd5-3}), and also comparing
(\ref{bd4-4})
with (\ref{bd55_5}), we find that (\ref{bd4-4}) implies going back
to the original J frame, as it should because it is the frame in which mass $m_{0}$ is taken to be constant.
We now try to solve the cosmological equations in J frame, in which we also suppress tildes to simplify the notation. It is also interesting to find that the behavior $V_{1}\sim \phi^{-4}$ as indicated in (\ref{bd5-6}) shows that this potential in E frame can be derived from a cosmological constant in J frame as given by
\begin{equation}
\Lambda =C_{1}r^2 m_{0}^4.
\label{bd8_4}
\end{equation}
{\em The quantum effect computed in E frame amounts to introducing $\Lambda$ back in the original conformal frame.} The field equations in J frame are given by
\begin{eqnarray}
2\varphi G_{\mu\nu}&=& T_{\mu\nu}+T_{\mu\nu}^{\phi}-g_{\mu\nu}\Lambda
-2\left( g_{\mu\nu}\mbox{\raisebox{-0.2em}{\large$\Box$}} -\nabla_{\mu}\nabla_{\nu} \right)\varphi,
\label{bd8_6}\\
\mbox{\raisebox{-0.2em}{\large$\Box$}}\varphi &=& \beta^2 (T -4\Lambda),\label{bd8_6a}\\
\nabla_{\mu}T^{\mu\nu}&=& 0.\label{bd8_7}
\end{eqnarray}
Notice also that $T_{\mu\nu}$ is the matter energy-momentum tensor while
\begin{equation}
T_{\mu\nu}^{\phi}=\epsilon\left( \partial_{\mu}\phi\partial_{\nu}\phi
-\frac{1}{2} g_{\mu\nu}\left( \partial \phi \right)^2\right) .
\label{bd8_8}
\end{equation}
Assuming spatially uniform $\phi$, we derive the cosmological equations:
\begin{eqnarray}
6\varphi H^2&=& \epsilon\frac{1}{2} \dot{\phi}^2 +\Lambda +\rho -6 H\dot{\varphi},\label{bd8_9z}\\
\ddot{\varphi}+3H\dot{\varphi}&=&4\beta^2 \Lambda,\label{bd8_9}\\
\dot{\rho} +4 H{\rho}&=&0,\label{bd8_10}
\end{eqnarray}
where we have chosen $T=0$ confining ourselves to the radiation-dominated era even if $\Lambda$ comes from nonzero $m_{0}$.
As a heuristic approach, let us choose
\begin{equation}
H=0.
\label{bd8_11}
\end{equation}
Then (\ref{bd8_10}) leads to
\begin{equation}
\rho = {\rm const}.
\label{bd8_12}
\end{equation}
Using (\ref{bd8_11}) in (\ref{bd8_9}) we obtain $\ddot{\varphi}= 4\beta^2 \Lambda,$ which allows a solution
\begin{equation}
\varphi(t)=2 \beta^2 \Lambda t^2 +\varphi_{1}t +\varphi_{0}.
\label{bd8_14}
\end{equation}
Notice that we have chosen $\Lambda >0$ hence $C_{1}>0$ so that
$\sigma$ falls off the potential slope toward infinity in E frame. This implies that
$\varphi$ increases also in J frame as indicated in (\ref{bd5-2}) if
$\beta^2 >0$. This is the very condition, however, which is in contradiction with the observation, as already discussed in connection with (\ref{bd5-13b}). Taking aside this drawback for the moment again, we expect
\begin{equation}
\phi\approx \sqrt{\frac{4\Lambda}{6\xi -1}}\,t,
\label{bd8_15}
\end{equation}
at sufficiently late times. Using this together with (\ref{bd8_11}) and (\ref{bd8_12}) in (\ref{bd8_9z}) gives
\begin{equation}
\rho = 3\Lambda \frac{1-2\xi}{6\xi -1},
\label{bd8_15a}
\end{equation}
which is positive if (\ref{bd5-13a}) and (\ref{bd5-13b}) are obeyed.
An example is shown in Fig. 2, in which we see how $H$ approaches zero, much faster than $t^{-1}$; the Universe quickly becomes stationary after alternate occurrences of expansion ($H>0$) and contraction ($H<0$). This together with other similar examples indicate strongly that there is an ``attractor" to
which solutions of different initial conditions would approach
asymptotically. Fig. 3, which is a 2-dimensional cross section of the 3-dimensional phase space of $\varphi,\;\dot{\varphi}$ and $\rho$, illustrates how different solutions are attracted to a common destination given by (\ref{bd8_15}) and (\ref{bd8_15a}), which represent in fact a curve in the whole phase space as one finds because of the relation $\dot{\phi}^2=\dot{\varphi}^2/(2\xi\varphi)$. A trajectory for a set of initial values proceeds along, spiraling around and coming ever closer to this curve.
One might be tempted to compare our solutions with those in Einstein's model with a negative cosmological constant $\Lambda <0$, but of course without $\phi$. This model allows a static solution $\rho = -\Lambda$ and $H=0$, but the Universe would never become stationary in contrast to our solutions; $\rho$ with a sufficiently large initial value decreases toward the minimum $-\Lambda$ but bounces back to increase in a touch-and-go fashion.
In this way we come to conclude that the BD model corrected for an important quantum effect should result in a steady state Universe, which is totally unacceptable in view of the success of the standard model in understanding primordial nucleosynthesis.
The constant scale factor in J frame may be interpreted also from the analysis in E frame, in which the length unit is provided by $\tau\sim\phi\sim
t^{1/2}$ which increases {\em in the same rate} as $a\sim t^{1/2}$ shown in
(\ref{bd5-10}) [\cite{nth}]. We also find that $\phi\sim t$ as given by
(\ref{bd8_15}) in J frame and $\phi\sim t_{*}^{1/2}$ in
(\ref{bd5-15}), in which the symbol $*$ is restored in E frame, are
consistent with each other, since $t\sim t_{*}^{1/2}$ is a consequence of the relation (\ref{bd5-20}).\footnote{Apply the replacement,
$\tilde{t}\rightarrow t, t\rightarrow t_{*}.$ For the dust-dominated Universe with $a_{*}\sim t_{*}^{2/3}$, we find $a\sim t^{1/6}.$} These observations seem
to support (\ref{bd5-9}) which is only approximate unlike its counter
part (\ref{bd8_10}) in J frame.
On the other hand, one may ask if there is any sensible solution with
$a\sim t^{1/2}$ in J frame. In (\ref{bd8_10}) we substitute
\begin{equation}
H=(1/2)t^{-1},
\label{bd10_5}
\end{equation}
thus obtaining
\begin{equation}
\rho =\rho_{0}t^{-2}.
\label{bd10_6}
\end{equation}
Then (\ref{bd8_9}) becomes
\begin{equation}
\ddot{\varphi}+\frac{3}{2}t^{-1}\dot{\varphi} =4\beta^2 \Lambda,
\label{bd10_7}
\end{equation}
which is solved asymptotically:
\begin{equation}
\varphi\approx \frac{4}{5}\beta^2 \Lambda t^2, \quad
{\rm or}\quad \phi\approx \sqrt{\epsilon\frac{8}{5}\left( 1
-6\beta^2 \right) \Lambda }\,t.
\label{bd10_8}
\end{equation}
We then find that the right-hand side of (\ref{bd8_9z}) is given by
\begin{equation}
\rho +\frac{3}{5}\left( 3-16\beta^2 \right)\Lambda.
\label{bd10_9}
\end{equation}
Now from (\ref{bd10_5}) and (\ref{bd10_8}), the left-hand side of
(\ref{bd8_9z}) should be time-independent. This can be matched with the situation in which $\rho$ given by (\ref{bd10_6}) decreases rapidly to be negligible compared with the second term of (\ref{bd10_9}); implying that the Universe becomes asymptotically ``vacuum dominated,'' again an unrealistic conclusion. Even
worse, ignoring $\rho$ in the right-hand side of (\ref{bd10_9}) and
using (\ref{bd10_8}) on the left-hand side of (\ref{bd8_9z}) yields
\begin{equation}
\beta^2 =\frac{1}{6},
\label{bd10_10}
\end{equation}
which on substituting into (\ref{bd5-2a}) gives
\begin{equation}
\xi^{-1}=0,
\label{bd10_11}
\end{equation}
hardly a realistic result.\footnote{The same analysis applied to the dust matter results in $\epsilon =-1$ and $\xi =-2/3$, being inconsistent with (\ref{bd55_5a}).}
\section{Discussions}
We add that our argument of choosing J frame is independent of whether we literally use atomic clocks to measure something during
the epoch in question. It is simply in accordance with realistic situations that analyses are based on quantum mechanics in which mass of every particle is taken to be truly constant.
We admit that there should be some other quantum effects to be included. The result obtained here is, however, so remote from what would be expected from the standard scenario that it is highly unlikely that those ``other" effects conspire miraculously to restore the success in the nucleosynthesis, among other things. It seems that we need some more fundamental modification of the model.
A possible way out is to abandon the assumption (iv) about the absence of the {\em direct} $\phi$ coupling to the matter in J frame. As an extreme counter example, we may replace the matter Lagrangian (\ref{bd55_4}) in J frame by
\begin{equation}
L_{\psi}=-\overline{\psi}\left( D\hspace{-.65em}/ +f\phi \right)\psi,
\label{bd5-33}
\end{equation}
where $f$ is a dimensionless Yukawa coupling constant. $\psi$ is massless in J frame, while in E frame we obtain the mass $m= f\xi^{-1/2}$ (in units of $(8\pi G)^{-1/2}$) which is {\em independent of} $\sigma$. The scalar field is {\em decoupled} from $\psi$ {\em in E frame}, hence is left {\em invisible} through the matter coupling; it plays a role {\em only in cosmology}, most likely as a form of dark matter. With {\em constant} mass in E frame, which is now physical, we may reasonably adjust parameters such that the standard scenario is reproduced to a good approximation.
There might be intermediate choices between the prototype model and this extreme model. Then we may expect the matter coupling generically {\em much weaker} than
\begin{equation}
H'_{\sigma_{1}}=-\beta m(t)\overline{\psi}\psi\sigma,
\label{bd5-30}
\end{equation}
in the prototype model, hence evading immediate conflicts with the
test of WEP and the constraint from the solar-system
experiments.\footnote{See Ref. [\cite{fn}] for a model of this type.} Needless to say, $G$ is predicted to be constant, by construction.
With modifications of this nature in mind, we add a comment on the mass term of the scalar field, which is ought to arise from $V_{1}$ as given by (\ref{bd5-6}). We should be interested here in a {\em fluctuating component} $\sigma_{1}(x)$ which is responsible for the force between local mass distributions, to be separated from the spatially uniform component $\sigma_{0}(t)$ evolving as the cosmic time $t$;
\begin{equation}
\sigma(x)=\sigma_{0}(t)+\sigma_{1}(x),
\label{bd5-24}
\end{equation}
which satisfies
\begin{equation}
\mbox{\raisebox{-0.2em}{\large$\Box$}}\sigma -V'(\sigma)=0,
\label{bd5-25}
\end{equation}
from which (\ref{bd5-8}) derives.
The cosmological component $\sigma_{0}$ is a solution of (\ref{bd5-25}) with $\sigma_{1}$ dropped;
\begin{equation}
\mbox{\raisebox{-0.2em}{\large$\Box$}}\sigma_{0}+4\beta V_{1}(0)e^{-4\beta\sigma_{0}}=0.
\label{bd5-26}
\end{equation}
If we use this in (\ref{bd5-25}) for the entire field (\ref{bd5-24}), we obtain
\begin{equation}
\mbox{\raisebox{-0.2em}{\large$\Box$}}\sigma_{1}-4\beta V_{1}(0)e^{-4\beta\sigma_{0}}
\left( 1-e^{-4\beta\sigma_{1}} \right)=0.
\label{bd5-27}
\end{equation}
Expanding the terms in the last parenthesis, we find
\begin{equation}
\mbox{\raisebox{-0.2em}{\large$\Box$}}\sigma_{1}=\mu^2\sigma_{1} +\cdots,
\label{bd5-28}
\end{equation}
where, using (\ref{bd5-11}) for $\sigma_{0}(t)$,\footnote{For dust-dominated Universe, the right-hand side is doubled.}
\begin{equation}
\mu^2 =16\beta^2 V_{1}(0)t^{-2},
\label{bd5-29}
\end{equation}
which is $V''(\sigma)$ at $\sigma = \sigma_{0}$.
This shows that the scalar field does acquire a ``mass" even though the potential has {\em no stationary point}, but the range of the force mediated by $\sigma_{1}$ is basically given by $t$, which is the size of the visible part of the Universe at each epoch. The force-range at the present epoch can be as ``short" as $10^5$ ly if $16\beta^2 V_{1}(0)\sim 10^{10}$, in contrast to (\ref{bd5-12a}). It is rather likely that the force can be considered to be infinite-range in any practical use.
We may relax the assumption $F\sim \phi^2$ in the prototype model. We recognize, however, that the relation of the type $V(\sigma)\sim m^4(\sigma)$ as in (\ref{bd5-6}) is quite generic and so is $m\sim t^{-1/2}$ according to the argument following (\ref{bd5-16}). This makes the conclusion (\ref{bd8_11}) almost inevitable, as long as the assumption (iv) is maintained.
As another aspect of more general $F(\phi)$, we point out that the factor $\xi^{-1/2}\phi^{-1}$ in (\ref{bd5-1}) is in fact $F^{-1/2}$. It then follows that the potential as given by (\ref{bd5-6}) generalizes to
\begin{equation}
V_{1}(\sigma)\sim m^{4}\sim F^{-2}.
\label{bd5-34}
\end{equation}
The relation (\ref{bd5-2}) is also traced back to
\begin{equation}
\frac{d\sigma}{d\phi}= F^{-1}\sqrt{\epsilon F +\frac{3}{2}F'\:^2}.
\label{bd5-35}
\end{equation}
We may then expect that the potential as a function of $\sigma$ would have a minimum if $F(\phi)$ has a maximum, the same conclusion as in Refs. [\cite{dm}].
The potential minimum should act, however, as an effective cosmological constant, which might present another serious conflict with observations unless it remains below the level of $\sim t^{-2}_{0}\sim 10^{-120}$ at the present epoch. In this respect we have an advantage in the potential having no stationary point.
As the last comment we point out that the occurrence of a ghost which
was required to give positive matter density is not entirely unnatural from the point of view of unified theories. Consider, for example, Kaluza-Klein approach to $4+n$ dimensional spacetime. The size of compactified {$n$}-dimensional space behaves as a 4-dimensional scalar field, which is shown to have {\em wrong sign} in the kinetic term. This model provides also one of the natural origins of the nonminimal coupling.
\begin{center}
{\Large\bf Acknowledgments}
\end{center}
I thank Akira Tomimatsu and Kei-ichi Maeda for valuable discussions.
\begin{center}
{\Large\bf References}
\end{center}
\begin{enumerate}
\item\label{bd}C. Brans and R.H. Dicke, Phys. Rev. {\bf 124}(1961)925.\item\label{yf1}Y. Fujii, Mod. Phys. Lett. {\bf A9}(1994)3685.
\item\label{dicke}R.H. Dicke, Phys. Rev. {\bf 125}(1962)2163.
\item\label{fn}Y. Fujii and T. Nishioka, Phys. Rev. {\bf D42}(1990)361.
\item\label{cho}Y.M. Cho, Phys. Rev. Lett. {\bf 68}(1992)3133.
\item\label{jd}P. Jordan, {\sl Schwerkraft und Weltalle}, (Friedrich
Vieweg und Sohn, Braunschweig, 1955).\item\label{will}See, for example, C. Will, {\sl Theory and Experiment in Gravitational Physics,} rev. ed., Cambridge University Press, Cambridge,1993.
\item\label{nth}T. Nishioka, Thesis, University of Tokyo, 1991.
\item\label{dm}T. Damour and K. Nordtvedt, Phys. Rev. Lett. {\bf 70}(1993)2217; T. Damour and A.M. Polyakov, Nucl. Phys. {\bf B423}(1994)532.
\end{enumerate}
\newpage
\epsfverbosetrue
\begin{figure}[h]
\hspace*{1cm}
\epsfxsize=12cm
\epsfbox{abc4.eps}
\caption{Some of the one-$\psi$-loop diagrams, shown only up to the third order in $\phi^{-1}$. Each blob represents $\phi^{-1}$
appearing in (12), a collection of many vertices of $\sigma$'s, while a solid line is for the $\psi$ line.}
\end{figure}
\begin{figure}[h]
\centering
\hspace*{1cm}
\epsfxsize=12cm
\epsfbox{bdd2f6c.eps}
\caption{A solution of (38)-(40) with the initial values $\varphi_{0}=0.25.\; \dot{\varphi}_{0}=0.0,\; \rho_{0}=0.1$ at $\ln t=1$. We chose $\Lambda = 0.5$ and $\xi =0.4$.}
\end{figure}
\begin{figure}[h]
\hspace{2cm}
\epsfxsize=10cm
\epsfbox{bdd2f5cc.eps}
\caption{Trajectories ($\ln t=1-30$) of the solutions of different initial values $\dot{\varphi}_{0}$ and $\rho_{0}$, as shown in the parentheses, with other parameters the same as in Fig. 2. The point of convergence ($\dot{\phi}^2=1.42857, \rho =0.21429$) is given by (44) and (45). The dotted line is for $H=0$; its left-side for $H>0$.}
\end{figure}
\end{document}
| proofpile-arXiv_065-490 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
Recent advances in the physics of single-electron charging of macroscopic
conductors (for general reviews see, e.g., Refs. \onlinecite{mes,sct}) have
led to proposals for several new analog and digital electronic devices.
Such devices are considered, in particular, to be the most likely
candidates to replace silicon transistors in future ultra-dense electronic
circuits -- see, e.g., Refs.~\onlinecite{pasct,sasha}.
Single-electronics is presently one of the most active areas of solid state
physics and electronics, with hundreds of experimental and theoretical
works being published annually. We are not aware, however, of any previous
attempts to quantitatively compare experimental data for a particular
device with results of theoretical analysis including geometrical
modeling\cite{knoll}. Such a comparison was the main objective of
this work. To that end, we selected one of the simplest devices, the
single-electron trap\cite{pasct,fulton}.
Figure~\ref{schematic} shows the schematic layout of the circuit we discuss
in this paper, which consists of a trap coupled to a single-electron
electrometer. We will distinguish two types of conductors (``nodes'') in
the circuit: {\em externals}, wires which extend to the edges of the chip
and connect to the external measuring devices; and {\em islands}, small
metallic segments that are connected each other and to the externals by
tunnel junctions.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=6cm
\epsffile{schematic.eps}
\end{center}
\caption{Schematic view of the 8-junction single-electron trap/electrometer
circuit. Islands 12 and 13 are strongly coupled and together provide the
energy well for an extra electron. Islands 6-11 form the array separating
the well from the drive external 1.}
\label{schematic}
\end{figure}
The trap consists of a larger island, providing the potential well for the
extra electron, separated from a voltage-biased ``drive'' external by an
$N$-island array. The islands of the array are linked by $(N-1)$ tunnel
junctions with low capacitance $C_j$ and tunnel conductance $G_j$:
\begin{eqnarray}
C_j&\ll&\frac{e^2}{k_B T}\,, \label{lowc} \\
G_j&\ll&\frac{e^2}{h}\,. \label{lowg}
\end{eqnarray}
\noindent
Under condition (\ref{lowg}), each electron is localized inside a single
island at any given time. As Fig.~\ref{profile}a shows, the array creates
an electrostatic energy barrier $\Delta W\sim~e^2/C_j$ between the
drive electrode and the trap island. To inject an additional electron
into the trap, a bias voltage $V=V_1-V_2$ is applied to the device. At a
certain value $V=V_{+}$ the energy barrier is suppressed: an electron
tunnels from the drive external through the array and into the trap island.
To extract the electron from the trap island, a voltage $V=V_{-}$ is
applied, causing a hole to tunnel from the drive external to the trap
island, annihilating the trapped electron.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsffile{profile.eps}
\end{center}
\caption{Energy of a typical circuit calculated for three values of $V_1$
($V_+$, $V_{eq}$, and $V_-$), with $V_2$ = 8 mV: (a) as a function of the
position of one electron in the array; (b) as a function of the position of
one hole in the array, with one electron in the trap (node 13).}
\label{profile}
\end{figure}
Electrons can also overcome the energy barrier by thermal activation and by
macroscopic quantum tunneling of charge (``cotunneling''). At sufficiently
low temperatures (\ref{lowc}), the rate of thermally activated hopping over
the barrier is roughly~\cite{mes,pasct}
\begin{equation}
\label{gammat}\Gamma_T \sim \frac{G_j}{C_j}
\exp\left(\frac{-\Delta W}{k_B T}\right)\,,
\end{equation}
\noindent
while the rate of spontaneous cotunneling through the barrier scales as
\cite{mqt}
\begin{equation}
\label{gammaq}\Gamma _Q\sim \frac{G_j}{C_j}\left(\frac{G_jh}{4\pi ^2e^2}\right)
^{N-2}\,.
\end{equation}
\noindent
If conditions (\ref{lowc}) and (\ref{lowg}) are satisfied, and the number
$N$ of junctions in the array is large enough, the rates of thermal
activation and cotunneling may be very low. Thus, the lifetime $\tau_L =
(\Gamma_T+\Gamma_Q)^{-1}$ of both the zero-electron and the one-electron
states of the trap may be quite long, and the device may be considered
bistable.
When the voltage $V$ is driven beyond the threshold $V_+$ or $V_-$, the
electron or hole tunnels through the array in time $\tau \sim~C_j/G_j$,
which may be many orders of magnitude shorter than $\tau _L$. Thus, in
principle, the trap can serve as a memory cell. Its contents can be read
out non-destructively by capacitive coupling of the trap to the
single-electron electrometer\cite {mes,sct,pasct} (see Section
\ref{results}).
Early attempts to trap single electrons were made by Fulton {\it et al.}
\cite{fulton}, using systems with two and four Al/AlO$_x$ junctions of area
$\sim~100\times 100$ nm$^2$ at temperatures down to 0.3 K. Their results
implied trapping times $\tau _L\simeq 1$ sec. Similar experiments by
Lafarge {\it et al.}\cite{lafarge} yielded $\tau _L<1$ sec, much shorter
than could be anticipated from formulas (\ref{gammat}) and
(\ref{gammaq}). A later attempt\cite{nakazato} used a semiconductor
(GaAs) structure with a narrow 2DEG channel instead of a well-defined
tunnel junction array. A bistability loop was observed, but its size was
not clearly quantized, implying that the number of trapped electrons was
much larger than one (the authors estimated this number to be 80-100).
Finally, Al/AlO$_x$ trap circuits designed and fabricated at Stony Brook\cite
{haus,liji} yielded trapping times of over $10^4$ sec (limited only by
observation time). The main goal of the present work was to compare the
experimental data obtained for these traps with a quantitative theoretical
analysis of the circuits. For this purpose, we have constructed a
geometrical model of the circuit, calculated the full matrix of self- and
mutual capacitances for the conducting nodes in the model, and simulated
static and dynamic properties of the trap using these capacitances.
\section{Fabrication}
\label{fabrication}
Circuits consisting of two layers of partially overlapping nodes were
fabricated using the standard shadow mask technique\cite{dolan,ful-dol}.
The process begins with a Si substrate, either stripped of oxide or covered
by a layer of SiO$_2$ of thickness $H$=500 nm. The substrate is coated with
a PMMA/copolymer double layer mask. The circuit pattern is written onto
the mask using a scanning electron microscope. Then the mask is developed,
the Al circuit elements are deposited onto the substrate, and the mask is
lifted off. The fabrication process is described fully in
Ref. \onlinecite{liji}. Here we present the essential details.
\subsection{Mask}
The circuit layout, consisting of a set of line segments, is first
specified in a ``mask file'' (Fig.~\ref{mask-big}). A version of the same
file is also used to start the computational modeling process (see Section
\ref{geometric}). Wide lines in Fig.~\ref{mask-big} represent the parts of
the externals that extend from the trap and electrometer to contact pads at
the edge of the chip. The narrow lines extending inward from the wide
lines (Fig.~\ref{mask-med}) represent the inner parts of the externals.
The short, narrow line segments (Fig.~\ref{layout}a) represent islands.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsffile{mask-big.eps}
\end{center}
\caption{Mask for complete chip containing several circuits. Pattern for
circuits discussed in this paper is circled.}
\label{mask-big}
\end{figure}
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsffile{mask-med.eps}
\end{center}
\caption{Closeup of layout of circuits discussed in this paper, showing 20
$\mu$m cutoff radius used in simulations, and external node numbers.
Externals have wide ($W = 1 \mu$m) and narrow ($w \simeq$ 50 nm) sections.}
\label{mask-med}
\end{figure}
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsffile{layout.eps}
\end{center}
\caption{(a) Central part of mask. (b) AFM image of central part of
fabricated circuit. (c) 3-D outline of central part of geometrical model
used for capacitance calculation.}
\label{layout}
\end{figure}
The pattern of lines is written on the mask using the electron beam. Upon
chemical development, each line in the PMMA becomes a window opening into a
larger cavity in the copolymer, which is more susceptible to the electrons.
This procedure results in a mask, shown schematically in Fig.~\ref{shadow},
with several suspended bridges.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsffile{shadow.eps}
\end{center}
\caption{Schematic view of shadow mask fabrication: (a) first-layer
evaporation, (b) oxidation (oxide thickness exaggerated for illustration
purposes), (c) second layer evaporation.}
\label{shadow}
\end{figure}
\subsection{Deposition}
The aluminum islands are deposited in two layers. The first layer is
resistively evaporated in high vacuum directly onto the room-temperature
substrate (Fig.~\ref{shadow}a). This layer is then oxidized at $\sim$~10
mTorr O$_2$ for $\sim$~10 min (Fig.~\ref{shadow}b), covering the Al islands
with a $\sim~1$ nm layer of AlO$_x$. Before depositing the second layer,
the chamber is re-evacuated and the substrate is tilted relative to the Al
source. The tilt creates a shift $s$ between these two groups of islands,
so that they partially overlap (Fig.~\ref{shadow}c). The AlO$_x$ creates
tunnel barriers between the first and second layer islands. In our
circuits, the shift $s$ was about 120 nm along the vertical direction in
Fig.~\ref{layout}a,b. The second aluminum layer is made thicker than the
first, to allow reliable step coverage.
An AFM image of the resulting circuit is shown in Fig.~\ref{layout}b. This
image exaggerates the island widths because of the finite angle of the AFM
tip. Other observations (including SEM imaging) show that the islands
oriented perpendicular to the direction of the shift were in fact spatially
separated, in the successful samples.
Figure~\ref{layout}c shows a simplified model of the central part of the
circuit, with externals and islands numbered. There are two islands for
each corresponding window in the mask. For example, islands 6 and 7 are
the first- and second-layer products of the same window (see also
Fig.~\ref{model}c.). Since the two layers of each external overlap each
other extensively and are connected to the same voltage/current source,
they effectively serve as one conductor. Thus, there is only one external
for each corresponding window in the mask.
\section{Geometrical Modeling}
\label{geometric} The essential electrostatics of a group of conductors can
be described by their mutual capacitance matrix, {\boldmath $C$}. A program
known as FastCap\cite{fastcap} can calculate {\boldmath $C$} for an
arbitrary collection of conductors, given the geometry of the conductors as
input. The conductor surfaces are presented to FastCap as a set of discrete
elements, or ``panels''. We wrote a program called Conpan (for {\it
conductor panels}) to generate a 3D paneling of a simplified model of the
experimental system, starting from a 2D mask file. We will first explain
the Conpan algorithm, then how its input parameters were derived from
experiments.
\subsection{Conpan Algorithm}
Conpan represents circuit nodes by means of data structures called
``sections''. Each section is a collection of data about a node or part of
a node. The data include parameters such as node number, layer number, and
limits in the $xy$ plane. Sections may be recursively divided into
subsections to represent overlaps and to facilitate paneling.
Consider two line segments from the larger mask file (Fig.~\ref{model}a).
These two segments eventually produce four islands separated by three
tunnel junctions. Conpan expands each segment into a first-layer section
(Fig.~\ref{model}b) using the line-width $w$. The second-layer sections
(Fig.~\ref{model}c) are initially identical to the first-layer sections
except for a uniform translation $s$ that results in overlaps.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsffile{model.eps}
\end{center}
\caption{Geometrical model construction in Conpan: (a) Two segments of the
array. (b) First-layer sections generated by Conpan for the two segments.
(c) Second-layer island sections (shaded), partially overlapping
first-layer sections. (d) Shape of first-layer islands. (e) Shape of
second-layer islands overlapping first-layer. (f) First- and second-layer
islands paneled for capacitance calculation.}
\label{model}
\end{figure}
\subsubsection{Overlap Detection}
Since a single second-layer section can overlap more than one first-layer
section, Conpan detects the overlaps using a recursive detection algorithm.
To begin, each second layer section is compared against each first layer
section to detect overlaps. When an overlap is found, the second layer
section spawns two daughter sections, one overlapping and one not. The
axis and coordinate of the split are stored in the the mother section,
along with pointers to the daughter sections. The mother section
becomes a placeholder, used only to keep track of the relationship among
its daughter sections.
The non-overlapping daughter is then is compared against the remaining
first-layer sections to find other overlaps. If there are more overlaps,
the daughter spawns a pair of sub-daughters, and so on. The recursive
process stops when no new overlaps are found. The daughter sections that
remain undivided are called ``final daughters''. Figure~\ref{divide}
gives a schematic view of the recursive overlap detection process for a
single island.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsffile{divide.eps}
\end{center}
\caption{Recursive overlap detection in Conpan for a circuit fragment where
a second-layer island $S$ overlaps two first-layer islands $F^1$ and $F^2$.
(a) Top view, with overlap areas shaded. (b) Comparing $S$ to all
first-layer islands, overlap of $F^1$ is detected. $S$ is split into two
daughter sections, $S_a$ and $S_b$, at $y = y^*$. (c) Comparing $S_b$ to
all first-layer islands, overlap of $F^2$ is detected. $S_b$ is split into
$S_{ba}$ and $S_{bb}$ at $y = y^*_b$. (d) Daughter sections $S_a$, $S_{ba}$,
and $S_{bb}$, having no further overlaps, are used to build the 3D
structure shown in (e).}
\label{divide}
\end{figure}
In addition to dividing up the second-layer sections to account for
overlaps, Conpan also splits first-layer sections along the line where they
are overlapped (see Fig.~\ref{model}f). This allows the edges of panels
facing each other across a junction to line up, facilitating convergence
in capacitance calculations.
\subsubsection{3D Representation}
Once all the overlaps have been found, Conpan can begin to create the 3-D
model of the circuit. Each final daughter section becomes the base
of a ``block'', a rectilinear solid representing part of a conductor. The
heights of the two layers are specified by the parameters $h_1$ and $h_2$.
First-layer blocks and non-overlapping second-layer blocks have their base
at $z = 0$. Overlapping second-layer blocks have their base at $z = h_1 +
t$, where $t$ is the thickness of the gap between overlapping islands that
represents the tunnel junction.
Finally, each block surface is divided into panels. The goal is to divide
the surfaces in such a way that an acceptably accurate capacitance
calculation can be performed, within the limits of available computer
memory and calculation time. The division process is guided by an input
parameter $a$, the goal panel length. The surface of a block with length
$L_i$ along axis $i$ is divided into the number $n_i$ of divisions that
brings $L_i/n_i$ closest to $a$. Once the block surface has been divided
along both its axes, the resulting grid of panels is written to a panel
file for input to FastCap. Each panel is stored simply as a quartet of
${x,y,z}$ coordinates, one for each of the four corners, together with the
number of the node it belongs to.
\subsection{Conpan Input Parameters}
\subsubsection{Junction Thickness}
In the physical circuit, the tunnel barriers separating the islands consist
of AlO$_x$, with unknown $x$ and thickness $t_j$. From literature data on
similar junctions \cite{maezawa}, we expect a dielectric constant $\epsilon
_j \sim~4$ and $t_j \sim~1$ nm. FastCap can handle dielectric surfaces much
as it handles conductors -- by dividing them into panels. However, each
additional dielectric panel demands more computer memory and calculation
time. Since $t_j$ is much smaller than the transversal dimensions in all
junctions, the electric field configuration outside the junctions does not
depend strongly on their internal geometry. Therefore, we avoided modeling
the junction dielectrics explicitly by replacing them with uniform
free-space gaps ($\epsilon =1$) with the effective thickness
$t=t_j/\epsilon _j$. This effective thickness was adjusted to make the
junction specific capacitance match the standard experimental value
$4.5\mu$F/cm$^2$ typical for the Al/AlO$_x$/Al junctions with tunnel
conductivity in our range ($\sim~10^5$ S/cm$^2$)\cite{maezawa,mager}.
\subsubsection{Line Widths}
The effective line width $w$ of islands (and of the narrow parts of
externals) is difficult to measure directly, because of its small magnitude
(see Fig.~\ref{layout}b). We determined $w$ by requiring that the
simulated inverse self-capacitance of the electrometer island
($C^{-1}_{14,14}$) match its experimentally measured value. We derive
$C^{-1}_{14,14}$ from the maximum value of the electrometer Coulomb
blockade threshold voltage $U_t$, as seen in electrometer I-V plots
(Fig.~\ref {elect_i-u}):
\begin{equation}
\label{c_sigma}(U_t)_{max} = eC^{-1}_{14,14} \simeq e/C_{14,14}\,.
\end{equation}
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsffile{elect_i-u.eps}
\end{center}
\caption{Current-voltage traces for the electrometer, (SiO$_2$/Si
substrate, T=35mK, superconductivity suppressed by magnetic field), current
biased, taken while rapidly varying $V_5$. The Coulomb blockade voltage
$U_t$ varies with $V_5$ from 0 to a maximum value of e$C^{-1}_{14,14}$.}
\label{elect_i-u}
\end{figure}
$C_{14,14}$ (conventionally known as $C_\Sigma$) in turn depends on $w$
because it is is dominated by the electrometer junction capacitances, which
increase monotonically with $w$. The experimentally measured values for
$C^{-1}_{14,14}$ were $(4.6\times 10^{-16}{\rm F})^{-1}$ for the circuit on
Si (sample \#LJS011494B) and $(2.7\times 10^{-16}{\rm F})^{-1}$ for the
circuit on SiO$_2$/Si substrate (sample \#LJS011494A). The island width $w$
used in simulation, as determined from Eq. (\ref{c_sigma}), was 30 nm for
Si and 42 nm for SiO$_2$/Si. Both of these values are consistent with the
values expected from fabrication parameters and from AFM and SEM imaging of
the samples.
$W$, the width of the wide parts of the externals, is specified as 1 $\mu$m
in our mask files, and can be accepted at ``face value'' because it is
large compared to the scale of geometrical uncertainty in the circuit, and
because the wide parts of the externals are all far (several $\mu$m) from
the islands.
\subsubsection{Layer Heights}
The heights of the two layers ($h_1$ and $h_2$) are determined with a
quartz monitor in the deposition unit during fabrication. In our case,
these heights were measured to be 30 and 50 nm ($\pm$ 10\%), respectively.
\subsection{Substrate}
To calculate the effects of the substrate on circuit capacitances, FastCap
requires a paneling of the complementary image of the ``footprint'' of the
nodes, because panels representing the dielectric/metal interface (parts of
the substrate covered by nodes) must be treated differently than panels
representing the dielectric/air interface (the exposed substrate). In a
manner analogous to that used for conductor panels (see
Section~\ref{matrices}), one could investigate various methods of paneling
the complementary substrate image in order to minimize the number of
panels, while yet retaining an acceptable level of accuracy. Such a
paneling algorithm itself is not simple to create.
We avoided this problem through an old calculational trick in electrostatics
-- the image method. A modified version of FastCap was created, called
ImageCap, which can simulate the effects of a single- or double-layer
substrate by creating a set (or multiple sets, in the double-layer case) of
image panels.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsffile{image.eps}
\end{center}
\caption{Image panels in ImageCap: (a) single substrate, (b) double
substrate.}
\label{image}
\end{figure}
In the single substrate case, each image panel is formed by reflecting the
original panel about the plane of the surface of the substrate
(Fig.~\ref{image}a). For the purposes of calculating the electrostatic
potential above the substrate, the charge on the image panel is
\begin{equation}
\label{image1q}q_1=-\frac{\epsilon -1}{\epsilon +1}q\,,
\end{equation}
where $q$ is the charge on the original panel, and $\epsilon $ is the
relative dielectric constant of the substrate.
For a substrate covered by an oxide of thickness $H$, an infinite series of
image charges is required for an exact representation of the electrostatic
effect of the substrate (Fig.~\ref{image}b). However, the distance from the
original charge to each successive image charge increases linearly,
\begin{equation}
z_{2,i}=-2Ti-d\,,\quad i=1,2...
\end{equation}
\noindent
while the value of each successive image charge decreases exponentially,
\begin{equation}
q_{2,i}=4\beta \frac{\epsilon _1\epsilon _2}{(\epsilon _1+\epsilon _2)^2}
(\alpha \beta )^{(i-1)}q\,,\quad i=1,2...
\end{equation}
\begin{equation}
\alpha =\frac{\epsilon _1-1}{\epsilon _1+1}\,,\quad\beta =\frac{\epsilon
_1-\epsilon _2}{\epsilon _1+\epsilon _2}\,.
\end{equation}
Here $\epsilon _1$ and $\epsilon _2$ are the dielectric constants of the
surface oxide layer and the bulk substrate, respectively. For our circuits,
we have accepted the table values $\epsilon =12.1$ for the bare Si
substrate and $\epsilon _1=4.5$, $\epsilon _2=12.1$ for the SiO$_2$/Si
substrate ($\alpha =0.64$, $\beta =-0.46)$. The resulting expression for
the double-layer image charges,
\begin{equation}
q_{2,i}= -0.36 \times (0.29)^{(i-1)} q \,,\quad i=1,2...
\end{equation}
\noindent
shows that $q_{2,4}$ is already down by three orders of magnitude from the
original charge. In our calculations, adding image levels beyond $q_{2,4}$
made no difference to the result, within a relative error (of the largest
self-capacitances) below $\sim~10^{-4}$ .
\section{Capacitance Matrices}
\label{matrices}
\subsection{Matrix Structure}
Using the circuit panels generated by Conpan, ImageCap generates the
capacitance matrix for the circuit. ImageCap adds the effect of image
panels when calculating potentials, and uses no multipole acceleration;
otherwise, its algorithms are the same as in FastCap\cite{fastcap}. First,
the inverse capacitance matrix for {\it panels} is calculated and inverted.
Each element $\widehat C_{ij}$ in the capacitance matrix for {\it nodes} is
then formed by summing all the panel capacitance matrix elements linking
nodes $i$ and $j$. The charges and potentials on the nodes are related by
\begin{equation}
\vec q = \mbox{\boldmath $\widehat C$} \vec \phi\,,
\label{mat_eq}
\end{equation}
\noindent
so that $\widehat C_{ij}$ is numerically equal to the amount of charge
induced on node $j$ when node $i$ is held at unit potential and all other
nodes have zero potential.
{\boldmath $\widehat C$} is an $N \times N$ matrix, where $N = N_e + N_i$,
and $N_e$ and $N_i$ are the numbers of external nodes and island nodes in
the circuit, respectively. Ordering all the external nodes before the
island nodes, we can write {\boldmath $\widehat C$} in terms of
submatrices:
\begin{equation}
\mbox{\boldmath $\widehat C$}=\left(
\begin{array}{c|c}
\bigotimes&\mbox{\boldmath $-\tilde C$}\\
\hline
\mbox{\boldmath $-\tilde C^T$}&\mbox{\boldmath $C$}\\
\end{array}\right)\,.
\label{quad_mat}
\end{equation}
\noindent
Here {\boldmath $C$} is the symmetric $N_i \times N_i$ matrix of
island-island capacitances and {\boldmath $\tilde C$} is the $N_e \times
N_i$ matrix of external-island capacitances (with elements defined positive,
by convention). External-external capacitances (represented above by the
$\bigotimes$) are not needed for our simulations.
The matrices calculated for our circuits are shown in Tables I and II.
Note the up/down alternation of mutual capacitances along the array for the
circuit on Si -- e.g., $\tilde C_{1,i}$, the capacitances linking
external node 1 to the islands. For example, although island 7 is closer
to external 1 than is island 8, $\tilde C_{1,7} \simeq 0.030 \times
10^{-16}$F is smaller than $\tilde C_{1,8} \simeq 0.045 \times 10^{-16}$F
(similarly for islands 9 and 10). In {\boldmath $C$}, we see that
$\|C_{6,9}\|$ is smaller than $\|C_{6,10}\|$, etc. This phenomenon
reflects the influence of the silicon substrate, which, due to its high
dielectric constant ($\epsilon \simeq 12$), links externals to the
first-layer islands (which lie flat on the substrate) more strongly than to
the second-layer islands (which lie partly on top of the first-layer
islands). The capacitances for the circuit on SiO$_2$/Si do not show these
oscillations as strongly, as we would expect from the smaller permittivity
($\sim~4.5$) of SiO$_2$.
\subsection{Model Accuracy}
Our model contains three main simplifications related to computational
constraints, each of which introduces error into our capacitance matrix
calculations.
\subsubsection{Free-space Junctions}
As noted above, we calculate {\boldmath $\widehat C$} using free-space
junctions of thickness $t$ instead of dielectric junctions of thickness
$t_j$. (Although initially this approximation was intended for
convenience, it later became a necessity as ImageCap does not handle
explicit dielectric panels.) The error involved in this approximation was
estimated by using FastCap to model a chain of islands in two ways: with
explicit dielectric junctions and with free-space junctions. Results for
an 8-island chain, with effective dielectric thickness chosen to make the
island self-capacitances in both models the same, indicate that the error
involved in this approximation is below 1\% for junction-linked islands,
and between 1\% and 4\% for non-junction-linked islands.
\subsubsection{Paneling}
In calculating capacitances, FastCap/ImageCap assigns a uniform charge
distribution to each panel. Hence, its accuracy depends on how well the
paneling follows changes in change distribution on the node surfaces.
Clearly, the denser the paneling, the better the representation of changes
in charge distribution. However, panel density is effectively limited by
available computer memory. For example, a FastCap simulation with 5000
panels typically requires more than 128 MB. ImageCap uses even more
memory, since it calculates all panel interactions directly. We
investigated the dependence of calculated capacitance on paneling density
for a simple two cube system (Fig.~\ref{2cube}a).
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsffile{2cube.eps}
\end{center}
\caption{Two cube system with $L/t=10$: (a) Panelings with 1, 3, and 9
panels/side. (b) Capacitances as a function of panel density ($\vec q =
\mbox{\boldmath $C$} \vec \phi$).}
\label{2cube}
\end{figure}
The results (Fig.~\ref{2cube}b) suggest that a non-uniform $3 \times 3$
grid (with a 1/10 ratio of edge panel length to central panel length,
reflecting the peak in surface charge near the edges) for the smaller,
roughly square-shaped node faces (Fig.~\ref {model}f) is sufficient to
calculate capacitances with an error below 10\%. This is essentially how
we paneled roughly square-shaped island surfaces. For longer faces
(Fig.~\ref{paneling}a) we used a larger number of divisions along their
length. In an islands-only test circuit, increasing the total number of
panels from $\sim 2000$ (corresponding to the $3 \times 3$ grid for roughly
square-shaped surfaces) to 6000 resulted in less than 1\% changes in
island-island capacitances. Thus we believe that the total error in
island-island capacitances due to finite panel density is perhaps only
$\sim$~1\%.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=7cm
\epsffile{paneling.eps}
\end{center}
\caption{Paneling used in our calculations: (a) Trap islands (nodes 12 and
13). (b) Thin parts of externals. (c) Wide parts of externals.}
\label{paneling}
\end{figure}
To reduce the number of panels in the model, the two layers of an external
are fused into one where they overlap. The error involved in this
simplification is negligible. In addition, the narrow parts of the
externals were divided along their length without edge panels
(Fig.~\ref{paneling}b). This simplification was found to cause an error in
island-external capacitance of $\sim$~5\% when the island and the external
are connected by a junction (Fig.~\ref{c-ext}), and $\sim$~1\% otherwise.
Finally, wide parts of the external leads were represented by only their
top and bottom surfaces (Fig.~\ref{paneling}c), again to save panels.
Since the width to height ratio $W/h \simeq 12$, the error introduced by
this simplification is negligible. The top and bottom surfaces are divided
according to the $3 \times 3$ type scheme described above for islands.
Despite the large size of the resulting panels, the error involved in this
simple paneling appears to be $\sim$~1\%.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsffile{c-ext.eps}
\end{center}
\caption{Effect of panel density on the calculated island-external
capacitance. The test circuit had only islands 14 and 15 and narrow parts
of externals.}
\label{c-ext}
\end{figure}
\subsubsection{Lengths of Externals}
\label{lengths}
The calculated capacitance values depend on the lengths of the external
wires used in the model. In general, island-external capacitances increase
with external length, at the expense of island stray capacitance
(capacitance to a ground at infinity); the self-capacitance of islands does
not change appreciably. To measure the error introduced by cutting off the
externals at a given length, we have calculated capacitance matrices for test
circuits with varying external lengths (Fig.~\ref{c-wire}). These circuits
consisted of only one island and only the wide parts of the five externals.
As a result, the error induced by cutting off externals in these test
circuits should be proportionately larger than the error in the complete
circuits. Still, the test circuits indicate that the error involved in
cutting of the circuit at a radius of 20 $\mu$m (as in our final versions
of the complete circuits) was less than 2\%.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsffile{c-wire.eps}
\end{center}
\caption{Effect of lead length cutoff on the calculated island-external
capacitance. The test circuit had only island 12 and wide parts of
externals.}
\label{c-wire}
\end{figure}
\subsubsection{Total Error}
Considering the error caused by the above simplifications in the
calculation of {\boldmath $\widehat C$} itself, it seems safe to say that
the the combined error for any given calculated capacitance matrix element
was less than 10\%. Note that we are not yet considering how well the
geometrical model corresponds to the physical circuit (see
Section~\ref{discussion}).
\section{Simulated and Experimental Results}
\label{results}
We have calculated most properties of our circuits using {\sc moses}, the
single-electron circuit simulation program\cite{moses}. This program uses a
Monte Carlo algorithm to simulate arbitrary SET circuits within the
framework of the orthodox theory of single-electron tunneling
\cite{mes,sct}. {\sc moses} needs to know the capacitance sub-matrices
{\boldmath $C$} and {\boldmath $\tilde C$} and the conductances of all
tunnel junctions. The resistance of two electrometer junctions connected in
series can be extracted from the slope of the experimental dc $I-V$ curve
of the electrometer at high voltage ($V\gg e/C_\Sigma$). From this
measurement, we calculated tunnel conductance per unit area. Conductances
of all other junctions in the circuit were then calculated by assuming that
their conductance is proportional to their nominal area. This
assumption may only be accurate to an order of magnitude; however, most of
the results discussed below pertain to stationary properties of the system,
and are thus unaffected by deviations in conductance.
\subsection{General Electrostatic Relations}
Solving the matrix equation (\ref{mat_eq}) for the island potentials
$\phi_i$, with our definition (\ref{quad_mat}) of the capacitance matrix we
get
\begin{equation}
\phi_i=\sum_{j\in isl}{C^{-1}_{ij}(q_j + \tilde q_j)}\,,\quad
\tilde q_j \equiv \sum_{k\in ext}{\tilde C_{kj}V_k}\,,
\end{equation}
\noindent
or, in a different form,
\begin{equation}
\phi_i=\sum_{j\in isl}{C^{-1}_{ij}q_j}+\sum_{k\in ext}{\alpha_{ik}V_k}\,,\quad
\alpha_{ik} \equiv \sum_{j\in isl}{C^{-1}_{ij}\tilde C_{kj} }\,,
\label{phi_i_def}
\end{equation}
\noindent
where $V_k$ are the external potentials. These relations allow us to
establish useful relations between changes in the external potentials
$\{V_k\}$ and the charge state of the islands $\{q_i\}$, and the dynamics
of the system as determined by the island potentials $\{\phi_i\}$.
\subsection{Electrometer}
Let us apply these relations, in particular, to the island of the
single-electron transistor (number 14 in our notation, see
Fig.~\ref{schematic}) serving as the electrometer. Experimentally, we
measure the dc voltage $U_{3-4}$ between the ``source'' and ``drain'' of
the transistor (externals 3 and 4) under a small ($\sim$~100 pA) dc
current bias. If the temperature is small enough ($k_B T \ll
e^2C^{-1}_{14,14}$), the voltage $U_{3-4}$ in such an experiment closely
follows the threshold $U_t$ of the Coulomb blockade of the transistor --
see Fig.~\ref{elect_i-u}.
It is well known (see, e.g., Refs.~\onlinecite{mes,sct}) that the threshold
is determined by the effective background charge $Q_o$ of the transistor
island, which may be defined as
\begin{equation}
\phi_{14}|_{U_3=U_4=0} = C^{-1}_{14,14}(q_{14}+Q_o)\,.
\label{qo_intro}
\end{equation}
\noindent
Comparing (\ref{qo_intro}) and (\ref{phi_i_def}) above, we obtain in our
notation
\begin{equation}
Q_o = \frac{1}{C^{-1}_{14,14}}
\left[
\sum_{j\in isl}^{j\ne 14}{C^{-1}_{14,j}q_j}+\sum_{k\in ext}{\alpha_{ik}V_k}
\right]\,.
\label{qo_sums}
\end{equation}
Eq. (\ref{qo_sums}) allows us to find the theoretically expected variation
of $Q_o$ due to any changes in the system. On the other hand, the
threshold voltage is an e-periodic function of $Q_o$, and its maximum
amplitude is expressed by Eq. (\ref{c_sigma}) (for the case when the two
transistor junction capacitances are the same). Thus, after we measure the
experimental value of $(U_t)_{max}$, we can express the change in the
effective charge $Q_o$ via the observed variation in $U_t$:
\begin{equation}
\Delta Q_o = \frac{e\Delta U_t}{2(U_t)_{max}}\,.
\label{qo-ut}
\end{equation}
We have applied this approach to compare experiment and theory for two
samples (\#LJS011494A with SiO$_2$/Si substrate and \#LJS011494B with Si
substrate).
\subsection{High-$T$ electrometer response}
We can readily measure $\Delta V_i$ (i = 1,2,5), the change in external
voltage corresponding to one period of the oscillating threshold voltage
(Fig.~\ref{elect_u-v5}). At $k_BT\geq 0.1e^2/C_\Sigma$ (experimentally,
$T\geq 0.5K$), thermal activation of electrons smears the Coulomb blockade
effects and makes the junctions essentially transparent to tunneling, while
the periodic response of the electrometer is still visible up to $k_BT \sim
0.3e^2/C_\Sigma$ ($T \sim~1.5$K). Thus, the measured values of $\Delta V_i$
depend only on the circuit geometry and are essentially independent of the
properties of the junctions.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsffile{elect_u-v5.eps}
\end{center}
\caption{A typical experimental dependence of electrometer dc voltage
$U_{3-4}$ on gate voltage $V_5$ for a circuit comprising only an
electrometer.}
\label{elect_u-v5}
\end{figure}
{\sc moses} is not a useful tool for directly modeling high-$T$ behavior,
as the number of jumps involved would be extremely high. However, we can
simulate high-$T$ behavior in {\sc moses} by specifying very high external
voltages $V_i$ while keeping temperature low (say, $T=0$). Under these high
voltage conditions, the islands are flooded with extra electrons, and the
tunnel junctions become effectively transparent to tunneling, just as in
the high temperature case. Thus we simply apply an external voltage $V_i
\gg e/C_\Sigma$, measure how many electrons enter the electrometer island,
and find the ratio of voltage $V_i$ to electrometer charge $q_{14}$.
Table III shows values of the ratio $\Delta V_i$ for simulated circuits and
for experimental circuits averaged over several nominally identical
samples. For the experimental values, the uncertainties given reflect the
spread of among the samples. For the simulated values, the uncertainties
given reflect the $\sim$~10\% error in calculated values, as described in
Section~\ref{matrices}. The simulated values are all lower than the
experimental ones (with the exception of $\Delta V_5$ on SiO$_2$/Si),
differing by as much as 50\%. The agreement is somewhat better for the
circuits on SiO$_2$/Si.
\subsection{Trap phase diagram}
The simplest measurable characteristic involving single-electron charging
of the trap is its phase diagram (Fig.~\ref{phase}), which reflects changes
in the charge states of the array and trap as a function of the drive
voltage $V_1$. In {\sc moses}, we can directly view the charge state of
each island in the array and trap as we vary $V_1$, as well the resulting
change in $Q_o$. In the physical circuit, however, we can only measure the
response $U_{3-4}$ of the electrometer and reduce it to the changes in
$Q_o$ using Eq.~\ref{qo-ut}.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsffile{phase.eps}
\end{center}
\caption{Electrometer phase diagram: $Q_0$ as a function of trap drive
voltage. $V_2$=8.2mV, sample on SiO$_2$/Si. Crosstalk from drive voltage
to electrometer has been subtracted, leaving only influence of trap charge
state. In experiment, superconductivity in aluminum is suppressed by a 2T
magnetic field.}
\label{phase}
\end{figure}
Figure ~\ref{phase} shows experimental and simulated electrometer response
to ramping the trap drive voltage $V_1$ up and down over a period of
several minutes. In both cases, the effects of the crosstalk between
external 1 and the electrometer have been removed. In the experiment, the
crosstalk is cancelled by feeding the electrometer gate (node 5) with a
voltage $V_5=-\alpha V_1$, with the coefficient $\alpha$ adjusted to make
the phase diagram plateaus horizontal. In simulations, {\sc moses}
accomplishes the same effect by subtracting $\Delta \phi_{14} =
\alpha_{1,14}V_1$ from the electrometer island potential.
Horizontal plateaus in Fig.~\ref{phase} correspond to particular charge
states of the system (trap~+~array), while vertical jumps correspond to
changes of charge state. Thus, the hysteretic loops are regions of
bi/multi-stability. The blow-up of the theoretical curve (Fig.~\ref{loops})
indicates the states for several plateaus. In particular, notice that the
largest plateaus correspond to states that are most stable because the array
is either charged uniformly (one electron on each island, for example) or
in a regular alternating pattern such as 1-0-1-0 (Fig.~\ref{hi-loops}).
The smaller plateaus correspond to more complex charge states which are
less stable.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsffile{loops.eps}
\end{center}
\caption{Closeup view of experimental and simulated phase loops. Charge
vectors for islands 6-13 on various plateaus in simulated phase diagram are
indicated in brackets. In simulation, $\vec q_o = 0$ was assumed.}
\label{loops}
\end{figure}
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsffile{hi-loops.eps}
\end{center}
\caption{Simulated phase loops at a higher range in $V_1$, showing wider
plateaus around [1010123].}
\label{hi-loops}
\end{figure}
The experimental phase diagram bears a qualitative resemblance to the
theoretical one, with somewhat shorter plateaus, though the order of
magnitude is the same ($\sim$~2 mV for major plateaus). Simulated phase
diagrams with randomly selected $\vec q_0$ show shorter plateaus than the
$\vec q_0 = 0$ phase diagram (see Sec.\ref{discussion} below).
In Fig.~\ref{phase}, the large jumps in $Q_o$ correspond to a single
electron entering the trap: $\delta Q_o \equiv (\Delta Q_o)|_{e\to tr}$.
Using Eq.~(\ref{qo_sums}), we can also express the simulated value of
$\delta Q_o$ as
\begin{equation}
\delta Q_o^{sim} = \frac{C^{-1}_{tr,14}}{C^{-1}_{14,14}}e\,,
\end{equation}
\noindent
where $tr$ = 12 or 13, depending on which trap island the electron stops
in. For comparison with experimental results, we take the average of the
two possible values. The results are shown in Table IV. The difference
between simulated and experimental values for Si is within the estimated
geometric calculation error (10\%), while the value for
SiO$_2$/Si is not.
\subsection{Plateau dependence on $V_2$}
For a given plateau, the switching voltages $V_1$ = $V_\pm$ depend on the
``ground'' voltage $V_2$ (see Fig.~\ref{schematic}). In the simplest
model, with no stray capacitances, (see, e.g., Ref. \onlinecite {pasct})
the charge state of the system depends only on the voltage $V=V_1-V_2$. In
that model, the dependences $V_\pm(V_2)$, corresponding to changes in the
charge state, would form parallel 45$^{\circ}$ lines in the $[V_1,V_2]$
plane. In reality, however, stray capacitances of the islands to
``infinity'' (i.e. to a distant common ground) make the average potential
$(V_1+V_2)/2$ of the system relevant as well. As a result, the region
corresponding to each charge state acquires a shape similar to a stretched
diamond (Fig.~\ref{diamond}).
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsffile{diamond.eps}
\end{center}
\caption{Experimental and simulated threshold voltages $V_\pm$ as functions
of $V_2$: Si substrate, superconductivity suppressed.}
\label{diamond}
\end{figure}
Simulations using {\sc moses} show that the diamond shape results from the
alternation of two types of electron transport that switch the charge
state. At the low-$V_2$ end of the diamond, the charge state switches
with the transfer of an electron in/out of the trap (see the energy diagram
in Fig.~\ref{profile}a). However, at the high-$V_2$ side, the barrier for
holes to enter or exit is lower than for electrons
(cf. Fig.~\ref{profile}b). Near the sharp ends of the diamond, the
critical transport may be even more complex (e.g., creation of an
electron-hole pair inside the array, with the sequential motion of its
components apart, one into the trap, and another into the external
electrode). In these regions, however, the plateau corresponding to the
charge state of the trap is already small and virtually disappears among
numerous plateaus corresponding to various internal charge states of the
array (Fig.~\ref{loops}). Figure~\ref{diamond} shows that while the
diamond shape of the charge state in $[V_1,V_2]$ is well reproduced in
experiment, the simulated width \mbox{$\|V_{+}-V_{-}\|$} of the bistability
region in $V_1$ is roughly twice the experimental value.
For each $V_2$, there is one value of $V_1$, called $V_{eq}$, at which the
energy barrier is the same for an electron to tunnel into or out of the
trap\cite{seneca}. A good measure of the relative influence of the two
external voltages on the trap is the derivative
\begin{equation}
\frac{dV_{eq}}{dV_2} = \frac{\alpha_{2,tr}}{1 - \alpha_{1,tr}}\,,
\end{equation}
\noindent
where $tr$ = 12 or 13, depending on which trap island actually traps the
electron for a given $(V_1,V_2)$. The two values are typically within 5\%
of each other, and we take their average when comparing simulated and
experimental results. In the experimental data, we define the average
$V_{eq}$ by bisecting the diamond shape in the graph (Fig.~\ref{diamond}).
$dV_{eq}/dV_2$ is essentially a geometric property of the circuit and
should not depend on thermal activation or cotunneling. As
Fig.~\ref{diamond} shows, the simulated and experimental values are very
close.
\subsection{Energy barrier}
At $V_1=V_{eq}(V_2)$, we can measure the energy barrier $\Delta W$
experimentally by measuring trapping lifetime as a function of temperature
(for experimental details, see Ref.~\onlinecite{haus}). The Arrhenius law
for lifetimes gives
\begin{equation}
\tau_L \propto \exp (\Delta W/kT),
\label{arrhen}
\end{equation}
\noindent
so that plotting $\log (\tau_L)$ vs. $1/T$ gives us $\Delta W$. Dynamical
simulations\cite{seneca} have shown that (\ref{arrhen}) is virtually
unaffected by cotunneling for relatively high temperatures ($\sim~100$ mK
and above). In simulation, {\sc moses} allows us to measure $\Delta W$
directly. Figure~\ref{dW_v2} shows the dependence of the trap energy
barrier $\Delta W$ on the bias voltage $V_2$, for the circuit on the
SiO$_2$/Si substrate. The simulated energy barrier profile peaks at roughly
the same value of $V_2$ as in the experiment, and the peak barrier value is
within $\sim~10\%$ of the experimental value. However, the simulated peak
is sharper.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsffile{dW_v2.eps}
\end{center}
\caption{Experimental and simulated dependence of barrier height on $V_2$
(SiO$_2$/Si substrate).}
\label{dW_v2}
\end{figure}
\section{Discussion}
\label{discussion}
Let us first discuss the results independent of the single-electron
charging effects: the transistor response $\delta Q_o$ to a single electron
entering the trap, the oscillation periods $\Delta V_i$, and the slope
$dV_{eq}/dV_2$. The differences between simulated and experimental values
for $\delta Q_o$ are 6\% and 23\% for the Si and SiO$_2$/Si substrates,
respectively. Values for $\Delta V_i$ do not agree as well: differences
between simulated and experimental values range from 15 to 38\% for the
trap on SiO$_2$/Si, and from 29 to 50\% for the trap on Si. We had
experimental data for $dV_{eq}/dV_2$ only on Si. Here the difference
between experimental and simulated results was $\sim~12\%$. These numbers
suggest how well our geometrical model corresponds to the physical circuit
(the accuracy of the orthodox theory and of the {\sc moses} simulator is
presumably much higher).
The most obvious idealization involved in our geometric modeling is that
the islands created by Conpan are rectilinear and uniform. Even at the
limited resolution of an AFM image (Fig.~\ref{layout}b), the contours of
the fabricated circuits appear rounded and irregular on a scale of $\sim$
10 nm. This is to be expected, due to the relatively large grain size of
evaporated Al ($\sim~50$ nm, comparable to the line width $w$) and the
stochastic nature of the grain growth process. Most capacitance matrix
elements should not depend strongly on small details of the island shape.
However, irregularities in the shape of overlapping islands may change the
area, and thus the capacitance, of the junctions linking them.
All other results involve single-electron charging effects. Here the
difference between the theory and experiment is larger - typically by a
factor of 2, and sometimes larger. We believe that the most important
origin of this difference is the set of background charges $\vec{q}_0$.
The Si substrate is capable of trapping charged impurities near the circuit
islands. The result of these impurities is that the charge on island $i$
effectively changes from $\tilde{q}_i$ to $\tilde{q}_i + q_{0i}$. These
charges may furthermore be capable of thermal migration over time.
Simulated plots of the electrometer response to trap charging with three
randomly selected $\vec q_0$ are shown in Fig.~\ref{multi_q0}. It appears
that the wide ($\sim$~4 mV) steps near $V_1 = 0$ in the $\vec q_0 = 0$ plot
are not stable to variations in $\vec q_0$: in most plots with random $\vec
q_0$, as in the experimental plot, all step widths are less than 3 mV.
\begin{figure}
\begin{center}
\leavevmode
\epsfxsize=8cm
\epsffile{multi_q0.eps}
\end{center}
\caption{Simulated response of electrometer to ramps of $V_1$ for $\vec q_0
= 0$ (solid line) and three randomly selected background charge vectors
$\vec q_0$. Plots are shifted vertically because background charge on
the electrometer island (node 14) effectively shifts $Q_o$.}
\label{multi_q0}
\end{figure}
To summarize: we have developed an automated way to construct simplified
models of experimental single-electron devices and circuits with metallic
islands, and we have compared the properties of model single-electron traps
with those of real traps. The observed differences between simulation and
experiment may be attributed to random deviations of the physical
structures from their nominal size and shape, and to random background
charges created by charged impurities. Future work of interest may include
more precisely defined single-electron devices, using better fabrication
technology, and the extension of quantitative modeling to
semiconductor-based single-electronic circuits and hybrid single-electronic
/ conventional transistor logic circuits. As single-electronics evolves
into a mature technology, such modeling will be essential.
\section{Acknowledgments}
We greatly appreciate numerous fruitful discussions with D. Averin,
R. Chen, L. Fonseca, A. Korotkov, W. Zheng, and K. Nabors. This work was
supported in part by AFOSR grants \#F49620-1-0044 and \#F49620-96-1-0320.
| proofpile-arXiv_065-491 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
After the observation of the top quark signal at the Tevatron,
the mechanism of the spontaneous electroweak symmetry breaking remains the
last untested property of the Standard Model ($\cal{SM}${}).
Although the recent global fits to the precision electroweak data from LEP,
SLC and Tevatron seem to indicate a preference to a light Higgs boson
$m_H=149^{+148}_{-82}$~GeV, $m_H<450$~GeV ($95\%C.L.$) \cite{LEPEWWG},
it is definitely premature to exclude the heavy Higgs scenario. The reason is
that a restrictive upper bound for $m_H$ is dominated by the result on
$A_{LR}$, which differs significantly from the $\cal{SM}${} predictions \cite{ALR}.
Without $A_{LR}$ upper bound on $M_H$ becomes larger than 600~GeV \cite{ALR},
which is not far in the logarithmic scale from the value of the order of
1~TeV, where perturbation theory breaks down. In order to estimate the
region of applicability of the perturbation theory, the leading two--loop
${\cal O}(g^4 m_H^4/M_W^4)$ electroweak corrections were under intense study
recently. In particular, the high energy weak--boson scattering
\cite{scattering}, corrections to the heavy Higgs line shape
\cite{GhinculovvanderBij}, corrections to the partial widths of the Higgs
boson
decay to pairs of fermions \cite{fermi_G,fermi_DKR} and intermediate vector
bosons \cite{vector_G,vector_FKKR} at two--loops have been calculated.
All these calculations resort at least partly to numerical methods. Even for
the two--loop renormalization constants in the Higgs scalar sector of the $\cal{SM}${}
\cite{GhinculovvanderBij,MDR} complete analytical expressions are not known.
In this paper we present our analytic results for these two--loop
renormalization constants, evaluated in the on--mass--shell renormalization
scheme in terms of transcendental functions $\zeta(3)$ and the maximal value
of the Clausen function $\mbox{Cl}(\pi/3)$.
\section{Lagrangian and renormalization}
The part of the $\cal{SM}${} Lagrangian describing the Higgs scalar sector of the $\cal{SM}${}
in terms of the bare quantities is given by:
\begin{eqnarray}
&&{\cal L} = \frac{1}{2}\partial_\mu H_0\partial^\mu H_0
+ \frac{1}{2}\partial_\mu z_0\partial^\mu z_0
+ \partial_\mu w^+_0\partial^\mu w^-_0
\nonumber\\
&& -\, \frac{{m_H^2}_0}{2v_0^2}\left(w_0^+w_0^- + \frac{1}{2}z_0^2
+ \frac{1}{2}H_0^2 + v_0 H_0 + \frac{1}{2}\delta v^2
\right)^2.
\label{lagrangian}
\end{eqnarray}
Here the tadpole counterterm $\delta v^2$ is chosen in such a way, that
a Higgs field vacuum expectation value is equal to $v_0$
\begin{equation}
v_0 = \frac{2\, {M_W}_0}{g}
\end{equation}
to all orders.
Renormalized fields are given by
\begin{equation}
H_0 = \sqrt{Z_H}\,H, \quad z_0 = \sqrt{Z_z}\,z, \quad
w_0 = \sqrt{Z_w}\,w.
\end{equation}
At two--loop approximation the wave function renormalization constants,
tadpole and mass counterterms take the form
\begin{eqnarray}
\sqrt{Z} &=& 1 + \frac{g^2}{16\pi^2}\,\delta Z^{(1)}
+ \frac{g^4}{(16\pi^2)^2}\,\delta Z^{(2)};
\nonumber \\
\delta v^2 &=& \frac{1}{16\pi^2}\,{\delta v^2}^{(1)}
+ \frac{g^2}{(16\pi^2)^2}\, {\delta v^2}^{(2)};
\label{constants} \\
{M_W}_{0} &=& M_W + \frac{g^2}{16\pi^2}\, \delta M_W^{(1)}
+ \frac{g^4}{(16\pi^2)^2}\, \delta M_W^{(2)};
\nonumber \\
{m_H^2}_{0} &=& m_H^2 + \frac{g^2}{16\pi^2}\, {\delta m_H^2}^{(1)}
+ \frac{g^4}{(16\pi^2)^2}\, {\delta m_H^2}^{(2)}.
\nonumber
\end{eqnarray}
Since weak coupling constant $g$ is not renormalized at the leading order in
$m_H^2$, the $W$--boson mass counterterm is related to the Nambu--Goldstone
wave function renormalization constant by the Ward identity ${M_W}_0=Z_w M_W$.
In the on--mass--shell renormalization scheme all the counterterms are fixed
uniquely by the requirement that the pole position of the Higgs and
$W$--boson
propagators coincide with their physical masses and the residue of the
Higgs boson pole is normalized to unity.
The one--loop counterterms equivalent to those used in
\cite{GhinculovvanderBij,MDR} are given by
\begin{eqnarray}
{\delta v^2}^{(1)}&=&
m_H^{2}{\xi^\epsilon_H}\,\Biggl\{-{\frac {6}{\epsilon}}+3
-\, \epsilon \biggl(\frac{3}{2}+\frac{\pi^2}{8}\biggr)
\Biggr\};
\nonumbe
\\
\frac{{\delta m_H^2}^{(1)}}{m_H^2}&=&
\frac{m_H^{2}{\xi^\epsilon_H}}{M_W^2}
\,\Biggl\{-{\frac {3}{\epsilon}}+3-{\frac {3\,\pi\,\sqrt {3}}{8}}
+\, \epsilon \biggl(-3+\frac{\pi^2}{16}+\frac{3\pi\sqrt{3}}{8}
+\frac{3\sqrt{3}C}{4}-\frac{3\pi\sqrt{3}\log{3}}{16}\biggr)
\Biggr\};
\nonumbe
\\
\delta Z_H^{(1)}&=&{\frac {m_H^{2}{\xi^\epsilon_H}\,}{M_W^{2}}}
\Biggl\{\frac{3}{4}-{\frac {\pi\,\sqrt {3}}{8}}
+\, \epsilon \biggl(-\frac{3}{4}+\frac{3\pi\sqrt{3}}{32}
+\frac{\sqrt{3}C}{4}-\frac{\pi\sqrt{3}\log{3}}{16}\biggr)
\Biggr\};
\label{zh}
\\
\frac{\delta M_W^{(1)}}{M_W}&=&\delta Z_w^{(1)}=
-{\frac {m_H^{2}{\xi^\epsilon_H}}{16\,M_W^{2}}}\Biggl\{1
-\frac{3}{4}\epsilon\Biggr\}.
\nonumbe
\end{eqnarray}
Here the dimension of space--time is taken to be $d=4+\epsilon$ and
\begin{equation}
\xi^\epsilon_H=e^{\gamma\,\epsilon/2}\,
\left(\frac{m_H}{2\,\pi}\right)^\epsilon.
\end{equation}
In contrast to papers \cite{GhinculovvanderBij,MDR} we prefer not to keep the
one--loop
counterterms of ${\cal O}(\epsilon)$ order, {\it i.e.} unlike in the
conventional on--mass--shell scheme used in \cite{GhinculovvanderBij,MDR},
we require that the one-loop normalization conditions are fulfilled only in
the limit $\epsilon\to 0$, where the counterterms of the order
${\cal O}(\epsilon)$ do not contribute. Such a modified on--mass-shell scheme
is equally consistent as the conventional one or
the standard scheme of minimal dimensional renormalization, which assumes
only the subtraction of pole terms at $\epsilon=0$ \cite{BR}.
(Moreover, in general one cannot subtract all the nonsingular
${\cal O}(\epsilon)$
terms in the Laurent expansion in $\epsilon$, as they are not polynomial
in external momenta.) The ${\cal O}(\epsilon)$ one--loop counterterms
considered in \cite{GhinculovvanderBij,MDR} do can really combine with the
$1/\epsilon$ terms at two--loop order to give finite contributions, but these
contributions are completely canceled by the additional finite parts of the
two--loop counterterms, fixed through the renormalization conditions in the
on--mass--shell renormalization scheme.
The reason is that after the inclusion of the one--loop counterterms all the
subdivergences are canceled and only the overall divergence remains, which
is to be canceled by the two--loop counterterms. The account of finite
contributions coming from the combination of ${\cal O}(\epsilon)$ one--loop
counterterms with $1/\epsilon$ overall divergence just redefines the finite
parts of the two--loop counterterms. An obvious advantage of this modified
on--mass--shell scheme is that the lower loop counterterms once calculated
could be directly used in higher loop calculations, while in the conventional
on--mass--shell scheme for $l$-loop calculation one needs to recalculate the
one-loop counterterms to include all the terms up to
${\cal O}(\epsilon^{l-1})$,
two-loop counterterms to include ${\cal O}(\epsilon^{l-2})$ terms and so on.
\section{Analytic integration}
The calculation of the Higgs and $W$--boson (or Nambu--Goldstone $w$, $z$
bosons) two--loop self energies, needed to evaluate
the renormalization constants (\ref{constants}), reduces to the evaluation
of the two--loop massive scalar integrals
\begin{eqnarray}
&&J(k^2;
n_1\, m_1^2,n_2\, m_2^2,n_3\, m_3^2,n_4\, m_4^2,n_5\, m_5^2)
= -\frac{1}{\pi^4}\int\,D^{(d)}P\,D^{(d)}Q \,\biggl(P^2-m_1^2\biggr)^{-n_1}\\
&& \times
\biggl((P+k)^2-m_2^2\biggr)^{-n_2}
\biggl((Q+k)^2-m_3^2\biggr)^{-n_3}\biggl(Q^2-m_4^2\biggr)^{-n_4}
\biggl((P-Q)^2-m_5^2\biggr)^{-n_5}
\nonumber
\end{eqnarray}
and their derivatives $J'$ at $k^2=m_H^2$ or at $k^2=0$.
The most difficult is a calculation of the all--massive scalar master
integral corresponding to the topology shown in the Fig.~1.
\vspace*{0.4cm}
\setlength{\unitlength}{1cm}
\begin{picture}(15,3)
\put(5,0){\epsfig{file=a.eps,height=3cm}}
\end{picture}
\begin{center}
\parbox{6in}{\small\baselineskip=12pt Fig.~1.
The two loop all--massive master graph. Solid line represents Higgs bosons.
}
\end{center}
\vspace*{0.4cm}
This integral has a discontinuity that is an elliptic integral, resulting from
integration over the phase space of three massive particles, and is not
expressible in terms of polylogarithms. However one can show
\cite{ScharfTausk} that on--shell $k^2=m_H^2$ or at the threshold
$k^2=9 m_H^2$ this is not the case. We use the dispersive method
\cite{Broadhurst,BaubergerBohm} to evaluate
this finite integral on the mass shell:
\begin{equation}
m_H^2\, J(k^2;m_H^2,m_H^2,m_H^2,m_H^2,m_H^2)=
\sigma_a(k^2/m_H^2)+\sigma_b(k^2/m_H^2),
\label{master}
\end{equation}
where $\sigma_{a,b}$ correspond to the dispersive integrals
calculated, respectively, from the two-- and three--particle discontinuities,
which are itself reduced to one--dimensional integrals.
The $\tanh^{-1}$ functions entering $\sigma_{a,b}$ can be removed
integrating by parts either in the dispersive integral \cite{Broadhurst},
or in the discontinuity integral \cite{BaubergerBohm}. By interchanging the
order of integrations the latter representation gives the three--particle cut
contribution $\sigma_b$ as a single integral of logarithmic functions
\cite{BaubergerBohm}. After some rather heavy analysis we obtain at
$k^2=m_H^2$:
\begin{eqnarray}
\sigma_a(1)&=&\int_0^1 dy \, \frac{8}{y^{4}-y^{2}+1}
\log \left({\frac {\left (y^{2}+1\right )^{2}}{y^{4}+y^{2}+1}}\right)
\left [{\frac {\left (y^{4}-1\right )\log (y)}{y}}
-{\frac {\pi\,y}{\sqrt {3}}}\right ] \nonumber \\
&=&{\frac {17}{18}}\,\zeta(3)-{\frac {10}{9}}\,\pi \,C+\pi ^{2}\,\log {2}
-{\frac {4}{9}}\,\pi ^{2}\,\log {3},
\\
\sigma_b(1)&=&\int_0^1 dy \, 2\,
\log \left({\frac {y^{2}+y+1}{y}}\right) \nonumber \\
&&\left [{\frac {\log (y+1)}{y}}
+{\frac {{\frac {\pi\,}{\sqrt {3}}}\left (y^{2}-3\,y+1\right )
-2\,\left (y^{2}-1\right )
\log (y)}{y^{4}+y^{2}+1}}-{\frac {\log (y)}{y+1}}\right ] \nonumber\\
&=&{\frac {1}{18}}\zeta(3)+{\frac {4}{9}}\,\pi \,C-\pi ^{2}\,\log {2}
+{\frac {4}{9}}\pi ^{2}\,\log {3}.
\end{eqnarray}
Here
$
C = \mbox{Cl}(\pi/3) = \mathop{\mathrm{Im}} \,\mbox{li}_2\left(\exp\left(i\pi/3\right)\right)=
1.01494\: 16064\: 09653\: 62502\dots
$
As a result we find
\begin{eqnarray}
m_H^2\, J(m_H^2;m_H^2,m_H^2,m_H^2,m_H^2,m_H^2)&=&
\zeta(3)-{\frac {2}{3}}\,\pi\,C \nonumber \\
&=&-0.92363\: 18265\: 19866\: 53695 \dots
\label{HHHHH}
\end{eqnarray}
The numerical value is in agreement with the one, calculated using the
momentum expansion \cite{DavydychevTausk}, and with the numerical values
given in \cite{MDR,Adrian}.
Given the value (\ref{HHHHH}), the simplest way to calculate the derivative
of the integral (\ref{master}) is to use Kotikov's method of differential
equations \cite{Kotikov,MDR}
\begin{eqnarray}
m_H^4\, J'(m_H^2;m_H^2,m_H^2,m_H^2,m_H^2,m_H^2)&=&
{\frac {2}{3}}\,\pi\,C-\zeta(3)-{\frac {\pi^{2}}{9}}
\nonumber \\
&=&-0.17299\: 08847\: 12284\: 42069 \dots
\end{eqnarray}
All the other two--loop self energy scalar integrals contain ``light''
particles (Nambu--Goldstone or $W$, $Z$--bosons) and some of them are
IR divergent in the limit $M_{W,Z}\to 0$. In principle, one can
calculate these IR divergent integrals in Landau gauge, where masses
of Nambu--Goldstone bosons are equal to zero and IR divergences are
represented as (double) poles at $\epsilon=0$ \cite{MDR}. However, in order to
have an additional check of the cancellation in the final answer of all the
IR divergent $\log(M_{W,Z}^2)$--terms, we work in 't~Hooft--Feynman gauge.
For the infra--red finite integrals the correct answer in the leading order in
$m_H^2$ is obtained just by
setting $M_{W,Z}=0$. We agree with the results for these integrals given in
\cite{MDR,ScharfTausk}. The two--loop IR divergent integrals correspond
to the topologies shown in the Fig.~2, which contain ``massless''
propagators squared.
\vspace*{0.4cm}
\setlength{\unitlength}{1cm}
\begin{picture}(15,3)
\put(1,0){\epsfig{file=b.eps,height=3cm}}
\put(10,0){\epsfig{file=c.eps,height=3cm}}
\end{picture}
\begin{center}
\parbox{6in}{\small\baselineskip=12pt Fig.~2.
The two--loop IR divergent graphs. Dashed line represents ``light''
particles.}
\end{center}
\vspace*{0.4cm}
The relevant technique to handle these diagrams follows from the so--called
asymptotic operation method \cite{Tkachov}.
According to the recipe of As--operation, the formal Taylor
expansion in small mass $M_{W}$ entering propagator (or its powers) should
be accomplished by adding the terms, containing the
$\delta$--function or its derivatives. The additional terms counterbalance
the infra--red singularities, arising in the formal expansion of propagator.
In our case we have
\begin{eqnarray}
\frac{1}{(P^{2} - M_{W}^{2})^{2}} & = & \frac{1}{(P^{2})^{2}}
+ 2\frac{M_{W}^{2}}{(P^{2})^{3}} + ... \nonumber \\
&+& C_{1}(M_{W})\delta^{(d)}(P) + C_{2}(M_{W})\partial^{2}\delta^{(d)}(P)+ ...
\end{eqnarray}
Here the first coefficient functions $C_{i}(M_{W})$ read
\begin{eqnarray}
C_{1}(M_{W}) & = & \int D^{(d)}P \frac{1}{(P^{2} - M_{W}^{2})^{2}}
\sim {\cal O}(M_{W}^{0}), \\
C_{2}(M_{W}) & = & \frac{1}{2d}\int D^{(d)}P
\frac{P^{2}}{(P^{2} - M_{W}^{2})^{2}}
\sim {\cal O}(M_{W}^{2}). \nonumber
\end{eqnarray}
This equality is to be understood in the following sense \cite{Tkachov}.
One should integrate both parts of the equation multiplied by a test function
and then take the limit $d \rightarrow 4$. If one keeps in the expansion all
terms up to order $M_{W}^{2 n}$, the resulting expression will represent a
correct expansion of the initial integral to order $o(M_{W}^{2 n})$.
To obtain the leading contribution to the diagrams Fig.~2,
it suffices just to take the first term of the Taylor expansion
and, correspondingly, the first ``counterterm'' $C_{1}\delta^{(d)} (P)$.
Finally, the combination of Mellin--Barnes representation and Kotikov's
method gives the following answer, corresponding to the first graph in
Fig.~2 neglecting the terms of order ${\cal O}(M_W^2/m_H^2)$:
\begin{eqnarray}
m_H^2\, J(m_H^2;2\,M_W^2,M_W^2,0,M_W^2,m_H^2)&=&
{\xi^{2\epsilon}_H}\Biggl ({\frac {2\,i\,\pi}{\epsilon}}-{\frac {i\,\pi}{2}}
+\frac{2}{\epsilon}\,\log (\frac{M_W^2}{m_H^2})
\\
&&-\frac{1}{2}+{\frac {5\,\pi^{2}}{6}}
-\log (\frac{M_W^2}{m_H^2})
+\frac{1}{2}\log^{2} (\frac{M_W^2}{m_H^2})\Biggr );
\nonumber \\
m_H^4\, J'(m_H^2;2\,M_W^2,M_W^2,0,M_W^2,m_H^2)&=&
{\xi^{2\epsilon}_H}\Biggl (-{\frac {2\,i\,\pi}{\epsilon}}+2\,i\,\pi
-\frac{2}{\epsilon}\left(1\,+\,\log (\frac{M_W^2}{m_H^2})\right) \\
&&+2-{\frac {5\,\pi^{2}}{6}}
+\log (\frac{M_W^2}{m_H^2})
-\frac{1}{2}\log^{2} (\frac{M_W^2}{m_H^2})\Biggr ).
\nonumber
\end{eqnarray}
The integral diverges as $1/\epsilon$, while if we would set $M_W=0$ from the
very beginning, it would diverge as $1/\epsilon^2$. The integral corresponding
to the second graph in the Fig.~2 up to ${\cal O}(M_W^2/m_H^2)$ is
\begin{eqnarray}
J(m_H^2;2\,M_W^2,0,0,M_W^2,m_H^2)&=&
{\xi^{2\epsilon}_H}\Biggl ({\frac {2}{\epsilon^{2}}}
+\frac{1}{\epsilon}\left(2\,\log (\frac{M_W^2}{m_H^2})-1\right) \\
&&+\frac{1}{2}-{\frac {\pi^{2}}{12}}
-\log (\frac{M_W^2}{m_H^2})
+\frac{1}{2}\log^{2} (\frac{M_W^2}{m_H^2})\Biggr) \nonumber
\end{eqnarray}
The two--loop vacuum integrals needed to evaluate the tadpole counterterm
${\delta v^2}^{(2)}$ have been calculated in \cite{vanderBijVeltman}.
\section{Results}
The analytic results for the two--loop renormalization constants are:
\begin{eqnarray}
\delta {v^2}^{(2)}&=&
\frac{m_H^{4}{\xi^{2\epsilon}_H}}{16\,M_W^2}
\left (
{\frac {72}{\epsilon^{2}}}
+{\frac {36\,\pi\,\sqrt {3}-84}{\epsilon}}
-162-3\,\pi^{2}+60\,\sqrt {3}C
\right );
\label{dv2}
\\
\frac{{\delta m_H^2}^{(2)}}{m_H^2}&=&
\frac{m_H^{4}{\xi^{2\epsilon}_H}}{64\,M_W^4}\Biggl (
{\frac {576}{\epsilon^{2}}}
+\frac {144\,\pi\,\sqrt {3}-1014}{\epsilon}
\nonumber \\
&&+{\frac {99}{2}}-252\,\zeta(3)
+87\,\pi^{2}
-219\,\pi\,\sqrt {3}
\nonumber \\
&&+156\,\pi\,C
+204\,\sqrt {3}C
\Biggr);
\label{mh2}
\\
\delta Z_H^{(2)}&=&\frac{m_H^{4}{\xi^{2\epsilon}_H}}{64\,M_W^4}
\Biggl ( {\frac {3}{\epsilon}} - {\frac {75}{4}} - 126\,\zeta(3)
\nonumber \\
&&+{\frac {25\,\pi^{2}}{2}}-76\,\pi\,\sqrt {3}+78\,\pi\,C+108\,\sqrt {3}C
\Biggr );
\label{zh2}
\\
\frac{\delta M_W^{(2)}}{M_W}&=&\delta Z_w^{(2)}=
\frac{m_H^{4}{\xi^{2\epsilon}_H}}{64\,M_W^4}\left (
{\frac {3}{\epsilon}}-\frac{3}{8}-{\frac {\pi^{2}}{6}}+{\frac {3\,
\pi\,\sqrt {3}}{2}}-6\,\sqrt {3}C
\right ).
\label{zw2}
\end{eqnarray}
For comparison we have also calculated these counterterms following the
renormalization scheme \cite{GhinculovvanderBij,fermi_G} and keeping the
${\cal O}(\epsilon)$ terms and found complete agreement with their (partly
numerical) results. In this scheme the renormalization constants
(\ref{dv2}), (\ref{mh2}) look a bit more complicated due to the presence of
the additional $\pi \sqrt{3}\log{3}$ terms
\begin{eqnarray}
\delta {v^2}^{(2)}&=&
\frac{m_H^{4}{\xi^{2\epsilon}_H}}{16\,M_W^2}
\Biggl (
{\frac {72}{\epsilon^{2}}}
+{\frac {36\,\pi\,\sqrt {3}-84}{\epsilon}}
+90-12\,\pi^{2}-12\,\sqrt {3}C
\\
&&-36\,\pi\,\sqrt{3}+18\,\pi\,\sqrt{3}\,\log{3}
\Biggr );
\label{dv2_e}\nonumber
\\
\frac{{\delta m_H^2}^{(2)}}{m_H^2}&=&
\frac{m_H^{4}{\xi^{2\epsilon}_H}}{64\,M_W^4}\Biggl (
{\frac {576}{\epsilon^{2}}}
+\frac {144\,\pi\,\sqrt {3}-1014}{\epsilon}
\nonumber \\
&&+{\frac {2439}{2}}-252\,\zeta(3)
+63\,\pi^{2}
-363\,\pi\,\sqrt {3}
\nonumber \\
&&+156\,\pi\,C
-84\,\sqrt {3}C+72\,\pi\sqrt{3}\,\log{3}
\Biggr).
\label{mh2_e}
\end{eqnarray}
The wave function renormalization constants $Z_{H,w}$ are identical in these
two schemes.
As an example of the physical quantity, for which all the schemes should give
the same result, we consider the two--loop heavy Higgs correction to the
fermionic Higgs width \cite{fermi_G,fermi_DKR}. The correction is given by the
ratio
\begin{eqnarray}
\frac{Z_H}{{M_W^2}_0/M_W^2}&=&
1 + 2\frac{g^2}{16\,\pi^2}
\left(\delta Z_H^{(1)}-\frac{\delta M_W^{(1)}}{M_W} \right)
\nonumber \\
&&+ \frac{g^4}{(16\,\pi^2)^2}\Biggl[
2\,\frac{\delta M_W^{(1)}}{M_W}\,
\left (\frac{\delta M_W^{(1)}}{M_W}-\delta Z_H^{(1)}\right )
+\left (\delta Z_H^{(1)}-\frac{\delta M_W^{(1)}}{M_W}\right )^{2}
\nonumber \\
&&+2\,\delta Z_H^{(2)}-2\,\frac{\delta M_W^{(2)}}{M_W}
\Biggr].
\end{eqnarray}
Substituting (\ref{zh}), (\ref{zh2})--(\ref{zw2}) we find
\begin{eqnarray}
&&\frac{Z_H}{{M_W^2}_0/M_W^2}=
1\, +\, \frac{1}{8}
\frac{g^2}{16\, \pi^2}\frac{m_H^2}{M_W^2}\left( 13 - 2 \pi \sqrt{3}\right)
\\
&&+\, \frac{1}{16}\biggl(\frac{g^2}{16\, \pi^2}\frac{m_H^2}{ M_W^2}\biggr)^2
\left(3-63\,\zeta(3)-{\frac {169\,\pi\,\sqrt {3}}{4}}
+{\frac {85\,\pi^{2}}{12}}+39\,\pi\,C+57\,\sqrt {3}\,C\right).
\nonumber
\end{eqnarray}
Again, we find complete agreement with the numerical result \cite{fermi_G}
and exact agreement with the result \cite{fermi_DKR}, taking into account
that their numeric constant $K_5$ is just minus our integral (\ref{HHHHH}).
\section*{Acknowledgments}
G.J. is grateful to J.J.~van~der~Bij and A.~Ghinculov for valuable
discussions. This work was supported in part by the Alexander von Humboldt
Foundation and the Russian Foundation for Basic Research grant 96-02-19-464.
| proofpile-arXiv_065-492 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
The observed structures of the universe are thought
to originate from the gravitational
condensation of primordial mass-energy fluctuations
whose evolution depends on the power spectrum
of the initial fluctuations, the amount and
nature of the various matter components present in the early universe
and the eventual existence of a cosmological constant. Since
most of the acting gravitational masses remain invisible,
constraining cosmological
scenario from astronomical observations has been one of the most
fascinating challenge of the last decades. From
the observations of the spatial galaxy distribution (see de Lapparent,
this conference), and from dynamical studies of gravitational systems
like galaxies or clusters of galaxies, it is possible to infer the
amount of dark matter as well as its distribution, or
to have insights into the clustering properties of
the visible matter in the universe.
However, none of these observations measures {\sl directly} the total
amount and the distribution of matter: angular or redshift surveys give the
distribution of {\sl light} associated with galaxies, and mass estimations of
gravitational structures can only be obtained within the assumption that
they are simple
relaxed or virialised dynamical systems. Dynamics of galaxy velocity fields
seems an efficient and promising approach to map directly the potentials
responsible from large-scale flows, but catalogs are still poor and some
assumptions are still uncertain or poorly understood on a physical
basis.
Gravitational lensing effects are fortunately a direct probe of
deflecting masses in the universe. They allow to determine
directly the amount of matter present along the line-of-sight
from observed or reconstructed deflection angles. After the
discovery of the first multiply imaged quasar (Walsh et al. 1979) and
the first observations of gravitational arcs (Soucail et al. 1987, 1988;
Lynds \& Petrossian 1986) and arclets (Fort et al. 1988),
gravitational lensing rapidly becomes one of the most useful tool
for probing dark matter on
all scales and cosmological parameter as well. In the following, we
summarize what we have learned from strong and weak lensing regimes
in clusters
of galaxies and how constraints
on the cosmological parameters can, or could, be obtained.
We will also discuss a technique
for measuring gravitational shear that has been developed
recently and which opens new perspectives for the
observations and the analysis of lensing effect by large-scale structures, as
those we discuss in the last section.
\begin{figure}
\vskip -2cm
\hskip 3truecm
\psfig{figure=ms2137_modele_color.ps,width=10. cm}
\caption{Model of MS2137-23. This cluster is at redshift 0.33 and shows
a tangential arc and the first radial arc ever detected. Up to now it is
the most constrained cluster and the first one where counter images were
predicted before being observed (Mellier et al. 1993). The external and
the dark internal solid lines are the critical lines. The internal grey
ellipse and the diamond are the caustic lines. The thin isocontours
shows the positions of the arcs and their counter images.}
\end{figure}
\section{Definitions and lensing equations}
Let us remind the basic principles of the gravitational lensing effects.
A deflecting mass changes the apparent position of a source
$\vec \theta_S$ into the apparent image position $\vec \theta_I$ by
the quantity $\vec \alpha$:
\begin{equation}
\vec \theta_S=\vec \theta_I+\vec \alpha(\vec \theta_I) \ .
\end{equation}
The deflection angle $\vec \alpha(\vec \theta_I)$
is the gradient of the two-dimensional
(projected along the line of sight at the angular position $\vec \theta_I$)
gravitational potential $\phi$.
The gravitational distortion of background objects is
described by the Jacobian of the transformation, namely the amplification
matrix $\cal A$ between the source and the image plane (Schneider, Ehlers \&
Falco 1992):
\begin{equation}
{\cal A}=\pmatrix{ 1-\kappa-\gamma_1 & -\gamma_2 \cr
-\gamma_2 & 1-\kappa+\gamma_1 \cr },
\end{equation}
where $\kappa$ is the convergence, $\gamma_1$ and $\gamma_2$ are the
shear components. They are related to the Newtonian gravitational potential
$\phi$ by:
\begin{equation}
\kappa={1\over 2} \nabla^2 \phi={\Sigma\over \Sigma_{\rm crit.}}; \ \ \ \gamma_1={1\over 2}(\phi_{,11}-\phi_{,22}) \ ; \ \ \ \gamma_2=\phi_{,12} \ ,
\end{equation}
where $\Sigma$ is the projected mass density and $\Sigma_{\rm crit.}$ is the
critical mass density which would exactly focus a light beam originating
from the source on the observer plane. It depends on the angular diameter
distances $D_{ab}$, (where $a,b=[o(bserver),l(ens) or s(ource)]$)
involved in the lens configuration:
\begin{equation}
\Sigma_{\rm crit.}={c^2\over 4\pi G}{D_{os}\over D_{ol} D_{ls}} \ .
\end{equation}
Depending on the ratio $l=\Sigma / \Sigma_{\rm crit.}$
we most often distinguish
for data analysis the strong
lensing ($l\gsim1$) and the weak lensing ($l\ll1$) regimes.
Note that the values of the cosmological parameters enter in the
relationship between the angular distance and the redshift. This is this
dependence that, in some cases, can be used to constrain $\Omega$ or
$\lambda$.
\section{Arcs and mass in the central region of clusters of galaxies }
Arcs and arclets correspond to strong lensing cases with $l\gg1$ and
$l \approx 1$ respectively. Giant arcs form at the points where the
determinant of the magnification matrix is (close to) infinite. To these
points (the critical lines) correspond the caustic lines in the source plane.
From the position of the source with respect to the caustic line, one can
easily define typical lensing configuration with formation of giant arcs
from the merging of two or three images, or radial arcs and
``straight arc'' as
well (see Fort \& Mellier 1994, Narayan \& Bartelmann 1996).
\begin{table}
\hskip 0.3 truecm
\psfig{figure=summary_mass_ARC.proc.ps,width=16. cm}
\caption{Summary of mass from giant arcs. We only give some typical
examples. More details can be found in Fort \& Mellier (1994) and
Narayan \& Bartelmann (1996). The scales are expessed in $h_{100}^{-1}$
Kpc.}
\end{table}
Some of these typical lensing cases have been observed in rich clusters
of galaxies. For MS2137-23 (Mellier et al. 1993; see figure 1), A370
(Kneib et al. 1993), Cl0024+1654 (Wallington et al. 1995), A2218 (Kneib et
al. 1995), very accurate models with image predictions have been done which
provide among the most precise informations we have about the
central condensation of dark matter
of clusters of galaxies on scale of $\approx 300$$h^{-1}$kpc
(table 1). In particular, giant arcs definitely demonstrate that
clusters of galaxies are dominated by dark matter, with
mass-to-light ratio (M/L) larger than 100, which closely follows the
geometry of the diffuse light distribution associated with the brightest
cluster members. Furthermore, since arcs
occur when $\Sigma /\Sigma_{\rm crit.}\gg1$, clusters of galaxies must be much
more concentrated, e.g. with a smaller core radius,
than it was expected from their galaxy and X-ray gas
distributions. Note that the occurrence of arcs is also enhanced by the
existence of additional clumps observed in most of rich clusters which
increases the shear substantially (Bartelmann 1995). The direct
observations of substructures by using giant arcs confirm that clusters
are dynamically young systems, therefore pointing towards a rather
high value of $\Omega$.
\section{The weak lensing regime and lensing inversion in clusters}
\subsection{Lensing inversion and mass reconstruction}
Beyond the region of strong lensing, background galaxies are still
weakly magnified. The light deviation induces a small increase of their
ellipticity in the direction perpendicular to the gradient of the
projected potential (shear). Despite the intrinsic ellipticity of the sources
and some observational or instrumental effects it is possible to
make statistical analysis of this coherent polarization of the
images of background galaxies and
to recover the mass distribution of the lens.
These weakly distorted galaxies are like a
background distorted grid which can be used to probe the projected mass
density $\Sigma$ of the foreground lens. The shape parameters
of the images $M^I$ are related to the shape parameters of the sources
$M^S$ by the equation,
\begin{figure}
\hskip 1truecm
\psfig{figure=a1942BMetACF.eps,width=15. cm}
\caption{Shear maps around the rich lensing cluster A1942. The CCD images
is a 4 hours exposure obtained at the Canada-France-Hawaii Telescope
in excellent seeing conditions (0.65"). North is top and east is on the
left. On the left panel,
the shear is measured by using
the Bonnet \& Mellier's method which consists is computing shape
parameters of an annular aperture centered of each individual galaxies.
On the right panel, the shear is obtained by computing the
autocorrelation (ACF) of the images (see section 4.2). Though the shape is
almost the same, the signal to noise is higher with the ACF.
Note the
central pattern which shows the bimodal nature of the mass distribution
located on the two brightest cluster members, and the eastern extension
of the shear pattern probably due to a substructure.}
\end{figure}
\begin{equation}
M^{S}= {\cal A} M^I {\cal A},
\end{equation}
where $M$ are the second order momenta of objects and ${\cal A}$
the magnification matrix. We thus have a
relation between the ellipticity of the sources
$\vec \epsilon_S$ and the observed ellipticity of the images
$\vec \epsilon_I$:
\begin{equation}
\vec \epsilon_S=
{\vec \epsilon_I-\vec g \over 1-\vec g \cdot \vec \epsilon_I} \ ;
\ \vec g={\vec \gamma\over 1-\kappa}\ .
\end{equation}
In particular, if the sources are randomly distributed then their
averaged intrinsic ellipticity verifies
$\big <\vec \epsilon_S\big >=0$ and we have
\begin{equation}
\vec g=\big <\vec \epsilon_I\big >\ .
\end{equation}
Since the main domain of application of the weak lensing is
the large-scale distribution of the Dark Matter, at scale above
$>0.5$ $h^{-1}$Mpc , the analysis has been mainly focussed
on the weak lensing regime, where $(\kappa,
\gamma) \ll 1$. The relations between the physical ($\vec\gamma$) and
observable ($\vec\epsilon_I$) quantities are then simpler,
since we have,
\begin{equation}
\big <\vec\epsilon_I\big >= \vec\gamma\ .
\end{equation}
The projected mass density $\Sigma$ of the lens can be obtained from the
distortion field by using Eq.(8) and the integration of Eq. (3)
(Kaiser \& Squires 1993):
\begin{equation}
\kappa(\vec \theta_I)={-2\over \pi} \int {\rm d}^2{\vec \theta}\ {\vec\chi(\vec
\theta-\vec \theta_I) \over (\vec \theta-\vec
\theta_I)^2} \cdot\vec\gamma(\vec \theta_I) +\kappa_0 \ ; \ \ \ \vec \chi(\vec \theta)=({\theta_1^2-
\theta_2^2\over \theta^2} \ , \ {2\theta_1 \theta_2 \over \theta^2}),
\end{equation}
where $\kappa_0$ is the integration constant. In the weak lensing regime,
Eqs.(9) provides a mapping of the total projected mass, using the
distortion of the background objects.
\begin{table}
\hskip 2.0 truecm
\psfig{figure=summary_mass_WL.proc.ps,width=14. cm}
\caption{Summary mass obtained from weak lensing inversion in the
literature. The averaged mass-to-light ratio is higher than the one
inferred from giant arc (see table 1) which is interpreted as a real
increase with distance from the cluster center. Note the strong
uncertainty in the case of MS1054 which is due to the strong dependence
of mass with the redshift of sources as the redshift of the lens
increases. The scales are expressed in $h_{100}^{-1}$ Kpc.}
\end{table}
Several improvements of the basic theory of lensing inversion have been
discussed in details by Seitz \& Schneider (1995, 1996), Schneider (1995) and
Kaiser (1996). Note that
the objects are also magnified by a factor $\mu$ which, in case of the
weak lensing regime, has the form,
\begin{equation}
\mu=1+2\kappa\ .
\end{equation}
This gives potentially
another independent way to measure the projected mass density
$\Sigma$ of a lens using the magnification instead of the distortion.
Table 2 gives a summary of the clusters for which lensing
inversion have been attempted. When compare with mass distribution
inferred from strong lensing, there is a clear trend towards higher $M/L$
when the scale increases. Bonnet et al. (1994) found $M/L$ close to
600 at 2.5 $h^{-1}$Mpc
from the cluster center. Note also the remarkable results from
Kaiser \& Luppino on a cluster at redshift 0.83, for which the
estimated mass strongly depends on the redshifts of the sources.
Indeed in general the estimation of the cluster mass using Eq. (9) requires the
knowledge of $\Sigma_{\rm crit.}$, which depends on the
usually poorly known redshifts of the
sources. Though this is not a critical issue for
nearby clusters ($z_l<0.2$), because then ${D_{os}/ D_{ls}}\simeq 1$, it could
lead to large mass uncertainties for more distant clusters it is the case
for MS1054.
Actually, even if the redshift of the sources were known, it would
still not be possible to get
the absolute value of the mass distribution, because possible
mass planes of constant density intercepting the line of sight
do not change the shear map.
Mathematically, this corresponds to the
unknown integration constant $\kappa_0$ in Eq.(10).
This degeneracy may be broken if one
measures the magnification $\mu$ which depends on the mass quantity inside
the light beam (Eq.(3)). While the
shear measurement does not require any information in the source plane, the
magnification measurement needs the observation of a reference (unlensed)
field to calibrate the magnification. Broadhurst et al. (1995) proposed to
compare the number count
$N(m,z)$ and/or $N(m)$ in a lensed and an unlensed field to measure
$\mu$. Depending on
the value of the slope $S$ of the number count in the reference field, we
observe a bias (more objects) or an anti-bias (less objects) in the
lensed field. The particular value $S=0.4$ corresponds to the case where
the magnification of faint objects is exactly compensated by the
dilution of the number count (Eq.(18)). This method was applied on
the cluster A1689 (Broadhurst, 1995), but the signal to noise of the
detection remains 5 times lower than with the distortion method for a given
number of galaxies. The magnification may also be determined
by the changes of the image sizes at fixed surface brightness
(Bartelmann \& Narayan 1995).
The difficulty with these methods is that they
required to measure the shape, size and magnitude of very faint
objects up to B=28
which depends on the detection threshold, the
convolution mask and the local statistical properties of the noise.
These remarks led us to propose a new method to analyze
the weak lensing effects, based on the auto-correlation function of
the pixels in
CCD images, which avoids shape parameter measurements of individual
galaxies (Van Waerbeke et al. 1996a). It is described in the next
subsection.
\subsection{The Auto-correlation method}
The CCD image is viewed as a density field rather than an image containing
delimited objects. The surface brightness, $I(\vec \theta)$,
in the image plane in the direction
$\vec \theta$ is related to the surface brightness in the source plane
$I^{(s)}$ by the relation,
\begin{equation}
I(\vec \theta)=I^{(s)}({\cal A}\vec \theta),
\end{equation}
which can be straightforwardly extended
to the auto-correlation function (ACF) (e.g. the
two-point autocorrelation function of the light distribution in a given area),
\begin{equation}
\xi(\vec \theta)=\xi^{(s)}({\cal A}\vec \theta)\ .
\end{equation}
This equation is more meaningful when it is written in the
weak lensing regime,
\begin{equation}
\xi(\vec \theta)=\xi^{(s)}(\theta)-\theta \ \partial_{\theta} \xi^{(s)}(\theta)
[1-{\cal A}]
\end{equation}
since the local ACF, $\xi(\vec \theta)$, now writes as
the sum of an isotropic unlensed term,
$\xi^{(s)}(\theta)$, an isotropic lens
term which depends on $\kappa$, and an anisotropic term which depends on
$\gamma_i$.
Let us now explore the gravitational lensing information that can be extracted
from the shape matrix $\cal M$ of the ACF,
\begin{equation}
{\cal M}_{ij}={\int {\rm d}^2\theta\ \xi (\vec \theta)\ \theta_i\ \theta_j\over \int
{\rm d}^2\theta\ \xi (\vec \theta)}\ .
\end{equation}
The shape matrix in the image plane is simply related to the shape
matrix in the source plane ${\cal M}^{(s)}$ by ${\cal M}_{ij}={\cal
A}_{ik}^{-1} {\cal A}_{jl}^{-1} {\cal M}^{(s)}_{kl}$. If the galaxies
are isotropically distributed in the source plane,
$\xi^{(s)}$ is isotropic, and in that case ${\cal
M}^{(s)}_{ij}=M\delta_{ij}$, where $\delta_{ij}$ is the identity matrix.
Using the expression of the amplification matrix $\cal A$ we get the
general form for $\cal M$,
\begin{equation}
{\cal M}={M(a+|g|^2)\over (1-\kappa)^2(1-|g|^2)} \pmatrix{
1+\delta_1 & \delta_2 \cr \delta_2 & 1-\delta_1 \cr }\ .
\end{equation}
The observable quantities (distortion $\delta_i$ and magnification
$\mu$) are given in terms of the components of the shape matrix,
\begin{equation}
\delta_1={{\cal M}_{11}-{\cal M}_{22}\over {\rm tr}({\cal M})} \ ; \ \ \
\delta_2={2{\cal M}_{12}\over {\rm tr}({\cal M})} \ ; \ \ \
\mu=\sqrt{{\rm det}({\cal M})\over M},
\end{equation}
where ${\rm tr}({\cal M})$ is the trace of $\cal M$ and
${\rm det}({\cal M})$ is the determinant of $\cal M$.
As for sheared galaxies, we see that the
distortion is available from a direct measurement in the image plane while
the magnification measurement requires to know the value of $M$ which
is related to the light distribution in the source plane, or in an
unlensed reference plane. The ACF provides a new and independent
way to measure
$\delta_i$ and $\mu$ which does not require shape, size or photometry
of individual galaxies. Furthermore, the signal to noise ratio is
proportional to the number density of background galaxies, $N$, instead
as $\sqrt{N}$ for the standard method (see figure 2).
A description of its practical implementation and first results are given
in Van Waerbeke et al. (1996a) and Van Waerbeke \& Mellier (1996b). Clearly,
the ACF is the most powerful technique for measuring the orientation of
very weak shear as the one we
expect from large-scale structures.
\section{The matter distribution on very large scale}
The direct observation of the mass distribution on scale larger than
10 $h^{-1}$Mpc (or $\approx $ 1 square degree) is one of the great hopes of
the weak gravitational lensing approach. Two observational directions are now
being investigated. In the first one we search for the shear of dark
condensation of mass that can be responsible for the magnification bias of
luminous quasars and radio-sources. It probes the lumpiness of matter
distribution within large-scale structures. In the second one, we analyze
the statistical properties of weakly lensed background galaxies on
degree scales in order to
obtain constraints on the cosmological parameters and
on the projected power spectrum.
\subsection{Shear around radio-sources and the lumpiness of matter}
Fugmann (1990) and Bartelmann \& Schneider (1993a,b) have
demonstrated that there is a strong correlation between the presence of
galaxies or clusters and bright quasars. They interpreted this as a
magnification bias induced the galaxy over-densities along the line of
sight. However, the measured correlation is too strong to be only due to
individual galaxies, so Bartelmann \& Schneider suggested that the
magnification
bias originates from groups or even rich cluster of galaxies.
If this suggestion is correct, the deflector responsible for the
magnification bias should also induce gravitational shear onto background
sources.
\begin{figure}
\hskip 3truecm
\psfig{figure=q1622imageandshear.ps,width=10. cm}
\caption{Shear map around the bright quasar Q1622. The image obtained at
CFHT clearly shows a coherent shear. The ellipses shows the center of
the shear pattern which is very close to the quasar (dark dot). The
others solid lines are galaxy number isodensity contours. Clearly the quasars,
the shear pattern and a galaxy concentration are almost positioned at
the same place which reinforces the hypothesis of a magnification effect
on the quasar.}
\end{figure}
Bonnet et al. (1993), Mellier et al. (1994) and
Van Waerbeke et al. (1996a) found a strong gravitational shear
and a galaxy excess around the quasar pair Q2345+007, the shear
pattern being associated with a group or a small cluster.
In the same way Fort et al. (1996a) have measured the shear around a few
over-bright QSO-s that could be magnified by nearby lensing mass (figure
3). It is the first tentative to detect
gravitational structures from a mass density criteria rather than
luminosity excess. The field of view around each QSO is small
and up to now, there is not enough data to draw a synthetic view of the
global matter distribution. However, both weak lensing detection around
bright radio-sources and the correlations found by Bartelmann \&
Schneider seem to favor a model where clumps of dark matter are more
numerous that expected and concentrated
on groups and clusters of galaxies.
\subsection{Statistical analysis of shear on very large scale}
In the statistical studies of the shear at very large scale
the lenses are not individually identified, but viewed as
a random population affecting the shape of the galaxies with
an efficiency depending on their distances.
Indeed, in this approach, we
consider the statistical
properties of the shear measured on backgroung objects
for randomly chosen line-of-sights. The measured shears
are actually filtered at a given angular scale so that a signal of
cosmological interest can be extracted.
For a filtering scale of about one degree, the structures
responsible of the gravitational shear being at a redshift
of about 0.4 are expected to be
on scales above 10 $h^{-1}$Mpc , that is in a regime where their properties
can be easily predicted with the linear or perturbation theory.
Within this line of thought,
Blandford et al. (1991), Miralda-Escud\'e (1991) and Kaiser
(1992) argued that the projected power spectrum
should be measurable with such a method
provided shape parameters are averaged on the degree scale, as
it is illustrated on figure 4.
\begin{figure}
\hskip 2.0 truecm
\psfig{figure=simul.lss.ludo.2.ps,width=14. cm}
\caption{
Simulation of shear expected from large scale structures. The left
panel shows a 256$^3$ Mpc$^3$ box with dark dots indicating the
location of matter. The long filaments are large scale structures
which originated from a uniform distribution and an initial power
spectrum $P(k)=k^{-1}$ under the adhesion approximation. The right
panel shows a slice which represents the projected mass distribution
as it would be observed by an observer. The thin straight lines are the
local orientation and intensity of the shear. It illustrates what we
expect from the future observations of weak lensing with wide field
CCD.
}
\end{figure}
For such a scale, however, the shear is expected to be as low as 1\%.
Therefore
its detection requires high image quality to avoid uncontrolled errors,
and large angular coverage so that it is possible to separate the
gravitational shear from the intrinsic galactic ellipticities
by averaging over thousands of galaxies. So far such a detection has
not been completed due to these severe constraints, but
there are now projects to build very large CCD mosaic camera which could be
capable of doing large deep imaging surveys. The MEGACAM project conducted by
the French institutions, CEA (CE in Saclay) and INSU,
the Canadian CNRC and the CFHT Corporation consists in building a
16K$\times$16K camera at the prime focus of CFHT. This camera will
provide a total field of view of 1$^o\times$1$^o$, with image quality
lower than 0.5" on the whole field. Using a Pertubation Theory
approach, Bernardeau et al. (1996) have
analyzed the statistical properties of the gravitational shear averaged
on MEGACAM scales. They have shown that with the observation of
about 25 such fields,
one could recover the projected power spectrum and $\Omega$
independently by using the variance and the skewness (third moment)
of the one-point probability distribution function
of the local convergence in the sample
\footnote{detailed calculations have shown however that there is a slight
degeneracy with the cosmological constant.}.
Basically, these moments write,
\begin{equation}
\big <\kappa^2_{\theta}\big > \propto \ \ P(k)\ \Omega^{1.5} \ \ z_s^{1.5}
\end{equation}
and
\begin{equation}
{\big <\kappa^3_{\theta}\big > \over \big <\kappa^2_{\theta}\big >^2} \propto
\ \ \Omega^{-0.8} \ \ z_s^{-1.35}
\end{equation}
where $\theta =30'$ is the scale where the convergence $\kappa$ is
averaged, $P(k)$ is the projected power spectrum of the dark matter
and $z_s$ is the
averaged redshift of sources. Note that the skewness, expressed
as the ratio of the third moment by the square of the second,
does not depend on
$P(k)$ and provides a direct information on $\Omega$.
These important results show that a very large-scale survey of
gravitational shear can provide important cosmological results
which could be compared with those coming from large-scale flows and
observations of the cosmic microwave background anisotropy spectrum
that are expected to culminate with the
COBRAS/SAMBA satellite mission. However, there are two shortcomings that must
be handled carefully: first, it requires detection of very weak shear
from shape parameters of galaxies. Thus, the image quality of the survey is
a crucial issue. The ACF method proposed by Van Waerbeke et al. (1996a)
should provide accurate shape information with a high signal to noise
ratio. It is probably
the best available method for measuring shear on very large
scales. The second point is the strong dependence of the variance and the
skewness with the redshift of lensed sources. Since there are no hopes to
obtain redshifts of these galaxies even from spectroscopy with the VLT-s,
we face on a difficult and crucial problem. It may be overcome
if multicolor photometric data can provide
accurate redshifts for very faint galaxies. Another possible solution
could be
the analysis of the radial magnification bias of faint distant galaxies
around rich clusters which, indeed, probes the
redshift distribution of the sources (Fort et al. 1996b and see section 6).
\section{Measuring the cosmological constant}
One remarkable property of gravitational lensing effect is the local change
of the galaxy number density. The observable number density of sources
results of the competition between deflection
effect which tends to
enlarge the projected area and the magnification which increases
the number of faint sources. The expected galaxy number density is,
\begin{equation}
N(r) = N_0\,\mu(r)^{2.5 \alpha -1},
\end{equation}
where $\mu$ is the magnification at the position $r$ and
$\alpha= {\rm d}\ \log(N) / {\rm d} m$
is the slope of the galaxy counts. When $\alpha$ is larger
than 0.4, the galaxy number density increases. At faint limiting magnitude,
$\alpha$ becomes significantly smaller than 0.4 and a clear galaxy depletion
can be detected (Broadhurst 1995, Fort et al. 1996b).
\begin{figure}
\hskip .5 truecm
\psfig{figure=depletion.blois.full.proc.eps,width=16. cm}
\caption{Principle of measurement of redshifts from depletion. The
right panel shows the depletions curves expected by a singular
isothermal sphere. If the lens is perfectly known
the minimum of each depletion curve (right panel) depends only
of the redshift of the source
and the radial position of the critical line is actually
equivalent to a redshift.
In a realistic case, the redshift
distribution is broad and the left panel shows the depletion as it would be
observed: instead of
the single peaked depletion we expect a more pronounced minimum between
two radii (= two redshifts ) whose angular positions strongly depend on the
cosmological constant for high redshift sources. Thus, if the mass
distribution of the lens is well known as in the rich lensing cluster
Cl0024+1654, $\lambda$ can be inferred from the shape of the depletion curve.
}
\end{figure}
\begin{figure}
\psfig{figure=depletion.cl0024.lambda.blois.proc.eps,width=14. cm}
\caption{Measuring the cosmological constant from the depletion observed
in Cl0024. The left panel shows the depletion curve observed in
Cl0024+1654. From the redshift of the first minimum and the second
minimum, one can constrain the cosmological constant in order to
position the second minimum at its right angular position. Whatever the
redshift of the most distant sources visible on the images, we see that
the angular position where the depletion curve raises again imposes that
$\lambda>0.65$}
\end{figure}
The shape of the depletion curves depends on the magnification $\mu$
which is a complicated function of the lensing potential, the redshift
distribution of sources and the cosmology. For a singular isothermal
sphere, the amplification at radius $r$ writes,
\begin{equation}
\mu(r) = {4 \pi \sigma^2 \over c^2}\,{D_{ls} \over D_{os}}\,{r \over r-1},
\end{equation}
and the depletion curve for a single redshift of sources looks like
those shown in figure 5. For a redshift distribution, the depletion curve
shows
a plateau between two extremum points corresponding to the lowest
and highest redshift of the sources. Once the redshift of one source
is known, the radial position of the highest redshift strongly depends
on the cosmological constant.
Fort et al. (1996b) tentatively measured depletions curves
in Cl0024+1654 (see figure 6) and A370 of faint galaxies close to the noise
level.
With the hypothesis of a single lens along the line of sight they
found that the angular position of the highest-redshift sources is
very high and imposes that the most distant galaxies visible in the
field have redshifts larger than $2$, while the width of the depleted areas
extend as far as 60 arcseconds which is incompatible with a low-$\lambda$
universe. In fact, the observations provide a lower limit $\lambda >0.6$
(figure 6).
\section{Conclusion}
In the last ten years,
the gravitational lensing effects turned out to be among the most
promising tools for cosmology. It is indeed a direct probe
of the large-scale cosmic mass
distribution and some observable quantities revealed to be
extremely sensitive to
the cosmological parameters. We summarize in table 3
what we learned about $\Omega$ from the mass distribution in rich clusters
of galaxies and what we expect in the near future. It shows that we
can hope for strong and reliable constraints on $\Omega$ and $\lambda$
from the developing observational tools:
we are now aiming at a determination of the
cosmological parameters within 10\% accuracy.
\begin{table}
\hskip 3.0 truecm
\psfig{figure=conclu.blois.proc.ps,width=10. cm}
\caption{A summary of the values of the cosmological parameters as
they are inferred from gravitational lensing. They are still some
uncertainties and many hopes for the future. But a real trend toward
$\Omega >0.3$ seems well established.}
\end{table}
\acknowledgements
We thanks P. Schneider, for discussions and
enthusiastic support.
F. Bernardeau is grateful to IAP, where most of this work has been completed,
for its hospitality.
\section{References}
{\parindent=0pt
\parskip=3 pt
Bartelmann, M., Schneider, P. (1993a) A\&A 259, 413.
Bartelmann, M., Schneider, P. (1993b) A\&A 271, 421.
Bartelmann, M. (1995) A\&A 299, 661.
Bartelmann, M., Narayan, R. (1995) ApJ 451, 60.
Bernardeau, F., Van Waerbeke, L., Mellier, Y. (1996) in preparation
Blandford, R.D., Saust, A.B., Brainerd, T.G., Villumsen, J.V. (1991),
MNRAS, 251, 600
Bonnet, H., Fort, B., Kneib, J-P., Mellier, Y., Soucail, G. (1993) A\&A
280, L7
Bonnet, H., Mellier, Y., Fort, B. (1994) ApJ 427, L83.
Bonnet, H., Mellier, Y. (1995) A\&A 303, 331.
Broadhurst, T. (1995) SISSA preprint astro-ph/9511150.
Broadhurst, T., Taylor, A.N., Peacock, J. (1995) ApJ 438, 49.
Fahlman, G., Kaiser, N., Squires, G., Woods, D. (1994) ApJ 437. 56.
Fort, B., Prieur, J.-L., Mathez, G., Mellier, Y., Soucail, G. (1988)
A\&A 200, L17.
Fort, B., Mellier, Y., (1994) A\&A Review 5, 239, 292.
Fort, B., Mellier, Y., Dantel-Fort, M., Bonnet, H., Kneib, J.-P. (1996a)
A\&A 310, 705.
Fort, B., Mellier, Y., Dantel-Fort (1996) SISSA preprint
astro-ph/9606039.
Fugmann, W. (1990) A\&A 240, 11.
Kaiser, N. (1992) ApJ 388, 272.
Kaiser, N. (1996) SISSA preprint astro-ph/9509019.
Kaiser, N., Squires, G. (1993) ApJ 404, 441
Kneib, J.-P., Mellier, Y., Fort, B., Mathez, G. (1993) A\&A 273, 367.
Kneib, J.-P., Mellier, Y., Pell\'o, R., Miralda-Escud\'e, J., Le
Borgne, J.-F., Boehringer, H., Picat, J.-P. (1995) A\&A 299, 168.
Kassiola, A., Kovner, I., Fort, B. (1992) ApJ 400, 41.
Luppino, G., Kaiser, N. (1996) SISSA preprint astro-ph/9601194.
Lynds, R., Petrossian, V. (1986) BAAS 18, 1014.
Mellier, Y., Fort, B., Kneib, J.-P. (1993) ApJ 407, 33.
Mellier, Y., Dantel-Fort, M., Fort, B., Bonnet, H. (1994) A\&A 289,
L15.
Miralda-Escud\'e, J., (1991) ApJ, 370, 1.
Narayan, R., Bartelmann, M. (1996) SISSA preprint astro-ph/9606001
Pierre, M., Le Borgne, J.-F., Soucail, G., Kneib, J.-P. (1996) A\&A 311,
413.
Schneider, P., Ehlers, J., Falco, E. E., (1992), {\it Gravitational
Lenses}, Springer.
Seitz, C., Kneib, J.P., Schneider, P., Seitz, S., (1996) in press.
Seitz, C., Schneider, P., (1995) A\&A 302, 9
Seitz, S., Schneider, P., (1996) A\&A 305, 388
Smail, I., Ellis, R.S., Fitchett, M. (1994) MNRAS, 270, 245.
Schneider, P. (1995), A\&A, 302, 639
Soucail, G., Fort, B., Mellier, Y., Picat, J.-P. (1987) A\&A 172, L14.
Squires, G., Kaiser, N., Babul, A., Fahlmann, G., Woods, D., Neumann,
D.M., B\"ohringer, H. (1996a) ApJ 461, 572.
Squires, G., Kaiser, N., Falhman, G., Babul, A., Woods, D. (1996b) SISSA
preprint astro-ph/9602105.
Tyson, J.A., Fisher, P. (1995) ApJL, 349, L1.
Van Waerbeke, L., Mellier, Y., Schneider, P., Fort, B., Mathez, G. (1996a)
A\&A in press. SISSA preprint astro-ph/9604137.
Van Waerbeke, L., Mellier, Y. (1996b). Proceedings of the XXXIst
Rencontres de Moriond, Les Arcs, France 1996. SISSA preprint
astro-ph/9606100.
Wallington, S., Kochanek, C. S., Koo, D. C. (1995) ApJ 441, 58.
Walsh, D., Carswell, R.F., Weymann, R.J., (1979), Nature, 279, 381
}
\newpage
\heading{PROGR\`ES R\'ECENTS EN LENTILLE GRAVITATIONNELLE}
\centerline{Y. Mellier$^{(1,4)}$, L. Van Waerbeke$^{(2)}$, F. Bernardeau$^{(3)}$,
B. Fort$^{(4)}$}
{\it
\centerline{$^{(1)}$Institut d'Astrophysique de Paris,}
\centerline{98$^{bis}$ Boulevard Arago,}
\centerline{75014 Paris, France.}
\centerline{$^{(2)}$ Observatoire Midi Pyr\'en\'ees,}
\centerline{14 Av. Edouard Belin,}
\centerline{31400 Toulouse, France.}
\centerline{$^{(3)}$ Service Physique Th\'eorique,}
\centerline{CE de Saclay,}
\centerline{91191 Gif-sur-Yvette Cedex, France.}
\centerline{$^{(4)}$Observatoire de Paris (DEMIRM),}
\centerline{61 Av. de l'Observatoire,}
\centerline{75014 Paris, France.}
}
\begin{abstract}{\baselineskip 0.4cm
L'effet de lentille gravitationelle est aujourd'hui consid\'er\'e comme
un des outils les plus prometteurs de la cosmologie. Il sonde
directement la mati\`ere distribu\'ee dans les grandes structures
et peut aussi fournir des informations importantes sur les param\`etres
cosmologiques, $\Omega$ et $\Lambda$. Dans cet article de revue, nous
r\'esumons les progr\'es observationels et th\'eoriques r\'ecents les plus
marquants obtenus dans ce domaine au cours des cinq derni\`eres
ann\'ees.
}
\end{abstract}
\end{document}
| proofpile-arXiv_065-493 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section*{Introduction}
Let $Y$ be a connected simplicial complex. Suppose that
$\pi$ acts freely and simplicially on $Y$ so that $X=Y/\pi$ is a finite
simplicial complex. Let ${\mathcal F}$ a finite subcomplex of $Y$, which
is a fundamental domain
for the action of $\pi$ on $Y$.
We assume that $\pi$ is amenable. The F\o{}lner criterion for amenability
of $\pi$ enables one to get, cf. \cite{Ad}, a {\em regular exhaustion}
$\big\{Y_{m}\big\}^{\infty}_
{m=1}$, that is a sequence of finite subcomplexes of $Y$
such that
(1) $Y_{m}$ consists of $N_{m}$ translates $g.{\mathcal F}$ of
${\mathcal F}$ for
$g\in\pi$.
(2) $\displaystyle Y=\bigcup^{\infty}_{m=1}Y_{m}\;$.
(3) If $\dot{N}_{m,\delta}$ denotes the number of translates of
${\mathcal F}$
which have distance (with respect to the word metric in $\pi$) less than or
equal to $\delta$ from a translate of ${\mathcal F}$ having a
non-empty intersection with the topological boundary $\partial{Y}_{m}$ of
$Y_{m}$ (we identify here $g.{\mathcal F}$ with
$g$)
then, for every $\delta > 0$,
$$
\lim_{m\rightarrow\infty}\;\frac{{\dot{N}}_{m,\delta}}{N_{m}}=0.
$$
One of our main results is
\begin{theorem}[Amenable Approximation Theorem]
$\;$Let $Y$ be a connected simplicial complex.
Suppose that $\pi$ is amenable and that $\pi$ acts freely and simplicially
on $Y$ so that $X=Y/\pi$ is a finite simplicial complex.
Let $\big\{Y_{m}\big\}^{\infty}_
{m=1}$ be a regular exhaustion of $Y$. Then
$$
\lim_{m\rightarrow\infty}\;\frac{b^{j}(Y_{m})}{N_{m}}=b_{(2)}^{j}(Y:\pi)
\;\;\mbox{ for all }\;\;j\ge 0.
$$
$$
\lim_{m\rightarrow\infty}\;\frac{b^{j}(Y_{m},
\partial Y_{m})}{N_{m}}=b_{(2)}^{j}(Y:\pi)
\;\;\mbox{ for all }\;\;j\ge 0.
$$
\end{theorem}
Here $b^{j}(Y_{m})$ denotes the $j^{th}$ Betti number of $Y_m$,
$b^{j}(Y_{m}, \partial Y_{m})$ denotes the $j^{th}$ Betti number of
$Y_m$ relative its boundary $\partial Y_m$ and $b_{(2)}^{j}(Y:\pi)$
denotes the $j$th $L^2$ Betti number of $Y$.
(See the next section for the definition
of the $L^2$ Betti numbers of a manifold)
\noindent{\bf Remarks.} This theorem proves the main conjecture in the
introduction of
an earlier paper \cite{DM}. The combinatorial techniques of this paper
contrasts
with the heat kernel approach used in \cite{DM}.
Under the assumption dim $H^{k}(Y)<\infty$, a special case of the
Amenable Approximation Theorem above
is obtained by combining proofs of
Eckmann \cite{Ec} and Cheeger-Gromov \cite{CG}.
The assumption dim $H^{k}(Y)<\infty$ is very restrictive and essentially
says
that $Y$ is a fiber bundle over a $B\pi$ with fiber a space with
finite fundamental group. Cheeger-Gromov use this to
show that the Euler
characteristic of a finite $B\pi$, where $\pi$ contains an infinite
amenable normal subgroup, is zero. Eckmann proved the same result
in the special
case when $\pi$ itself is an infinite amenable group.
There is a standing conjecture that any normal
covering space of a finite simplicial complex is of determinant class
(cf. section 4 for the definition of determinant class and for a
more detailed discussion of what follows).
Let $M$ be a smooth compact manifold, and $X$ triangulation of $M$.
Let $\widetilde M$ be any normal covering space of $M$, and $Y$ be the
triangulation of $\widetilde M$ which projects down to $X$.
Then on $\widetilde M$, there are two notions of determinant class,
one analytic and the other combinatorial. Using results of
Efremov \cite{E}, Gromov and Shubin
\cite{GS}, one observes as in \cite{BFKM} that the
combinatorial and analytic
notions of determinant class coincide.
It was proved in \cite{BFK}) using estimates of L\"uck \cite{L} that any
{\em residually finite} normal
covering space of a finite simplicial complex is of determinant class, which
gave evidence supporting the conjecture.
Our interest in this conjecture stems from work on $L^2$
torsion \cite{CFM}, \cite{BFKM}. The $L^2$ torsion is a well defined
element in the determinant line of the reduced $L^2$ cohomology, whenever
the covering space is of determinant class. Our next main theorem
says that any {\em amenable} normal
covering space of a finite simplicial complex is of determinant class,
which gives further evidence supporting the conjecture.
\begin{theorem}[Determinant Class Theorem]
$\;$Let $Y$ be a connected simplicial complex.
Suppose that $\pi$ is amenable and that $\pi$ acts freely and simplicially
on $Y$ so that $X=Y/\pi$ is a finite simplicial complex. Then $Y$
is of determinant class.
\end{theorem}
The paper is organized as follows. In the first section, some
preliminaries on $L^2$
cohomology and amenable groups are
presented. In section 2, the main abstract approximation theorem is
proved. We
essentially use the combinatorial analogue of the principle of not feeling
the boundary (cf. \cite{DM}) in Lemma 2.3 and a finite dimensional result
in \cite{L}, to prove this theorem.
Section 3 contains the proof of the Amenable Approximation Theorem and
some related
approximation theorems.
In section 4, we prove that an arbitrary {\em amenable} normal covering
space of a finite simplicial complex is of determinant class.
The second author warmly thanks Shmuel Weinberger for some useful
discussions. This paper has been inspired by L\"uck's work \cite{L} on
residually finite groups.
\section{Preliminaries}
Let $\pi$ be a finitely generated discrete group and ${\mathcal U}(\pi)$
be the von Neumann algebra
generated by the action of $\pi$ on $\ell^{2}(\pi)$ via the left regular
representation. It is the weak (or strong) closure of the complex group
algebra of
$\pi$, ${\mathbb C}(\pi)$ acting on $\ell^2(\pi)$ by left translation.
The left regular representation is then a unitary representation
$\rho:\pi\rightarrow{\mathcal U}(\pi)$. Let ${\text{Tr}}_{{\mathcal U}(\pi)}$
be the faithful normal trace on
${\mathcal U}(\pi)$ defined by the inner product
${\text{Tr}}_{{\mathcal U}(\pi)}(A) \equiv
(A\delta_e,\delta_e)$ for
$A\in{\mathcal U}(\pi)$ and where $\delta_e\in\ell^{2}(\pi)$ is given by
$\delta_e(e)=1$ and
$\delta_e(g)=0$ for $g\in\pi$ and $g\neq e$.
Let $Y$ be a simplicial complex, and $|Y|_j$ denote the set of all
$p$-simplices
in $Y$. Regarding the orientation of simplices, we use the following
convention. For
each $p$-simplex $\sigma \in |Y|_j$, we identify $\sigma$ with any other
$p$-simplex
which is obtained by an {\em even} permutation of the vertices in $\sigma$,
whereas
we identify $-\sigma$ with any other $p$-simplex
which is obtained by an {\em odd} permutation of the vertices in $\sigma$.
Suppose that
$\pi$ acts freely and simplicially on $Y$ so that $X=Y/\pi$ is a finite
simplicial complex. Let ${\mathcal F}$ a finite subcomplex of $Y$, which
is a fundamental domain
for the action of $\pi$ on $Y$. Consider the Hilbert space of square
summable cochains on $Y$,
$$
C^j_{(2)}(Y) = \Big\{f\in C^j(Y): \sum_{\sigma\; a\; j-simplex}|f(\sigma)|^2
<\infty \Big\}
$$
Since $\pi$ acts freely on $Y$, we see that there is an isomorphism of
Hilbert $\ell^2(\pi)$ modules,
$$
C^j_{(2)}(Y) \cong C^j(X)\otimes\ell^2(\pi)
$$
Here $\pi$ acts trivially on $C^j(X)$
and via the left regular representation on $\ell^2(\pi)$. Let
$$
d_{j}:C^{j}_{(2)}(Y)\rightarrow C^{j+1}_{(2)}(Y)
$$
denote the coboundary operator. It is clearly a bounded linear operator.
Then the (reduced) $L^2$ cohomology groups of $Y$
are defined to be
$$
H^j_{(2)}(Y) = \frac{\mbox{ker}(d_j)}{\overline{\mbox{im}(d_{j-1})}}.
$$
Let ${d_j}^*$ denote the Hilbert space adjoint of
$d_{j}$. One defines the combinatorial Laplacian
$\Delta_{j} : C^{j}_{(2)}(Y) \rightarrow C^{j}_{(2)}(Y)$ as
$\Delta_j = d_{j-1}(d_{j-1})^{*}+(d_{j})^{*}d_{j}$.
By the Hodge decomposition theorem in this context, there is an isomorphism
of Hilbert $\ell^2(\pi)$ modules,
$$
H^j_{(2)}(Y)\;\; \cong\;\; \mbox{ker} (\Delta_j).
$$
Let $P_j: C^{j}_{(2)}(Y)\rightarrow \mbox{ker} \Delta_j$ denote the
orthogonal projection to the kernel of the Laplacian. Then the $L^2$
Betti numbers $b_{(2)}^j(Y:\pi)$ are defined as
$$
b_{(2)}^j(Y:\pi) = {\text{Tr}}_{{\mathcal U}(\pi)}(P_j).
$$
In addition, let $\Delta_j^{(m)}$
denote the
Laplacian on the finite dimensional cochain space $C^j(Y_m)$ or
$C^j(Y_m,\partial Y_m)$. We do use the same notation for the two Laplacians
since all proofs work equally well for either case.
Let $D_j(\sigma, \tau) = \left< \Delta_j \delta_\sigma, \delta_\tau\right>$
denote the matrix coefficients of the Laplacian $\Delta_j$ and
${D_j^{(m)}}(\sigma, \tau) = \left< \Delta_j^{(m)} \delta_\sigma,
\delta_\tau\right>$
denote the matrix coefficients of the Laplacian $\Delta_j^{(m)}$. Let
$d(\sigma,\tau)$ denote the
{\em distance} between $\sigma$ and $\tau$ in the natural simplicial
metric on $Y$, and
$d_m(\sigma,\tau)$ denote the
{\em distance} between $\sigma$ and $\tau$ in the natural simplicial
metric on $Y_m$. This distance (cf. \cite{Elek}) is defined as follows.
Simplexes $\sigma$ and $\tau$ are one step apart, $d(\sigma,\tau)=1$, if
they have equal dimensions,
$\dim \sigma = \dim \tau =j$, and there exists either a simplex of
dimension $j-1$ contained in both $\sigma$ and $\tau$ or a simplex of
dimension $j+1$ containing both $\sigma$ and $\tau$. The distance between
$\sigma$ and $\tau$ is equal to $k$
if there exists a finite sequence $\sigma = \sigma_0, \sigma_1, \ldots ,
\sigma_k = \tau$, $d(\sigma_i,\sigma_{i+1})=1$ for $i=0,\ldots,k-1$, and
$k$ is the minimal length of such a sequence.
Then one has the following, which is an easy
generalization of Lemma 2.5 in \cite{Elek} and follows immediately from
the definition of combinatorial Laplacians and finiteness of the
complex $X=Y/\pi$.
\begin{lemma} $D_j(\sigma, \tau) = 0$ whenever $d(\sigma,\tau)>1$ and
$ {D_j^{(m)}}(\sigma, \tau) = 0$ whenever $d_m(\sigma,\tau)>1$.
There is also a positive constant $C$ independent of $\sigma,\tau$ such that
$D_j(\sigma, \tau)\le C$ and ${D_j^{(m)}}(\sigma, \tau)\le C$.
\end{lemma}
Let $D_j^k(\sigma, \tau) = \left< \Delta_j^k \delta_\sigma,
\delta_\tau\right>$
denote the matrix coefficient of the $k$-th power of the Laplacian,
$\Delta_j^k$, and
$D_j^{(m)k}(\sigma, \tau) = \left< \left(\Delta_j^{(m)}\right)^k
\delta_\sigma, \delta_\tau\right>$
denote the matrix coefficient of the $k$-th power of the Laplacian,
$\Delta_j^{(m)k}$. Then
$$D_j^k(\sigma, \tau) = \sum_{\sigma_1,\ldots\sigma_{k-1} \in |Y|_j}
D_j(\sigma, \sigma_1)\ldots D_j(\sigma_{k-1}, \tau)$$
and
$$D_j^{(m)k}(\sigma, \tau) = \sum_{\sigma_1,\ldots\sigma_{k-1} \in |Y_m|_j}
{D_j^{(m)}}(\sigma, \sigma_1)\ldots {D_j^{(m)}}(\sigma_{k-1}, \tau).$$
Then the following lemma follows easily from Lemma 1.1.
\begin{lemma} Let $k$ be a positive integer. Then $D_j^k(\sigma, \tau) = 0$
whenever $d(\sigma,\tau)>k$ and
$D_j^{(m)k}(\sigma, \tau) = 0$ whenever $d_m(\sigma,\tau)>k$.
There is also a positive constant $C$ independent of $\sigma,\tau$ such that
$D_j^k(\sigma, \tau)\le C^k$ and $D_j^{(m)k}(\sigma, \tau)\le C^k$.
\end{lemma}
Since $\pi$ commutes with the Laplacian $\Delta_j^k$, it follows that
\begin{equation}\label{inv}
D_j^k(\gamma\sigma, \gamma\tau) = D_j^k(\sigma, \tau)
\end{equation}
for all $\gamma\in \pi$ and for all $\sigma, \tau \in |Y|_j$. The
{\em von Neumann
trace} of $\Delta_j^k$ is by definition
\begin{equation} \label{vNt}
{\text{Tr}}_{{\mathcal U}(\pi)}(\Delta_j^k) =
\sum_{\sigma\in |X|_j} D_j^k(\sigma, \sigma),
\end{equation}
where $\tilde{\sigma}$ denotes an arbitrarily chosen lift of $\sigma$ to
$Y$. The trace is well-defined in view of (\ref{inv}).
\subsection{Amenable groups}
Let $d_1$ be the word metric on $\pi$. Recall the following
characterization of amenability due
to F\o{}lner, see also \cite{Ad}.
\begin{definition} A discrete group $\pi$ is said to be {\em amenable}
if there is a sequence
of finite subsets $\big\{\Lambda_{k}\big\}^{\infty}_{k=1}$ such that for
any fixed $\delta>0$
$$
\lim_{k\rightarrow\infty}\;\frac{\#\{\partial_{\delta}\Lambda_{k}\}}{\#
\{\Lambda_{k}\}}=0
$$
where $\partial_{\delta}\Lambda_{k}=
\{\gamma\in\pi:d_1(\gamma,\Lambda_{k})<\delta$
and $d_1(\gamma,\pi-\Lambda_{k})<\delta\}$ is a $\delta$-neighborhood of
the boundary of $\Lambda_{k}$. Such a sequence
$\big\{\Lambda_{k}\big\}^{\infty}_
{k=1}$ is called a {\em regular sequence} in $\pi$. If in addition
$\Lambda_{k}\subset\Lambda_{k+1}$ for all $k\geq 1$ and
$\displaystyle\bigcup^
{\infty}_{k=1}\Lambda_{k}=\pi$, then the sequence
$\big\{\Lambda_{k}\big\}^{\infty}_
{k=1}$ is called a {\em regular exhaustion} in $\pi$.
\end{definition}
Examples of amenable groups are:
\begin{itemize}
\item[(1)]Finite groups;
\item[(2)] Abelian groups;
\item[(3)] nilpotent groups and solvable groups;
\item[(4)] groups of subexponential growth;
\item[(5)] subgroups, quotient groups and extensions of amenable groups;
\item[(6)] the union of an increasing family of amenable groups.
\end{itemize}
Free groups and fundamental groups of closed negatively curved manifolds are
{\em not} amenable.
Let $\pi$ be a finitely generated amenable discrete group, and
$\big\{\Lambda_{m}\big\}^{\infty}_{m=1}$ a regular exhaustion in $\pi$.
Then it defines a regular exhaustion $\big\{Y_m\big\}^{\infty}_{m=1}$ of
$Y$.
Let $\{P_j(\lambda):\lambda\in[0,\infty)\}$ denote the right continuous
family of spectral projections of the Laplacian $\Delta_j$.
Since $\Delta_j$ is $\pi$-equivariant, so are $P_j(\lambda) =
\chi_{[0,\lambda]}(\Delta_j)$,
for $\lambda\in [0,\infty)$. Let
$F:[0,\infty)\rightarrow[0,\infty)$ denote the spectral
density function,
$$
F(\lambda)={\text{Tr}}_{{\mathcal U}(\pi)}(P_j(\lambda)).
$$
Observe that the $j$-th $L^{2}$ Betti number of $Y$ is also given by
$$
b_{(2)}^j(Y:\pi)=F(0).
$$
We have the spectral density function for every dimension $j$ but we do not
indicate explicitly this dependence. All our arguments are performed with a
fixed value of $j$.
Let $E_{m}(\lambda)$ denote the number of eigenvalues $\mu$ of
$\Delta_j^{(m)}$ satisfying $\mu\leq\lambda$ and which are counted with
multiplicity. We may sometimes omit the subscript $j$ on $\Delta_j^{(m)}$
and $\Delta_j$ to simplify the notation.
We next make the following definitions,
$$
\begin{array}{lcl}
F_{m}(\lambda)& = &\displaystyle\frac{E_{m}(\lambda)}{N_{m}}\;\\[+10pt]
\overline{F}(\lambda) & = & \displaystyle\limsup_{m\rightarrow\infty}
F_{m}(\lambda) \;\\[+10pt]
\mbox{\underline{$F$}}(\lambda) & = & \displaystyle\liminf_{m\rightarrow
\infty}F_{m}(\lambda)\; \\[+10pt]
\overline{F}^{+}(\lambda) & = & \displaystyle\lim_{\delta\rightarrow +0}
\overline{F}(\lambda+\delta) \;\\[+10pt]
\mbox{\underline{$F$}}^{+}(\lambda) & = & \displaystyle\lim_{\delta
\rightarrow +0}\mbox{\underline{$F$}}(\lambda+\delta).
\end{array}
$$
\section{Main Technical Theorem}
Our main technical result is
\begin{theorem} Let $\pi$ be countable, amenable group.
In the notation of section 1, one has
\begin{itemize}
\item[(1)]$\;F(\lambda)=\overline{F}^{+}(\lambda)=
\mbox{\underline{$F$}}^{+}(\lambda)$.
\item[(2)] $\;\overline{F}$ and {\underline{$F$}} are right
continuous at zero and we have the equalities
\begin{align*}
\overline{F}(0) & =\overline{F}^{+}(0)=F(0)=\mbox{\underline{$F$}}
(0)=\mbox{\underline{$F$}}^{+}(0) \\
\displaystyle & =\lim_{m\rightarrow\infty}F_{m}(0)=\lim_{m\rightarrow
\infty}\;\frac{\#\{E_{m}(0)\}}{N_{m}}\;.
\end{align*}
\item[(3)] $\;$Suppose that $0<\lambda<1$. Then there is a
constant $K>1$ such
that
$$
F(\lambda)-F(0)\leq-a\;\frac{\log K^{2}}{\log\lambda}\;.
$$
\end{itemize}
\end{theorem}
To prove this Theorem, we will first prove a number of preliminary lemmas.
\begin{lemma} There exists a positive number $K$ such that the operator
norms of
$\Delta_j$ and of $\Delta_j^{(m)}$ for all $m=1,2\ldots$
are smaller than $K^2$.
\end{lemma}
\begin{proof}
The proof is similar to that in \cite{L}, Lemma 2.5 and uses Lemma 1.1
together with uniform local finiteness of $Y$. More precisely we use the
fact that the number of $j$-simplexes in $Y$ at distance at most one from a
$j$-simplex $\sigma$ can be bounded \emph{independently} of $\sigma$,
say
$\#\{\tau \in |Y|_j : d(\tau, \sigma) \leq 1\} \leq b$.
\emph {A fortiori} the same is true (with the same constant $a$) for $Y_m$
for all $m$. We now estimate the $\ell^2$ norm of $\Delta \kappa$ for a
cochain $\kappa = \sum_\sigma a_\sigma \sigma$ (having identified a
simplex $\sigma$ with the dual cochain). Now
$$
\Delta \kappa =
\sum_\sigma \left ( \sum_\tau D(\sigma, \tau ) a_\tau \right ) \sigma
$$
so that
$$
\sum_\sigma \left ( \sum_\tau D(\sigma, \tau) a_\tau \right )^2 \leq
\sum_\sigma \left ( \sum_{d(\sigma, \tau )\leq 1} D(\sigma, \tau )^2 \right
) \left ( \sum_{d(\sigma, \tau )\leq 1} a_\tau^2 \right )
\leq C^2 b \sum_\sigma \sum_{d(\sigma, \tau ) \leq 1} a_\tau^2,
$$
where we have used Lemma 1.1 and Cauchy-Schwartz inequality. In the last
sum above, for every simplex $\sigma$, $a_{\sigma}^2$ appears at most
$b$ times. This proves that $\|\Delta \kappa \|^2
\leq C^2 b^2 \|\kappa \|^2$.
Identical estimate holds (with the same proof) for $\Delta^{(m)}$
which yields the lemma if we set $K=\sqrt{C b}$.
\end{proof}
Observe that $\Delta_j$ can be regarded as a matrix with entries in
${\mathbb Z} [\pi]$, since
by definition, the coboundary operator $d_j$ is a matrix with entries in
${\mathbb Z} [\pi]$, and
so is its adjoint $d_j^*$ as it is equal to the simplicial boundary
operator.
There is a natural trace for matrices with entries in
${\mathbb Z} [\pi]$, viz.
$$
{\text{Tr}}_{{\mathbb Z} [\pi]}(A)= \sum_i {\text{Tr}}_{{\mathcal U} [\pi]}(A_{i,i}).
$$
\begin{lemma} $\;$Let $\pi$ be an amenable group and
let $p(\lambda) = \sum_{r=0}^d a_r \lambda^r$ be a polynomial. Then,
$$
{\text{Tr}}_{{\mathbb Z} [\pi]}(p(\Delta_j))=
\lim_{m\rightarrow\infty}\frac{1}{N_{m}}\;
{\text{Tr}}_{{\mathbb C}}\Big(p\Big(\Delta_j^{(m)}\Big)\Big).
$$
\end{lemma}
\begin{proof} First observe that if $\sigma\in |Y_m|_j$ is such that
$d(\sigma , \partial Y_m) > k$, then Lemma 1.2 implies that
$$
D_j^k(\sigma, \sigma) = \left<\Delta_j^k \delta_\sigma, \delta_\sigma\right>
= \left<\Delta_j^{(m)k} \delta_\sigma, \delta_\sigma\right> =
D_j^{(m)k}(\sigma, \sigma).
$$
By (\ref{inv}) and (\ref{vNt})
$$
{\text{Tr}}_{{\mathbb Z} [\pi]}(p(\Delta_j))= \frac{1}{N_m} \sum_{\sigma\in |Y_m|_j}
\left< p(\Delta_j)\sigma , \sigma \right>.
$$
Therefore we see that
$$
\left| {\text{Tr}}_{{\mathbb Z} [\pi]}(p(\Delta_j)) - \frac{1}{N_{m}}\;
{\text{Tr}}_{{\mathbb C}}\Big(p\Big(\Delta_j^{(m)}\Big)\Big)\right| \le
$$
$$
\frac{1}{N_{m}} \, \sum_{r=0}^d \, |a_r|
\sum_{
\begin{array}{lcl} & \sigma\in |Y_m|_j \\
& d(\sigma, \partial Y_m) \leq d
\end{array}}
\, \left( D^r(\sigma, \sigma) + D^{(m)r}(\sigma, \sigma)\right).
$$
Using Lemma 1.2, we see that there is a positive constant $C$ such that
$$
\left| {\text{Tr}}_{{\mathbb Z} [\pi]}(p(\Delta_j)) - \frac{1}{N_{m}}\;
{\text{Tr}}_{{\mathbb C}}\Big(p\Big(\Delta_j^{(m)}\Big)\Big)\right| \le
2\, \frac{\dot{N}_{m,d}}{N_{m}} \, \sum_{r=0}^d \, |a_r| \, C^r.
$$
The proof of the lemma is completed by taking the limit as
$m\rightarrow\infty$.
\end{proof}
We next recall the following abstract lemmata of L\"uck \cite{L}.
\begin{lemma} $\;$Let $p_{n}(\mu)$ be a sequence of polynomials
such that for
the characteristic function of the interval $[0,\lambda]$,
$\chi_{[0,\lambda]}
(\mu)$, and an appropriate real number $L$,
$$
\lim_{n\rightarrow\infty}p_{n}(\mu)=\chi_{[0,\lambda]}(\mu)\;\;\mbox{ and }
\;\;|p_{n}(\mu)|\leq L
$$
holds for each $\mu\in[0,||\Delta_j||^{2}]$. Then
$$
\lim_{n\rightarrow\infty}{\text{Tr}}_{{\mathbb Z} [\pi]}(p_{n}(\Delta_j))=F(\lambda).
$$\end{lemma}
\begin{lemma} $\;$Let $G:V\rightarrow W$ be a linear map of
finite dimensional
Hilbert spaces $V$ and $W$. Let $p(t)=\det(t-G^{*}G)$ be the characteristic
polynomial of $G^{*}G$. Then $p(t)$ can be written as $p(t)=t^{k}q(t)$
where $q(t)$ is a polynomial with $q(0)\neq 0$. Let $K$ be a real number,
$K\geq\max\{1,||G||\}$ and $C>0$ be a positive constant with
$|q(0)|\geq C>0$.
Let $E(\lambda)$ be the number of eigenvalues
$\mu$ of $G^{*}G$, counted with multiplicity,
satisfying
$\mu\leq\lambda$. Then for $0<\lambda<1$,
the following
estimate is satisfied.
$$
\frac{ \{E(\lambda)\}- \{E(0)\}}{\dim_{{\mathbb C}}V}\leq
\frac{-\log C}{\dim_{{\mathbb C}}V(-\log\lambda)}+
\frac{\log K^{2}}{-\log\lambda}\;.
$$\end{lemma}
\begin{proof}[Proof of theorem 2.1]
Fix $\lambda\geq 0$ and define for $n\geq 1$ a continuous function
$f_{n}:{\mathbb R}\rightarrow {\mathbb R}$ by
$$
f_{n}(\mu)=\left\{\begin{array}{lcl}
1+\frac{1}{n} & \mbox{ if } & \mu\leq\lambda\\[+7pt]
1+\frac{1}{n}-n(\mu-\lambda) & \mbox{ if } &
\lambda\leq\mu\leq\lambda+\frac{1}{n} \\[+7pt]
\frac{1}{n} & \mbox{ if } & \lambda+\frac{1}{n}\leq \mu
\end{array}\right.
$$
Then clearly $\chi_{[0,\lambda]}(\mu)<f_{n+1}(\mu)<f_{n}(\mu)$ and $f_{n}
(\mu)\rightarrow\chi_{[0,\lambda]}(\mu)$ as $n\rightarrow\infty$ for all
$\mu\in[0,\infty)$. For each $n$, choose a polynomial $p_{n}$ such that
$\chi_{[0,\lambda]}(\mu)<p_{n}(\mu)<f_{n}(\mu)$ holds for all
$\mu\in[0,K^{2}]$.
We can always find such a polynomial by a sufficiently close approximation of
$f_{n+1}$. Hence
$$
\chi_{[0,\lambda]}(\mu)<p_{n}(\mu)<2
$$
and
$$
\lim_{n\rightarrow\infty}p_{n}(\mu)=\chi_{[0,\lambda]}(\mu)
$$
for all $\mu\in [0,K^{2}]$. Recall that $E_{m}(\lambda)$ denotes the number
of eigenvalues $\mu$ of $\Delta_j^{(m)}$ satisfying $\mu\leq\lambda$
and counted with multiplicity. Note that
$||\Delta_j^{(m)} || \leq K^{2}$
by Lemma 2.2.
$$
\begin{array}{lcl}
\displaystyle\frac{1}{N_{m}}\;
{\text{Tr}}_{{\mathbb C}}\big(p_{n}(\Delta_j^{(m)})\big)
&=&\displaystyle \frac{1}{N_{m}}\sum_{\mu\in [0,K^2]}p_{n}(\mu)\\[+12pt]
\displaystyle & =&
\displaystyle\frac{ E_{m}(\lambda)}{N_{m}}+\frac{1}{N_{m}}\left\{
\sum_{\mu\in [0,\lambda ]}(p_{n}(\mu)-1)+\sum_{\mu\in (\lambda ,
\lambda + 1/n]}p_{n}(\mu)\right.\\[+12pt]
\displaystyle& &\displaystyle \hspace*{.5in}\left.+\;\sum_{\mu\in (\lambda
+ 1/n, K^2]}p_{n}(\mu)\right\}
\end{array}
$$
Hence, we see that
\begin{equation} \label{A}
F_{m}(\lambda)=\frac{ E_{m}(\lambda)}{N_{m}}
\leq\frac{1}{N_{m}}\;{\text{Tr}}_{{\mathbb C}}\big(p_{n}(\Delta_j^{(m)})\big).
\end{equation}
In addition,
$$
\begin{array}{lcl}
\displaystyle\frac{1}{N_{m}}\;
{\text{Tr}}_{{\mathbb C}}\big(p_{n}(\Delta_j^{(m)})\big)& \leq &
\displaystyle\frac{ E_{m}(\lambda)}{N_{m}}
+\;\frac{1}{N_{m}}\sup\{p_{n}(\mu)-1:\mu\in[0,\lambda]\}\;
E_{m}(\lambda) \\[+16pt]
\displaystyle &+&\displaystyle\;\frac{1}{N_{m}}\sup\{p_{n}(\mu):
\mu\in[\lambda,\lambda+1/n]\}\;
(E_{m}(\lambda+1/n)-E_{m}(\lambda)) \\[+16pt]
\displaystyle &+&\displaystyle\;\frac{1}{N_{m}}\sup\{p_{n}(\mu):
\mu\in[\lambda+1/n,\;K^{2}]\}\;
(E_{m}(K^{2})-E_{m}(\lambda+1/n)) \\[+16pt]
\displaystyle &\leq &\displaystyle\frac{ E_{m}(\lambda)}{N_{m}}+
\frac{ E_{m}(\lambda)}{nN_{m}}+
\frac{(1+1/n) (E_{m}(\lambda+1/n)-E_{m}(\lambda))}{N_{m}}
\\[+16pt]
\displaystyle & &\displaystyle\hspace*{.5in}+\;
\frac{(E_{m}(K^{2})-E_{m}(\lambda+1/n))}
{nN_{m}} \\[+16pt]
\displaystyle &\leq &\displaystyle
\frac{ E_{m}(\lambda+1/n)}{N_{m}}+\frac{1}{n}\;
\frac{ E_{m}(K^{2})}{N_{m}} \\[+16pt]
\displaystyle &\leq & \displaystyle F_{m}(\lambda+1/n)+\frac{a}{n}
\end{array}
$$
since $E_m(K^2)=\dim C^j(Y_m) \leq aN_m$ for a positive constant
$a$ independent
of $m$.
It follows that
\begin{equation} \label{B}
\frac{1}{N_{m}}\;{\text{Tr}}_{{\mathbb C}}\big(p_{n}(\Delta_j^{(m)})\big)\leq F_{m}
(\lambda+1/n)+\frac{a}{n}.
\end{equation}
Taking the limit inferior in (\ref{B}) and the limit superior in (\ref{A}),
as $m\rightarrow\infty$, we get that
\begin{equation} \label{C}
{\overline{F}}(\lambda)\leq {\text{Tr}}_{{\mathbb Z} [\pi]}\big(p_{n}(\Delta_j)\big)
\leq\mbox{\underline{$F$}}(\lambda+1/n)+\frac{a}{n}.
\end{equation}
Taking the limit as $n\rightarrow\infty$ in (\ref{C}) and
using Theorem 2.4, we see that
$$
{\overline{F}}(\lambda)\leq F(\lambda)
\leq\mbox{\underline{$F$}}^{+}(\lambda).
$$
For all $\varepsilon>0$ we have
$$
F(\lambda)\leq\mbox{\underline{$F$}}^{+}(\lambda)\leq\mbox{\underline{$F$}}
(\lambda+\varepsilon)\leq {\overline{F}}(\lambda+\varepsilon)
\leq F(\lambda+\varepsilon).
$$
Since $F$ is right continuous, we see that
$$
F(\lambda)={\overline{F}}^{+}(\lambda)=\mbox{\underline{$F$}}^{+}
(\lambda)
$$
proving the first part of theorem 2.1.
Next we apply theorem 2.5 to $\Delta_j^{(m)}$. Let $p_{m}(t)$ denote the
characteristic polynomial of $\Delta_j^{(m)}$ and
$p_{m}(t)=t^{r_{m}}q_{m}(t)$ where
$q_{m}(0)\neq 0$.
The matrix describing
$\Delta_j^{(m)}$ has integer entries. Hence $p_{m}$ is a polynomial
with integer
coefficients and $|q_{m}(0)|\geq 1$. By Lemma 2.2 and Theorem 2.5 there
are constants $K$ and $C=1$ independent of $m$, such that
$$
\frac{F_{m}(\lambda)-F_{m}(0)}{a}\leq\frac{\log K^{2}}
{-\log\lambda}
$$
That is,
\begin{equation}\label{D}
F_{m}(\lambda)\leq F_{m}(0)-\frac{a\log K^{2}}{\log\lambda}.
\end{equation}
Taking limit inferior in (\ref{D}) as $m\rightarrow\infty$ yields
$$
\mbox{\underline{$F$}}(\lambda)\leq\mbox{\underline{$F$}}(0)
-\frac{a\log K^{2}}{\log\lambda}.
$$
Passing to the limit as $\lambda\rightarrow +0$, we get that
$$
\mbox{\underline{$F$}}(0)=\mbox{\underline{$F$}}^{+}(0)
\qquad \qquad
\mbox{and } \qquad\qquad {\overline{F}}(0)={\overline{F}}^{+}(0).
$$
We have seen already that ${\overline{F}}^{+}(0)=F(0)=\mbox{\underline{$F$}}
(0)$, which proves part ii) of Theorem 2.1. Since $\displaystyle-
\frac{a\log K^{2}}{\log\lambda}$ is right continuous in $\lambda$,
$$
{\overline{F}}^{+}(\lambda)\leq F(0)-\frac{a\log K^{2}}{\log\lambda}.
$$
Hence part iii) of Theorem 2.1 is also proved.
\end{proof}
We will need the following lemma in the proof of Theorem 0.2 in
the last section. We
follow the proof of Lemma 3.3.1 in \cite{L}.
\begin{lemma}
$$
\int_{0+}^{K^2}\left\{\frac{F(\lambda)-F(0)}{\lambda}\right\} d\lambda\le
\liminf_{m\to\infty}
\int_{0+}^{K^2}\left\{\frac{F_m(\lambda)-F_m(0)}{\lambda}\right\} d\lambda
$$
\end{lemma}
\begin{proof}
By Theorem 2.1, and the monotone convergence theorem, one has
\begin{align*}
\int_{0+}^{K^2}\left\{\frac{F(\lambda)-F(0)}{\lambda}\right\}d\lambda
& = \int_{0+}^{K^2}\left\{\frac{\underline{F}(\lambda)-
\underline{F}(0)}{\lambda}\right\}d\lambda\\
& = \int_{0+}^{K^2}\liminf_{m\to\infty}\left\{\frac{F_m(\lambda)-
F_m(0)}{\lambda}\right\}d\lambda \\
& = \int_{0+}^{K^2}\lim_{m\to\infty}\left(\inf\left\{\frac{F_n(\lambda)-
F_n(0)}{\lambda}|n \ge m\right\}\right)d\lambda \\
& = \lim_{m\to\infty}\int_{0+}^{K^2}\inf\left\{\frac{F_n(\lambda)-
F_n(0)}{\lambda}|n \ge m\right\}d\lambda \\
& \le \liminf_{m\to\infty} \int_{0+}^{K^2}\left\{\frac{F_m(\lambda)-
F_m(0)}{\lambda}\right\}d\lambda.
\end{align*}
\end{proof}
\section{Proofs of the main theorems}
In this section, we will prove the Amenable Approximation Theorem (Theorem
0.1)
of the introduction. We will also prove some related spectral results.
\begin{proof}[Proof of Theorem 0.1 (Amenable Approximation Theorem)]
Observe that
\begin{align*}
\frac{b^j(Y_{m})}{N_{m}} & =
\frac{\dim_{\mathbb C}\Big(\ker(\Delta_j^{(m)})\Big)}{N_{m}} \\
&=F_{m}(0).
\end{align*}
Also observe that
\begin{align*}
b_{(2)}^j(Y:\pi) & = \dim_{\pi}\Big(\ker(\Delta_j)\Big)\\
&=F(0).
\end{align*}
Therefore Theorem $0.1$ follows from Theorem 2.1 after taking the
limit as $m\to\infty$.
\end{proof}
Suppose that $M$ is a compact Riemannian manifold and $\Omega^{j}_{(2)}
(\widetilde{M})$ denote the Hilbert space of square integrable $j$-forms on
a normal covering space $\widetilde{M}$, with transformation group $\pi$.
The Laplacian ${\widetilde{\Delta}}_{j}:\Omega^{j}_{(2)}
(\widetilde{M})\rightarrow\Omega^{j}_{(2)}(\widetilde{M})$ is essentially
self-adjoint and has a spectral decomposition
$\{{\widetilde P}_{j}(\lambda):\lambda\in
[0,\infty)\}$ where each ${\widetilde P}_{j}(\lambda)$ has finite
von Neumann trace.
The associated von Neumann spectral density function,
${\widetilde F}(\lambda)$ is
defined as
$$
{\widetilde F}:[0,\infty)\rightarrow [0,\infty),\;\;\;
{\widetilde F}(\lambda)=
{\text{Tr}}_{{\mathcal U}(\pi)}
({\widetilde P}_{j}(\lambda)).
$$
Note that ${\widetilde F}(0)=b_{(2)}^{j}(\widetilde{M}:\pi)$ and that
the spectrum of
$\widetilde{\Delta}_{j}$ has a gap at zero if and only if there is a
$\lambda>0$ such that
$$
{\widetilde F}(\lambda)={\widetilde F}(0).
$$
Suppose that $\pi$ is an amenable group. Fix a triangulation
$X$ on $M$. Then the normal cover $\widetilde{M}$ has an induced
triangulation
$Y$. Let $Y_{m}$ denote be a subcomplex of $Y$ such that
$\big\{Y_{m}\big\}^{\infty}_{m=1}$
is a regular exhaustion of $Y$. Let $\Delta^{(m)}_j:C^{j}(Y_{m},
{\mathbb C})\rightarrow C^{j}(Y_{m},{\mathbb C})$ denote the
combinatorial Laplacian, and
let $E^{(m)}_j(\lambda)$ denote
the number of eigenvalues $\mu$ of $\Delta^{(m)}_j$ which are less than or
equal to $\lambda$. Under the hypotheses above we prove the following.
\begin{theorem}[Gap criterion] $\;$The spectrum of ${\widetilde{\Delta}}_j$
has a gap at zero if and only if there is a $\lambda>0$ such that
$$
\lim_{m\rightarrow\infty}\;
\frac{E^{(m)}_j(\lambda)-E^{(m)}_j(0)}{N_{m}}=0.
$$\end{theorem}
\begin{proof}
Let $\Delta_j:C^{j}_{(2)}(Y)\rightarrow
C^{j}_{(2)}(Y)$ denote the combinatorial Laplacian acting on $L^2$
j-cochains on $Y$.
Then by \cite{GS}, \cite{E}, the von Neumann spectral density function $F$
of the combinatorial Laplacian $\Delta_j$ and the von Neumann
spectral density
function $\widetilde F$ of the
analytic Laplacian ${\widetilde{\Delta}}_j$ are dilatationally
equivalent, that is, there are constants $C>0$ and $\varepsilon>0$
independent
of $\lambda$ such that for all $\lambda\in(0,\varepsilon)$,
\begin{equation} \label{star}
F(C^{-1}\lambda)\leq {\widetilde F}(\lambda)\leq F(C\lambda).
\end{equation}
Observe that $\frac{E^{(m)}_j(\lambda)}{N_{m}} = F_m(\lambda)$.
Therefore the theorem also follows from Theorem 2.1.
\end{proof}
There is a standing conjecture that the Novikov-Shubin invariants of a
closed manifold are positive (see \cite{E}, \cite{ES} and \cite{GS}
for its definition). The next theorem gives evidence supporting
this conjecture, at least in the case of amenable fundamental groups.
\begin{theorem}[Spectral density estimate] $\;$There are constants $C>0$
and $\varepsilon>0$ independent of $\lambda$, such that for all $\lambda\in
(0,\varepsilon)$
$$
{\widetilde F}(\lambda)-{\widetilde F}(0)\leq\frac{C}{-\log(\lambda)}\;.
$$\end{theorem}
\begin{proof}
This follows from Theorem 2.1 and Theorem 3.1
since $\widetilde{\Delta}_j$ has a gap at zero if and only if
$
{\widetilde F}_j(\lambda)={\widetilde F}_j(0)
$ for some $\lambda>0$.
\end{proof}
\section{On the determinant class conjecture}
There is a standing conjecture that any normal
covering space of a finite simplicial complex is of determinant class.
Our interest in this conjecture stems from our work on $L^2$
torsion \cite{CFM}, \cite{BFKM}. The $L^2$ torsion is a well defined
element in the determinant line of the reduced $L^2$ cohomology, whenever
the covering space is of determinant class. In this section, we use
the results
of section 2 to prove that any {\em amenable} normal
covering space of a finite simplicial complex is of determinant class.
Recall that a covering space $Y$ of a finite simplicial complex $X$
is said to be of {\em determinant class} if, for $0 \le j \le n,$
$$ - \infty < \int^1_{0^+} \log \lambda d F (\lambda),$$
where $F(\lambda)$ denotes the von Neumann
spectral density function of the combinatorial Laplacian $\Delta_j$
as in Section 2.
Suppose that $M$ is a compact Riemannian manifold and $\Omega^{j}_{(2)}
(\widetilde{M})$ denote the Hilbert space of square integrable $j$-forms on
a normal covering space $\widetilde{M}$, with transformation group $\pi$.
The Laplacian ${\widetilde{\Delta}}_{j}:\Omega^{j}_{(2)}
(\widetilde{M})\rightarrow\Omega^{j}_{(2)}(\widetilde{M})$ is essentially
self-adjoint and the associated von Neumann spectral density function,
${\widetilde F}(\lambda)$ is
defined as in section 3.
Note that ${\widetilde F}(0)=b_{(2)}^{j}(\widetilde{M}:\pi)$
Then $\widetilde{M}$ is said to be of {\em analytic-determinant class},
if, for $0 \le j \le n,$
$$ - \infty < \int^1_{0^+} \log \lambda d {\widetilde F} (\lambda),$$
where ${\widetilde F}(\lambda)$ denotes the von Neumann spectral density
function
of the analytic Laplacian ${\widetilde{\Delta}}_{j}$ as above. By results
of
Gromov and Shubin \cite{GS},
the condition that $\widetilde{M}$ is of analytic-determinant class
is independent of the choice of Riemannian metric on $M$.
Fix a triangulation
$X$ on $M$. Then the normal cover $\widetilde{M}$ has an induced
triangulation
$Y$. Then $\widetilde{M}$ is said to be of {\em combinatorial-determinant
class}
if $Y$ is of determinant class. Using results of Efremov \cite{E}, and
\cite{GS} one sees that
the condition that $\widetilde{M}$ is of combinatorial-determinant class
is independent of the choice of triangulation on $M$.
Using again results of \cite{E} and \cite{GS}, one observes as in
\cite{BFKM}
that the combinatorial and analytic
notions of determinant class coincide, that is
$\widetilde{M}$ is of combinatorial-determinant class if and only if
$\widetilde{M}$ is of analytic-determinant class. The appendix of \cite{BFK}
contains a proof that every residually finite covering of a compact manifold
is of determinant class. Their proof is based on L\"uck's
approximation of von Neumann spectral density functions \cite{L}. Since
an analogous approximation holds in our setting (cf. Section 2),
we can apply the argument of \cite{BFK} to prove Theorem 0.2.
\begin{proof}[Proof of Theorem 0.2 (Determinant Class Theorem)]
Recall that the \emph{normalized} spectral density functions
$$
F_{m} (\lambda) = \frac 1 {N_m} E_j^{(m)} (\lambda)
$$
are right continuous.
Observe that $F_{m}(\lambda)$ are step functions and
denote by
${\det}' \Delta_j^{(m)}$ the modified determinant of $\Delta_j^{(m)}$,
i.e. the product
of all {\em nonzero} eigenvalues of $\Delta_j^{(m)}$. Let $a_{m, j}$
be the
smallest nonzero eigenvalue and $b_{m, j}$ the largest eigenvalue of
$\Delta_j^{(m)}$. Then, for any $a$ and $b$, such that
$0 < a < a_{m, j}$ and
$b > b_{m,j}$,
\begin{equation}\label{one}
\frac 1 {N_m} \log {\det}' \Delta_j^{(m)} = \int_a^b \log \lambda d F_{m}
(\lambda).
\end{equation}
Integration by parts transforms the Stieltjes integral
$\int_a^b \log \lambda d F_{m}
(\lambda)$ as follows.
\begin{equation}\label{two}
\int_a^b \log \lambda d F_{m} (\lambda) = (\log b) \big( F_{m} (b)
- F_{m} (0) \big) - \int_a^b \frac {F_{m} (\lambda) - F_{m} (0)} \lambda d
\lambda.
\end{equation}
As before, $F(\lambda)$ denotes the spectral density function of the operator
$\Delta_j$ for a fixed $j$.
Recall that $F(\lambda)$ is continuous to the right in $\lambda$. Denote
by ${\det}'_\pi\Delta_j$ the modified Fuglede-Kadison determinant
(cf. \cite{FK})
of $\Delta_j$,
that is, the Fuglede-Kadison determinant of $\Delta_j$ restricted to the
orthogonal
complement of its kernel. It is given by the
following Lebesgue-Stieltjes integral,
$$
\log {\det}^\prime_\pi \Delta_j = \int^{K^2}_{0^+} \log
\lambda d F (\lambda)
$$
with $K$ as in Lemma 2.2, i.e. $ || \Delta_j || < K^2$,
where $||\Delta_j||$ is the operator norm of $\Delta_j$.
Integrating by parts, one obtains
\begin{align} \label{three}
\log {\det}^\prime_\pi (\Delta_j) & = \log K^2 \big( F(K^2) - F(0) \big)
\nonumber\\
& + \lim_{\epsilon \rightarrow 0^+}
\Big\{(- \log \epsilon) \big( F (\epsilon)
-F(0) \big) - \int_\epsilon^{K^2} \frac {F (\lambda) -
F (0)} \lambda d \lambda
\Big\}.
\end{align}
Using the fact that $ \liminf_{\epsilon \rightarrow 0^+} (- \log \epsilon)
\big( F (\epsilon) - F (0) \big) \ge 0$
(in fact, this limit exists and is zero)
and $\frac {F (\lambda) - F (0)}
\lambda \ge 0$ for $\lambda > 0,$ one sees that
\begin{equation}\label{four}
\log {\det}^\prime_\pi (\Delta_j) \ge
( \log K^2) \big(F (K^2) - F(0) \big) -
\int_{0^+}^{K^2}
\frac {F(\lambda) - F (0)} \lambda d \lambda.
\end{equation}
We now complete the proof of Theorem 0.2.
The main ingredient is the estimate of $\log {\det}'_\pi(\Delta_j)$ in terms
of
$\log {{\det}}^\prime \Delta_j^{(m)}$ combined with the fact that
$\log {\det}'
\Delta_j^{(m)}\ge 0$ as the determinant $\det' \Delta_j^{(m)}$ is a
positive integer.
By Lemma 2.2, there exists a positive number $K$, $1 \le K < \infty$,
such that, for $m \ge 1$,
$$
|| \Delta_j^{(m)} || \le K^2 \quad {\text{and}}\quad || \Delta_j ||
\le K^2.$$
By Lemma 2.6,
\begin{equation}\label{five}
\int_{0^+}^{K^2} \frac {F (\lambda) - F (0)} \lambda d \lambda \le
\liminf_{m \rightarrow \infty} \int_{0^+}^{K^2} \frac
{F_{m} (\lambda) - F_{m} (0)} \lambda d \lambda.
\end{equation}
Combining (\ref{one}) and (\ref{two}) with the inequalities
$\log {\det}' \Delta_j^{(m)}
\ge 0$, we obtain
\begin{equation}\label{six}
\int_{0^+}^{K^2} \frac {F_{m} (\lambda) - F_{m} (0)} \lambda d \lambda
\le (\log K^2) \big(F_{m} (K^2) - F_{m} (0)\big).
\end{equation}
From (\ref{four}), (\ref{five}) and (\ref{six}), we conclude that
\begin{equation}\label{seven}
\log {\det}'_\pi \Delta_j \ge (\log K^2) \big( F(K^2) - F(0) \big)
- \liminf_{m \rightarrow \infty}(\log K^2) \big( F_{m} (K^2) - F_{m}
(0) \big).
\end{equation}
Now Theorem 2.1 yields
$$
F (\lambda) = \lim_{\epsilon \rightarrow 0^+} \liminf_{m \rightarrow
\infty}
F_{m} (\lambda + \epsilon)
$$
and
$$
F (0) = \lim_{m \rightarrow \infty} F_{m} (0).
$$
The last two equalities combined with (\ref{seven}) imply that
$\log {\det}'_\pi \Delta_j \ge 0$.
Since this is true for all $j=0,1,\ldots,\dim Y$,
$Y$ is of determinant class.
\end{proof}
| proofpile-arXiv_065-494 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Topological invariants in general metric-affine space}
\renewcommand{\thesection}{\arabic{section}}
\markright{Pontryagin and Euler forms and Chern-Simons...}
\setcounter{equation}{0}
In this section we consider a metric-affine space ($L_{4}$,$g$) that
is a connected 4-dimensional oriented differentiable manifold ${\cal M}$
equipped with a linear connection $\Gamma$ and a metric $g$ of index 1.$^{12}$
We shall use an anholonomic local vector frame $e_{a}$
($a=1,2,3,4$) and a 1-form coframe $\theta^{a}$ with $e_{a}\rfloor\theta^{b}
=\delta^{b}_{a}$ ($\rfloor$ means the interior product). The vector basis
$e_{a}$ can be chosen to be pseudo-orthonormal with respect to a metric
\begin{equation}
g=g_{ab}\theta^{a}\otimes\theta^{b}\;.\label{eq:0}
\end{equation}
In this case one gets,
\begin{equation}
g_{ab} := g(e_{a},e_{b}) = diag(+1,+1,+1,-1)\;. \label{eq:1}
\end{equation}
\par
In ($L_{4}$,$g$) a metric $g$ and a connection $\Gamma$ are not
compatible in the sense that the $GL(4,R)$-covariant exterior differential
(${\cal D}:=d + \Gamma\wedge\ldots$) of the metric does not vanish,
\begin{equation}
{\cal D}g_{ab} = dg_{ab} - \Gamma^{c}\!_{a}g_{cb} - \Gamma^{c}\!_{b}g_{ac}
=: - Q_{ab}\; , \label{eq:2}
\end{equation}
where $\Gamma^{a}\!_{b}$ is a connection 1-form and $Q_{ab}$ is a
nonmetricity 1-form.
\par
A curvature 2-form $\Omega^{a}\!_{b}$ and a torsion 2-form
${\cal T}^{a}$,
\begin{equation}
\Omega^{a}\!_{b}=\frac{1}{2}R^{a}\!_{bcd}\theta^{c}\wedge\theta^{d}\;,
\qquad {\cal T}^{a}=\frac{1}{2}T^{a}\!_{bc}\theta^{b}\wedge\theta^{c}\;,
\label{eq:3}
\end{equation}
are defined by virtue of the Cartan's structure equations,
\begin{eqnarray}
\Omega^{a}\!_{b}=d\Gamma^{a}\!_{b}+\Gamma^{a}\!_{c}\wedge\Gamma^{c}\!_{b}\;,
\label{eq:4}\\ {\cal T}^{a}={\cal D}\theta^{a}=d\theta^{a}+\Gamma^{a}\!_{b}
\wedge\theta^{b}\;. \label{eq:5}
\end{eqnarray}
\par
Let us consider the 4-form
\begin{equation}
\Pi= B^{b}\!_{a}\!^{q}\!_{p}\Omega^{a}\!_{b}\wedge\Omega^{p}\!_{q}
\;, \label{eq:6}
\end{equation}
where $B^{b}\!_{a}\!^{q}\!_{p}$ is an unknown $GL(4,R)$-invariant tensor.
From the form of (\ref{eq:6}) it is easy to get the following symmetry
property of this tensor,
\begin{equation}
B^{b}\!_{a}\!^{q}\!_{p}=B^{q}\!_{p}\!^{b}\!_{a}\;. \label{eq:06}
\end{equation}
The 4-form (\ref{eq:6}) is proportional to the volume 4-form $\eta$ of the
4-dimensional manifold ${\cal M}$, where
\begin{equation}
\eta = \frac{1}{4!}\eta_{abcd}\theta^{a}\wedge\theta^{b}\wedge \theta^{c}
\wedge\theta^{d}\;, \quad \eta_{abcd}=\sqrt{-det \Vert g_{kl}\Vert}\,
\epsilon_{abcd}\;.
\end{equation}
Here $\epsilon_{abcd}$ is the components of the totaly antisymmetric
$GL(4,R)$-invariant Levi-Civita 4-form density $^{12}$
($\epsilon_{1234}=-1$).
\par
Since ${\cal D}\eta = d\eta = 0$ as a 5-form on the 4-dimensional
manifold ${\cal M}$, one has the identity$^{13}$
\begin{equation}
{\cal D}\eta_{abcd} = -\frac{1}{2} Q \eta_{abcd}\;, \qquad Q:=g^{pq} Q_{pq}
\;. \label{eq:290}
\end{equation}
\par
The explicit form of the tensor (\ref{eq:06}) should be determined on
the basis of the condition that the integral
\begin{equation}
\int_{{\cal M}}\Pi \label{eq:7}
\end{equation}
over the oriented 4-demensional manifold {\cal M} without boundary
does not depend on the choise of a metric and a connection and therefore
the variation of the integrand of (\ref{eq:7}) with respect to a metric and a
connection should be equal to an exact form. Here we consider the manifold
${\cal M}$ without boundary for the simplicity. For the manifolds with
boundary some additional surface terms should be taken into account.$^{11}$
\par
As a consequence of (\ref{eq:0}) the variation with respect to a
metric $g$ is determined only by variations of 1-forms $\theta_{a}$ because
of the fact that the variation $\delta g_{ab}=0$ when one chooses the
pseudo-orthonormal basis $e_{a}$ and gets the condition (\ref{eq:1}). The
tensor $B^{b}\!_{a}\!^{q}\!_{p}$ in (\ref{eq:6}) also should not to be varied
when the local vector basis $e_{a}$ is chosen to be anholonomic and
pseudo-orthonormal because it can be constructed (as an $GL(4,R)$-invariant
tensor) only from the metric tensor, Kronecker delta $\delta^{b}_{a}$ and the
$GL(4,R)$-invariant totaly antisymmetric Levi-Civita density
$\epsilon_{abpq}$.
\par
The variation of (\ref{eq:6}) yields the expression,
\begin{equation}
\delta\Pi=2\delta\Gamma^{a}\!_{b}\wedge ({\cal D} B^{b}\!_{a}\!^{q}\!_{p})
\wedge\Omega^{p}\!_{q}+d(2\delta\Gamma^{a}\!_{b}\wedge
B^{b}\!_{a}\!^{q}\!_{p} \Omega^{p}\!_{q})\;.\label{eq:8}
\end{equation}
Here the following relation has been used,
\begin{equation}
\delta\Omega^{a}\!_{b}\wedge\Phi^{b}\!_{a}=d(\delta\Gamma^{a}\!_{b}\wedge
\Phi^{b}\!_{a}) + \delta\Gamma^{a}\!_{b}\wedge {\cal D}\Phi^{b}\!_{a}\; ,
\label{eq:9}
\end{equation}
that valids for an arbitrary 2-form $\Phi^{a}\!{_b}$.
\par
One can see that the variation (\ref{eq:8}) is equal to an exact
form, if the tensor $B^{b}\!_{a}\!^{q}\!_{p}$ satisfies the condition,
\begin{equation}
{\cal D} B^{b}\!_{a}\!^{q}\!_{p}= 0 \;. \label{eq:11}
\end{equation}
\par
In a general metric-affine space ($L_{4}$,$g$) there are only two
possibilities to satisfy (up to constant factors) the condition (\ref{eq:11}),
\begin{equation}
(a)\quad B^{b}\!_{a}\!^{q}\!_{p} =\delta^{b}_{a}\delta^{q}_{p}\;, \qquad
(b)\quad B^{b}\!_{a}\!^{q}\!_{p} =\delta^{b}_{p}\delta^{q}_{a}\;.\label{eq:12}
\end{equation}
\par
In the case (a) the 4-form $\Pi$ (\ref{eq:6}) reads,
\begin{equation}
\Pi_{\Omega}=\Omega^{a}\!_{b}\wedge\Omega^{b}\!_{a} = \mbox{\rm Tr}(\Omega
\wedge\Omega )\;, \label{eq:13}
\end{equation}
and in the case (b) one has,
\begin{equation}
\Pi_{tr\Omega}=\Omega^{a}\!_{a}\wedge \Omega^{b}\!_{b}= \mbox{\rm Tr}\Omega
\wedge\mbox{\rm Tr}\Omega \;. \label{eq:15}
\end{equation}
We see that the 4-forms (\ref{eq:13}) and (\ref{eq:15}) are equal up to
constant factors to the well-known Pontryagin forms.$^{11,12}$
\renewcommand{\thesection}{\Roman{section}.}
\section{The topological invariants in a Weyl- \newline Cartan space}
\renewcommand{\thesection}{\arabic{section}}
\markright{Pontryagin and Euler forms and Chern-Simons...}
\setcounter{equation}{0}
A Weyl-Cartan space $Y_{4}$ is a space with a metric,
curvature, torsion and nonmetricity which obeys the constraint,
\begin{equation}
Q_{ab} = \frac{1}{4}g_{ab}Q\;. \label{eq:21}
\end{equation}
This constraint can be introduced into the variational approach with the help
of the method of Lagrange multipliers. In this case the integral (\ref{eq:7})
has to be modified,
\begin{equation}
\int_{{\cal M}}\left (\Pi + \Lambda^{ab}\wedge (Q_{ab} - \frac{1}{4}g_{ab}Q)
\right )\;, \label{eq:22}
\end{equation}
where the Lagrange multiplier $\Lambda^{ab}$ is a tensor-valued
3-form with the properties,
\begin{equation}
\Lambda^{ab}=\Lambda^{ba}\;, \qquad \Lambda^{a}\!_{a}=0 \;. \label{eq:220}
\end{equation}
\par
The variation of (\ref{eq:22}) with respect to $\theta^{a}$,
$\Gamma^{a}\!_{b}$ and the Lagrange mutiplier yields that the following
variational derivatives have to vanish identically,
\begin{eqnarray}
\delta \Gamma^{a}\!_{b}\;: &{\cal D}(B^{b}\!_{a}\!^{q}\!_{p})\wedge
\Omega^{p}\!_{q} - \Lambda^{b}\!_{a} = 0 \;, & \label{eq:23}\\
\delta\Lambda^{ab}\;: &Q_{ab} - \frac{1}{4}Q g_{ab} = 0 \;.
&\label{eq:24}
\end{eqnarray}
As in the previous section the variational derivative with respect to
$\theta^{a}$ is absent because of the fact that there is no an explicit
dependence on $\theta^{a}$ of the integrand expression in (\ref{eq:22}).
\par
The identity (\ref{eq:23}) in $Y_{4}$ is equivalent to the following
identities,
\begin{eqnarray}
&\Lambda^{ba}= ({\cal D} - \frac{1}{4}Q)B^{(ba)q}\!_{p} \wedge
\Omega^{p}\!_{q}\; , \label{eq:25}\\
&({\cal D} - \frac{1}{4}Q)B^{[ba]q}\!_{p} \wedge
\Omega^{p}\!_{q}= 0\; , \label{eq:26}\\
&{\cal D}B^{a}\!_{a}\!^{q}\!_{p} \wedge\Omega^{p}\!_{q} = 0\;. \label{eq:27}
\end{eqnarray}
\par
The identities (\ref{eq:26}), (\ref{eq:27}) are satisfied in
the following four cases:
\begin{eqnarray}
&(a)\quad B^{baq}\!_{a}\!^{q}\!_{p} =g^{ba}\delta^{q}_{p}\;, \qquad
(b)\quad B^{baq}\!_{p} =\delta^{b}_{p}g^{qa}\;,
\label{eq:28}\\
&(c)\quad B^{baq}\!_{p} =g^{bq}\delta^{a}_{p}\;, \qquad
(d)\quad B^{baq}\!_{p} =\eta^{baq}\!_{p}\;.\label{eq:29}
\end{eqnarray}
In the case (d) one has to use (\ref{eq:2}), (\ref{eq:21}) and
(\ref{eq:290}).
\par
The equality (\ref{eq:25}) determines the Lagrange miltiplier.
In all cases (a)-(d) one has $\Lambda^{ab}=0$. This means that the
Weyl-Cartan constraint (\ref{eq:24}) can be imposed both before and after the
variational procedure.
\par
The cases (a) and (b) coinside with (\ref{eq:12}) and yield for a
Weyl-Cartan space $Y_{4}$ the Pontryagin forms (\ref{eq:13}) and (\ref{eq:15})
of the previous section. The cases (c) and (d) appear in $Y_{4}$ but not in
($L_{4}$,$g$).
\par
In the case (c) one has the Pontryagin form,
\begin{equation}
\Pi_{CW}=\Omega^{ab}\wedge \Omega_{ab} = \mbox{\rm Tr}(\Omega \wedge \Omega
^{T})\;,\label{eq:30}
\end{equation}
where $\Omega^{T}$ means the transpose of $\Omega$. In $Y_{4}$ with the
help of the relation,
\begin{equation}
\Omega_{ab}= \Omega_{[ab]} + \frac{1}{4}g_{ab}\mbox{\rm Tr}\Omega\;,
\label{eq:31}
\end{equation}
(\ref{eq:30}) can be decomposed as follows,
\begin{equation}
\Omega^{ab}\wedge \Omega_{ab}=\Omega^{[ab]}\wedge \Omega_{[ab]} + \frac{1}{4}
\mbox{\rm Tr}\Omega \wedge \mbox{\rm Tr}\Omega\;. \label{eq:32}
\end{equation}
On the other hand the Pontryagin form (\ref{eq:13}) in $Y_{4}$ has the
decomposition,
\begin{equation}
\Omega^{a}\!_{b}\wedge \Omega^{b}\!_{a}= -\Omega^{[ab]}\wedge\Omega_{[ab]}
+\frac{1}{4}\mbox{\rm Tr}\Omega\wedge \mbox{\rm Tr}\Omega \;. \label{eq:33}
\end{equation}
Therefore in a Weyl-Cartan space $Y_{4}$ one has two fundamental Pontryagin
forms, which are equal up to constant factors to
\begin{equation}
\Pi_{C}=\Omega^{[ab]}\wedge \Omega_{[ab]}\;, \qquad
\Pi_{W}=\mbox{\rm Tr}\Omega\wedge \mbox{\rm Tr}\Omega \;.
\label{eq:34}
\end{equation}
The former form is the volume preserving Pontryagin form and the latter one
is the dilatonic Pontryagin form.
\par
In the case (d) we get the Euler form in a Weyl-Cartan space $Y_{4}$,
\begin{equation}
{\cal E}= \eta^{b}\!_{a}\!^{q}\!_{p}\Omega^{a}\!_{b}\wedge \Omega^{p}\!_{q}
\;. \label{eq:35}
\end{equation}
One can use the holonomic coordinate basis $e_{\alpha} = \partial_{\alpha}$
and express the topological invariant corresponding to (\ref{eq:35}) in the
component form,
\begin{equation}
\int_{{\cal M}}{\cal E} = \int_{{\cal M}}E \sqrt{-g}dx^{1}\wedge dx^{2}
\wedge dx^{3}\wedge dx^{4}\;, \label{eq:36}
\end{equation}
\begin{equation}
E = R^{2} - (R_{\alpha\beta} +\tilde{R}_{\alpha\beta})(R^{\beta\alpha} +
\tilde{R}^{\beta\alpha}) + R_{\alpha\beta\mu\nu}R^{\mu\nu\alpha\beta}\; ,
\label{eq:37}
\end{equation}
where $R^{\alpha}\!_{\beta\mu\nu}$ are the components of the curvature 2-form
in a holonomic basis, the following notations being used,
$R_{\alpha\beta} = R^{\sigma}\!_{\alpha\sigma\beta}$,
$\tilde{R}_{\alpha\beta} = R_{\alpha\sigma\beta}\!^{\sigma}$,
$R = R_{\sigma}\!^{\sigma}$.
\par
The Gauss-Bonnet-Chern Teorem$^{10}$ states the relation of the
integral (\ref{eq:36}) over the oriented compact manifold ${\cal M}$ without
boundary with the Euler characteristic of this manifold.
The explicit proof using a holonomic basis of the independence of
(\ref{eq:36}) on the choice of a metric and a connection of
a Weyl-Cartan space $Y_{4}$ is explained in Ref. 14.
\renewcommand{\thesection}{\Roman{section}.}
\section{Chern-Simons terms in a Weyl- \newline Cartan space}
\renewcommand{\thesection}{\arabic{section}}
\markright{Pontryagin and Euler forms and Chern-Simons...}
\setcounter{equation}{0}
It is well known that in ($L_{4}$,$g$) the Pontryagin forms can be
represented as the exterior derivatives of the $GL(4,R)$ Chern-Simons
terms,$^{12}$
\begin{eqnarray}
&\Pi_{\Omega}=d{\cal C}_{\Omega}\;, \qquad {\cal C}_{\Omega}=\Gamma^{b}\!_{a}
\wedge \Omega^{a}\!_{b} - \frac{1}{3}\Gamma^{b}\!_{a}\wedge\Gamma^{a}\!_{c}
\wedge \Gamma^{c}\!_{b}\;, \label{eq:40}\\
&\Pi_{W}=d{\cal C}_{W}\;, \qquad {\cal C}_{W} = \frac{1}{2}Q\wedge \Omega^{a}
\!_{a}\;. \label{eq:41}
\end{eqnarray}
\par
It is easy to see that Pontryagin form $\Pi_{C}$ (\ref{eq:34}) in a
Weyl-Cartan space $Y_{4}$ can be represented in an analogous manner,
\begin{equation}
\Pi_{C} = d{\cal C}_{C}\;, \qquad {\cal C}_{C}=
\Gamma^{[b}\!_{a]}\wedge \Omega^{[a}\!_{b]} - \frac{1}{3}\Gamma^{[b}\!_{a]}
\wedge\Gamma^{[a}\!_{c]}\wedge \Gamma^{[c}\!_{b]}\;. \label{eq:42}
\end{equation}
\par
As it was pointed out in Ref. 12, the Euler form (\ref{eq:35})
in the framework of a Riemann-Cartan space can be expressed in terms of the
corresponding Chern-Simons type construction,
\begin{equation}
{\cal E} = d{\cal C}_{{\cal E}}\;, \qquad {\cal C}_{{\cal E}} =
\eta^{b}\!_{a}\!^{q}\!_{p} \left (\Omega^{a}\!_{b}\wedge \Gamma^{p}\!_{q} -
\frac{1}{3}\Gamma^{a}\!_{b}\wedge \Gamma^{p}\!_{f}\wedge\Gamma^{f}\!_{q}
\right ) \;. \label{eq:43}
\end{equation}
\par
Let us prove that formula (\ref {eq:43}) is also valid in a
Weyl-Cartan space $Y_{4}$. The proof is based on the two Lemmas.
\par
{\it Lemma 1}. If the equality
\begin{equation}
{\cal D}\eta^{b}\!_{a}\!^{q}\!_{p}=0\;, \label{eq:44}
\end{equation}
is valid, then the identity (\ref{eq:43}) is fulfilled.
\par
{\it Proof}. In anholonomic orthonormal frames one has
$d\eta^{b}\!_{a}\!^{q}\!_{p}=0$, and therefore (\ref{eq:44}) yields,
\begin{equation}
\Gamma^{b}\!_{f}\eta^{f}\!_{a}\!^{q}\!_{p}-\Gamma^{f}\!_{a}\eta^{b}\!_{f}\!
^{q}\!_{p}+\Gamma^{q}\!_{f}\eta^{b}\!_{a}\!^{f}\!_{p}-\Gamma^{f}\!_{p}
\eta^{b}\!_{a}\!^{q}\!_{f}=0\;. \label{eq:45}
\end{equation}
After multiplying (\ref{eq:45}) externally by the 3-form $\Gamma^{a}\!_{s}
\wedge\Gamma^{s}\!_{b}\wedge\Gamma^{p}\!_{q}\wedge$, one gets the
$Y_{4}$-identity,
\begin{equation}
\eta^{b}\!_{a}\!^{q}\!_{p}\Gamma^{a}\!_{s}\wedge\Gamma^{s}\!_{b}\wedge
\Gamma^{p}\!_{f}\wedge\Gamma^{f}\!_{q}=0\;. \label{eq:46}
\end{equation}
After multiplying (\ref{eq:45}) externally by the 3-form $\Omega^{a}\!_{b}
\wedge\Gamma^{p}\!_{q}\wedge$, one gets the second $Y_{4}$-identity,
\begin{equation}
\eta^{b}\!_{a}\!^{q}\!_{p}(2\Omega^{a}\!_{b}\wedge\Gamma^{p}\!_{f}\wedge
\Gamma^{f}\!_{q}-\Omega^{a}\!_{f}\wedge\Gamma^{f}\!_{b}\wedge\Gamma^{p}\!_{q}
+\Gamma^{a}\!_{f}\wedge\Omega^{f}\!_{b}\wedge\Gamma^{p}\!_{q})=0\;.
\label{eq:47}
\end{equation}
Now using the identities (\ref{eq:46}) and (\ref{eq:47}), the Cartan's
structure equation (\ref{eq:4}) and the Bianchi identity,
\begin{equation}
{\cal D}\Omega^{a}\!_{b}=d\Omega^{a}\!_{b}+\Gamma^{a}\!_{f}\wedge
\Omega^{f}\!_{b}-\Omega^{a}\!_{f}\wedge\Gamma^{f}\!_{b}=0\;,
\end{equation}
let us perform the exterior differentiation of the Chern-Simons term
${\cal C}_{{\cal E}}$ (\ref{eq:43}) and get,
\begin{eqnarray}
&d{\cal C}_{{\cal E}}-{\cal E}=\frac{1}{3}\eta^{b}\!_{a}\!^{q}\!_{p}
\Gamma^{a}\!_{s}\wedge\Gamma^{s}\!_{b}\wedge\Gamma^{p}\!_{f}\wedge
\Gamma^{f}\!_{q}\nonumber \\
&-\frac{2}{3}\eta^{b}\!_{a}\!^{q}\!_{p}
(2\Omega^{a}\!_{b}\wedge\Gamma^{p}\!_{f}\wedge\Gamma^{f}\!_{q}
-\Omega^{a}\!_{f}\wedge\Gamma^{f}\!_{b}\wedge\Gamma^{p}\!_{q}
+\Gamma^{a}\!_{f}\wedge\Omega^{f}\!_{b}\wedge\Gamma^{p}\!_{q})=0\;,
\end{eqnarray}
as was to be proved.
\par
{\it Lemma 2}. The equality (\ref{eq:44}) is valid if and only if
the space under consideration is a Weyl-Cartan space $Y_{4}$.
\par
{\it Proof}. In a general ($L_{4}$,$g$) space one has,
\begin{equation}
{\cal D}\eta^{b}\!_{a}\!^{q}\!_{p}=\eta^{q}\!_{map}\tilde{Q}^{bm} -
\eta^{b}\!_{map}\tilde{Q}^{qm}\;, \label{eq:48}
\end{equation}
where $\tilde{Q}^{bm} := Q^{bm}- \frac{1}{4}g^{bm}{Q}$ is
the tracefree part of the nonmetricity 1-form, $\tilde{Q}^{b}\!_{b}
=0$. For a Weyl-Cartan space $Y_{4}$ one has $\tilde{Q}^{bm}=0$ and
the sufficient condition of the Lemma is evident. The necessary condition of
the Lemma is the consequence of the fact that the vanishing of (\ref{eq:48})
leads to the equality,
\begin{equation}
g^{ab}\tilde{Q}^{pq}-g^{bp}\tilde{Q}^{aq}-g^{aq}
\tilde{Q}^{bp}+g^{pq}\tilde{Q}^{ab}=0\;,
\end{equation}
which yields $\tilde{Q}^{pq}=0$, as was to be proved.
\renewcommand{\thesection}{\Roman{section}.}
\section{Conclusions}
\renewcommand{\thesection}{\arabic{section}}
\markright{Pontryagin and Euler forms and Chern-Simons...}
We have proved the existence of the Pontryagin and Euler forms in a
Weyl-Cartan space on the basis of the variational method with Lagrange
multipliers. It has been discovered that the Pontryagin form,
$\Pi_{C}=\Omega^{[ab]}\wedge \Omega_{[ab]}$, and Euler form,
${\cal E}= \eta^{b}\!_{a}\!^{q}\!_{p}\Omega^{a}\!_{b}\wedge \Omega^{p}\!_{q}$,
which are specific for a Riemann-Cartan space, also exist in a Weyl-Cartan
space. With the help of these forms the topological invariants of a
Weyl-Cartan space which do not depend on the choice of a metric and a
connection are constructed. It has been proved that these forms can be
expressed via the exterior derivatives of the corresponding Chern-Simons
terms in a Weyl-Cartan space (see (\ref{eq:42}) and (\ref{eq:43}),
respectively). From the Lemma 2 proved it follows that the relation
(\ref{eq:43}) is not valid in the more general geometry than the Weyl-Cartan
one.
\newpage
\vskip 0.6cm
\begin{description}
\item{$^{1}$}
M.B. Green,J.H. Schwarz and E. Witten, {\em Superstring Theory}, 2 volumes
(Cambridge University Press, Cambridge, 1987).
\item{$^{2}$}
G. 't Hooft, M. Veltman, {\em Ann. Inst. H. Poincar\'{e}} {\bf 20}, 69 (1974).
\item{$^{3}$}
K.S. Stelle, {\em Phys. Rev.} {\bf D16}, 953 (1977).
\item{$^{4}$}
R. Bach, {\em Math. Z.} {\bf 9}, 110 (1921).
\item{$^{5}$}
C. Lanczos, {\em Ann. Math.(N.Y.)} {\bf 39}, 842 (1938).
\item{$^{6}$}
J.R. Ray, {\em J. Math. Phys.} {\bf 19}, 100 (1978).
\item{$^{7}$}
V.N. Tunjak, {\em Izvestija vyssh. uch. zaved. (Fizika)} N9, 74 (1979) [in
Russian].
\item{$^{8}$}
H.T. Nieh, {\em J. Math. Phys.} {\bf 21}, 1439 (1980).
\item{$^{9}$}
K. Hayashi, T. Shirafuji, {\em Prog. Theor. Phys.} {\bf 65}, 525 (1981).
\item{$^{10}$}
R. Sulanke und P. Wintgen, {\em Differentialgeometrie und faserb\"undel}
(Hoch\-schulb\"ucher f\"ur mathematik, band 75)(VEB Deutscher Verlag der
Wis\-senshaften, Berlin, 1972).
\item{$^{11}$}
T. Eguchi, P.B. Gilkey and A.J. Hanson, {\em Phys. Reports} {\bf 66}, 213
(1980).
\item{$^{12}$}
F.W. Hehl, J.D. McCrea, E.W. Mielke and Yu. Ne'eman, {\em Phys. Reports}
{\bf 258}, 1 (1995).
\item{$^{13}$}
R. Tresguerres, {\em J. Math. Phys.} {\bf 33}, 4231 (1992).
\item{$^{14}$}
O.V. Babourova, B.N. Frolov, {\em Gauss-Bonnet type identity in Weyl-Cartan
space-time}, LANL e-archive gr-qc/9608... (1996).
\end{description}
\end{document}
| proofpile-arXiv_065-495 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}\label{sec:intr}
A typical quantity to analyze the nature of the
perturbative expansion in Quantum Field Theory is the partition function
\begin{equation}
Z ( \lambda ) = {1 \over Z_0} \int \left[ {\rm d} \phi \right]
e^{- S \left[ \phi \right]}
\label{eq:partfunc}
\end{equation}
with
\begin{equation}
S \left[ \phi \right] = \int {\rm d^d} x \left[ {1 \over 2} \left( \partial_{\mu} \phi \right)^2
+ {1 \over 2} m^2 \phi^2 + {\lambda \over 4} \phi^4 \right].
\label{eq:action}
\end{equation}
The normalization factor $1/Z_0$ is the partition function of the free field
($Z \rightarrow 1 $ when $ \lambda \rightarrow 0 $). The analysis of the
perturbative expansion of any Green's function goes along similar lines as
in the case of $Z$. In the example above we consider a scalar field
theory for simplicity.
The traditional argument for understanding the divergent nature of the
perturbative expansion can be traced back to Dyson~\cite{dy}. Although the
form was different, the content of his argument is captured by the
following statement:
``If the perturbative series were to converge to the exact result, the function being
expanded would be analytic in $\lambda$ at $\lambda = 0$.
But the function (Z for example) is not analytic in $\lambda$ at that
value. Therefore, as a function of $\lambda$, the perturbative series
is either divergent or converges to the wrong answer."
Estimations of the large order behavior of the coefficients
of the perturbative series showed that the first possibility is the one
actually realized \cite{lo,bw}.
That Z, as a function of $\lambda$, is not analytic at $\lambda = 0$,
can be guessed by simply noting that if in its functional
integral representation (Eq.~(\ref{eq:partfunc})) we make the real part of
$\lambda$ negative, the integral diverges.
In fact, there is a branch cut in the first Riemann sheet that can be
chosen to lie along the negative real axis, extending from
$\lambda = - \infty$ to $ \lambda = 0$ \cite{bw0,bs}.
The above argument is very powerful and extends to the perturbative
series of almost all other nontrivial field theories. It
has also motivated a series of very important calculations of the
large order behavior of the perturbative coefficients \cite{lo}, general analysis
on the structure of field theories \cite{tHooft}, as well as improvements over
perturbative computations of different physical quantities \cite{lo}.
For all its power, it is fair to say that the argument,
as almost any other {\it reductio ad absurdum} type of argument,
fails to point towards a solution of the problem of divergence.
It is only through the indirect formalism of Borel transforms that
questions of recovery of the full theory from its perturbative
series can be discussed \cite{bs,gh,zinn}.
In this paper an alternative way of understanding the divergent
nature of the perturbative series is presented. This way of
understanding the problem complements the traditional argument
briefly described above, hopefully illuminating aspects that
the traditional approach leaves obscure. In particular,
as we will see, the arguments in this paper point directly
towards the aspects of the perturbative series that need to
be modified to achieve a convergent series. It is hoped that the
way of understanding the problem presented here will help to provide
new insights into the urgent problem of extracting non-perturbative
information out of Quantum Field Theories.
In section~\ref{sec:leb} we develop our analysis of the divergence of
perturbation theory. In Sec.~\ref{sec:converg} we point out the ingredients
that, according to the analysis of Sec.~\ref{sec:leb}, a modification
of perturbation theory would need to achieve convergence. We also present
a remarkable formula~(\ref{win}) that allows us to implement such modifications
in terms of Gaussian integrals, paving the way to the application of this
convergent modified perturbative series to Quantum Field Theories. The proof
of the properties of the function~(\ref{win}) is done in Appendix 1.
In section~\ref{Improvement} we analyze recent work on the convergence of
various optimized expansions~[12-19] in terms of the ideas
presented here.
In Sec.~\ref{sec:concl} we summarize our results and mention directions
of the work currently in preparation. Finally, in Appendix 2, we apply
the ideas of this paper in a simple but illuminating example for which we
actually develop a convergent series by modifying the aspects of the
perturbative series pointed out by our analysis as the source of divergence.
\section{Lebesgue's Dominated Convergence Theorem and Perturbation
Theory}\label{sec:leb}
\subsection{The wrong step in perturbation theory}\label{sec:wrong}
Although the notation will not always be explicit, we work in an Euclidean
space of dimension smaller than 4 and in a finite volume.
Let's remember how the perturbative series is generated in the
functional integral formalism for a quantity like $Z$:
\begin{eqnarray}\label{eq:pertZ}
Z ( \lambda ) & = & \int \left[ {\rm d} \phi \right] e^{-
\int {\rm d^d} x \left[ {1 \over 2} \left( \partial_{\mu} \phi \right)^2
+ {1 \over 2} m^2 \phi^2 \right] - {\lambda \over 4} \int {\rm d^d} x \phi^4 } \\
\label{eq:first}
& = & \int \left[ {\rm d} \phi \right] \sum_{n=0}^{\infty} {\left( -1 \right)^n \over n!}
\left( {\lambda \over 4} \int {\rm d^d} x \phi^4 \right)^n e^{- \int {\rm d^d} x
\left[ {1 \over 2} \left( \partial_{\mu} \phi \right)^2 + {1 \over 2} m^2 \phi^2 \right] } \\
\label{eq:second}
& = & \sum_{n=0}^{\infty} \int \left[ {\rm d} \phi \right] {\left( -1 \right)^n \over n!}
\left( {\lambda \over 4} \int {\rm d^d} x \phi^4 \right)^n e^{- \int {\rm d^d} x
\left[ {1 \over 2} \left( \partial_{\mu} \phi \right)^2 + {1 \over 2} m^2 \phi^2 \right] }
\end{eqnarray}
The final sum is in practice truncated at some finite order $N$.
The functional integrals that give the contribution of every
order $n$ are calculated using Wick's theorem and Feynman's
diagram techniques with the corresponding renormalization.
We see then that the generation of the perturbative series in the
functional integral formalism is a two step process. First
(\ref{eq:first}) the integrand is expanded
in powers of the coupling constant, and then (\ref{eq:second}) the sum is
interchanged with the integral\footnote{ In this paper we will often use the
familiar word ``integrand" to refer to $e^{-S}$ or any functional inside the
functional integration symbol. It would be more precise to preserve this word
for $e^{-S_{\rm Int}}$ in the measure defined by the free field. The terminology
used here is, however, common practice in the Quantum Field Theory
literature and also helps to emphasize the similarities with the intuitive finite
dimensional case presented below. }
It will be convenient to have a simpler example in which the arguments
of this paper become very transparent. Consider the simple integral
\begin{eqnarray}\label{eq:pertsimple}
z ( \lambda ) & = & { 1 \over \sqrt{\pi} } \int_{- \infty}^{\infty} {\rm d} x
e^{- \left( x^2 + {\lambda \over 4} x^4 \right)}
\end{eqnarray}
and its corresponding perturbative expansion
\begin{eqnarray}\label{eq:pertsimple1}
z ( \lambda ) & = & { 1 \over \sqrt{\pi} } \int_{- \infty}^{\infty} {\rm d} x \sum_{n=0}^{\infty}
{\left( -1 \right)^n \over n! } \left( {\lambda \over 4} x^4 \right)^n e^{- x^2 } \\
\label{eq:pertsimple2}
& = & { 1 \over \sqrt{\pi} } \sum_{n=0}^{\infty} \int_{- \infty}^{\infty} {\rm d} x
{\left( -1 \right)^n \over n! } \left( {\lambda \over 4} x^4 \right)^n e^{- x^2 } \\
\label{eq:simplecoef}
& \equiv & \sum_{n=0}^{\infty} \left( -1 \right)^n c_n \ \lambda^n.
\end{eqnarray}
This simple integral has been used many times in the past as a paradigmatic
example of the divergence of perturbation theory \cite{zinn}. It is then specially
suited for a comparison between the traditional arguments and the ones
presented in this paper.
Again we see the two step process to generate the perturbative
series. First the integrand is expanded in powers of $\lambda$~(\ref{eq:pertsimple1})
and then the sum is interchanged with the integral~(\ref{eq:pertsimple2}) .
In this simple example the perturbative coefficients can be
calculated exactly for arbitrary $n$. In the large $n$ limit they become:
\begin{equation}\label{eq:simplecoeflargen}
c_n \sim {\sqrt{2} \over 2 \pi } \left( n -1 \right)! \quad {\rm when}
\quad n \rightarrow \infty.
\end{equation}
With such factorial behavior, the series diverges for all $ \lambda$
different from zero as is well known. On the other hand the function $z (
\lambda )$, as defined in Eq.~(\ref{eq:pertsimple}), gives a well defined
positive real number for every positive real $\lambda$. Therefore one or
both of the two steps done to generate the perturbative series must be wrong.
Similarly, in the functional integral case normalized with respect to the free
field (\ref{eq:partfunc}), $Z$ is a well defined number while its
perturbative series diverges. One or both of the two steps must be wrong.
The first step, the expansion of the integrand in powers of $ \lambda$,
is clearly correct. As the integrand (not the integral!) is analytic
in $ \lambda$ for
every finite $ \lambda$, the expansion merely corresponds to a Taylor
series. The second step, the interchange of sum and integral, must therefore
be the wrong one.
The next obvious step is then to recall the theorems that govern
the interchange between sums and integrals, to understand in detail
why this is wrong in our case. The most powerful theorem in this
respect is the well known theorem of Dominated Convergence of
Lebesgue. In a simplified version, enough for our purposes, it
says the following:
\begin{quote}
Let $f_N$ be a sequence of integrable functions that converge
pointwisely to a function $f$
\begin{equation}\label{eq:pointconv}
f_N \longrightarrow f \quad {\rm as} \quad N \rightarrow \infty
\end{equation}
and bounded in absolute value by a positive integrable function $h$
(dominated)
\begin{equation}\label{eq:bound}
|f_N| \leq h \ ,\quad \forall N.
\end{equation}
Then, it is true that
\begin{equation}\label{eq:thesis}
\lim_{N \rightarrow \infty} \int f_N = \int \lim_{N \rightarrow \infty} f_N
= \int f .
\end{equation}
\end{quote}
As a special case, if the convergence~(\ref{eq:pointconv}) is uniform and
the measure of integration is finite, then the interchange is also valid. It
should be emphasized that Lebesgue's theorem follows from the axioms of
abstract measure theory. Therefore if the problem under consideration
involves a well defined measure, as is the case for the Quantum Field
Theories considered here~\cite{glja}, the theorem holds.
In our case we can write formally\footnote{See previous footnote.},
\begin{equation}\label{eq:fNfunctint}
f_N \left[ \phi (x) \right]= {1 \over Z_0} \sum_{n=0}^{N}
{\left( -1 \right)^n \over n!}
\left( {\lambda \over 4} \int {\rm d^d} x \phi^4 \right)^n e^{- \int {\rm d^d} x
\left[ {1 \over 2} \left( \partial_{\mu} \phi \right)^2 + {1 \over 2} m^2 \phi^2 \right] }
\end{equation}
for the functional integral case, and
\begin{equation}\label{eq:fNsimpltint}
f_N (x) = { 1 \over \sqrt{\pi} } \sum_{n=0}^{N} {\left( -1 \right)^n \over n!}
\left( {\lambda \over 4} x^4 \right)^n e^{- x^2 }
\end{equation}
for the simple integral example.
One important aspect of the dominated convergence theorem approach
to analyze the divergence of perturbation theory is that it
focuses on the integrands, objects relatively simple to analyze.
On the contrary, the analyticity approach briefly described in the
introduction focuses on the integrals, that are much more difficult
to analyze. So, before we try to understand what aspects of the
dominated convergence theorem fail in our case, let's see the
``phenomena" (the integrand) for the intuitive simple example.
\begin{figure}[htp]
\hbox to \hsize{\hss\psfig{figure=ex00204.eps,width=0.9\hsize}\hss}
\caption{Exact integrand, ${\rm zero}^{{\rm th}}$, second and fourth perturbative
approximations. $\lambda = 1$.}
\label{fig1}
\end{figure}
In figure ~\ref{fig1}, the exact integrand, together with some perturbative
approximations,
are displayed. We can appreciate the way in which the successive
approximations behave. For small $x$, and up to some
critical value that we call $x_{c,N}$, where the subindex $c$ stands
for {\it critical} while the subindex $N$ indicates that this value changes with the
order, the perturbative integrands approximate very well the exact integrand.
Even more, $x_{c,N}$ grows with $N$. But for $x$ bigger that $x_{c,N}$ a ``bump''
begins to emerge. The height of these bumps, as we will see in detail shortly,
grows factorially with the order, while the width remains approximately constant.
So, the larger the order in perturbation theory, the larger the region in which
the perturbative integrands approximate very well the exact integrand, but
the stronger the upcoming deviation. As we will see shortly, it is precisely
this deviation that is responsible for the divergence of the perturbative series
and the famous factorial growth. We will also see that an exactly analogous
phenomena happens in the functional integral case and is again the responsible
for the divergence of the perturbative series.
Returning to the problem of understanding the aspect of the dominated convergence
theorem that fail in the perturbative series we will now show that the sequence
of integrands of Eq.~(\ref{eq:fNfunctint}) and Eq.~(\ref{eq:fNsimpltint})
converge respectively to the exact integrands
\begin{equation}\label{eq:exIntd}
F = {1 \over Z_0} \ e^{- \int {\rm d^d} x \left[ {1 \over 2}
\left( \partial_{\mu} \phi \right)^2
+ {1 \over 2} m^2 \phi^2 \right] - {\lambda \over 4} \int {\rm d^d} x \phi^4 }
\end{equation}
and
\begin{equation}\label{eq:exintd}
f = {1 \over \sqrt{\pi}} e^{- \left( x^2 + {\lambda \over 4} x^4 \right)}
\end{equation}
but {\it not} in a {\it dominated} way. This is, there is no positive integrable
function $h$ satisfying the property of Eq.~(\ref{eq:bound}).
\subsection{Failure of domination in the simple example}\label{sec:simpleint}
That the sequence of integrands of Eq.~(\ref{eq:fNfunctint}) and
Eq.~(\ref{eq:fNsimpltint}) converge respectively to the exact integrands
(\ref{eq:exIntd}) and (\ref{eq:exintd})
is obvious, since, as mentioned before, for finite $\lambda$ they are
analytic functions of $\lambda$ and so their Taylor
expansion converge (at least for finite field strength). To see the failure of the domination hypothesis it is convenient to analyze
the ``shape" of every term of $f_N$. Namely, for the field theory case,
\begin{equation}\label{eq:cnFT}
c_n \left[ \phi (x) \right] \equiv {1 \over Z_0} {\left( -1 \right)^n \over n!}
{1 \over 4^n}
\left( \lambda \int {\rm d^d} x \phi^4 \right)^n e^{- \int {\rm d^d} x
\left[ {1 \over 2} \left( \partial_{\mu} \phi \right)^2 + {1 \over 2} m^2 \phi^2 \right] }
\end{equation}
while for the simple integrand
\begin{equation}\label{eq:cnSI}
c_n (x) = {1 \over \sqrt{\pi}} {\left( -1 \right)^n \over n!} \left( {\lambda \over 4} \right)^n x^{4n}
e^{- x^2 }.
\end{equation}
In this section we analyze the failure of the domination hypothesis for the
simple example~(\ref{eq:pertsimple}) because, as it turns out, it is
remarkably similar to the Quantum Field Theory example analyzed in the
next section.
In Fig.~\ref{fig2} we can inspect the functions $c_3 (x)$ and $c_4(x)$ for
$\lambda = 1$ corresponding to the
simple integrand case that we analyze first. The maximum of $c_n (x)$ is reached
at
\begin{equation}\label{eq:xmax}
x_{{\rm max}} = \pm (2 n)^{1/2}.
\end{equation}
There, for large $n$, the function takes the value
\begin{equation}\label{eq:cnmax}
c_n (x_{{\rm max}}) = {1 \over 2 \pi^{3/2}} (-1)^n (n-1)! \ \lambda^n .
\end{equation}
On the other hand, the width remains constant as n increases
as can be seen by a Gaussian approximation around the maximum
$x_{{\rm max}} = (2 n)^{1/2}$:
\begin{equation}\label{gaussapp}
c_n (x) \approx {1 \over 2 \pi^{3/2}} (-1)^n (n-1)! \ {\rm exp}
\left[ -2 (x - (2n)^{1/2})^2 \right] \ \lambda^n .
\end{equation}
The integration of this Gaussian approximation gives, for large $n$
\begin{equation}\label{gaussapp2}
\int {\rm d} x \ c_n (x) \approx {1 \over 2} { \sqrt{2} \over 2 \pi} (-1)^n (n-1)! \
\lambda^n
\end{equation}
in accordance with Eq.~(\ref{eq:simplecoeflargen}) if we take into account
the factor of 2 coming from the two maxima $\pm (2 n)^{1/2}$.
\begin{figure}[htp]
\hbox to \hsize{\hss\psfig{figure=ex04c3c4.eps,width=0.9\hsize}\hss}
\caption{Exact integrand, and fourth perturbative approximation together with
the third, and fourth terms. $\lambda = 1$.}
\label{fig2}
\end{figure}
The mechanism of convergence of the $f_N $'s to $f$ becomes clear now.
The $f_N $'s are made out of a pure Gaussian (the ``free" term) plus
``bumps" (the perturbative corrections) oscillating in sign (see
Fig.~\ref{fig2}). The maxima of these bumps grows factorially with the order
while their width remain approximately constant (more specifically, the
Gaussian approximation around the maxima (Eq.~(\ref{gaussapp})), that becomes
exact
when the order goes to infinity, has a variance independent of the order). For
fixed $N$, and for $x$ smaller than a certain value, the bumps delicately
``almost" cancel each other, leaving only a small remnant that modifies the
free integrand into the interacting one. However, for $x$ larger than that
value, the last bump begins to emerge and, being the last, does not have the
next one to get cancelled (in the $N \rightarrow \infty$ limit, there is no
last bump and the convergence is achieved for every x). Consequently, after
a certain value $x_{c,N}$, the function $f_N$ deviates strongly from $f$ and
is governed by the uncanceled $N^{{\rm th}}$ bump only, of height
proportional to $(N-1)!$ and finite variance. This is so because, since
the height of the bump grows factorially with the order, for $N$ large enough
the last bump is far greater than all the previous ones, so it remains almost
completely uncanceled. Since the variance of the bumps is independent
of the order, this means that for every finite order, there is a region of
{\it finite measure} in which the perturbative integrand is of the order
of the height of the last bump. In figure ~\ref{fig2} we can
see how the function $c_4 (x)$ is left almost completely uncanceled by
$c_3 (x)$ and dominates the deviation of $f_4$ from $f$.
That $x_{c,N}$ (the value of $|x|$ up to which the perturbative integrand very
accurately approximate the exact one) grows with $N$, going to infinity when
$N \rightarrow \infty$, is a simple consequence of Taylor's theorem applied
to the analytic function $e^{- \lambda x^4 /4}$.
The above analysis makes clear the failure of domination of the sequence
of Eq.~(\ref{eq:fNsimpltint}) towards $f$ (Eq.~(\ref{eq:exintd})). Indeed, any positive
function $h(x)$ with the property
\begin{equation}\label{eq:domSI}
|f_N (x)| \leq h (x)\ ,\quad \forall N
\end{equation}
fails to be integrable, since it has to ``cover" the bump, whose area grows factorially
with $N$. So, although the sequence of $f_N (x)$'s converges to $f(x)$, the
convergence is not dominated as we wanted to show.
Eq.~(\ref{gaussapp2}), together with the above comments, indicate
that the same reason for which the sequence of integrands (\ref{eq:cnSI})
fails to be dominated, is the one that produces the factorial growth in
the perturbative series.
In the Field Theory case, although we can not rely on figures
like~\ref{fig1} and \ref{fig2} to guide our intuition, as we will show now,
the analogy with the simple integral example is so close that the interpretation
is equally transparent.
\subsection{Failure of domination in Quantum Field Theory}\label{sec:QM}
For Quantum Field Theory, as for the simple example analyzed above,
it is convenient to consider every term $c_n [ \phi (x) ]$
(Eq.~\ref{eq:cnFT}) of the perturbative approximation $f_N$
(Eq.~\ref{eq:fNfunctint}) of the exact integrand (Eq.~\ref{eq:exIntd}),
\begin{equation}\label{eq:cnFT2}
c_n \left[ \phi (x) \right] = {1 \over Z_0}{\left( -1 \right)^n \over n!}
e^{- \int {\rm d^d} x \left[ {1 \over 2} \left( \partial_{\mu} \phi \right)^2 +
{1 \over 2} m^2 \phi^2 \right] + n \ln{\left[ \left( \lambda/4 \right)
\int {\rm d^d} x \ \phi^4 \right] } }
\end{equation}
where we have written the $n^{{\rm th}}$ power of the interaction in exponential
form. The mathematical analysis below follows closely the discussions in
chapter 38 Ref.~\cite{zinn}. Although the problem treated there is
different from the one treated here, many techniques used in~\cite{zinn}
can be directly borrowed here.
For $n$ large enough, the analysis of its ``shape" reduces to the familiar
procedure of finding its maxima, as in the case of the simple integrand. The
equation determining the maxima of $c_n [ \phi (x) ]$ is the equation that minimizes the exponent, and can be thought of as the equation of motion of the
effective action
\begin{equation}\label{eq. effactQM}
S \left[ \phi \right] = \int {\rm d^d} x \left[ {1 \over 2} \left( \partial_{\mu}
\phi \right)^2 + {1 \over 2} m^2 \phi^2 \right] - n \ln{ \left[
{\lambda \over 4} \int {\rm d^d} x \ \phi^4 \right] },
\end{equation}
which is
\begin{equation}\label{eq:eqQM}
- \nabla^2 \phi + m^2 \phi - { 4 n \over \int {\rm d^d} x \phi^4 } \phi^3 = 0.
\end{equation}
Making the change of variables
\begin{equation}\label{eq:chvarQM1}
\phi ( x ) = m \left({ \int {\rm d^d} x \phi^4 \over 4n} \right)^{1/2} \varphi ( m x ) =
m^{ {\rm d}/2 -1} \left( {4n \over \int {\rm d^d} u \varphi^4 (u) }
\right)^{1/2} \varphi ( m x ),
\end{equation}
$ \varphi $ satisfies the equation
\begin{equation}\label{eq:numeqQM}
- \nabla^2 \varphi (u) + \varphi (u) - \varphi^3 (u) = 0 \quad ,
\qquad u \equiv m x.
\end{equation}
This equation corresponds to the instanton equation of the negative mass
$\lambda \phi^4$ theory. The analysis of their solutions can be found in many
places. We are interested in the solutions with minimal, finite action. For
these solutions, in the infinite volume limit, scaling arguments provide very
interesting information. We mentioned at the beginning of
section~\ref{sec:wrong} that we work in a finite volume. However, if the
volume is large enough, the following infinite volume arguments remain
valid up to errors that go to zero exponentially fast when the volume
goes to infinity.
Since the solution $\phi_{{\rm max}} (x)$ (the subindex ``max" indicates
that, in functional space, $c_n \left[ \phi (x) \right]$ reaches its maximum
at $\phi_{{\rm max}} (x)$, this should not be confused with the fact that the
the action (\ref{eq. effactQM}) reaches its {\it minimum} there) is a minimum
of the action (\ref{eq. effactQM}), given an arbitrary constant $\alpha$,
$S \left[ \alpha \phi_{{\rm max}} (x) \right]$ should have a minimum at $\alpha = 1$ \cite{zinn,derr}.
This implies the equation
\begin{equation}\label{Salphphi1}
\int {\rm d^d} x \left( \partial_{\mu} \phi_{{\rm max}} \right)^2 + m^2 \int {\rm d^d} x \phi_{{\rm max}}^2
- 4n = 0.
\end{equation}
Similarly, $S \left[ \phi_{{\rm max}} (\alpha x) \right]$ should also have a minima
at $\alpha = 1$, implying,
\begin{equation}\label{Salphphi2}
{ \left( 2 - d \right) \over d} \int {\rm d^d} x \left( \partial_{\mu} \phi_{{\rm max}} \right)^2 -
m^2 \int {\rm d^d} x \phi_{{\rm max}}^2 + 2 n = 0.
\end{equation}
Solving the system of equations (\ref{Salphphi1}) and (\ref{Salphphi2}) we obtain,
\begin{eqnarray}\label{solusysteq1}
\int {\rm d^d} x \left( \partial_{\mu} \phi_{{\rm max}} \right)^2 &=& n \ d \\ \label{solusysteq2}
m^2 \int {\rm d^d} x \phi_{{\rm max}}^2 &=& n \left( 4 - d \right).
\end{eqnarray}
From which we conclude in particular
\begin{equation}\label{eq. effactQMFree}
\int {\rm d^d} x \left[ {1 \over 2} \left( \partial_{\mu}
\phi_{{\rm max}} \right)^2 + {1 \over 2} m^2 \phi_{{\rm max}}^2 \right] = 2n,
\end{equation}
independent of the dimension. The relations (\ref{solusysteq1}) and (\ref{solusysteq2})
can be explicitly checked in the case $d=1$ (Quantum Mechanics), in which the
solutions to eq.(\ref{eq:eqQM}) are known analytically. They are
\begin{equation}\label{d=1sol}
\phi_{{\rm max}}^{{\rm d} =1} (t) = \left( {3n \over 2 m} \right)^{1/2} {1 \over
\cosh{ \left[ m \left( t - t_0 \right) \right]} }
\end{equation}
giving
\begin{eqnarray}\label{solusysteq1D1}
\int {\rm d} t \left( \dot{ \phi}_{{\rm max}}^{{\rm d} =1} \right)^2 &=& n \\ \label{solusysteq2D1}
m^2 \int {\rm d} t \left( \phi_{{\rm max}}^{{\rm d} =1} \right)^2 &=& 3 n .
\end{eqnarray}
Since $\varphi (u)$, introduced in eq.(\ref{eq:chvarQM1}) and satisfying
eq.(\ref{eq:numeqQM}), is dimensionless (remember that $u = m x$ is also
dimensionless), and the corresponding $\phi_{{\rm max}} (x)$ has finite action,
the quantity
\begin{equation}\label{Adef}
A \equiv { 1 \over 4} \int {\rm d^d} u \varphi^4 (u)
\end{equation}
is a finite, pure number greater than zero~\cite{zinn}. For the Quantum
Mechanical case
mentioned above, $A = 4/3$. For the cases $d>1$, $A$ is not explicitly known
but, as just said, it must be a finite, positive, pure number. With the
definition~(\ref{Adef}), Eq.~(\ref{eq:chvarQM1}) becomes,
\begin{equation}\label{relphimaxvarphi}
\phi_{{\rm max}} (x) = m^{d/2 - 1} \left( { n \over A} \right)^{1/2} \varphi (m x)
\end{equation}
Since $\varphi (m x)$ satisfies the n-independent Eq.~(\ref{eq:numeqQM}), we
conclude that the field strength of $\phi_{{\rm max}}$ grows with the square
root of the order $n$.
Equation (\ref{eq. effactQMFree}), together with the definition (\ref{Adef}) and
the relation (\ref{relphimaxvarphi}), allow us to write an expression for the
action (\ref{eq. effactQM}) at $\phi = \phi_{{\rm max}}$,
\begin{equation}\label{Sphimax}
S \left[ \phi_{{\rm max}} \right] = 2 n - n \ln{ \left[ {\lambda \ m^{d-4} \over A} n^2
\right] }
\end{equation}
The value of $c_n \left[ \phi (x) \right]$ at $\phi = \phi_{{\rm max}}$ then
becomes, for large n,
\begin{equation}\label{cnphimax}
c_n \left[ \phi_{{\rm max}} (x) \right] \approx {1 \over Z_0}
{ \left( -1 \right)^n \over 2 \pi}
\left( n - 1 \right) ! \left( {\lambda \ m^{d-4} \over A} \right)^n.
\end{equation}
With the change of variables
\begin{equation}\label{chvarphiq}
\phi (x) = \phi_{{\rm max}} (x) + m^{d/2 - 1} \phi_{\rm q} (m x),
\end{equation}
the Gaussian approximation of $c_n \left[ \phi \right]$ around $\phi_{{\rm max}}$ is,
\begin{eqnarray}\label{eq:gaussapp}
c_n \left[ \varphi (u) \right] & \approx & {1 \over Z_0}
{ \left( -1 \right)^n \over 2 \pi}
\left( n - 1 \right) ! \left( {\lambda \ m^{d-4} \over A} \right)^n \cdot \nonumber \\
\label{local}
& & e^{- {1\over 2} \int {\rm d^d} u_1 {\rm d^d} u_2 \phi_{\rm q} (u_1) \left[ \left( -
\nabla^2_{u_1} + 1 - 3 \varphi^2 (u_1) \right)
\delta (u_1 - u_2)\right] \phi_{\rm q} (u_2) } \cdot \\ \label{nonlocal}
& & e^{- {1\over 2} \int {\rm d^d} u_1 {\rm d^d} u_2 \phi_{\rm q} (u_1) \left[
\left( 1 / A \right) \varphi^3 (u_1) \varphi^3 (u_2) \right] \phi_{\rm q} (u_2) }
\end{eqnarray}
where $u = mx$ and $\varphi (u)$, solution of Eq.(\ref{eq:numeqQM}),
is related to $\phi_{{\rm max}}$ through Eq.(\ref{relphimaxvarphi}). This
Gaussian approximation becomes exact in the limit $n \rightarrow \infty$.
The second derivative operator, that we call $D$, is then,
\begin{equation}\label{D}
D = D_{{\rm local}} + D_{{\rm non-local}}
\end{equation}
with
\begin{equation}\label{Dlocal}
D_{{\rm local}} = - \nabla^2 + 1 - 3 \varphi^2
\end{equation}
and
\begin{equation}\label{Dnonlocal}
D_{{\rm non-local}} = { 1 \over A} |v> <v| , \quad {\rm with}\quad
< u | v > = \varphi^3 (u)
\end{equation}
and $A$ given in Eq.(\ref{Adef}).
The operator $D_{{\rm local}}$ is well known (see for example~\cite{zinn}).
It has $d$ eigenvectors
$| 0_{\mu} >$ with zero eigenvalues given by
\begin{equation}\label{zeroeig}
< u | 0_{\mu} > = {\partial \over \partial u^{\mu}}{ \varphi (u) }.
\end{equation}
These vectors are also zero-eigenvectors of $D$, as can be seen by noting
that $|v>$ is orthogonal to them,
\begin{equation}\label{orthv0}
< v | 0_{\mu} > = 0.
\end{equation}
They reflect the translation invariance of the action (\ref{eq. effactQM}).
$D_{{\rm local}}$ is also known to have one and only one negative eigenvector.
The proof of this fact given in Appendix 38 of reference~\cite{zinn}, that uses
Sobolev inequalities, can be repeated line by line to prove that, on the
contrary, $D$ is a positive semi-definite operator,
\begin{equation}\label{Dpossemidef}
D \ge 0
\end{equation}
in the operator sense.
Projecting out the d-dimensional eigenspace of eigenvalue zero, the resulting
operator, that we call $D'$, is positive definite.
\begin{equation}\label{Dposdef}
D' = D'_{{\rm local}} + D_{{\rm non-local}} > 0
\end{equation}
This equation explicitly states that the projection over the strictly positive
eigenvectors modifies only $D_{{\rm local}}$. The non-local part, as we saw,
is a projector orthogonal to the zero modes and is therefore not modified under
that operation.
Equations (\ref{Dpossemidef}) and (\ref{Dposdef}) suggest that the operator
$D$, with the corresponding renormalization for $d>1$, generate a well defined Gaussian measure in a finite volume (remember $d < 4$). In fact, the
determinant of $D'_{{\rm local}}$ was calculated many times in the
past~\cite{zinn}, and a generalization of a Quantum
Mechanical argument of ref.~\cite{au} indicates that this is all we need to compute the
determinant of $D'$. The argument goes as follows,
\begin{eqnarray}\label{det1}
{\rm Det} \left[ D' \right] &=& {\rm Det} \left[ D'_{{\rm local}} + { 1 \over A} |v> <v|
\right] \nonumber \\
&=& {\rm Det} \left[ D'_{{\rm local}} \right]
\left( 1 + { 1 \over A} <v| D^{' -1}_{{\rm local}} |v> \right).
\end{eqnarray}
Since $\varphi (u)$ is orthogonal to $\partial_{\mu} \varphi (u)$ (the zero modes
of $D$ and $D_{{\rm local}}$),
\begin{equation}\label{det2}
D'_{{\rm local}} \varphi = D_{{\rm local}} \varphi = -2 \varphi^3.
\end{equation}
The last equality follows from the definition of $D_{{\rm local}}$ in Eq.(\ref{Dlocal})
and the equation (\ref{eq:numeqQM}) satisfied by $\varphi$. Inverting
$D'_{{\rm local}}$, and remembering the definition of $|v>$ and $A$ in Eqs.
(\ref{Dnonlocal}) and (\ref{Adef}), we obtain
\begin{equation}\label{det3}
<v| D^{' -1}_{{\rm local}} |v> = -2 A.
\end{equation}
Replacing this result in Eq.(\ref{det1}), we arrive at the result
\begin{equation}\label{det4}
{\rm Det} \left[ D' \right] = - {\rm Det} \left[ D'_{{\rm local}} \right].
\end{equation}
As already mentioned, $D'_{{\rm local}}$ has one and only one negative
eigenvector, consequently its determinant is negative. Eq. (\ref{det4}) indicates
then that ${\rm Det} \left[ D' \right]$ is positive, as it should be according to
(\ref{Dposdef}). The effect of the nonlocal part is to change the sign of the
determinant of the local part.
The preceding equations allow us to integrate the Gaussian approximation
of $c_n \left[ \varphi (u) \right]$ given in Eqs.~(\ref{local},\ref{nonlocal}).
Using the method of collective coordinates to project out the zero modes,
the Jacobian of the corresponding change of variables is, at leading order
in $1/ n$,
\begin{equation}\label{Jac}
J = \prod_{\mu = 1}^{d}{ \left[ \int \left( \partial_{\mu} \phi_{\rm max} \right)^2
{\rm d^d} x \right]^{1/2} }
\end{equation}
where no sum over $\mu$ is implied.
It can be shown that the solutions of Eq.~(\ref{eq:eqQM}) corresponding to
minimal action are spherically symmetric~\cite{zinn}, then (\ref{Jac}) can be
written as
\begin{equation}\label{Jac2}
J = \left[ {1 \over {\rm d} } \int \left( \partial_{\mu} \phi_{\rm max} \right)^2
{\rm d^d} x \right]^{{\rm d} /2}
\end{equation}
where now, sum over $\mu$ is implied. Using Eq.~(\ref{solusysteq1}) we
then find
\begin{equation}\label{Jac3}
J = n^{ {\rm d} /2}.
\end{equation}
With this expression, the functional integral of $c_n \left[ \varphi (u) \right]$
can be written as
\begin{eqnarray}\label{CalcFIcn1}
{1 \over Z_0} \int \left[ {\rm d} \phi \right] \ c_n \left[ \phi \right] &=&
{ \left( -1 \right)^n \over 2 \pi} \left( n - 1 \right) ! \left( {\lambda \ m^{d-4}
\over A} \right)^n \cdot \\ \label{CalcFIcn2}
& & \left( {\rm Vol} \ m^d \right) n^{d/2}
\left( - {\rm Det} \left[ {D'_{{\rm local}} \over D_0} \right] \right)^{-1/2}
\end{eqnarray}
where$D_0 \equiv - \nabla^2 + 1$. The factors
in the line~(\ref{CalcFIcn1}) correspond to the value of $c_n \left[ \phi
\right]$ at $\phi_{\rm max}$ up to the normalization $1 / Z_0$ as can be
seen in Eq.~(\ref{cnphimax}). The factor ``Vol" arises after the integration
over the flat coordinates corresponding to the center of $\phi_{\rm max}$.
The $n^{d/2}$ comes from the Jacobian of the change of variables as
mentioned before. The factor $m^d$ arises after the rescaling of the
fields that makes them dimensionless in both $c_n \left[ \phi \right]$ and
$Z_0$. This happens because there are $d$ more integration variables in
$Z_0$ due to the integration over the collective coordinated in the numerator.
Finally, the factor
$\left( - {\rm Det} \left[ D'_{{\rm local}} \right] \right)^{-1/2}$ is the result
of the integration over the coordinates orthogonal to the zero modes of $D$,
while $\left( {\rm Det} \left[ D_0\right] \right)^{1/2}$ is the dimensionless
normalization factor (the mass dimension of both, the numerator and the
denominator, was already taking care of in the term $m^d$). The minus sign is
due to the non-local part of $D$ that, as proved above, simply changes the
sign of the determinant of the local part, making it positive.
In the case $d=1$,
$- {\rm Det} \left[ D'_{{\rm local}} / D_0 \right] = 1/ 12$~\cite{zinn,au}, and Eqs.~(\ref{CalcFIcn1},\ref{CalcFIcn2}) (with $A= 4/3 $ as already
mentioned) become identical to the
corresponding result of Ref.~\cite{au} if we take into account the different
normalization here and a factor of 2 that is taken
care of by remembering that the sign of the solution of Eq.~(\ref{eq:eqQM})
is undetermined, therefore both, positive and negative solutions contribute
equally to the functional integral.
For $d = 2$ or 3, the formal expression~(\ref{CalcFIcn1},\ref{CalcFIcn2})
needs of course to be renormalized. All the arguments in this section remain
valid for the theory with a Pauli-Villars regularization~\cite{zinn}. The
action~(\ref{eq:action}) becomes
\begin{equation}
S \left[ \phi \right] = \int {\rm d^d} x \left[ {1 \over 2} \phi \left(
- \nabla^2 + {\nabla^4 \over \Lambda^2} + m^2 \right) \phi
+ {\lambda \over 4} \phi^4 + {1 \over 2} \delta m^2 (\Lambda) \
\phi^2 \right].
\label{eq:actionregul}
\end{equation}
The modification of the kinetic part of the action affects both the
equation~(\ref{eq:eqQM}) and the scaling arguments, but by an amount
that becomes small like $\Lambda^{-2}$ when the ultra-violet cut-off
$\Lambda$ becomes large.
As shown in Ref.~\cite{zinn}, although the counterterm increases with
the cut-off, since it is proportional to at least one power of $\lambda$,
taking the small $\lambda$ limit before the large cut-off limit justifies
to ignore it in the equation~(\ref{eq:eqQM}) and the scaling arguments.
On the other hand it contributes to the result~(\ref{CalcFIcn1},\ref{CalcFIcn2})
an amount that exactly cancels the divergence in the
${\rm Det} \left[ D'_{{\rm local}} \right]$ making the final expression finite
as it should be.
In the large $n$ limit, where the Gaussian
approximation~(\ref{local},\ref{nonlocal}) becomes exact, the
expression~(\ref{CalcFIcn1},\ref{CalcFIcn2}) gives the large order
behavior of the perturbative series of $Z$ (up to the factor of 2 mentioned
above) {\it without any assumption about the analytic structure in
$\lambda$}~\cite{au}. A completely analogous procedure would give
the large order behavior of any Green's function.
Eqs.~(\ref{relphimaxvarphi}),~(\ref{cnphimax}),~(\ref{local},\ref{nonlocal})
and~(\ref{CalcFIcn1},\ref{CalcFIcn2}) , allow us to draw an accurate picture
of the mechanism underlying the lack of domination (in the sense of the
Lebesgue's theorem) of the convergence of the sequence of perturbative
integrands~(\ref{eq:fNfunctint}) towards~(\ref{eq:exIntd}), and consequently
of the mechanism underlying the divergence of the perturbative series. In
fact, perhaps not surprisingly given the similarity of their large order
behavior, this picture is very similar to the one described in the previous
section for the simple integral example.
In a finite volume, there is a region of finite measure in field space in which
the perturbative approximation $f_N [ \phi (x) ]$ of Eq.~(\ref{eq:fNfunctint})
approximate the exact integrand~(\ref{eq:exIntd}) with an error smaller
than a given prescribed number. This region grows with $N$, becoming the
full field space in the $N \rightarrow \infty$ limit. As in the simple example,
this is a consequence of Taylor's theorem applied to the (analytic)
integrand~(\ref{eq:exIntd}).
The problem is that, for any finite $N$, outside that region the approximate
integrand $f_N [ \phi (x) ]$ strongly deviates from the exact one. This can be
seen by noting that the maxima of every term of $f_N$ grow factorially
with the order. Therefore, for large enough $N$, the last term is far greater
than the previous ones at its maxima. Even more, as shown above, the
Gaussian approximation around that maxima (that becomes exact for
$N \rightarrow \infty$) defines a measure that does not go to zero as
$N \rightarrow \infty$ (in fact, it is independent of
$N$~(\ref{local},\ref{nonlocal})). This means that for every finite $N$,
there is a region of finite measure in field space (and this measure does not
go to zero as $N \rightarrow \infty$) in which the deviation between the
perturbative integrand and the exact one is of the order of the maxima of the
last term of $f_N$, i.e., of the order of $\left( N - 1 \right)!$. No integrable
functional can therefore satisfy the property (\ref{eq:bound}) of the
Lebesgue's theorem.
That is the mechanism that makes the sequence of perturbative integrands,
although convergent to the exact one, non-dominated in the sense of
Lebesgue's theorem. That is therefore the mechanism that makes the sequence
of integrals (i.e., the perturbative series) divergent. In fact, as
Eqs.~(\ref{CalcFIcn1},\ref{CalcFIcn2}) show, the famous factorial
behavior of the large order coefficients of the perturbative series is a
consequence, after integration, of the above mechanism.
\section{Steps Towards a Convergent Series}\label{sec:converg}
It was mentioned in the introduction that the analysis of the divergence
of perturbation theory presented in this paper would point directly towards
the aspects of the perturbative series that need to be modified in order
to generate a convergent series. This is the topic of the present section.
In the previous section we analyzed perturbation theory from the point of view of
the Dominated Convergence Theorem. We have detected the precise way in
which the convergence of the sequence of perturbative integrands to the exact
one takes place, and the way this convergence fails to be dominated. We have
learned that for any finite order $N$, the field space naturally divides into two
regions. In the first one, that grows with the order, eventually becoming the full
field space (in the $N \rightarrow \infty$ limit), the perturbative integrands very
accurately approximate the exact one. In the other one, however, the deviation
between the perturbative and exact integrands is so strong, that the sequence
of integrals diverge.
It is then clear that
{\it if we could somehow modify the integrands, order by order, in the region
where they deviate from the exact one, while preserving them as they
are in the other region, then, with a ``proper" modification, such modified
sequence of integrands would converge in a dominated way. According to the
Dominated Convergence Theorem, their integrals would then converge to the exact
integral, achieving the desired goal of a convergent modified perturbation
theory.}
\\
Let us call $\Omega_N$ to the region of field space in which the ${\rm N^{th}}$
perturbative integrand approximate with a given prescribed error the exact
integrand (\ref{eq:exIntd}). The {\it characteristic function},
${\rm Ch} (\Omega_N, \left\{ \phi (x) \right\})$, of that region is equal to 1 for
field configurations belonging to it, and zero otherwise:
\begin{equation}\label{chfunctome}
{\rm Ch} (\Omega_N, \left\{ \phi (x) \right\}) \equiv
\cases{
1 & for $ \left\{ \phi (x) \right\} \in \Omega_N $ \cr
0 & for $ \left\{ \phi (x) \right\} \not\in \Omega_N $}
\end{equation}
One possible realization of the above strategy of modifying the integrands
(\ref{eq:fNfunctint}) in the ``bad" region of field space is to make them zero
there. We would have
\begin{equation}\label{fprimach}
f_N' \left[ \phi (x) \right]= {1 \over Z_0} \sum_{n=0}^{N}
{\left( -1 \right)^n \over n!} e^{- S_0 }
\left( {\lambda \over 4} \int {\rm d^d} x \phi^4 \right)^n
{\rm Ch} (\Omega_N, \left\{ \phi (x) \right\})
\end{equation}
According to the analysis of the previous section, choosing $\Omega_N$
appropriately, the sequence of $f_N' \left[ \phi (x) \right]$ will converge
dominatedly, and the corresponding interchange between sum and integral
will now be allowed. A rigorous proof of this is
left for a paper currently in preparation. For the purposes of the present
argument, it is sufficient to rely on the analysis of the previous section
to assume its validity. Also, in the next section we will analyze, along
the general ideas of this paper, some resummation schemes for which rigorous
proofs of convergence have recently been given~[12-19]. As that
analysis will show,
these methods strongly rely on the general notions underlying the
formula~(\ref{fprimach}). Their convergence supports, then, the validity of the
dominated nature of the convergence of~(\ref{fprimach})
towards~(\ref{eq:exIntd}).
An urgent issue, however, is the practical applicability of the above
strategy. To implement it, we need a functional representation of the
characteristic function~(\ref{chfunctome}) (or an approximation to it) that
only involves
{\it Gaussian and polynomial} functionals. In the same way in which a
functional representation of the Dirac delta function allow us to perform functional integrals with constraints, the Fadeev-Popov quantization of Gauge
theories being the most famous example, a functional representation of the
characteristic function~(\ref{chfunctome}) would allow us to functionally
integrate only the
desired region of functional space. Since, basically, the functionals we know
how to integrate reduce to Gaussians multiplied by polynomials, the desired
representation of the characteristic function should {\it only} involve those
functionals. Conversely, if it only involves those functionals, all the
sophisticated machinery developed for perturbation theory (including all
the perturbative renormalization methods) would automatically be applicable.
With this in mind, consider the following function,
\begin{equation}\label{win}
W (M, u) \equiv e^{-Mu} \sum_{j=0}^{M} {\left( Mu \right)^j \over j! }
\end{equation}
where $M$ is a positive integer. Note that $W (M,u)$ arises from
$1 = e^{-Mu} e^{+Mu}$ by expanding the second exponential up to order M.
$W (M, u)$ has the following remarkable properties,
\begin{enumerate}
\item $W(M,u) \rightarrow 1 $ when $M \rightarrow \infty$ for $0<u < 1$.
The convergence is uniform, with the error going to zero as
\begin{equation}\label{item1}
R (M,u) \le e^{M \left( \ln{u} - (u - 1) \right)} {1 \over \sqrt{2 \pi M}}
{u \over 1 - u + 1/M}.
\end{equation}
\item $W(M,u) \rightarrow 0 $ when $M \rightarrow \infty$ for $1<u $. The
convergence is also uniform , with an error of the form
\begin{equation}\label{item2}
W (M, u) \le e^{M \left( \ln{u} - (u - 1) \right)}.
\end{equation}
\end{enumerate}
As we see, the exponent corresponds to the same function in both cases.
For $u > 0$, this function is always negative except at its maxima, at $u =1$,
where it is 0. Therefore the convergence is in both cases exponentially fast
in $M$, with the exponent becoming more and more negative, for a fixed $M$,
when $u$ differs more and more from 1. The proof of properties 1 and 2 is in
the Appendix 1.
If we replace $u$ by a positive definite quadratic form $< \phi | D | \phi
>/C_N$, then the insertion of Eq.~(\ref{win}) into the functional integral would
effectively cut off the region of integration $< \phi | D | \phi > \ > C_N$
\begin{eqnarray}\label{fprimaw}
Z_N' \left[ \phi (x) \right] &=& {1 \over Z_0} \int \left[ {\rm d} \phi \right]
\sum_{n=0}^{N} {\left( -S_{\rm Int} \right)^n \over n!} e^{- S_0 }
\lim_{M \rightarrow \infty} W \left( M, {< \phi | D | \phi> \over C_N}
\right) \\
&=& {1 \over Z_0} \sum_{n=0}^{N} {\left( -1 \right)^n \over n!}
\lim_{M \rightarrow \infty} \int \left[ {\rm d} \phi \right] e^{- S_0 }
\left( S_{\rm Int} \right)^n W
\left( M, {< \phi | D | \phi> \over C_N} \right) \nonumber \\ \label{fprimaw2}
\end{eqnarray}
$C_N$ is a constant that changes with the order $N$ of the expansion in
$\lambda$, increasing with $N$ but in such a way that in the region
$< \phi | D | \phi> < C_N$ the difference between the perturbative and the
exact integrands is smaller than a given prescribed error. Since the
convergence of $W$ is uniform according to properties
1 and 2, with errors given in Eqs.~(\ref{item1}) and~(\ref{item2}), the
corresponding interchange between the sum in Eq.~(\ref{fprimaw2})
and the functional
integral is justified. The fact that $u$ becomes a {\it quadratic} form implies that
the resulting integrands are Gaussians multiplied by monomials, therefore the
familiar Feynman diagram techniques can be used to integrate them. It
also implies that no new loops appear and the sum in $j$
from~(\ref{win}) becomes an algebraic problem. A typical functional integral
to compute has the form
\begin{equation}\label{typicalfunctint}
\int \left[ {\rm d} \phi \right] e^{- \int {\rm d^d} x \left[ {1 \over 2}
\left( \partial_{\mu} \phi \right)^2 + {1 \over 2} m^2 \phi^2
+ ( \phi D \phi / C_N ) \right] } \left( \int {\rm d^d} x \phi^4 \right)^n
\left( \int {\rm d^d} x \ \phi D \phi \right)^m
\end{equation}
as can be seen by replacing the definition~(\ref{win}) into~(\ref{fprimaw2})
with $u = < \phi | D | \phi > / C_N$.
Note that at any given order in $\lambda$, it is not necessary in principle
to go to infinity in $M$. That would amount to replace the
perturbative integrands by zero in the region $< \phi | D | \phi > \ > C_N$,
realizing the strategy mentioned before. But since the convergence in $W$
is uniform, a finite, large enough $M$ (depending on the order in the expansion
in the coupling constant), would suffice to tame the behavior
of the perturbative integrands and transform them into a {\it dominated}
convergent sequence. In fact, as we will see, many methods of improvement
of perturbation theory use effectively formula~(\ref{win}) without sending
$M \rightarrow \infty$ for any given finite order in perturbation theory.
In any case, as already mentioned, that limit is in principle computable, since
it does not involves new loops. Work in this direction is in progress.
The convergence of the sequence~(\ref{fprimaw2}) towards $Z (\lambda)$
may be thought, at first sight, to be in conflict with our well established
knowledge about the non-analyticity of this function at $\lambda = 0$. In fact,
Eq.~(\ref{fprimaw2}) seems to be a power series in $\lambda$ (the powers of
$\lambda$ coming from the powers of $S_{\rm Int}$), therefore, if convergent,
that power series would define a function of $\lambda$ analytic at
$\lambda = 0$. It must be recognized, however, that the
validity of Lebesgue's Dominated Convergence Theorem is completely
independent of any analyticity consideration. Therefore, if its hypothesis are
satisfied, its conclusions must be valid. This being said, the question of how
does the convergence of~(\ref{fprimaw2}) fits with the non-anlayticity of $Z (
\lambda )$ deserves an answer. To begin with, even at finite order in $\lambda$,
the function~(\ref{fprimaw2}) is not necessarily analytic at $\lambda =0$ despite
its analytic appearance. This is because the constant $C_N$ may have an
implicit nonanalytic dependence on $\lambda$. In Appendix 2 this is actually
the case in the context of a simple example to which the present ideas are
applied. But the mechanism that ultimately introduces the proper non-analyticity
in $\lambda$ is the limit process $N \rightarrow \infty$. Given a non-analytic
function like $Z (\lambda)$ one can always construct a sequence of analytic
functions that converge to it. Satisfying the hypothesis of the Dominated
Convergence Theorem is a way of achieving that, avoiding all the
complicated and {\it model dependent} issues of non-analyticity. Note that the
validity of these hypothesis for a given sequence of integrands can be
checked independently of any analyticity consideration.
In the Appendix 2 we prove the convergence of the general strategy
discussed here for the simple integral example analyzed in
section~(\ref{sec:simpleint}). For that case, making $u = (x/x_{c,N})^2$, the
function $W (M, u)$ becomes in the limit the characteristic function of the
interval $|x|< x_{c,N}$. We use this to explicitly compute the non-analytic
function $z (\lambda)$ (Eq.~(\ref{eq:pertsimple})) calculating only Gaussian
integrals. We also show explicitly how a non-analytic dependence of
$x_{c,N}$ on $\lambda$ naturally arises just by demanding the validity
of the Lebesgue's hypothesis and how the $N \rightarrow \infty$ limit
process captures the full non-analyticity of $z (\lambda)$. The same
method also works for the ``negative mass case", where the Borel
resummation method fails. In figure~\ref{fig3}, we can appreciate the
convergence of $W$ towards the characteristic function of the interval
$|x|< x_{c,N}$ for $x_{c,N} = 1$ for two different values of $M$.
\begin{figure}[htp]
\hbox to \hsize{\hss\psfig{figure=win.eps,width=0.7\hsize}\hss}
\caption{Function $W(M,x,x_{c,N})$ with $x_{c,N}=1$ for $M=3$ (segmented
line) and $M=60$ (continuous line). The convergence towards the
characteristic function of the interval $|x|< x_{c,N}$ is apparent.}
\label{fig3}
\end{figure}
\section{Improvement methods of perturbation theory.}\label{Improvement}
The analysis of the mechanism of divergence of the perturbative series
presented in this paper, together with the formula~(\ref{win}) and its
properties, offer a large range of possibilities to construct a convergent
series. In the previous section we have shown how that formula can be
used to effectively cut-off the region of field space where the strong
deviation between perturbative and exact integrands take place. But as we will
see, this is only one possible way, among many, to use the formula~(\ref{win})
to transform the sequence of perturbative integrands into a dominated one.
Another example of its possible use is the so called ``optimized delta
expansion"~\cite{gral, stev}. In a series of papers~\cite{tony1}-\cite{kle},
it was proved that such an expansion converges for the partition function
of the anharmonic oscillator in finite Euclidean time. The problem of
convergence in the infinite Euclidean time (or zero temperature) limit for the
free energy or any connected Green's function is still under investigation,
as well as its extension to Quantum Field Theories~\cite{tony3,arv}.
The method was proved to generate a convergent series for the energy
eigenvalues~\cite{gui1, gui2}, although such studies make heavy use of
analyticity properties valid specifically in the models studied. In these
works, it was realized that many methods of improvement of perturbations
theory, such as the order dependent mappings of references~\cite{zinn2, leg},
posses the same general structure as the linear delta expansion.
A considerable amount of work has been dedicated to investigate the virtues
and limitations of the method and extensions of it~\cite{kle, nev}.
It is not the place here to give a detailed analysis of these methods. But we
would like to briefly indicate how they can be
understood in terms of the ideas presented here. In what follows, our analysis
is restricted to $d=1$ (Quantum Mechanics) where rigorous results
about the convergence of the methods considered here are available.
Let's consider the case of the anharmonic oscillator. Its action is given
in Eq.~(\ref{eq:action}) for $d=1$. The idea of the method is to replace it
by an interpolating action
\begin{equation}\label{sdelta}
S_{\delta} = \int {\rm d} t \left[ {1 \over 2} \left( {\rm d}_t \phi \right)^2
+ {1 \over 2} \left( m^2 + {\lambda \over 2 m} \alpha \right) \phi^2 +
\delta {\lambda \over 4} \left( \phi^4 -{ \alpha \over m} \phi^2 \right) \right].
\end{equation}
Clearly, the dependence on the parameter $\alpha$ in $S_{\delta}$ is lost
when $\delta = 1$. For that value, the action~(\ref{sdelta}) reduces
to~(\ref{eq:action}). However, if we expand up to a finite order in $\delta$
and then make $\delta=1$, the result still depends on $\alpha$. The idea is
to tune $\alpha$, order by order in the expansion in $\delta$, so that the result
is a convergent series. It was shown in the references mentioned above that
the methods works if $\alpha$ is tuned properly. For example, in
reference~\cite{tony2}, the asymptotic scaling $\alpha \simeq N^{2/3}$
was used to prove the convergence of the method for the partition
function at finite Euclidean time.
It is interesting to note that originally~\cite{tony1, tony2}, $\alpha$ was
tuned according to heuristic prescriptions such as the ``principle of minimal
sensitivity"~\cite{stev} (at any given order in $\delta$, choose $\alpha$ so
that the result is insensitive to small changes in it), or the criterion of
``fastest apparent convergence" (the value of $\alpha$ at which the next order
in delta vanishes). But later~\cite{tony3, arv},
it was realized that the best strategy was simply to leave $\alpha$
undetermined, find an expression for the error (that obviously depends on
$\alpha$), and then choose $\alpha$ so that the error goes to zero when
the order in $\delta$ goes to infinity. It is clear that a {\it structural}
understanding of the convergence of the method can help to construct the
necessary generalizations to overcome the difficulties associated with the
convergence in the infinite volume limit for connected Green's functions, as
well as the extensions to general Quantum Field Theories.
To understand the ``optimized delta expansion" in terms of the ideas presented
in this paper, let us expand the functional integral corresponding to the
action~(\ref{sdelta}) in powers of $\delta$ up to a finite order $N$, and make
$\delta =1$ as the method indicates,
\begin{eqnarray}\label{delta1}
Z (m, \lambda, \alpha, N) &=& {1 \over Z_0} \int \left[ {\rm d} \phi \right]
e^{- \int {\rm d} t \left[ {1 \over 2} \left( {\rm d}_t \phi \right)^2
+ {1 \over 2} \left( m^2 + {\lambda \alpha \over 2 m} \right) \phi^2 \right] }
\cdot \nonumber \\
& & \left[ \sum_{n=0}^N {\left( - 1 \right)^n \over n!} \left(
{\lambda \over 4} \int \phi^4 - {\lambda \alpha \over 4 m}
\int \phi^2 \right)^n \right]
\end{eqnarray}
The general analysis of the mechanism of divergence of perturbation theory
of section~\ref{sec:leb} indicates that if the function~(\ref{delta1}) generates
a convergent series with $\alpha$ scaling properly with $N$, then, barring
miraculous coincidences, the corresponding integrands should converge
dominatedly (or, even better, uniformly) towards the exact
integrand~(\ref{eq:fNfunctint}). We want to obtain a qualitative understanding
on how this method achieves that.
Expanding the binomial and making some elementary changes of variables in the
indices of summation, we obtain the expression
\begin{eqnarray}\label{delta2}
Z (m, \lambda, \alpha, N) &=& {1 \over Z_0} \int \left[ {\rm d} \phi \right]
e^{- \int {\rm d} t \left[ {1 \over 2} \left( {\rm d}_t \phi \right)^2
+ {1 \over 2} \left( m^2 + {\lambda \alpha \over 2 m} \right) \phi^2 \right] }
\cdot \nonumber \\
& & \left[ \sum_{i=0}^N {\left( - 1 \right)^i \over i!} \left(
{\lambda \over 4} \int \phi^4 \right)^i
\left( \sum_{k=0}^{N-i} {1 \over k!} \left( {\lambda \alpha \over 4 m}
\int \phi^2 \right)^k \right) \right]
\end{eqnarray}
This equation already shows some of the distinctive characteristics of
the method. As we see, the ${\rm i^{th}}$ power of the interacting action in the
expansion of $e^{-S_{\rm Int}}$ up to order $N$, is multiplied by
\begin{equation}\label{delta3}
{\cal W} \left( N - i \right) \equiv e^{- \left( \lambda \alpha / 4 m \right)
\int \phi^2} \left( \sum_{k=0}^{N-i} {1 \over k!} \left( {\lambda \alpha \over 4 m}
\int \phi^2 \right)^k \right).
\end{equation}
Note that ${\cal W} \left( N \right)$ corresponds to the function $W (M, u)$
with $M = N$ ($N$ is the order in the expansion of $e^{-S_{\rm Int}}$), and
the variable $u$ replaced by the quadratic form $\left( \left( \lambda / 4 m
\right) \int \phi^2 \right) / C_N$, where $C_N = N / \alpha$. Making for example
$\alpha \simeq N^{2/3}$ as in Ref.~\cite{tony2} (where it was proved that
with such scaling
the method generates a convergent series), we see then, that, according to the
previous section, ${\cal W} \left( N \right)$ is an approximation of the theta
function in the region of field space characterized by
\begin{equation}\label{delta4}
{\lambda \alpha \over 4 m} \int {\rm d} x \ \phi^2 \le N^{1/3}.
\end{equation}
Equation~(\ref{delta2}), however, shows that the mechanism used to
achieve dominated convergence can not be reduced to a simple insertion of
the function $W(M,u)$ with $M=N$ and
$u = \left( \left( \lambda / 4 m \right) \int \phi^2 \right) / C_N$.
That would be the case if all the powers of the expansion of $e^{-S_{\rm Int}}$
up to order $N$ were multiplied by ${\cal W} \left( N \right)$. But
equation~(\ref{delta2}) shows that the ${\rm i^{th}}$ power of the interacting
action is in fact multiplied by ${\cal W} \left( N - i \right)$.
\\
At this point it is convenient to pause for a moment in our study of the
``optimized delta expansion" to give some useful definitions.
Let us call {\bf passive} mechanisms (to achieve dominated, or uniform
convergence of a sequence of integrands to the exact one) to those that
can be reduced to the product of the $N^{\rm th}$ perturbative integrand
and the characteristic function of a region $\Omega_N$ of field space for
some sequence $\left\{ \Omega_N \right\}$.
Passive methods use only information that is already available in the
perturbative integrands, they just get rid of the ``noise" inherent to perturbation
theory. Because of that, in addition to define a convergent series, they can
also be very useful to study perturbation theory itself. The function $W (N,u)$,
with $u$ replaced by a properly selected quadratic operator, was specially
designed to make passive methods practical. In a sense, section~\ref{sec:converg} is a discussion of passive methods.
{\bf Active} mechanisms are those that are not passive, as defined above.
\\
What kind of mechanism is the one underlying the ``optimized delta expansion"
method?
A trivial generalization of the proof, in the previous section, of the convergence
of $W(M,u)$ towards the theta function for $u>0$, shows that the function
\begin{equation}\label{wbarra}
\overline{W} (M,u,i) \equiv e^{-Mu} \sum_{n=0}^{M-i}
{\left( M u \right)^i \over i!}
\end{equation}
also converges towards the theta function for $u>0$ in the limit
\begin{equation}\label{limit}
M \rightarrow \infty, \quad i \ {\rm fixed}.
\end{equation}
In this sense, the ``optimized delta expansion" method
does have passive aspects. As Eqs.~(\ref{delta2}, \ref{delta3}) show, it
amounts to multiplying the ${\rm i^{th}}$ power of the expansion up to
order $N$ of $e^{S_{\rm Int}}$ by $\overline{W}$ with $u = \left( \left(
\lambda / 4 m \right) \int \phi^2 \right) / C_N$ and $C_N = N / \alpha$.
Since this function converges to the characteristic function of the region
characterized by Eq.~(\ref{delta4}), this means that the first $i$ terms, of
the expansion up to order $N$ of $e^{S_{\rm Int}}$ are {\it effectively}
multiplied by the same function (an approximate characteristic function) for
$i \ll N$ . Therefore, the first $i$ terms, with $i \ll N$, use only the
information available in the perturbative series to converge to the exact
integrand.
What about the other terms?, the ones characterized by $i {\ \lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ } N$?
Surprisingly, these terms produce a convergence of the corresponding
integrands towards the exact one that is faster than possible with only passive
components!
It is not the place here to study this aspect in detail, so, let us simply see
this ``faster than passive" convergence for the simple integral example.
Applied to the ``mass-less" version of the integral~(\ref{eq:pertsimple}), the
optimized delta expansion method was proved to generate a rapidly convergent
sequence in Ref.~\cite{tony1}. That is, the sequence given by
\begin{equation}\label{simODE}
I_N \equiv \sum_{n=0}^N {(-1)^n \over n!} \int_{- \infty}^{\infty} {\rm d} x \
e^{ - \alpha (N) \ x^2} ({\lambda \over 4} x^4- \alpha (N) \ x^2)^n
\end{equation}
was proved to converge to
\begin{equation}\label{sim}
I \equiv \int_{- \infty}^{\infty} {\rm d} x \ e^{ - \lambda x^4 / 4}
\end{equation}
when $\alpha (N) \simeq \sqrt{N}$ with an error that goes to zero at the
very fast rate of $R_N < C N^{1/4} e^{-0.663 N}$ when $N \rightarrow \infty$.
$C$ is a numerical constant.
We are interested in understanding whether the corresponding convergence
of the integrands is faster than passive.
For our qualitative purposes, it is
enough to observe, in figure~\ref{fig4}, the convergence towards the exact
integrand
\begin{equation}\label{exinm0}
I_{\rm exa} (x) = e^{ - {\lambda \over 4} x^4}
\end{equation}
of both, the perturbative integrand
\begin{equation}\label{pertintm0}
I_{\rm pert} = \sum_{n=0}^N { ( - \lambda x^4 / 4)^n \over n!}
\end{equation}
and the optimized delta expansion integrand
\begin{equation}\label{odeinm0}
I_{\rm ode} = \sum_{n=0}^N {(-1)^n \over n!}
e^{ - \alpha (N) \ x^2} ({\lambda \over 4} x^4- \alpha (N) \ x^2)^n
\end{equation}
with $\alpha (N) \simeq \sqrt{N}$, for $N=4$.
\begin{figure}[htp]
\hbox to \hsize{\hss\psfig{figure=odelexp.eps,width=0.9\hsize}\hss}
\caption{In the main plot, the superiority of the convergence of the fourth
order optimized delta expansion (ode) with respect to the same order
perturbative approximation is evident. In the sub-graph, the difference
between the ode and the exact integrand is plotted. Note the difference in
the scales of the y axis of the main and sub graph.}
\label{fig4}
\end{figure}
We can see how accurate the convergence of $I_{\rm ode} (x)$ is, already
at this low order. In particular, when the perturbative integrand begins to
diverge, $I_{\rm ode} (x)$ continues to approximate remarkably well the
exact integrand. In the sub-figure, we can appreciate the difference
between $I_{\rm exa} (x)$ and $I_{\rm ode} (x)$. Note the difference in the
$y$ axis scale of graph and sub-graph.
It is then clear that the optimized delta expansion method, with its subtle
combination of passive and active components, manages to generate
a sequence of integrands that (uniformly) converges towards the exact
one at a rate that far exceeds the possibilities within a purely passive method.
We stop here the qualitative discussion of the optimized delta expansion
method expressing two general lessons:
\begin{enumerate}
\item Any method of improvement of the perturbative series in a given
quantum theory, where a functional integral representation of the quantity
under study exists, must rely, at the level of the integrands, on an
improvement over the pointwise convergence of the Taylor series in the
coupling constants of $e^{-S}$.
\item The problem of finding a convergent series reduces to the problem
of finding a dominatedly convergent sequence of integrands towards $e^{-S}$.
\end{enumerate}
This second simple statement, not only provides a guide to the construction
of convergent schemes, but also emphasizes the fact that, in principle, a
dominatedly convergent sequence of integrands, do not have to have
any relation whatsoever with the corresponding Taylor expansion. In order
to be able to use the usual techniques of Quantum Field Theory, it is
reasonable to restrict the search for a convergent scheme to a sequence of
integrands of the general form:
\begin{equation}\label{generic}
f_N = e^{-S_0} \sum_{n=0}^N a_n \ S_{\rm Int}^n \ f_n (< \phi | D_n | \phi >)
\end{equation}
where the functional $f_n$ of the quadratic form $< \phi | D_n | \phi >$,
should take care of the non-dominated convergence that is bound to
appear with only powers of the interacting action. The function $W$ of
section~\ref{sec:converg}, with its possible generalizations, is an ideal
candidate for this purpose. But the selection of the coefficients $a_n$
amounts to a pure problem in optimization of the convergence of the
integrands $-$ no a priori connection with any Taylor series is necessary.
\section{Conclusions}\label{sec:concl}
In this paper we have exposed the mechanism, at the level of the integrands,
that make the perturbative expansion of a functional integral divergent.
We have seen in detail how the sequence of integrands violate the domination
hypothesis of the Lebesgue's Dominated Convergent Theorem. That theorem,
as is well known, establishes the conditions under which it is allowed
to interchange an integration and a limit, in particular the one that takes
place in the generation of perturbation series.
It was shown that at any finite order in perturbation theory, the field
space divides into two regions. One, that grows with the order, in which
the perturbative integrands very accurately approximate the exact integrand.
In the other however, a strong deviation takes place. It was shown that the
behavior in this second region violates the hypothesis of Lebesgue's
theorem, and, consequently, generates the divergence of perturbation theory.
The famous factorial growth of the large order coefficients of the perturbative
series was shown to be an effect, after integration, of the very mechanism that
violates the hypothesis of the theorem.
All of the above was done explicitly without relying in the particular
analytic properties of the models studied. It is therefore natural to
assume that similar mechanisms of violation of the Lebesgue's hypothesis
are present in any other Quantum Field Theory, although for just
renormalizable theories other mechanisms are responsible for renormalons.
Studies in this direction are in progress.
The mechanism of divergence presented here points towards a simple way
to achieve a convergent series: integrate only in the ``good" region of
field space. Since this region grows with the order, becoming in the limit
the whole field space, integrating in a correspondingly increasing region
we would obtain a convergent series. A step forward towards a practical
implementation of this program was made with the construction of the
function $W$~(\ref{win}). This function allows us to introduce a Gaussian
representation of the characteristic function of regions of field space,
very much like the imposition of constraints in the functional integral
was allowed by a functional representation of the Dirac's delta function.
A rigorous proof of the convergence of this practical implementation of the
above mentioned strategy is in progress. In Appendix 2 it was applied to
a simple integral example.
Finally, a qualitative analysis of the optimized delta expansion method
of improvement of perturbation theory in terms of the ideas of this paper
was done. Some general properties of improvement methods, useful to
generates new schemes, as well as to understand and improve old ones,
have been established.
\newpage
\section*{Acknowledgments}
S. P. was supported in part by U.S. Dept. of Energy Grant DE-FG
02-91ER40685. He would like to thank A. Duncan for numerous very
useful discussions, and also to A. Das and S. Rajeev.
\section*{Appendix 1}
In this appendix we will prove the two properties of formula~(\ref{win}).
Because for $u>0$ all the terms of the sum defining $W (M, u)$ are positive,
we have trivially $W (M, u) > 0$. On the other hand, since in the Taylor
expansion of $e^{Mu}$ all the terms are positive, we have
$\sum_{n=0}^{M} \left( Mu \right)^n / n! \le e^{Mu}$.
Therefore $W (M, u) \le 1$. So for every $M$ and positive or zero $u$ we
have,
\begin{equation}\label{app:bound}
0 \le W(M,u) \le 1.
\end{equation}
Consider first the case $0<u < 1$.
\begin{equation}\label{app:rest}
1 - W (M, u) = e^{-Mu} \sum_{n=M+1}^{\infty} {\left( Mu \right)^n \over n! }
\equiv R(M,u),
\end{equation}
we will prove that $R(M,u) \rightarrow 0$ when $M \rightarrow \infty$.
Changing variables to $j = n -M$, we get
\begin{eqnarray}\label{app:rest2}
R(M,u) &=& e^{-Mu} {\left( Mu \right)^M \over M! } \sum_{j=1}^{\infty}
\left( Mu \right)^j {M! \over \left( j+M \right)! } \\
&\le& e^{-Mu} {\left( Mu \right)^M \over M! } \sum_{j=1}^{\infty}
{\left( Mu \right)^j \over \left( M + 1 \right)^j } \\
&\le& e^{-Mu} {\left( Mu \right)^M \over M! } {u \over 1 - u + 1/M}.
\end{eqnarray}
But $M^M / M! \rightarrow e^M / \sqrt{2 \pi M}$ for large $M$, so
\begin{equation}\label{app:rest3}
R (M,u) \le e^{M \left( \ln{u} - (u - 1) \right)} {1 \over \sqrt{2 \pi M}}
{u \over 1 - u + 1/M}.
\end{equation}
The exponent is negative in the region $0<u < 1$ since, being both,
$\ln{u}$ and $(u - 1)$ negative there, $| \ln{u} | > |u - 1|$ in this region.
Therefore
\begin{equation}\label{app:proved}
R (M,u) \rightarrow 0 \ , {\rm when} \quad M \rightarrow \infty
\end{equation}
in the region $0<u < 1$ and property 1 is proved with an exponentially fast
convergence.
In the region $u > 1$, we have
\begin{eqnarray}\label{app:uge1}
W (M, u) &=& e^{-Mu} \sum_{n=0}^{M} {\left( Mu \right)^n \over n! } \le
e^{-Mu} u^M \sum_{n=0}^{M} {M^n \over n! } \\ \label{app:uge2}
&\le& e^{M \left( \ln{u} - (u - 1) \right)}.
\end{eqnarray}
The first inequality is valid because $u>1$ and the second because \\
$e^M > \sum_{n=0}^{M} M^n / n! $. The exponent is again negative.
For $u>1$, both, $ \ln{u}$ and $( u-1)$ are positive, but now
$| \ln{u} | < |u - 1|$. So property 2 is also valid with an exponentially fast
convergence.
For $u=1$ all we know is that $W$ is bounded by Eq.(\ref{app:bound}). That is
all we need. Numerics suggest $W (M, 1) \rightarrow 1/2$ when $M \rightarrow
\infty$.
This finishes our proof.
\section*{Appendix 2}
In this appendix we apply the strategy discussed in section~\ref{sec:converg}
to generate a series convergent to the function $z (\lambda)$
(Eq.~(\ref{eq:pertsimple})). This is done using the function $W$ of
Eq.~(\ref{win}) and computing {\it exclusively} Gaussian integrals, therefore
we restrict ourselves to using only those techniques that are also available in
Quantum Field Theory.
As mentioned in section~\ref{sec:converg}, the simplest possible modification
of the perturbative integrand (\ref{eq:fNsimpltint}) that would transform the
corresponding sequence into a dominated one, amounts to keep them as they
are for $|x| < x_{c,N}$ and replacing them by zero for $|x| > x_{c,N}$. That is,
\begin{equation}\label{convseq}
f'_N = \cases{ \pi^{-1/2} \sum_{n=0}^{N} {\left( -1 \right)^n \over n! }
\left( {\lambda \over 4} x^4 \right)^n e^{- x^2 } & for $|x| < x_{c,N} $ \cr
0 & for $|x| > x_{c,N}$}.
\end{equation}
In fact, choosing $x_{c,N}$ so as to properly avoid the region in which the
deviation takes place, the sequence of $f'_N$ converges {\it uniformly} towards
the exact integrand (\ref{eq:exintd}) as we will show shortly. Consequently, the
corresponding sequence of integrals
\begin{eqnarray}\label{exint2}
\int_{- \infty}^{\infty} {\rm d} x f'_N & = & \pi^{-1/2} \int_{- x_{c,N} }^{x_{c,N}} {\rm d} x
\sum_{n=0}^{N} {\left( -1 \right)^n \over n! }
\left( {\lambda \over 4} x^4 \right)^n e^{- x^2 } \\ \label{exint3}
& = & \pi^{-1/2} \sum_{n=0}^{N} \int_{- x_{c,N} }^{x_{c,N}} {\rm d} x
{\left( -1 \right)^n \over n! } \left( {\lambda \over 4} x^4 \right)^n e^{- x^2 }
\end{eqnarray}
will converge to the desired integral
\begin{equation}\label{desint}
z (\lambda) = \pi^{-1/2} \int_{- \infty}^{\infty} {\rm d} x
e^{ - \left( x^2 + {\lambda \over 4} x^4 \right) }.
\end{equation}
In (\ref{exint2}) the change in the limits of integration from $\pm \infty$ to
$\pm x_{c,N}$ is just due to the definition of $f'_N$ in Eq.(\ref{convseq}).
The interchange between sum and integral in (\ref{exint3}) is now allowed
because in the region $\left[ - x_{c,N}, x_{c,N} \right]$ we have uniform
convergence (this is a stronger condition than dominated convergence). The
resulting integrals are not Gaussian due to the finite limits of integration. We will
show how they can be calculated using only Gaussian integrals.
A trivial way to achieve convergence of the sequence of integrals of the
$f'_N$ of Eq.~(\ref{convseq}) towards~(\ref{desint}) amounts to keep
$x_{c,N}$ equal to a finite constant $``a"$ independent of N, while taking the limit
$N \rightarrow \infty$. In this limit, Eq.~(\ref{exint3}) becomes identical to
$\pi^{-1/2} \int_{-a}^{a} {\rm d} x e^{ - \left( x^2 + {\lambda \over 4} x^4 \right) }$,
since for finite $a$ the Taylor series of the integrands converge uniformly.
Therefore, as already said, the interchange between sum and integral
is legal. Finally, taking the limit $a \rightarrow \infty$, we would obtain the
desired convergence towards $z (\lambda)$.
However, better use can be made of the information available in $f'_N$ for finite
$N$. For example, for every finite $N$, we can choose $x_{c,N}$ so that
\begin{equation}\label{diff}
|f'_N (x) - f (x) | \le {\epsilon_{T,N} \over 2 x_{c,N}} \quad {\rm for} \quad
|x| < x_{c,N},
\end{equation}
with $\epsilon_{T,N}$
going to zero as $N \rightarrow \infty$. Then, since we have
\begin{equation}\label{diff2}
|f'_N (x) - f (x) | \le e^{- \left( x_{c,N}^2 + {\lambda \over 4} x_{c,N}^4 \right)}
\equiv {\epsilon_{c,N} \over 2} \quad {\rm for} \quad |x| > x_{c,N},
\end{equation}
the $f'_N (x)$ will uniformly converge towards the exact integrand $f(x)$
if~(\ref{diff}) is consistent with $x_{c,N} \rightarrow \infty$ when
$N \rightarrow \infty$. Indeed, if this happens, we would have
\begin{equation}
| \int_{- \infty}^{\infty} \left( f(x) - f_N (x) \right) {\rm d} x | \le
\epsilon_{T,N} + \epsilon_{c,N} \rightarrow 0\quad {\rm when} \quad
N \rightarrow \infty.
\end{equation}
The term $\epsilon_{T,N}$ comes trivially from~(\ref{diff}), while $\epsilon_{c,N}$
comes from~Eq.(\ref{diff2}) and the inequality
\begin{equation}\label{esterr}
\int_{ x_{c,N} }^{\infty} e^{- \left( x^2 + {\lambda \over 4} x^4 \right)} {\rm d} x
\le e^{- \left( x_{c,N}^2 + {\lambda \over 4} x_{c,N}^4 \right)} = \epsilon_{c,N},
\end{equation}
valid for $x_{c,N}>1$.
Applying Taylor's theorem to the function $e^{- \lambda x^4 / 4}$ one can
easily show that the condition~(\ref{diff}) is satisfied if
\begin{equation}\label{xcNasfunctN}
x_{c,N} = \left[ \left( N+1 \right) ! {\epsilon_{T,N} \over 2 }
\left( {4 \over \lambda} \right)^{(N+1)} \right]^{1/(4(N + 5/4))}.
\end{equation}
Note that the non-analytic dependence of $x_{c,N}$ on $\lambda$ arises
automatically from the imposition of Eq.~(\ref{diff}) to satisfy the hypothesis
of the Lebesgue's theorem.
Remember that the only condition on $\epsilon_{T,N}$ to achieve convergence
of the sequence of integrals is to go to zero when $N \rightarrow \infty$
consistently with $x_{c,N} \rightarrow \infty$ in that limit. Choosing for
example
\begin{equation}\label{prescrerr}
\epsilon_{T,N} = e^{- 4 N^{1/4}},
\end{equation}
we obtain asymptotically,
\begin{equation}\label{xcNassymp}
x_{c,N} \rightarrow \left( 4 N / e \lambda \right)^{1/4}.
\end{equation}
This implies (through Eq.~(\ref{diff2})),
\begin{equation}\label{calcerr}
\epsilon_{c,N} \rightarrow e^{- \left( 4 N / e \lambda \right)^{1/2} - N / e}.
\end{equation}
Equations~(\ref{prescrerr}) and~(\ref{calcerr}) show the exponential rate at
which the convergence of the sequence of integrals take place.
Clearly the form~(\ref{prescrerr}) for $\epsilon_{T,N}$ is not unique, not even
the most efficient one, but enough to achieve convergence.
In the table~(\ref{table}) one can appreciate the numerical convergence
\begin{table}[t]\caption{ Integration over the small field configurations only
produces a convergent series. In the last column the improvement
over the perturbative values can be appreciated.\label{table}}
\begin{tabular}{|c|c|c|c|}
\hline
Order & Exact value ($\lambda=4/10$) & Conv. series &
Pert. series \\
\hline
2 & 0.837043 & 0.803160 & 0.848839 \\
\hline
4 & 0.837043 & 0.830264 & 0.854087 \\
\hline
6 & 0.837043 & 0.835516 & 0.901897 \\
\hline
8 & 0.837043 & 0.836667 & 1.316407 \\
\hline
20 & 0.837043 & 0.837044 & 2.33755 $ 10^8$ \\
\hline
\end{tabular}
\end{table}
for $\lambda = 4/10$.
\\
Up to now we have proved that the general strategy of section~\ref{sec:converg}
does, in fact, generate a convergent sequence towards $z (\lambda)$.
However, the resulting integrals in~(\ref{exint3}) are not Gaussians, making
the applicability of the method in Quantum Field Theory dubious, to say the
least. We will show now that the integrals of Eq~(\ref{exint3}) can be computed,
using Eq.~(\ref{win}) with $u = (x/x_{c,N})^2$, calculating only Gaussian
integrals. The steps involved are
\begin{eqnarray}\label{calcnonGaussint1}
\int_{-x_{c,N}}^{x_{c,N}} x^r e^{-x^2} {\rm d} x &=&
\int_{- \infty}^{\infty} x^r e^{-x^2} \lim_{M \rightarrow \infty}
W(M,x,x_{c,N}) {\rm d} x \\ \label{calcnonGaussint2}
&=& \lim_{M \rightarrow \infty} \int_{-\infty}^{\infty} x^r e^{-x^2}
W(M,x,x_{c,N}) {\rm d} x \\
&=& \lim_{M \rightarrow \infty} \sum_{n=0}^{M} {1 \over n!}
\left( {M \over x_{c,N}^2} \right)^n \int_{-\infty}^{\infty}
e^{- \left( 1 + M / x_{c,N}^2 \right) x^2} x^{2n + r} {\rm d} x \nonumber \\
\label{calcnonGaussint3}
\end{eqnarray}
The two properties of $W$ validate both equality~(\ref{calcnonGaussint1})
and (because of the uniformity of the convergence in
$W$)~(\ref{calcnonGaussint2}). In the last line~(\ref{calcnonGaussint3})
we just make explicit the meaning of (\ref{calcnonGaussint2}). So it is clear
that these two properties are enough to prove the validity
of~(\ref{calcnonGaussint3}), where only Gaussian integrals are present.
But it is a good exercise to find a {\it direct} proof of it in the case at hand,
where everything can be computed exactly. We do this next.
For $r$ odd the integrals vanish, so let's consider the case $r$ even, that is,
$r = 2 t$, for any integer $t$.
On the one hand we have
\begin{equation}\label{left}
\int_{-x_{c,N}}^{x_{c,N}} x^{2t} e^{-x^2} {\rm d} x =
\left( x_{c,N} \right)^{2t+1} \sum_{k=0}^{\infty} {(-1)^k \over k!}
{\left( x_{c,N} \right)^{2k} \over \left( k+t+ 1/2 \right)}
\end{equation}
where the necessary interchange between sum and integral to arrive to the
result is allowed due to the uniform convergence of the Taylor series of
$e^{- x^2}$ in the finite segment~$\left[ -x_{c,N}, x_{c,N} \right]$.
On the other hand,
\begin{eqnarray}\label{right1}
& & \lim_{M \rightarrow \infty} \sum_{n=0}^{M} {1 \over n!}
\left( {M \over x_{c,N}^2} \right)^n \int_{-\infty}^{\infty}
e^{- \left( 1 + M / x_{c,N}^2 \right) x^2} x^{2 \left( n + t \right)} {\rm d} x
\qquad\qquad\qquad \\ \label{right2}
&=& \lim_{M \rightarrow \infty} \sum_{n=0}^{M} {1 \over n!} \Gamma
\left( n+t+ 1/2 \right) \left( {x_{c,N}^2 \over M} \right)^{t + 1/2}
\left(1 + {x_{c,N}^2 \over M} \right)^{- \left( n +t + 1/2 \right)} \\ \label{right3}
&=& \lim_{M \rightarrow \infty} \sum_{n=0}^{M} {1 \over n!}
\left( {x_{c,N}^2 \over M} \right)^{t + 1/2} \sum_{k=0}^{\infty} {(-1)^k \over k!}
\Gamma \left( n+t+k+ 1/2 \right) \left( {x_{c,N}^2 \over M} \right)^{k} \\
&=& \left( x_{c,N} \right)^{2t+1} \sum_{k=0}^{\infty} {(-1)^k \over k!}
{\left( x_{c,N} \right)^{2k} \over \left( k+t+ 1/2 \right)}
\left[ \lim_{M \rightarrow \infty} {(k+t+ 1/2) \over M^{(k+t+ 1/2)}}
\sum_{n=0}^{M} { \Gamma \left( n+t+k+ 1/2 \right) \over n!} \right] \nonumber \\
\label{right4}
\end{eqnarray}
In line~(\ref{right2}) we have used the equation
\begin{equation}\label{innt}
\int_{- \infty}^{\infty} x^{2n} e^{-p x^2} {\rm d} x = {\Gamma (n + 1/2) \over
p^{n+ 1/2}},
\end{equation}
in line (\ref{right3}) we have expanded the last term of (\ref{right2}) in powers
of $x_{c,N}^2 / M$ and carried out some cancellations, and finally in
(\ref{right4}) we have interchanged the $M \rightarrow \infty$ limit with
the infinite sum in $k$.
Comparing (\ref{left}) and (\ref{right4}), we see that the validity of Eq. (\ref{calcnonGaussint3})
depends on the validity of the equation
\begin{equation}\label{identity}
\lim_{M \rightarrow \infty} {(k+t+ 1/2) \over M^{(k+t+ 1/2)}}
\sum_{n=0}^{M} { \Gamma \left( n+k+t+ 1/2 \right) \over n!} = 1\quad
\forall \ {\rm integers} \ k, t > 0
\end{equation}
That this identity holds for every integer $t$ and $k$ can be seen by
considering the following analytic function of the complex variable $z$:
\begin{equation}\label{analfunct}
O (z) \equiv \lim_{M \rightarrow \infty} {(1/z) \over M^{(1/z)}}
\sum_{n=0}^{M} { \Gamma \left( n+1/z \right) \over \Gamma \left( n+1 \right)!}.
\end{equation}
If the identities (\ref{identity}) hold, this function must be identically 1, since for
$1/z_j = j + 1/2 $ with $j$ integer it reduces to them, and for ever increasing $j$,
we obtain a sequence accumulating at $z=0$ on which the function should be 1.
Conversely we will prove that $O(z)$ is indeed identically 1 as an analytic
function of $z$, proving in consequence the identities (\ref{identity}) for arbitrary
$t$ and $k$. Consider the sequence $1/z_j = j + 1$ for $j$ integer. This sequence
also accumulates at $z=0$, and for all its points we have
\begin{eqnarray}\label{proofident}
O \left( 1/ ( j + 1) \right) &=& \lim_{M \rightarrow \infty} {(j+1) \over M^{(j+1)}}
\sum_{n=0}^{M} { \Gamma \left( n+ j+1 \right) \over \Gamma \left( n+1 \right)!}
\\
&=& \lim_{M \rightarrow \infty} {(j+1) \over M^{(j+1)}} \sum_{n=0}^{M}
\Pi_{i=1}^j (i+n) \\
&=& \lim_{M \rightarrow \infty} {(j+1) \over M^{(j+1)}} \left[ \sum_{n=0}^{M}
n^j + {\cal O} (n^{j-1}) \right] \\
&=& \lim_{M \rightarrow \infty} {(j+1) \over M^{(j+1)}}
\left[ { M^{(j+1)} \over (j+1)} + {\cal O} (M^j) \right]
\stackrel{M \rightarrow \infty}{\longrightarrow} 1
\end{eqnarray}
Therefore $O(z) = 1$ for all $z$. This finishes the direct proof of
Eq.~(\ref{calcnonGaussint3}).
As was mentioned before, Eq.~(\ref{xcNasfunctN}), that was derived
independently of any analyticity consideration, and only with the purpose
of satisfying the hypothesis of Lebesgue's theorem, introduces a
non-analyticity in the sequence of integrals of $f_N'$ even for finite $N$.
But even for the case where $x_{c,N}$ is fixed to a constant $a$, discussed
before, in which the limit $N \rightarrow \infty$ is taken first, and then $a$ is sent
to infinity, and therefore the sequence is made out of truly analytic functions,
the convergence towards $z (\lambda)$ is perfectly compatible with analyticity
considerations. The functions
$\pi^{-1/2} \int_{-a}^{a} {\rm d} x e^{ - \left( x^2 + {\lambda \over 4} x^4 \right) }$
(the result of the $N \rightarrow \infty$ limit), are clearly analytic in $\lambda$.
But they converge to (in fact they define!) the nonanalytic function
$z (\lambda)$ when $a \rightarrow \infty$. The limit of an infinite sequence of
analytic functions does not have to be analytic.
Another important issue is that the same method works also for the ``negative
mass case", where the Borel resummation method fails. Indeed, from the discussion
of this section it must be obvious that, with a proper scaling of $x_{c,N}$, the
$f_N'$'s with negative quadratic part of the exponent also converges uniformly
towards $e^{ (x^2 - {\lambda \over 4} x^4) }$ for $x$ in $\left[ - x_{c,N} , x_{c,N}
\right]$. Therefore, the sequence of integrals is also convergent.
| proofpile-arXiv_065-496 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{INTRODUCTION}
The laser interferometers like LIGO\cite{LIGO}, VIRGO\cite{VIRGO},
GEO\cite{GEO} and TAMA\cite{TAMA} are currently constructed
and the detection of gravitational waves is expected
at the end of this century.
One of the most important sources of gravitational waves
for these detectors is
coalescing binary compact stars such as
NS-NS, NS-BH and BH-BH binaries.
Each mass, each spin and the distance of the binary
can be determined by applying the matched filter techniques\cite{CF}
to the gravitational wave form
of the so-called last three minutes of the binary\cite{C}.
In the end of the last three minutes two compact stars coalesce
and nonlinear character of the gravity
and the tidal effects become important,
which will be the most exciting part of the coalescing event.
It is known from the study of orbits of a test particle
in the Schwarzschild metric
that the innermost stable circular orbit (ISCO) exists at $r=6M$.
For binary cases,
Kidder, Will, \& Wiseman\cite{KWW} investigated a point-mass binary
using second post-Newtonian equation of motion,
and found that the ISCO of the comparable mass binary
is at $r_{gr} \simeq 7M_{tot}$
where $r_{gr}$ is the separation of the binary
in the Schwarzschild like radial coordinate.
As for the tidal effects, Chandrasekhar\cite{Ch69} studied
the Roche limit by using Newtonian gravity and
treating binary systems as incompressible homogeneous ellipsoids.
He found that the stars are tidally disrupted before contact
at $r_t=(2.25M_{tot}/\rho)^{1/3}$ for equal mass binary
where $\rho$ is the constant density of ellipsoids.
For the typical neutron star with $M_{tot}=2.8M_\odot$ and
$\rho =1 \times 10^{15}{\rm g/cm^3}$, $r_t$ becomes $5.6M_{tot}$.
Recently Lai, Rasio, \& Shapiro\cite{LRS1}\cite{LRS2} have
investigated the binary consist of finite size compressible stars
using approximate equilibria.
In a series of their papers, they took into account
the effect of the quadrupole order deviation of stars
and found that the hydrodynamic instability occurs
at $r_h=6\sim 7.5 M_{tot}$ for $n=0.5$ polytropic neutron star
with the radius $R_0=2.5M_{tot}$.
In order to know at what radius the final merging phase begins,
all three effects, that is,
the general relativistic, tidal and hydrodynamic effects
should be taken into account simultaneously
since $r_{gr}, r_t$ and $r_h$ have similar values of $\sim 7M_{tot}$.
For this purpose
we must solve the fully general relativistic equations,
which is a difficult 3D problem in numerical relativity\cite{N}
although the partly general relativistic results
have already been presented \cite{WM} \cite{Shibata}.
To know the qualitative feature of the ISCOs
we will use various approximations
to calculate equilibria of the binary in this paper.
The reward is that all the calculations can be done analytically
so that our results will contribute some physical aspects
to the final understanding of the ISCO.
In this paper, we will solve the Roche problem\cite{Ch69}
as a model of the binary
that consists of a finite size star and a point-like gravity source.
A gravitational potential which has unstable circular orbits
will be used as an interaction potential of the binary.
Specifically, we generalize the pseudo-Newtonian potential
proposed by Paczy$\acute{{\rm n}}$sky \& Wiita\cite{PW}
to mimic the general relativistic effects.
We think that this model mimics a neutron star-black hole binary,
and describes its qualitative behavior.
This paper is organized as follows.
In \S2 the basic equations necessary for constructing
equilibrium configurations of the Roche ellipsoids (REs)
and the Roche-Riemann ellipsoids (RREs)\cite{Aizenman}
are derived in the case that the interaction potential is general.
In \S3 circular orbits of the neutron star-black hole binary
are calculated using the generalized pseudo-Newtonian potential
to mimic the effects of general relativity
and the results are shown in \S4.
In \S5 our results are compared
with those using the Newtonian potential
and the second order post-Newtonian equation of motion .
We use the units of $c=G=1$ through this paper.
\section{GENERALIZED ROCHE-RIEMANN ELLIPSOIDS}
We first regard the black hole as a point particle of mass $m_2$
and denote the gravitational potential by the black hole as $V_2(r)$.
To mimic the effects of general relativity by $V_2(r)$,
we do not fix the form of $V_2(r)$ at the moment.
Next we treat the neutron star
as an incompressible, homogeneous ellipsoid
of the semi-axes $a_1, a_2$ and $a_3$,
mass $m_1$ and the density $\rho_1$.
We treat the self-gravity of the neutron star ($V_1$) as Newtonian.
Although for mathematical convenience
we assume the incompressibility for the equation of state,
the effect of the compressibility can be easily taken into account
in an approximate way\cite{LRS1}.
Following Chandrasekhar\cite{Ch69},
we call the neutron star the primary and the black hole the secondary.
\subsection{Second Virial Equations}
We use the tensor virial method\cite{Ch69},
and derive the equations necessary
for constructing equilibrium figures of this system.
Choose a coordinate system such that
the origin is at the center of mass of the primary,
the $x_1$-axis points to the center of mass of the secondary,
and $x_3$-axis coincides with the direction of
the angular velocity of the binary ${\bf \Omega}$.
In the frame of reference rotating with ${\bf \Omega}$,
the Euler equations of the primary are written as
\begin{eqnarray}
\rho_1 \frac{du_{i}}{dt} = -\frac{\partial P}{\partial x_{i}} +
\rho_1 \frac{\partial}{\partial x_{i}} \left[ V_{1} + V_{2} +
\frac{1}{2} {\Omega}^2 \left\{ {\left(
\frac{m_{2} R}{m_{1}+m_{2}}
- x_1 \right)}^2 + x_2^2 \right\} \right] +
2 \rho_1 \Omega \epsilon_{il3} u_{l}, \label{eom}
\end{eqnarray}
where $\rho_1, u_i, P$ and $R$ are the density, the internal velocity,
the pressure and the separation
between the neutron star and the black hole, respectively.
Let us expand the interaction potential $V_2$
in power series of $x_k$ up to the second order.
This approximation is justified
if $R$ is much larger than $a_1, a_2$ and $a_3$.
We assume that the potential $V_2$ is spherically symmetric
so that it depends only on the distance $r$
from the center of mass of the secondary as
\begin{eqnarray}
V_2 = V_2(r),
\end{eqnarray}
where $r$ is given by
\begin{eqnarray}
r=\left\{(R-x_1)^2 +x_2^2 +x_3^2\right\}^{1/2}.
\end{eqnarray}
The expansion of $ V_2(r)$ becomes
\begin{eqnarray}
V_2 = (V_2)_0 - \left( \frac{\partial V_2}{\partial r}
\right)_0 x_1 + \frac{1}{2} \left(
\frac{\partial^2 V_2}{\partial r^2} \right)_0 x_1^2 +
\frac{1}{2R} \left( \frac{\partial V_2}{\partial r}
\right)_0 \left( x_2^2 + x_3^2 \right), \label{V2}
\end{eqnarray}
where the subscript $0$ denotes the derivatives
at the origin of the coordinates.
In the case of the circular orbit, we have
from the force balance at the center
\begin{eqnarray}
\frac{m_2 R}{m_1+m_2} {\Omega}^2 = -{\left( \frac{\partial V_2}
{\partial r} \right)}_0 (1+\delta), \label{omefo}
\end{eqnarray}
where $\delta$ is the quadrupole term
of the interaction potential\cite{LRS1}.
Substituting Eqs.(\ref{V2}) and (\ref{omefo}) into Eq.(\ref{eom}),
we have
\begin{eqnarray}
\rho_1 \frac{du_{i}}{dt} = -\frac{\partial P}{\partial x_{i}}
&+& \rho_1 \frac{\partial}{\partial x_{i}} \left[ V_{1} + \delta
\left( \frac{\partial V_2}{\partial r} \right)_0 x_1+
\frac{1}{2} {\Omega}^2 \left( x_1^2 + x_2^2 \right) +
\frac{1}{2} \left( \frac{{\partial}^2 V_2}
{\partial r^2} \right)_0 x_1^2 \right. \nonumber \\
&+& \left. \frac{1}{2R} \left( \frac{\partial V_2}{\partial r}
\right)_0 \left( x_2^2 + x_3^2 \right) \right]
+ 2 \rho_1 \Omega \epsilon_{il3} u_l. \label{feom}
\end{eqnarray}
Multiplying $x_j$ to Eq.(\ref{feom})
and integrating over the volume of the primary,
we have
\begin{eqnarray}
\frac{d}{dt} \int \rho_1 u_i x_j d^3 {\bf x} &=&
2 T_{ij} + W_{ij} + \left\{ {\Omega}^2 + \left(
\frac{{\partial}^2 V_2}{\partial r^2} \right)_0 \right\}
\delta_{1i} I_{1j} \nonumber \\
& & + \left\{ {\Omega}^2 + \frac{1}{R}
\left( \frac{{\partial} V_2}{\partial r} \right)_0 \right\}
\delta_{2i} I_{2j}
+ \frac{1}{R} \left( \frac{{\partial} V_2}
{\partial r} \right)_0 \delta_{3i} I_{3j} \nonumber \\
& & + 2 \Omega \epsilon_{il3} \int \rho_1 u_l x_j
d^3 {\bf x} + \delta_{ij} \Pi, \label{sve}
\end{eqnarray}
where
\begin{eqnarray}
T_{ij} &\equiv& \frac{1}{2} \int \rho_1 u_{i} u_{j} d^3 {\bf x} :
{\rm Kinetic~Energy~Tensor}, \\
W_{ij} &\equiv& \int \rho_1 \frac{\partial V_1}{\partial x_i}
x_{j} d^3 {\bf x} : {\rm Potential~Energy~Tensor}, \\
I_{ij} &\equiv& \int \rho_1 x_{i} x_{j} d^3 {\bf x} ~~~:
{\rm Moment~of~Inertia~Tensor},
\end{eqnarray}
and
\begin{eqnarray}
\Pi &\equiv& \int P d^3 {\bf x}.
\end{eqnarray}
In Eq.(\ref{sve}) there is no terms related to $\delta$.
Since it is possible to take the coordinate system comoving
with the center of mass of the binary system,
the term proportional to $\delta$ in Eq.(\ref{feom}) vanishes
when we integrate over the volume of the primary.
Eq.(\ref{sve}) is the basic equation
to construct the equilibrium figures of the Roche ellipsoids (REs)
and the Roche-Riemann ellipsoids (RREs)
for the general potential $V_2(r)$.
\subsection{Equilibrium Roche-Riemann Sequence}
In this subsection,
we show how to construct the equilibrium figures of the RREs.
The Roche-Riemann ellipsoid is the equilibrium
in which the shape of the primary does not change in the rotating frame
although the uniform vorticity exists inside the primary.
We restrict the problem to the simplest case
where the uniform vorticity of the primary
is parallel to the rotation axis, i.e.
the primary is the Riemann S-type ellipsoid\cite{Ch69}.
We set the coordinate axes
to coincide with the principal axes of the primary.
For the uniform vorticity $\zeta$,
the internal velocity $u_i$ in the rotating frame is given by
\begin{eqnarray}
u_1 &=& Q_1 x_2, \\
u_2 &=& Q_2 x_1,
\end{eqnarray}
and
\begin{eqnarray}
u_3 &=& 0,
\end{eqnarray}
where
\begin{eqnarray}
Q_1 &=& - \frac{a_1^2}{a_1^2 + a_2^2} \zeta
\end{eqnarray}
and
\begin{eqnarray}
Q_2 &=& \frac{a_2^2}{a_1^2 + a_2^2} \zeta.
\end{eqnarray}
For the stationary equilibrium, Eq.(\ref{sve}) is rewritten as
\begin{eqnarray}
Q_{ik}Q_{jl}I_{kl} + W_{ij} &+& \left\{ {\Omega}^2 + \left(
\frac{{\partial}^2 V_2}{\partial r^2} \right)_0
\right\} \delta_{1i} I_{1j} + \left\{ {\Omega}^2
+ \frac{1}{R} \left( \frac{{\partial} V_2}{\partial r}
\right)_0 \right\} \delta_{2i} I_{2j} \nonumber \\
&+& \frac{1}{R} \left( \frac{{\partial} V_2}{\partial r}
\right)_0 \delta_{3i} I_{3j} + 2 \Omega \epsilon_{il3}
Q_{lk} I_{kj} = - \delta_{ij} \Pi, \label{stasve}
\end{eqnarray}
where $Q_{ij}$ is not zero only for
\begin{eqnarray}
Q_{12} &=& Q_1, \\
Q_{21} &=& Q_2.
\end{eqnarray}
Eq.(\ref{stasve}) has only diagonal components as
\begin{eqnarray}
Q_1^2 I_{22} + W_{11} + \left\{ {\Omega}^2 +
\left( \frac{{\partial}^2 V_2}
{\partial r^2} \right)_0 \right\} I_{11} +
2 \Omega Q_2 I_{11} &=& - \Pi, \label{xsve} \\
Q_2^2 I_{11} + W_{22} + \left\{ {\Omega}^2 + \frac{1}{R}
\left( \frac{{\partial} V_2}
{\partial r} \right)_0 \right\} I_{22} -
2 \Omega Q_1 I_{22} &=& - \Pi, \label{ysve}
\end{eqnarray}
and
\begin{eqnarray}
W_{33} + \frac{1}{R} \left( \frac{{\partial} V_2}
{\partial r} \right)_0 I_{33} &=& - \Pi. \label{zsve}
\end{eqnarray}
We assume for simplicity that the gravitational potential
of the primary is Newtonian.
In this case, the potential energy tensor
and the moment of inertia tensor
of the incompressible, homogeneous ellipsoids are calculated as
\begin{eqnarray}
W_{ij} &=& -2 \pi \rho_1 A_{i} I_{ij},
\end{eqnarray}
and
\begin{eqnarray}
I_{ij} &=& \frac{1}{5} m_1 a_{i}^2 \delta_{ij},
\end{eqnarray}
where
\begin{eqnarray}
A_{i} &=& a_{1} a_{2} a_{3} \int_0^{\infty} \frac{du}
{\Delta \left(a_{i}^2 +u \right)} ,
\end{eqnarray}
and
\begin{eqnarray}
\Delta^2 &=& \left( a_{1}^2 +u \right)
\left( a_{2}^2 +u \right) \left( a_{3}^2 +u \right).
\end{eqnarray}
Eliminating $\Pi$ from Eqs.(\ref{xsve})-(\ref{zsve}),
we have
\begin{eqnarray}
\left[ \left\{ 1+ 2\frac{a_2^2}{a_1^2+a_2^2} f_R +
\left( \frac{a_1a_2}{a_1^2+a_2^2} f_R \right)^2 \right\}
\Omega^2 + \left( \frac{\partial^2 V_2}{\partial r^2}
\right)_0 \right] a_{1}^2 &-& \frac{1}{R} \left(
\frac{{\partial} V_2}{\partial r} \right)_0 a_{3}^2 \nonumber \\
&=& 2 \pi \rho_1
\left( a_{1}^2 - a_{3}^2 \right) B_{13} \label{sve1}
\end{eqnarray}
and
\begin{eqnarray}
\left[ \left\{ 1+ 2\frac{a_1^2}{a_1^2+a_2^2} f_R +
\left( \frac{a_1a_2}{a_1^2+a_2^2} f_R \right)^2 \right\}
\Omega^2 + \frac{1}{R} \left( \frac{\partial V_2}{\partial r} \right)_0
\right] a_{2}^2 &-& \frac{1}{R} \left( \frac{\partial V_2}
{\partial r} \right)_0 a_{3}^2 \nonumber \\
&=& 2 \pi \rho_1 \left( a_{2}^2 - a_{3}^2 \right) B_{23},
\label{sve2}
\end{eqnarray}
where
\begin{eqnarray}
f_{R} \equiv \frac{\zeta}{\Omega}
\end{eqnarray}
and the following relations are used;
\begin{eqnarray}
a_i^2 A_i -a_j^2 A_j &=&
\left( a_i^2-a_j^2 \right) B_{ij},
\end{eqnarray}
where
\begin{eqnarray}
B_{ij} &=&a_1 a_2 a_3 \int_0^{\infty} \frac{u du}
{\Delta (a_i^2 +u)(a_j^2 +u)}.
\end{eqnarray}
Now from Eq.(2.5) $\Omega$ is given by
\begin{eqnarray}
\Omega^2 = -\frac{1+p}{R} \left( \frac{\partial V_2}{\partial r}
\right)_0 (1+\delta)
~~~~~~~~~~\left( p \equiv \frac{m_1}{m_2} \right).
\label{omega}
\end{eqnarray}
Dividing Eq.(\ref{sve1}) by Eq.(\ref{sve2}),
we have the equation to determine the Roche-Riemann sequences as
\begin{eqnarray}
\frac{\left[ (1+p)(1+\delta)
\left\{ 1+ 2\frac{a_2^2}{a_1^2+a_2^2} f_R
+\left( \frac{a_1 a_2}{a_1^2+a_2^2} f_R \right)^2 \right\}
-R \left( \frac{\partial^2 V_2}{\partial r^2} \right)_0/
\left( \frac{\partial V_2}{\partial r} \right)_0 \right]a_1^2 +a_3^2}
{\left[ (1+p)(1+\delta)
\left\{ 1+2 \frac{a_1^2}{a_1^2+a_2^2} f_R
+\left( \frac{a_1a_2}{a_1^2+a_2^2} f_R \right)^2 \right\}
-1 \right] a_2^2 + a_3^2}
= \frac{\left( a_1^2 - a_3^2 \right) B_{13}}
{\left( a_2^2 - a_3^2 \right) B_{23}}. \label{gaxis}
\end{eqnarray}
Using Eqs.(\ref{sve2}) and (\ref{omega}),
we can determine the orbital angular velocity $\Omega$ by
\begin{eqnarray}
\frac{\Omega^2}{\pi \rho_1}=\frac{2(1+p)(1+\delta)\left(a_2^2
-a_3^2 \right) B_{23}}{ \left[ (1+p)(1+\delta)\left\{ 1+
2 \frac{a_1^2}{a_1^2+a_2^2} f_R + \left(
\frac{a_1a_2}{a_1^2+a_2^2} f_R \right)^2 \right\}-1
\right] a_2^2 + a_3^2}. \label{gOme}
\end{eqnarray}
Note that $f_R$ is related to the circulation ${\cal C}$ as
\begin{eqnarray}
{\cal C} &=& \oint {\bf u}_{inertial} \cdot d{\bf l}=
\pi a_1 a_2 (2 + f_R) \Omega,
\end{eqnarray}
where
\begin{eqnarray}
{\bf u}_{inertial} &=& \left\{
\begin{array}{@{\,}ll}
(u_{inertial})_1 = (Q_1 -\Omega) x_2, \\
(u_{inertial})_2 = (Q_2 + \Omega) x_1-\frac{R}{1+p}
\Omega~ \\
(u_{inertial})_3 = 0.
\end{array}
\right.
\end{eqnarray}
If there is no viscosity inside the primary,
the circulation should be conserved from Kelvin's circulation theorem.
\subsection{Total Angular Momentum}
The total energy and the total angular momentum of the binary
are the decreasing functions of time
since the gravitational waves are emitted.
If the total angular momentum has its minimum at some separation
of the binary, we regard this point as the ISCO
\footnote{Lai, Rasio, \& Shapiro show in appendix D of \cite{LRS1}
that the true minimum point of the total energy coincides with
that of the total angular momentum.
Strictly speaking,
if the rotation includes only to the quadrupole order,
this coincidence fails.
However the difference is as small as the numerical accuracy
\cite{LRS1}.}.
The total angular momentum of our system
which is the sum of the orbital and the spin angular momentum
is given by
\begin{eqnarray}
J_{tot} &=& m_1 r_{cm}^2 \Omega + m_2 (R-r_{cm})^2 \Omega +
I \Omega + \frac{2}{5}m_1 \frac{a_1^2 a_2^2}
{a_1^2 + a_2^2} \zeta \nonumber \\
&=& \frac{m_1 m_2}{m_1 + m_2} R^2 \Omega
\left\{ 1 + \frac{1}{5} (1+p) \frac{1}{R^2}
\left( a_1^2 + a_2^2 + 2\frac{a_1^2 a_2^2}{a_1^2+a_2^2}
f_R \right) \right\} \label{angmom}
\end{eqnarray}
where
\begin{eqnarray}
r_{cm} = \frac{m_{2} R}{m_1+m_2}.
\end{eqnarray}
The first term in the braces
of the right hand side of Eq.(\ref{angmom}) comes
from the orbital angular momentum of the binary system
and the second does from the spin angular momentum of the primary.
\section{GENERALIZED PSEUDO-NEWTONIAN POTENTIAL}
There are variety of choices of $V_2(r)$
to mimic the general relativistic effects of the gravitation.
We generalize the so-called pseudo-Newtonian potential
proposed by Paczy$\acute{{\rm n}}$sky \& Wiita\cite{PW} originally.
This potential fits the effective potential
of the Schwarzschild black hole quite well as we will show later.
We will use the generalized pseudo-Newtonian potential defined by
\begin{eqnarray}
V_2(r) &=& \frac{m_2}{r-r_{pseudo}} \label{pnpot},\\
r_{pseudo} &=& r_s \left\{ 1+ g(p) \right\}, \\
g(p) &=& \frac{7.49p}{6(1+p)^2} - \frac{10.4 p^2}{3(1+p)^4}
+ \frac{29.3 p^3}{6(1+p)^6}, \label{gp} \\
r_s &\equiv& \frac{2GM_{tot}}{c^2}, \\
M_{tot} &=& m_1 + m_2,
\end{eqnarray}
where $p=m_1/m_2$ and $g(p)$ is the special term
to fit the ISCOs of the hybrid second post-Newtonian calculations
by Kidder, Will, \& Wiseman\cite{KWW}.
For $p=0$, the generalized pseudo-Newtonian potential
agrees with the pseudo-Newtonian potential
proposed by Paczy$\acute{{\rm n}}$sky \& Wiita\cite{PW}.
Fig.1(a) shows effective potentials (solid lines) and
locations of circular orbits (dots)
in our generalized pseudo-Newtonian potential
( $p=0$ \& $r_{pseudo}=r_s$) and in the Schwarzschild metric.
Although by this choice of the parameter ($r_{pseudo}=r_s$),
the locations of the ISCOs in the generalized pseudo-Newtonian potential
agree with those in the Schwarzschild metric,
the angular momenta at the ISCO are different,
that is, the angular momentum
in the generalized pseudo-Newtonian potential ($J_{pseudo}$)
for $p=0$ is $(9/8)^{1/2}$ times larger than
that in the Schwarzschild metric ($J_{Sch}$) at the ISCO.
Therefore in Fig.1(a) and (b) we compare circular orbits with
different angular momentum related as
\begin{eqnarray}
J_{pseudo} = \left( \frac{9}{8} \right)^{1/2} J_{Sch}.
\end{eqnarray}
From Fig.1(b) we see that the radii of the circular orbits of
the generalized pseudo-Newtonian potential agrees with
those of the effective potential
around Schwarzschild black hole within 10\% accuracy near the ISCO.
This is the reason why we believe that
our generalized pseudo-Newtonian potential
expresses the effect of general relativity within 10\% or so.
Using Eq.(\ref{pnpot}) and Eq.(\ref{omega}),
we can rewrite Eq.(\ref{gOme}) as
\begin{eqnarray}
\frac{p^2 r_s^3 (\bar{a}/m_1)^3}{12(1+p)^3 R(R-r_{pseudo})^2}
- \frac{(a_2^2-a_3^2) B_{23}}
{ \left[ (1+p)(1+\delta)
\left\{ 1+ 2 \frac{a_1^2}{a_1^2 + a_2^2} f_R +
\left( \frac{a_1a_2}{a_1^2 + a_2^2} f_R \right)^2 \right\}
-1 \right] a_2^2 + a_3^2 } = 0, \label{sequence}
\end{eqnarray}
where $\bar{a}$ is the mean radius of the primary.
In the generalized pseudo-Newtonian case,
the quadrupole term $\delta$ is written as
\begin{eqnarray}
\delta=\frac{3}{10} \left\{2a_1^2-
\frac{(3R-r_{pseudo})(R-r_{pseudo})}{3R^2}
\left(a_2^2+a_3^2 \right) \right\}
\frac{1}{(R-r_{pseudo})^2}. \label{quadru}
\end{eqnarray}
We also have the separation of the binary as
\begin{eqnarray}
R=\frac{E}{E-2} r_{pseudo}, \label{sob}
\end{eqnarray}
where
\begin{eqnarray}
E \equiv &-&(1+p)(1+\delta)
\left\{ 1+ 2\frac{a_2^2}{a_1^2+a_2^2} f_R +
\left( \frac{a_1 a_2}{a_1^2 + a_2^2} f_R \right)^2 \right\}
- \frac{a_3^2}{a_1^2} \nonumber \\
&+& \left\{ \left[ (1+p)(1+\delta) \left\{ 1 +
2\frac{a_1^2}{a_1^2+a_2^2} f_R
+ \left( \frac{a_1a_2}{a_1^2+a_2^2} f_R \right)^2 \right\}
-1 \right] \frac{a_2^2}{a_1^2} +
\frac{a_3^2}{a_1^2} \right\}
\frac{(a_1^2-a_3^2) B_{13}}{(a_2^2-a_3^2) B_{23}}.
\end{eqnarray}
For the given mass ratio $p$, mean radius $/bar{a}/m_1$,
circulation parameter $f_{R}$ and axial ratio $a_3/a_1$,
we can determine
the axial ratio $a_2/a_1$ by solving Eq.(\ref{sequence})
with Eqs.(\ref{quadru}) and (\ref{sob}).
Using the axial ratios ($a_2/a_1$, $a_3/a_1$),
we are able to calculate the orbital angular velocity by Eq.(\ref{gOme})
and the separation of the binary by Eq.(\ref{sob}).
The total angular momentum is calculated by Eq.(\ref{angmom}).
Finding the minimum of the total angular momentum,
we can determine the location of the ISCO.
When the viscosity inside the primary is so effective
that no internal motion exists,
$f_R=0$ in the above equation, i.e. the Roche ellipsoids (REs).
While if the primary is in-viscid and ${\cal C}=0$,
we have $f_R=-2$, i.e.
the irrotational Roche-Riemann ellipsoids (IRREs).
Note that if we substitute the Newtonian potential
as an interaction potential,
Eqs.(\ref{gaxis}) and (\ref{gOme}) agree with
the equations derived by Chandrasekhar\cite{Ch69}
in the REs ($f_R=0$) case and those by Aizenman\cite{Aizenman}
in the RREs case.
\section{RESULTS}
Kochanek\cite{Kocha} and Bildsten \& Cutler\cite{BC} showed
that the internal structure of a coalescing binary neutron star is
the irrotational Roche-Riemann ellipsoids (IRREs).
However we calculate both the REs and the IRREs for comparison.
We show the results in four cases;
1) the REs with $p=1$ or 0.1,
2) the IRREs with $p=1$ or 0.1.
Results are given in Table.I to Table.VI,
where $\tilde{\Omega}=\Omega/\sqrt{\pi \rho_1}$ represents
the normalized orbital angular velocity,
and $\tilde{J}$ denotes the normalized total angular momentum
defined by
\begin{eqnarray}
\tilde{J}=\frac{J_{tot}}{m_1 m_2 (r_s/M_{tot})^{1/2}}.
\end{eqnarray}
$\bar{a}$ is the mean radius of the primary defined by
\begin{eqnarray}
\bar{a} = \left( \frac{m_1}{\frac{4}{3} \pi \rho_1}
\right)^{1/3}.
\end{eqnarray}
In Tables I to VI, $\dagger$ means the point of the ISCO
defined in this paper,
and $\ddagger$ does the point of the Roche limit where
the Roche limit is defined by the distance of closest approach
for equilibrium to be possible\cite{Ch69}.
The values in the parentheses show the power of $10$.
\subsection{$p=1$ Case}
Fig.2(a) shows $\tilde{J}$
as a function of the normalized separation $R/r_s$.
In Fig.2(a), thin solid, dotted and dashed lines are
the REs with $\bar{a}/m_1$ being $3$, $5$ and $8$, respectively,
while thick solid, dotted and dashed lines are
the IRREs with $\bar{a}/m_1$ being $3$, $5$ and $8$, respectively.
We defined in \S\S II.C that the location of the ISCO is
the minimum point of $\tilde{J}$.
From Fig.2(a), we see that the separations of the binary at the ISCO
in the IRREs case are almost the same as those in the REs case.
Fig.2(b) shows the axial ratios $a_2/a_1$ and $a_3/a_1$
as a function of $R/r_s$.
The conventions of lines are the same as those in Fig.2(a).
The relation among the length of the axes is $a_1>a_2>a_3$ in the REs,
while $a_1>a_3>a_2$ in the IRREs.
In the REs, the tidal force makes the $a_1$ axis long
while it does $a_2$ and $a_3$ axes short.
The centrifugal force makes $a_1$ and $a_2$ axes long.
As a result we have $a_1>a_2>a_3$.
In the IRREs, in addition to the above effects,
the Coriolis force caused
by the internal motion of the primary exists.
This makes $a_1$ and $a_2$ axes short, which yields $a_1>a_3>a_2$ .
From Fig.2(b), it is found that in the IRREs the star
with $\bar{a}/m_1=3$ reaches the ISCO
at the point of $a_2/a_1 \simeq 0.912$ and $a_3/a_1 \simeq 0.918$.
On the other hand, the star with $\bar{a}/m_1=8$ terminates
when $a_2/a_1 \simeq 0.760$ and $a_3/a_1 \simeq 0.792$.
This means that
the primary with the smaller mean radius reaches the ISCO
before the shape of the primary deviates from the sphere considerably.
This tendency is the same for the REs.
\subsection{ $p=0.1$ Case}
Fig.3(a) shows $\tilde{J}$ as a function of $R/r_s$.
The conventions are the same as those in Fig.2(a)
except $\bar{a}/m_1$ being $3$, $5$ and $8$.
From Fig.3(a), we see that all lines with
the mean radii in the range of $3 \le \bar{a}/m_1 \le 8$
take their minimum at the points near $R/r_s \sim 3.25$.
This value is almost the same as the ISCO
by Kidder, Will, \& Wiseman\cite{KWW} for $p=0.1$.
This means that when $p$ is much less than $1$ and
the mean radius of the primary is less than $8m_1$ which corresponds
to 17km for $m_1=1.4M_\odot$,
the size of the primary has little effect on the ISCO.
Fig.3(b) shows the axial ratios $a_2/a_1$ and $a_3/a_1$
as a function of $R/r_s$.
The conventions are the same as those in Fig.3(a).
We see that the stars with smaller mean radii reach
the ISCO even when the deviation
from the spherical symmetry is very small.
Since the binary system enters an unstable circular orbit
before the primary is tidally deformed,
the tidal effects are not important.
We see that even $\bar{a}/m_1\sim 8$
the minimum values $a_2/a_1$ and $a_3/a_1$ are not small ($\sim 0.85$).
\section{DISCUSSIONS}
In this section, we discuss the differences between
the case of the generalized pseudo-Newtonian potential and
that of the Newtonian potential,
and between the REs and the IRREs.
We will also compare our results with other papers.
\subsection{The Case of $p=1$}
In Fig.4, the separation $R_{ISCO}/r_s$ is shown
as a function of $\bar{a}/m_1$.
Thick lines and thin lines represent
the case of the generalized pseudo-Newtonian potential and
that of the Newtonian potential, respectively.
Solid lines and dotted lines express the case of
the IRREs and the REs, respectively.
We see that for the Newtonian potential,
$R_{ISCO}/r_s$ increases in proportion to $\bar{a}/m_1$
regardless of types of ellipsoids,
while for the generalized pseudo-Newtonian potential,
the behavior of $R_{ISCO}/r_s$ changes around
$\bar{a}/m_1 \simeq 3.5$.
For $\bar{a}/m_1 \mathrel{\mathpalette\Oversim>} 3.5$,
$R_{ISCO}/r_s$ increases in proportion to $\bar{a}/m_1$ as
in the case of the Newtonian potential.
In this region, the tidal effect dominates the system and
the effects of the general relativity become less important.
In conclusion, for $\bar{a}/m_1 \mathrel{\mathpalette\Oversim>} 3.5$,
the tidal effect dominates the stability of the binary system and
for $\bar{a}/m_1 \mathrel{\mathpalette\Oversim<} 3.5$, the binary system is dominated
by the relativistic effect, i.e.
the fact that the interaction potential has an unstable orbit.
From Fig.4 it is also found that the location of the ISCOs
in the IRREs case is not so different from that in the REs case
for the same $\bar{a}/m_1$.
The orbital frequency of the IRREs at the ISCO
for $m_1=1.4M_{\odot}$ and $\bar{a}/m_1 = 5$ is estimated from
$\tilde{\Omega}$ in Table.IV as
\begin{eqnarray}
\Omega_{\rm ISCO} &=& 495~[{\rm Hz}].
\end{eqnarray}
This value is smaller than that in the Newtonian potential case
($\Omega_{\rm ISCO}^{Newton} = 599~[{\rm Hz}]$).
\subsection{The Case of $p=0.1$}
If we consider the primary
as the neutron star of mass $1.4M_{\odot}$,
then, the secondary is regarded
as the black hole of mass $14M_{\odot}$.
$R_{ISCO}/r_s$ is shown
as a function of $\bar{a}/m_1$ in Fig.5.
For the Newtonian potential,
$R_{ISCO}/r_s$ increases
in proportion to $\bar{a}/m_1$ like the case of $p=1$.
For the generalized pseudo-Newtonian potential,
in the range of $\bar{a}/m_1$ of Fig.5,
$R_{ISCO}/r_s$ converges to the value $3.25$ obtained
by Kidder, Will, \& Wiseman\cite{KWW}.
This is because the existence of the unstable orbit
in the generalized pseudo-Newtonian potential
influences $R_{ISCO}/r_s$,
and this effect dominates
when the radius of the primary is small.
This is clearer for the small mass ratio $p$.
Therefore if $p$ is small, for the range of the radius
($3 \le \bar{a}/m_1 \le 8$)
relevant to the neutron star,
the effect of the neutron star's size is very small.
This comes essentially from the fact
that the Newtonian estimate of the Roche radius is
smaller than the radius of the ISCO.
One can estimate the orbital frequency of the IRREs
at the ISCO for $m_1=1.4M_{\odot}$
and $\bar{a}/m_1 = 5$ from $\tilde{\Omega}$ in Table.V as
\begin{eqnarray}
\Omega_{\rm ISCO} &=& 187~[{\rm Hz}].
\end{eqnarray}
\subsection{Comparison with Other Works}
Kidder, Will, \& Wiseman\cite{KWW} studied
the motion of the point-particle binary systems using
{\it hybrid Schwarzschild second post-Newtonian equation of motion},
and obtained the separation at the ISCO ($r_{ISCO}$) expressed by
\begin{eqnarray}
\frac{r_{ISCO}}{M_{tot}} \simeq 6 + 7.49 \eta - 20.8 \eta^2
+29.3 \eta^3 \label{KWWISCO}
\end{eqnarray}
where
\begin{eqnarray}
\eta &=& \frac{m_1 m_2}{M_{tot}^2}.
\end{eqnarray}
Table.VII shows the comparison of Eq.(\ref{KWWISCO})
with our results for $\bar{a}/m_1 = 5$.
It is found that for $p=0.1$, the finite size effect is not important
because the primary is much lighter than the secondary,
so that the results are almost the same as \cite{KWW}.
On the other hand, when $p=1$ the finite size of the primary
increases the separation at the ISCO
by the general relativistic and tidal effects.
Lai, Rasio, \& Shapiro\cite{LRS3},
discussed the relativistic effects on the binary system
for compressible Darwin ellipsoids
using a simple approximate model\cite{LRS4}.
The Darwin ellipsoids are the equilibrium configurations
constructed by two identical synchronized finite-size stars
including the mutual tidal interactions.
In their approach the effects of general relativity
and the Newtonian tidal interactions
for finite-size compressible stars are combined by hand.
While we formulated the problem
using arbitrary interaction potentials of the secondary
for the incompressible primary.
We adopted the semi-relativistic potential called
the generalized pseudo-Newtonian potential
to mimic the general relativistic effects of gravitation.
We solved the equilibria of the REs and the IRREs in this potential.
Their results and ours are compared in Table.VIII,
where $r_m$ expresses the minimum separation obtained by
Lai, Rasio, \& Shapiro\cite{LRS3}.
From this table, we see that both results agree
rather well inspite of different approaches and approximations.
\acknowledgments
KT would like to thank H. Sato, K. Nakao and M. Shibata
for useful discussions and continuous encouragement.
This work was in part supported by a Grant-in-Aid for Basic Research
of Ministry of Education, Culture, Science and Sports (08NP0801).
| proofpile-arXiv_065-497 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
{\em Ab-initio} Hartree-Fock (HF) and configuration interaction (CI) methods are standard tools in computational
chemistry nowadays and various program packages are available for accurate
calculations of properties of atoms and molecules. For solids,
HF calculations have become possible, on a broad scale, with the advent of the program
package CRYSTAL \cite{CRYSTAL}. However, the problem of an accurate treatment of electron
correlation is not fully settled
(for a survey see \cite{FuldeBuch}).
Although the absolute value of the HF energy is usually much larger than
the correlation energy, the correlation energy is very important for
energy differences. For example, the O$^-$ ion is not stable at the HF level,
and correlations are necessary in order to obtain even qualitative agreement with the
experimental result for the electron affinity of oxygen. In solid state physics, NiO is a well known example
of a system which is insulating due to correlations.
The most widely used method to include correlations
in solids is density functional theory (DFT) \cite{DFT}.
DFT has also recently become quite popular for a computationally efficient
treatment of exchange and correlation in molecules.
However, a systematic improvement towards the exact results is currently not possible with DFT.
Wave-function-based methods are more suitable for this purpose.
In the last years,
Quantum Monte-Carlo
calculations have been performed for several
systems \cite{QMC}.
Correlations are included here by multiplying the HF
wavefunction with a Jastrow factor.
An approach more closely related
to quantum chemistry is the Local Ansatz \cite{Stollhoff,FuldeBuch}
where judiciously chosen local excitation operators are applied to
HF wavefunctions from CRYSTAL calculations.
Some years ago, an incremental scheme has been
proposed and applied in calculations for semiconductors
\cite{StollDiamant,Beate}, for graphite \cite{StollGraphit} and for
the valence band of diamond \cite{Juergen}; here information on the effect of local excitations on solid-state properties is drawn
from calculations using
standard quantum chemical program packages. In a recent paper \cite{DDFS}
we showed that this method can be successfully extended to ionic solids;
we reported results for the correlation contribution to the cohesive energy of MgO.
In the present article,
we apply the scheme to the
cohesive energy of CaO, as a second example. In addition, we show
how correlations affect the lattice constants of
MgO and CaO.
For these systems, several calculations have been performed at the HF level
with
the CRYSTAL code \cite{CausaMgO1,DRFASH,MackrodtCaO,CausaZupan,McCarthy,Catti}
as well as with inclusion of correlations using DFT
\cite{DRFASH,CausaZupan,McCarthy,Pyper}.
\section{The Method}
The method of increments can be used to build up correlation effects in solids
from local correlation contributions which in term may be obtained by
transferring results from suitably embedded finite clusters to the infinite crystal.
It has been fully described in \cite{StollDiamant,Beate,StollGraphit,DDFS}, and a formal derivation has been given
within the framework of the projection technique \cite{TomSchork}. Thus,
we will only briefly repeat the main
ideas.\\
(a) Starting from self-consistent field (SCF)
calculations localized orbitals are
generated which are assumed to be similar in the clusters and in the solid. \\
(b) One-body correlation-energy increments are calculated: in our specific case
these are the correlation energies $\epsilon (A)$,
$\epsilon (B)$, $\epsilon (C)$, ... of localized orbital groups which can be attributed to X$^{2+}$ (X = Mg, Ca) or O$^{2-}$ ions
at ionic positions $A$, $B$, $C$, ... Each localized orbital group is correlated separately. \\
(c) Two-body increments are defined as non-additivity corrections:
\begin{eqnarray*}
\Delta \epsilon {(AB)}=\epsilon (AB) - \epsilon (A) -\epsilon (B),
\end{eqnarray*}
where $\epsilon {(AB)}$ is the correlation energy of the joint
orbital system of $AB$.\\
(d) Three-body increments are defined as
\begin{eqnarray*}
\Delta\epsilon(ABC) = & \epsilon(ABC) -
[\epsilon(A) + \epsilon(B) + \epsilon(C)] - \nonumber \\
& [\Delta\epsilon(AB) + \Delta\epsilon(AC) + \Delta\epsilon(BC)].
\end{eqnarray*}
Similar definitions apply to higher-body increments. \\
(e) The correlation
energy of the solid can now be expressed as the sum of all possible increments:
\begin{equation}
\epsilon_{\rm bulk} = \sum_A \epsilon(A) + \frac{1}{2} \sum_{A,B} \Delta\epsilon(AB) + \frac{1}{3!} \sum_{A,B,C}
\Delta\epsilon(ABC) + ...
\end{equation}
Of course, this
only makes sense if the incremental expansion is well
convergent, i.e. if $\Delta \epsilon (AB)$ rapidly decreases with
increasing distance of the ions at position $A$ and $B$ and if the
three-body terms are significantly smaller than the two-body ones.
A pre-requisite is that the correlation method used for evaluating the increments must be size-extensive:
otherwise the two-body increment $\Delta \epsilon (AB)$
for two ions $A$ and $B$ at infinite distance would not vanish.
In our present work, we used three different size-extensive approaches, cf.
Sect. 2.1.
Finally, the increments must be transferable, i.e.\ they should only weakly depend
on the cluster chosen.
\subsection{Correlation Methods}
In this section we want to give a brief description of the correlation
methods used.
In the averaged coupled-pair functional (ACPF \cite{ACPF})
scheme, the correlation energy is expressed in the form
\begin{equation}
E_{\rm corr}[\Psi_c]=\frac{< \Psi_{SCF}+\Psi_c|H-E_{SCF} |\Psi_{SCF}+\Psi_c >}
{1+g_c<\Psi_c|\Psi_c>}
\end{equation}
with $\Psi_{SCF}$ being the SCF-wavefunction
(usually of the spin-restricted Hartree-Fock type)
and $\Psi_c$ the correlation part of the
wavefunction,
\begin{eqnarray}
|\Psi_c>= \sum_{a \atop r}c_{a}^{r}a^+_r a_a|\Psi_{SCF}> +
\sum_{a<b \atop r<s}c_{ab}^{rs}a^+_r a^+_s a_a a_b|\Psi_{SCF}>;
\end{eqnarray}
$g_c$ is chosen as $\frac{2}{n}$ in order to make the expression (2)
approximately size-consistent
($n$ being the number of correlated electrons).
For more details (and the extension to multi-reference cases), see \cite{ACPF}.\\
In the coupled-cluster singles and doubles (CCSD \cite{CCSD}) scheme,
the wavefunction is expressed with the help of an
exponential ansatz:
\begin{eqnarray}
|\Psi_{CCSD}>=\mbox{exp}({\sum_{a \atop r}c_a^r a^+_r a_a +
\sum_{a<b \atop r<s}c_{ab}^{rs}a^+_r a^+_s a_a a_b})|\Psi_{SCF}>.
\end{eqnarray}
$a^+$ ($a$) are creation (annihilation) operators of electrons in
orbitals which are occupied ($a$, $b$) or unoccupied ($r,$ $s$) in
the SCF wavefunction.\\
Finally, in the CCSD(T) scheme, three particle excitations are included
by means of perturbation theory as proposed in \cite{Raghavachari}.\\
We used these three methods to compare their quality
in applications to solids. It turns out that
ACPF and CCSD give very similar results, while CCSD(T) yields slighty improved energies
\cite{CCSDTWerner}.
Altogether, the results are not strongly dependent on the methods and no
problem arises, therefore, if only one method should be applicable (as is the case
for low-spin open-shell systems, where CCSD and CCSD(T) are not yet
readily available).
All calculations of this work were done by using the program package MOLPRO
\cite{KnowlesWerner,MOLPROpapers}.
\section{Cohesive Energy of CaO}
\subsection{Basis sets and test calculations}
For calculating the correlation
contribution to the cohesive energy of CaO, we closely follow the approach of Ref.\ \cite{DDFS}. For oxygen we choose
a $[5s4p3d2f]$ basis set \cite{Dunning}.
Calcium is described by a small-core pseudopotential replacing the
$1s$, $2s$ and $2p$ electrons \cite{KauppCa},
and a corresponding $[6s6p5d2f1g]$ valence basis set
(from ref. \cite{KauppCa}, augmented with polarization functions $f_1$=0.863492, $f_2$=2.142 and
$g$=1.66) is used.
All orbitals are correlated, with the exception of the O $1s$ core.
In particular, the correlation contribution of the outer-core Ca $3s$ and $3p$ orbitals
is explicitly taken into account.
We did not use a large-core (X$^{2+}$) pseudopotential and a core
polarization potential (CPP) for treating core-valence correlation as was done in the
case of
MgO \cite{DDFS}, because Ca is close to the transition metals and
excitations into
$d$-orbitals are important. The influence of the latter on the X$^{2+}$ core cannot be well represented
by a CPP since the $3d$ orbitals are core-like themselves (cf.\ the discussion in
\cite{MuellerFleschMeyer}). Correlating the Ca outer-core orbitals explicitly, using the small-core (Ca$^{10+}$) pseudopotential,
we circumvent this problem.
Using this approach, we performed test calculations for the
first and second ionization potential of the Ca atom (Table 1) and calculated
spectroscopic properties of the CaO molecule (Table 2).
In both cases, we obtain good agreement with experiment.
\subsection{Intra-atomic Correlation}
We first calculated one-body correlation-energy increments. For Ca$^{2+}$,
the results are virtually independent of the solid-state surroundings. This was tested by
doing calculations for a free $\rm {Ca}^{2+}$ and a $\rm {Ca}^{2+}$
embedded in point charges. (A cube of 7x7x7 ions was simulated by point
charges $\pm$2, with charges
at the surface planes/edges/corners reduced by factors 2/4/8, respectively.)
In the case of $\rm {O}^{2-}$, of course, the solid-state influence is decisive for stability, and we took it
into account by using an embedding similar to that of Ref.\
\cite{DDFS}: the Pauli repulsion of the 6 nearest $\rm {Ca}^{2+}$ neighbours was simulated by
large-core pseudopotentials \cite{Fuentealba}, while the rest
of a cube of 7x7x7 lattice sites was treated in point-charge approximation again.
A NaCl-like structure with a lattice constant of 4.81 \AA \mbox{ } was adopted.
(The experimental value for the
lattice constant is 4.8032 \AA \mbox{ } at a temperature of T=17.9 K
\cite{LandoltGitter}).
We performed similar calculations for various other finite-cluster approximations of the CaO crystal, in order
to insure that the results are not sensitive to lattice extensions beyond the cube mentioned above.
The results for the one-body correlation-energy increments are shown in Table 3.
It is interesting to note that the absolute value of the Ca $\rightarrow
\rm{Ca}^{2+}$
increment is larger than in the case
of Mg, although the electron density in the valence region of Ca is lower
than for Mg.
The larger correlation contribution for Ca can be rationalized by the fact that excitations into
low-lying unoccupied $d$-orbitals are much more important for Ca than for Mg.
This is a result which would be difficult to explain by
density functional theory: in a local-density framework, higher density leads to a higher
absolute value of the
correlation energy.
In Ref.\ \cite{DDFS} we argued that the increment in correlation energy
$\epsilon(\rm {embedded}$ $\rm O^{2-})-\epsilon(\rm {free}$ ${\rm O})$ is not
just twice the
increment $\epsilon(\rm {free}$ $\rm O^{-})-\epsilon(\rm {free}$ $\rm O)$.
However, comparing the increments
$\epsilon$ (embedded $\rm {O}^{2-})-\epsilon (\rm {embedded} $ $\rm O)$
and $\epsilon$ ( {embedded} $\rm {O}^{-})-\epsilon (\rm embedded $ ${\rm O})$
one finds a factor very close to two.
This can be seen from Table
4 where we compare the increments in the case of MgO.
Thus, for the embedded species linear scaling is appropriate as in the case of the gas-phase isoelectronic series
Ne, Ne$^+$, Ne$^{2+}$: there, the increments in correlation energy are
0.0608 H (Ne$^{2+}$ $\rightarrow$ Ne$^+$) and 0.0652 H (Ne$^{+}$ $\rightarrow$ Ne)
\cite{Davidson}.
Table 4 also shows that the correlation contribution to the electron affinity of the oxygen atom
is {\em smaller} for the embedded species than in the gas phase.
This is due to the fact that energy differences to excited-state configurations become larger
when enclosing O$^{n-}$ in a solid-state cage.
Once again, this is at variance with a LDA description
as the electron density in the case
of the embedded $\rm O^-$
is more compressed than in the case of a free $\rm O^-$.
In Figure 1 we show the charge density distribution of
O$^{2-}$, again in the case of MgO. We used basis functions on both O and Mg;
the Mg $1s$, $2s$ and $2p$-electrons are replaced by a pseudopotential.
One recognizes the minimum near the Mg$^{2+}$ cores, where the
Pauli repulsion prevents the oxygen electrons from penetrating into the Mg$^{2+}$
core region. This way, the solid is stabilized.
The 6$^{th}$ contour line, counting from Mg to O, is the line which
represents a density of 0.002 a.u. This is the density which
encloses about 95 \% of the charge and was proposed as an estimate
of the size of atoms and molecules \cite{Bader}.
The sum of the intra-ionic correlation-energy increments discussed in this subsection
turns out to yield only $\sim$ 60 \% of
the correlation contribution of the cohesive energy of CaO. This percentage is quite similar to that
obtained for MgO \cite{DDFS}, at the same level.
Thus, although MgO and CaO are to a very good approximation purely ionic solids,
the inter-atomic correlation effects to be dealt with in the next subsection play an important role.
\subsection{Two- and three-body increments}
When calculating two-body
correlation-energy increments, point charges or pseudopotentials surrounding
a given ion have to be replaced by
'real' ions. In the case of an additional 'real' $\rm {O}^{2-}$, its next-neighbour shell also
has to be replaced by a cage of pseudopotentials simulating $\rm Ca^{2+}$.
This way the increments shown in Table 3 are obtained.
It turns out that the Ca-O increments are much more important than
the O-O increments, while Ca-Ca increments are negligibly small. The changes with respect to
MgO \cite{DDFS} can easily be rationalized: On the one hand, the
lattice constant is larger than in the case of MgO (4.81 \AA \mbox{ }
vs. 4.21 \AA),
which reduces the van der Waals interaction and makes the O-O increments
smaller. On the other hand, the polarizability of $\rm Ca^{2+}$ is by a
factor of more than 6
higher than that of $\rm Mg^{2+}$
(see for example Ref. \cite{Fuentealba}),
which leads to large Ca-O increments.
We show the van der Waals-like decay in Figure 2
by plotting the two-body increments
O-O for CaO from CCSD calculations (without including weight
factors). By multiplying with the sixth power of the distance, one can
verify the van der Waals-law. Plots for the other two-body increments
are qualitatively similar.
Three-body increments contribute with less
than 2 \% to the correlation piece of the bulk cohesive energy and may safely be neglected, therefore.
A survey of the convergency pattern of the incremental expansion, for both CaO and MgO, is given in Figs.\ 3 and 4.
\subsection{Sum of increments}
Adding up the increments of sections 3.2 and 3.3
(cf.\ Table 5), we obtain between 71 and 78 \% of the
'experimental' correlation contribution to the cohesive energy
which we define as the difference of the experimental cohesive energy (11.0 eV,
\cite{CRC})
plus the zero-point energy (which is
taken into account within the Debye approximation
and is of the order 0.1 eV) minus
the HF binding energy (7.6 eV, Ref. \cite{MackrodtCaO}). The percentage
obtained is slightly
less compared to the case of MgO \cite{DDFS} where 79 to 86 \% were
recovered. One of the reasons for this difference is
that we used a CPP in the case of MgO which covers nearly 100\% of the
core-valence correlation contributions in Mg, while the
explicit treatment of that correlation piece for Ca was less exhaustive.
Another reason is that on the
Hartree-Fock level $f$-functions for Ca (which are not yet implemented in
CRYSTAL) would probably increase the cohesive energy and lower the
'experimental' correlation contribution.
Finally, as in the case of MgO, a significant part of the missing correlation
energy should
be due to basis set errors for the O atom. --
The total cohesive energy
recovered in our calculations is in the range of between 91 and 93 \% of the
experimental value.
Our results are compared in Table 5
to those from density functional calculations. We
choose the results from \cite{DRFASH} where a correlation-only functional
was used and 0.078 to 0.097 H of the correlation contribution to the
cohesive energy were obtained, depending on the specific correlation functional used.
\section{Lattice constants}
At the Hartree-Fock level, the lattice constant is in good agreement with
experiment for MgO
\cite{CausaMgO1,DRFASH,CausaZupan,McCarthy,Catti},
whereas there is a deviation of 0.05 \AA \mbox{ }
in the case of CaO \cite{MackrodtCaO}.
It is interesting, therefore, to study the influence of correlation effects on lattice
constants. In Tables 6 and 7, we give the necessary increments
for MgO and CaO, respectively.
We find two
main effects of correlations. On the one hand, the van der
Waals-interaction leads to a reduction of the lattice spacing since the
attractive interaction is of the form $-\frac{1}{r^6}$
and obviously stronger at shorter distance. On the other hand, we find
that the intra-ionic correlation of the $\rm O^{2-}$-ion forces a larger constant. This can be
understood from the argument that excited configurations are lower in energy
and mix more strongly with the ground-state
determinant if the
$\rm O^{2-}$ is less compressed as explained in section 3.2.
Adding up all these contributions (cf.\ Table 8), they are found to nearly cancel
in the case of MgO and to lead to a reduction of only 0.01 \AA \mbox{ }.
For obtaining this result, we applied a linear fit to the correlation energy
and superimposed it on the HF potential curve of Refs. \cite{Catti,MackrodtCaO}.
We checked the validity of the linear approximation by calculating
selected increments at other lattice constants.
In the case of CaO, the van der Waals-interaction is more important and
the lattice constant is reduced to 4.81 \AA \mbox{ } which is in nice
agreement with the experimental value.
The lattice constants seem to be in better agreement with the experimental
values than those calculated from density functional theory for MgO
\cite{CausaZupan,McCarthy} and CaO \cite{CausaZupan}, where deviations
of $\pm$ 2 \% are found. This is similar to earlier findings for
semiconductors \cite{Beate}.
\section{Conclusion}
We determined the correlation contribution to the cohesive energy of CaO
using an expansion into local increments
recently applied to MgO. Making use of quantum-chemical {\em ab-initio}
configuration-interaction
calculations for evaluating individual increments,
we obtain $\sim$ 80 \% of the
expected value. The missing energy is probably mainly due to
the lack of $g$ and higher polarization functions in our one-particle basis set.
The computed lattice constants show deviations of less than 1\%
from the experimental values. We found two correlation effects on the lattice constants:
the inter-atomic van der Waals-force leads to a reduction, whereas intra-atomic
correlations of the $\rm O^{2-}$ ions lead to an increase of the lattice constant.
The main difference between CaO and MgO is the reduced importance of
the inter-atomic O-O correlations in CaO (due to the larger lattice
constant) and the higher importance of the Ca-O correlations (due to
the higher polarizability of Ca$^{2+}$).
Compared to DFT, the numerical effort of our scheme is significantly higher. However, we feel
that the advantage of the present approach is the high quality and stability
of the results both for atoms, ions as well as for solids. Another advantage
is the possibility of a systematic improvement by using larger basis sets.
We think that the method of local increments is capable now of being routinely applied to ionic
systems, and a systematic study on alkali halides is underway. An
extension to open-shell systems such as NiO is also a project currently under
investigation.
\section*{Acknowledgments}
We would like to thank Prof.\ P.\ Fulde for supporting this work and
Prof.\ W.\ C.\ Nieuwpoort (Groningen)
for interesting suggestions.
We are grateful to Prof.\ H.-J.\ Werner (Stuttgart) for providing the
program package MOLPRO.
| proofpile-arXiv_065-498 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{INTRODUCTION}
The discovery of at least five new $\gamma$-ray pulsars by the Compton
Gamma-Ray Observatory (CGRO) and ROSAT has re-ignited theoretical work on
the physical processes and modeling of high-energy radiation from pulsars.
Including the previously known $\gamma$-ray pulsars, Crab (Nolan et al. 1993)
and Vela (Kanbach et al. 1994), the recent detection of pulsed $\gamma$-rays
from PSR B1706-44 (Thompson et al. 1992), Geminga (Halpern \& Holt 1992,
Bertsch et al. 1992), PSR1509-58 (Wilson et al. 1993), PSR B1055-52
(Fierro et al. 1993), PSR B1951+32 (Ramanamurthy et al. 1995) and possibly
PSR0656+14 (Ramanamurthy et al. 1996) bring the total to at least seven.
For the first time, it is possible to look for the similarities and patterns in
the $\gamma$-ray emission characteristics that may reveal clues to the origin
of this emission. Examples are a preponderance of double pulses, a
$\gamma$-ray luminosity vs. polar cap current correlation, a spectral
hardness vs. characteristic age correlation, and spectral cutoffs above a
few GeV (e.g. Thompson 1996). PSR1509-58 stands out among the known
$\gamma$-ray pulsars as having both an unusually low spectral cutoff energy
(somewhere between 2 and 30 MeV) and the highest inferred surface magnetic
field ($3 \times 10^{13}$ Gauss). It has been detected by the CGRO instruments
operating only in the lowest energy bands, BATSE (Wilson et al. 1993)
and OSSE (Matz et al. 1994), with the higher energy instruments, COMPTEL
(Bennett et al. 1993) and EGRET (Nel et al. 1996), giving upper limits that
require a cutoff or turnover between 2 and 30 MeV. There is no evidence
for pulsed TeV emission (Nel et al. 1992).
There are currently two types of models for $\gamma$-ray pulsars being
investigated in detail. Polar cap models assume that particles are
accelerated along open field lines near the neutron star by strong parallel
electric fields (e.g. Arons 1983). The primary particles induce
electromagnetic cascades through the creation of electron-positron pairs
by either curvature radiation (Daugherty \& Harding 1982, 1994, 1996) or
inverse-Compton radiation (Sturner \& Dermer 1994) $\gamma$-rays.
Outer gap models assume that the primary particles are accelerated along
open field lines in the outer magnetosphere, near the null charge surface,
where the corotation charge changes sign, and where strong electric fields
may develop (Cheng, Ho \& Ruderman 1986a,b; Chiang \& Romani 1992;
Romani \& Yadigaroglu 1995). Since the magnetic fields in the outer gaps
are too low to sustain one-photon pair production cascades, these models
must rely on photon-photon pair production of $\gamma$-rays, interacting
with either non-thermal X-rays from the gap or thermal X-rays from the
neutron star surface, to initiate pair cascades.
Magnetic one-photon pair production, $\gamma\to e^+e^-$, has so far been
the only photon attenuation mechanism assumed to operate in polar cap
cascade models. Another attenuation mechanism, photon splitting
\teq{\gamma\to\gamma\gamma}, will also operate in the high-field regions near pulsar
polar caps but has not yet been included in polar cap model
calculations. The rate of photon splitting increases rapidly with
increasing field strength (Adler 1971), so that it may even be the
dominant attenuation process in the highest field pulsars. There are
several potentially important consequences of photon splitting for
$\gamma$-ray pulsar models. Since photon splitting has no threshold, it
can attenuate photons below the threshold for pair production, $\varepsilon =
2/\sin\theta_{\rm kB}$, and can thus produce cutoffs in the spectrum at
lower energies. Here $\theta_{\rm kB}$ is the angle between the photon
momentum and the magnetic field vectors, and $\varepsilon$ is (hereafter)
expressed in units of $m_ec^2$. When the splitting rate becomes large
enough, splitting can take place during a photon's propagation through
the neutron star magnetosphere \it before \rm the pair production
threshold is crossed (i.e. before an angle \teq{\sim 2/\varepsilon} to the
field is achieved). Consequently, the production of secondary electrons
and positrons in pair cascades will be suppressed. Instead of {\it
pair} cascades, one could have {\it splitting} cascades, where the
high-energy photons split repeatedly until they escape the
magnetosphere. The potential importance of photon splitting in neutron
star applications was suggested by Adler (1971), Mitrofanov \it et al.
\rm (1986) and Baring (1988). Its attenuation and reprocessing
properties have been explored in the contexts of annihilation line
suppression in gamma-ray pulsars (Baring 1993), and spectral formation
of gamma-ray bursts from neutron stars (Baring 1991). Photon splitting
cascades have also been investigated in models of soft $\gamma$-ray
repeaters, where they will soften the photon spectrum very efficiently
with no production of pairs (Baring 1995, Baring \& Harding 1995a,
Harding \& Baring 1996, Chang et al. 1996).
In this paper we examine the importance of photon splitting in
$\gamma$-ray pulsar polar cap models (it presumably will not operate in
the low fields of outer gap models). Following a brief discussion of
the physics of photon splitting in Section~2, we present calculations of
the splitting attenuation lengths and escape energies in the dipole
magnetic field of a neutron star. A preliminary study (Harding, Baring
\& Gonthier 1996) has shown that splitting will be the primary mode of
attenuation of $\gamma$-rays emitted parallel to a magnetic field
\teq{B \gtrsim 0.3 B_{\rm cr} = 1.3\times 10^{13}} Gauss. We then
present, in Section~3, photon splitting cascade models for two cases:
(1) when only one mode of splitting ($\perp \rightarrow
\parallel\parallel$) allowed by the kinematic selection rules (Adler
1971, Shabad 1975) operates, suppressing splitting of photons of
parallel polarization (so that they can only pair produce), but still
permitting photons of perpendicular polarization to either split once or
produce pairs, and (2) when the three splitting modes allowed by CP
(charge-parity) invariance operate, producing mode switching and a
predominantly photon splitting cascade. In Section~4, model cascade
spectra are compared to the observed spectrum of PSR1509-58 to
determine the range of magnetic colatitude emission points (if any) that
can produce a spectral cutoff consistent with the data. These spectra
have cutoff energies that are decreasing functions of the magnetic
colatitude. It is found that a reasonably broad range of polar cap sizes
will accommodate the data, and that strong polarization signatures
appear in the spectra due to the action of photon splitting.
\section{PHOTON SPLITTING AND PAIR CREATION ATTENUATION}
The basic features of magnetic photon splitting \teq{\gamma\to\gamma\gamma} and
the more familiar process of single-photon pair creation \teq{\gamma\to e^+e^-}
are outlined in the next two subsections before investigating their role as
photon attenuation mechanisms in pulsar magnetospheres. Note that throughout
this paper, energies will be rendered dimensionless, for simplicity, using the
scaling factor \teq{m_ec^2}. Magnetic fields will also often be scaled by the
critical field \teq{B_{\rm cr}}; this quantity will be denoted by a prime:
\teq{B'=B/B_{\rm cr}}.
\subsection{Photon Splitting Rates}
The splitting of photons in two in the presence of a strong magnetic
field is an exotic and comparatively recent prediction of quantum
electrodynamics (QED), with the first correct calculations of the
reaction rate being performed in the early '70s (Bialynicka-Birula and
Bialynicki-Birula 1970; Adler \it et al. \rm 1970; Adler 1971). Its
relative obscurity to date (compared, for example, with magnetic pair
creation) in the astrophysical community stems partly from the
mathematical complexity inherent in the computation of the rate.
Splitting is a third-order QED process with a triangular Feynman
diagram. Hence, though splitting is kinematically possible, when
\teq{B=0} it is forbidden by a charge conjugation symmetry of QED known
as Furry's theorem (e.g. see Jauch and Rohrlich, 1980), which states
that ring diagrams that have an odd number of vertices with only
external photon lines generate interaction matrix elements that are
identically zero. This symmetry is broken by the presence of an
external field. The splitting of photons is therefore a purely quantum
effect, and has appreciable reaction rates only when the magnetic field
is at least a significant fraction of the quantum critical field
\teq{B_{\rm cr} = m_e^2c^3/(e\hbar )=4.413\times 10^{13}} Gauss.
Splitting into more than two photons is prohibited in the limit of zero
dispersion because of the lack of available quantum phase space
(Minguzzi 1961).
The reaction rate for splitting is immensely complicated by dispersive
effects (e.g. Adler 1971; Stoneham 1979) caused by the deviation of the
refractive index from unity in the strong field. Consequently,
manageable expressions for the rate of splitting are only possible in
the limit of zero dispersion, and are still then complicated triple
integrations (see Stoneham 1979, and also Ba\u{\i}er, Mil'shte\u{\i}n,
and Sha\u{\i}sultanov 1986 for electric field splitting) due to the
presence of magnetic electron propagators in the matrix element. Hence
simple expressions for the rate of splitting of a photon of energy
\teq{\omega} in a field \teq{B} were first obtained by Bialynicka-Birula
and Bialynicki-Birula (1970), Adler \it et al. \rm (1970) and Adler
(1971) in the low-energy, non-dispersive limit: \teq{\omega B/B_{\rm
cr}\lesssim 1}. The total rate in this limit, averaged over photon
polarizations (Papanyan and Ritus, 1972), is expressible in terms of an
attenuation coefficient
\begin{equation}
T_{\rm sp} (\omega )\;\approx\; {{\alpha^3}\over{10\pi^2}}\,
\dover{1}{\lambda\llap {--}}\, {\biggl({{19}\over{315}}\biggr) }^2\,
B'^6\, {\cal C}(B')\, \omega^5\,\sin^6\theta_{\rm kB} \quad ,
\label{eq:splitotrate}
\end{equation}
where \teq{\alpha =e^2/\hbar c\approx 1/137} is the fine structure
constant, \teq{\lambda\llap {--} =\hbar /(m_ec)} is the Compton wavelength of the
electron, and \teq{\theta_{\rm kB}} is the angle between the photon
momentum and the magnetic field vectors. Here \teq{{\cal C}(B')} is a
strong-field modification factor (derivable, for example, from Eq.~41 of
Stoneham, 1979: see Eq.~[\ref{eq:splitratecorr}] below) that
approximates unity when \teq{B\ll B_{\rm cr}} and scales as \teq{B^{-6}}
for \teq{B\gg B_{\rm cr}}.
The corresponding differential spectral rate for the splitting of photons of
energy \teq{\omega} (with \teq{\omega\ll 1}) into photons of energies
\teq{\omega'} and \teq{\omega -\omega'} is
\begin{equation}
T_{\rm sp}(\omega ,\omega')\;\approx\; 30\,\dover{\omega'^2
(\omega -\omega' )^2}{\omega^5}\, T_{\rm sp}(\omega )\quad .
\label{eq:splitdiffrate}
\end{equation}
Equations~(\ref{eq:splitotrate}) and~(\ref{eq:splitdiffrate}) are valid
(Baring, 1991) when \teq{\omega B' \sin\theta_{\rm kB}\lesssim 1}, which
for pulsar fields and \teq{\omega \sin\theta_{\rm kB} \lesssim 2}, generally corresponds to the regime of weak vacuum dispersion.
Reducing \teq{\theta_{\rm kB}} or
\teq{B} dramatically increases the photon energy required for splitting
to operate in a neutron star environment. The produced photons emerge
at an angle \teq{\theta_{\rm kB}} to the field since splitting is a
collinear process in the low-dispersion limit.
Adler (1971) observed that in the low-energy limit, the splitting rate
was strongly dependent on the polarization states of the initial and
final photons; this feature prompted the suggestion by Adler et al.
(1970) and Usov and Shabad (1983) that photon splitting should be a
powerful polarizing mechanism in pulsars. The polarization-dependent
rates can be taken from Eq.~(23) of Adler (1971), which can be related
to equations~(\ref{eq:splitotrate}) or~(\ref{eq:splitdiffrate}) via
\begin{equation}
T^{\rm sp}_{\perp\to\parallel\parallel}\; =\;
\dover{1}{2}\, T^{\rm sp}_{\parallel\to\perp\parallel}\; =\;
\biggl( \dover{{\cal M}_1^2}{{\cal M}_2^2} {\biggr)}^2\,
T^{\rm sp}_{\perp\to\perp\perp}\; =\; \dover{2 {\cal M}_1^2\,
T_{\rm sp} }{ 3{\cal M}_1^2 + {\cal M}_2^2 } \quad ,
\label{eq:splitpolrate}
\end{equation}
where the scattering amplitude coefficients
\begin{eqnarray}
{\cal M}_1 & = &\dover{1}{B'^4}\int^{\infty}_{0} \dover{ds}{s}\, e^{-s/B'}\,
\Biggl\{ \biggl(-\dover{3}{4s}+\dover{s}{6}\biggr)\,\dover{\cosh s}{\sinh s}
+\dover{3+2s^2}{12\sinh^2s}+\dover{s\cosh s}{2\sinh^3s}\Biggr\}\nonumber\\
{\cal M}_2 & = &\dover{1}{B'^4}\int^{\infty}_{0} \dover{ds}{s}\, e^{-s/B'}\,
\Biggl\{ \dover{3}{4s}\,\dover{\cosh s}{\sinh s} +
\dover{3-4s^2}{4\sinh^2s} - \dover{3s^2}{2\sinh^4s}\Biggr\}
\label{eq:splitcoeff}
\end{eqnarray}
are given in Adler (1971) and Eq.~41 of Stoneham (1979). In the limit
of \teq{B\ll B_{\rm cr}}, \teq{{\cal M}_1\approx 26/315} and \teq{{\cal
M}_2\approx 48/315}, while in the limit of \teq{B\gg B_{\rm cr}},
equation~(\ref{eq:splitcoeff}) produces \teq{{\cal M}_1\approx
1/(6B'^3)} and \teq{{\cal M}_2\approx 1/(3B'^4)}. The factor of two in
the numerator of the right hand side of
equation~(\ref{eq:splitpolrate}) accounts for the duplicity of photons
produced in splitting. The photon polarization labelling convention of
Stoneham (1979) is adopted here (this standard form was not used by
Adler, 1971): the label \teq{\parallel} refers to the state with the
photon's \it electric \rm field vector parallel to the plane containing
the magnetic field and the photon's momentum vector, while \teq{\perp}
denotes the photon's electric field vector being normal to this plane.
Summing over the polarization modes yields the relationship for the
strong-field modification factor in equation~(\ref{eq:splitotrate}):
\begin{equation}
{\cal C}(B')\; =\;\dover{1}{12}\,
{\biggl({{315}\over{19}}\biggr) }^2\, \Bigl( 3{\cal M}_1^2 + {\cal M}_2^2
\Bigr) \quad .
\label{eq:splitratecorr}
\end{equation}
Note that, in the absence of vacuum dispersion, the splitting modes
\teq{\perp\to\perp\parallel}, \teq{\parallel\to\perp\perp} and
\teq{\parallel\to\parallel\parallel} are forbidden by arguments of CP
(charge-parity) invariance in QED (Adler 1971); dispersive effects admit
the possibility of non-collinear photon splitting so that there is a
small but non-zero probability for the \teq{\perp\to\perp\parallel}
channel. Equations~(\ref{eq:splitotrate})--(\ref{eq:splitratecorr})
define the rates to be used in the analyses of this paper, and are valid
for \teq{\omega B'\sin\theta_{\rm kB}\ll 1}. The triple
integral expressions that Stoneham (1979) derives are valid (below pair
creation threshold) for a complete range (i.e. 0 to \teq{\infty}) of the
expansion parameter \teq{\omega B'\sin\theta_{\rm kB}}, but
are not presently in a computational form suitable for use here. Work
is in progress to address this deficiency (Baring \& Harding 1996), and preliminary results indicate that
equations~(\ref{eq:splitotrate})--(\ref{eq:splitratecorr}) approximate
Stoneham's (1979) formulae to better than two percent for \teq{\omega
B'\sin\theta_{\rm kB}\leq 0.2}, and differs by at most a
factor of around 2.5 for \teq{\omega B'\sin\theta_{\rm kB}
\sim 1.5}, the value relevant to the calculations of this paper;
the splitting rate given by Stoneham's formulae initially increase
above the low energy limits as \teq{\omega B'\sin\theta_{\rm kB}} increases.
Recently there has appeared a new result on the rates of photon
splitting. Mentzel, Berg \& Wunner (1994) presented an S-matrix
calculation of the rates for the three polarization modes permitted by
CP invariance that are considered here. While their formal development
is comparable to an earlier S-matrix formulation of splitting in
Melrose \& Parle (1983a,b), their presentation of numerical results
appeared to be in violent disagreement (see also their astrophysical
presentation in Wunner, Sang \& Berg 1995) with the splitting results
obtained via the Schwinger proper-time technique by Adler (1971) and
Stoneham (1979) that comprise
equations~(\ref{eq:splitotrate})--(\ref{eq:splitratecorr}) here. These
results have now been retracted, the disagreement being due to a sign error
in their numerical code (Wilke \& Wunner 1996). The revised results are in
much better agreement with the rates computed by Adler (1971).
However, the revised numerical splitting rates of Wilke \& Wunner (1996) still
differ by as much as a factor of 3 from Baring \& Harding's (1996)
computations of Stoneham's (1979) general formulae.
Ba\u{\i}er, Mil'shte\u{\i}n,
\& Sha\u{\i}sultanov (1996) generate numerical results from their
earlier alternative proper-time calculation (Ba\u{\i}er, Mil'shte\u{\i}n,
\& Sha\u{\i}sultanov 1986) that are in accord with Stoneham's and
Adler's (1971) results and also with those of Baring \& Harding (1996). The
numerical computation of the S-matrix formalism is a formidable task.
The proper-time analysis, though difficult, is more amenable, and has
been reproduced in the limit of \teq{B\ll B_{\rm cr}} by numerous
authors. As the S-matrix and
proper-time techniques should produce equivalent results, and indeed
have done so demonstrably in the case of pair production (see DH83 and
Tsai \& Erber 1974), we choose to use the amenable proper-time results
outlined above in the calculations of this paper.
The above results ignore the fact that the magnetized vacuum is
dispersive and birefringent, so that the phase velocity of light is less
than \teq{c} and depends on the photon polarization. Dispersion can
therefore alter the kinematics of QED processes such as splitting (Adler
1971), and further dramatically complicates the formalism for the rates
(Stoneham 1979). Extensive discussions of dispersion in a magnetized
vacuum are presented by Adler (1971) and Shabad (1975); considerations
of plasma dispersion are not relevant to the problem of gamma-ray
emission from pulsars because they become significant only for densities
in excess of around \teq{10^{27}}cm$^{-3}$, which are only attained at
the stellar surface. Adler (1971) showed that in the limit of \it weak
\rm vacuum dispersion (roughly delineated by $B'\sin\theta_{\rm kB}
\lesssim 1$), where the refractive indices for the polarization states
are {\it very} close to unity, energy and momentum could be simultaneously
conserved only for the splitting mode \teq{\perp\to\parallel\parallel}
(of the modes permitted by CP invariance) below pair production
threshold. This kinematic selection rule was demonstrated for subcritical
fields, where the dispersion is very weak, a regime that generally applies
to gamma-ray pulsar scenarios. Therefore, it is probable that only the
one mode (\teq{\perp\to\parallel\parallel}) of splitting operates in
gamma-ray pulsars. This result may be modified by subtle effects such
as those incurred by field non-uniformity. We adopt a dual scenario in
this paper for the sake of completeness: one in which all CP-permitted
modes of splitting operate, and one in which Adler's kinematic selection
rules are imposed. Note that in magnetar models of soft gamma repeaters
(e.g Baring 1995, Harding and Baring 1996), where supercritical fields
are employed, moderate vacuum dispersion arises. In such a regime, it is
not clear whether Adler's selection rules still endure, since his
analysis implicitly uses weak dispersion limits of linear vacuum
polarization results (e.g. see Shabad 1975), and omits higher order
contributions (e.g. see Melrose and Parle 1983a,b) to the vacuum
polarization (for example, those that couple to photon absorption via
splitting) that become prominent in supercritical fields. Furthermore,
plasma dispersion effects may be quite pertinent to soft gamma repeater
models (e.g. Bulik and Miller 1996), rendering them distinctly different
from pulsar scenarios.
\subsection{Pair Production Rate}
One-photon pair production is a first-order QED process that is quite
familiar to pulsar theorists. It is forbidden in field-free regions due
to the imposition of four-momentum conservation, but takes place in an
external magnetic field, which can absorb momentum perpendicular to \bf
B\rm . The rate (Toll 1952, Klepikov 1954) increases rapidly with
increasing photon energy and transverse magnetic field strength, becoming
significant for $\gamma$-rays above the threshold, $\omega =
2/\sin\theta_{\rm kB}$, and for fields approaching $B_{\rm cr}$. When
the photon energy is near threshold, there may be only a few
kinematically available pair states, and the rate will be resonant at
each pair state threshold, producing a sawtooth structure (Daugherty \&
Harding 1983, hereafter DH83). For photon energies well above
threshold, the number of pair states becomes large, allowing the use of
a more convenient asymptotic expression for the polarization dependent
attenuation coefficient (Klepikov 1954, Tsai \& Erber 1974):
\begin{equation}
T^{\rm pp}_{\parallel,\perp} = {1\over 2}{\alpha\over \lambda\llap {--}} B'
\sin\theta_{\rm kB}\Lambda_{\parallel,\perp}(\chi),
\label{eq:ppasymp}
\end{equation}
\begin{equation}
\Lambda_{\parallel, \perp}(\chi) \approx \left\{
\begin{array}{lr}
(0.31, 0.15)\, \exp \mbox{\Large $(-{4\over 3\chi})$} & \chi \ll 1
\\ \\
(0.72, 0.48) \, \chi^{-1/3} & \chi \gg 1
\end{array} \right. \label{eq:ppratlim}
\end{equation}
where $\chi \equiv (\omega/2)B'\sin\theta_{\rm kB}$.
In polar cap pulsar models (e.g. Sturrock 1971, Ruderman and Sutherland 1975),
high energy radiation is usually emitted at very small angles
to the magnetic field, well below pair threshold (see Harding 1995, for
review). The $\gamma$-ray photons will convert into pairs only after they have
traveled a distance $s$ comparable to the field line radius of curvature
$\rho$, so that $\sin\theta_{\rm kB} \sim s/\rho$. From the above expression,
the pair production rate will be vanishingly small until the argument of the
exponential approaches unity, i.e. when $\omega B'\sin\theta_{\rm kB} \gtrsim
0.2$. Consequently, pair production will occur well above threshold when $B
\ll 0.1 B_{\rm cr}$ and the asymptotic expression will be valid. However when
$B \gtrsim 0.1 B_{\rm cr}$, pair production will occur at or near threshold,
where the asymptotic expression has been shown to fall orders of magnitude
below the exact rate (DH83). In the present calculation, we approximate the
near-threshold reduction in the asymptotic pair production attenuation
coefficient by making the substitution, $\chi \rightarrow \chi/F$, where $F =
1 + 0.42(\omega\sin\theta_{\rm kB}/2)^{-2.7}$ in equation~(\ref{eq:ppratlim})
(DH83). Baring (1988) has derived an analytic expression for the one-photon
pair production rate near threshold which gives a result that agrees numerically
with the approximation of DH83.
Yet even the near-threshold correction to the asymptotic rate becomes poor when
$B \gg 0.1 B_{\rm cr}$ and the photons with parallel and perpendicular
polarization produce pairs only (DH83) in the ground (0,0) and first excited
(0,1) and (1,0) states respectively. Here \teq{(j,k)} denotes the Landau
level quantum numbers of the produced pairs. Therefore when the local $B>0.1
B_{\rm cr}$, instead of the asymptotic form in equation~(\ref{eq:ppratlim}), we
use the exact, polarization-dependent, pair production attenuation
coefficient (DH83), including only the (0,0) pair state for $\parallel$
polarization:
\begin{equation}
T^{\rm pp}_{\parallel} = \dover{\alpha\sin\theta_{\rm kB}}{\lambda\llap {--} \xi
|p_{\hbox{\sevenrm 00}}|}\,\exp(-\xi), \quad
\omega \ge 2/\sin\theta_{\rm kB}\quad ,
\label{eq:tpppar}
\end{equation}
and only the sum of the (0,1) and (1,0) states for $\perp$ polarization:
\begin{equation}
T^{\rm pp}_{\perp} = \dover{\alpha\sin\theta_{\rm kB}}{\lambda\llap {--}\xi
|p_{\hbox{\sevenrm 01}}|}\, (E_0 E_1 + 1 + p_{01}^2)\,\exp(-\xi), \quad
\omega \ge \dover{1+(1 + 2B')^{1/2}}{\sin\theta_{\rm kB}}
\label{eq:tppperp}
\end{equation}
where
\begin{displaymath}
E_0 = (1 + p_{01}^2)^{1/2}\quad , \quad
E_1 = (1 + p_{01}^2 + 2B')^{1/2}
\end{displaymath}
for
\begin{displaymath}
|p_{jk}| = \left[ \dover{\omega^2}{4} \sin^2\theta_{\rm kB} - 1 - (j+k)B' +
\left( \dover{(j-k)B'}{\omega\sin\theta_{\rm kB}} \right)^2\right]^{1/2}
\end{displaymath}
and
\begin{equation}
\xi = \dover{\omega^2}{2B'}\sin^2\theta_{\rm kB}\quad . \label{eq:xi}
\end{equation}
Actually, both the asymptotic and exact mean-free paths ($1/T_{\rm pp}$) are
so small in fields where photons pair produce at threshold that it is, in fact,
not important which rate is used at very high field strengths (i.e. $B' \gtrsim
1$). The pair production rate in this regime thus behaves like a wall at
threshold and photons will pair produce as soon as they satisfy the kinematic
restrictions on \teq{\omega} given in equations~(\ref{eq:tpppar})
and~(\ref{eq:tppperp}). The creation of bound pairs rather than free pairs
is possible in fields $B' \gtrsim 0.1$ (Usov \& Melrose 1995), but this should
not affect the present calculation since we do not model the full pair cascade.
\subsection{Attenuation Lengths}
To assess the relative importance of photon splitting compared to pair
production through a dipole magnetic field, we compute the attenuation length
\teq{L}, defined to be the path length over which the optical depth is unity:
\begin{equation}
\tau(\theta, \varepsilon)\; =\;\int_0^L T(\theta_{\rm kB},\, \omega )\,
ds\; =\; 1\quad , \label{eq:tau}
\end{equation}
where \teq{ds} is the pathlength differential along the photon momentum
vector \teq{{\bf k}}and $T$ is the attenuation coefficient for either
splitting, $T_{\rm sp}$, or pair production, $T_{\rm pp}$. In this
paper, attenuation lengths are computed as averages over polarizations
of the initial photon and, for splitting, sums over the final
polarization states. Here \teq{\theta_{\rm kB}} and the photon energy
\teq{k^{\hat o} = \omega} are functions of the position (e.g. see
equation~[\ref{eq:4momgr}]), specifically measured in the local inertial
frame, while \teq{\theta} is the colatitude of emission and \teq{\varepsilon}
is the photon energy to an observer at infinity; our treatment of curved
spacetime is discussed immediately below. In regions where the path
length is much shorter than both the scale length of the field strength
variation or the radius of curvature of the field, \teq{L} reduces to
the inverse of the attenuation coefficient. In the calculation of the
splitting attenuation lengths, all three CP-permitted modes are assumed to operate. The attenuation length behavior of the individual modes are similar.
\placefigure{fig:geometry}
We assume that test photons are emitted at the neutron star surface and
propagate outward, initially parallel or at a specified angle,
$\theta_{\rm kB,0}$ to the dipole magnetic field (see
Fig.~\ref{fig:geometry} for a depiction of the geometry). Photon
emission in polar cap models of gamma-ray pulsars can occur above the
stellar surface (but see the discussion in Section~5), which would
generate attenuation lengths somewhat longer than those determined here,
due to the \teq{r^{-3}} decay of the field. A surface origin of the
photons is chosen in this paper to provide a simple and concise
presentation of the attenuation properties. We have included the
general relativistic effects of curved spacetime in a Schwarzschild
metric, following the treatment of Gonthier \& Harding (1994, GH94) who
studied the effects of general relativity on photon attenuation via
magnetic pair production. GH94 included the curved spacetime photon
trajectories, the magnetic dipole field in a Schwarzschild metric and
the gravitational redshift of the photon energy. One improvement we
have made here to the treatment of GH94 is to explicitly keep track of
the gravitational redshift of the photon energy as a function of
distance from the neutron star (see Appendix for details). Our analysis
is confined to the Schwarzschild metric because the dynamical timescales
for gamma-ray pulsars are considerably shorter than their period (e.g.
\teq{P=0.15}sec. for PSR1509-58), so that rotation effects in the Kerr
metric can be neglected. We have taken a neutron star mass, $M =
1.4\,M_\odot$ and radius, $R = 10^6$ cm in these calculations.
Fig.~\ref{fig:attenl} illustrates how the attenuation lengths for photon
splitting and pair production vary with energy for different magnetic
colatitudes of the emission point, for surface fields of
\teq{B_0=0.1B_{\rm cr}}, and \teq{B_0=0.7B_{\rm cr}}. A field of $B_0 =
0.7B_{\rm cr}$ is the value of the polar surface field derived from the
magnetic dipole spin-down energy loss (Shapiro \& Teukolsky 1983), using
the measured $P$ and $\dot P$ for PSR1509-58. As noted by Usov \&
Melrose (1995), this is exactly twice the value of the surface field
given by formulae in other sources (Manchester \& Taylor 1977, Michel
1991), which assume (inaccurately) that the dipole magnetic moment
$\mu = B_0\,R^3$ rather than $\mu = B_0\,R^3/2$ for a uniformly
magnetized sphere of radius $R$. The other $\gamma$-ray pulsars have
surface field strengths in the range $1 - 9\times 10^{12}$ G, or
$0.02 - 0.2\,B_{\rm cr}$ (the Crab and Vela pulsars have fields around
$0.2\,B_{\rm cr}$). Note that the attenuation lengths in
Fig.~\ref{fig:attenl} are for unpolarized radiation; the curves for
$\parallel$ and $\perp$ polarization states look very similar.
\placefigure{fig:attenl}
The curves in Fig.~\ref{fig:attenl} have a power-law behavior at high
energies, i.e. for attenuation lengths much less than $10^{6}$ cm, where
the dipole field is almost uniform in direction and of roughly constant
strength. They also exhibit sharp increases at the low energy end,
where photons begin to escape the magnetosphere without attenuation. We
may estimate the behavior of the power-law portions of the attenuation
length curves in Fig~\ref{fig:attenl} as follows. Since the photons are
assumed to initially propagate parallel to the field, the field
curvature will give propagation oblique to the field only after
significant distances are traversed, so that the obliquity of the photon
to the field scales, to first order, as the distance travelled,
\teq{\sin\theta_{\rm kB}\propto s}. Inserting this in
equation~(\ref{eq:splitotrate}) gives a photon splitting attenuation
coefficient \teq{\propto s^6} i.e. an optical depth \teq{\propto \varepsilon^5
s^7}, since \teq{T_{\rm sp}\propto\varepsilon^5}. Inversion then indicates that
the attenuation length should vary as \teq{L \propto \varepsilon^{-5/7}}: this
is borne out in Fig.~\ref{fig:attenl}. For $B_0 \gtrsim 0.1B_{\rm cr}$,
pair production occurs as soon as the threshold $\varepsilon_{\rm th} =
2/\sin\theta_{\rm kB}$ is crossed (cf. Section 2.2) during the photon
propagation in the magnetosphere. Essentially, due to the enormous
creation rate immediately above the threshold, this energy serves as an
impenetrable ``wall'' to the photon. Again, since \teq{\sin\theta_{\rm
kB} \propto s} in the early stages of propagation, the pair production
attenuation length should scale as \teq{L\propto 2/\varepsilon}. These
proportionalities hold in both curved and flat spacetime since general
relativistic effects distort spacetime in a smooth and differentiable
manner (see the Appendix). However, the attenuation lengths computed in
the Schwarzschild metric are about a factor of 1.5 lower than those
computed in flat spacetime (Baring \& Harding 1995b).
The photon splitting attenuation coefficient we have used is strictly
valid only below pair threshold. Hence, the attenuation lengths for
splitting depicted in Fig.~\ref{fig:attenl} can be regarded as only
being symbolic when they exceed those for pair production, since then
pair threshold is reached before splitting occurs. No technically
amenable general expressions for the rate of splitting above pair threshold
exist in the physics literature. But the vicinity of parameter space
just below pair threshold is the regime of importance for $\gamma$-ray
pulsar models, where the emitted photons propagate until they either
split or they reach pair threshold, in which case they pair produce.
The attenuation length curves near the crossover points in
Fig.~\ref{fig:attenl} for $B_0 = 0.7B_{\rm cr}$ will require inclusion
of high energy corrections to the attenuation coefficient (Stoneham
1979) that arise as the $\gamma\to e^+e^-$ threshold is approached.
Currently work is in progress to compute these modifications (Baring and
Harding 1996, in preparation), and preliminary results indicate that the
rate in equation~(\ref{eq:splitotrate}) is quite accurate for
\teq{B\lesssim 0.2B_{\rm cr}}, but increases by factors of at most a few
for \teq{B=0.7B_{\rm cr}} and \teq{\omega =2}, as mentioned in
Section~2.1 above.
\subsection{Escape Energies}
The energy at which the attenuation length becomes infinite defines the
\it escape energy\rm, below which the optical depth is always \teq{\ll
1}, and photons escape the magnetosphere; the existence of such an
escape energy is a consequence of the \teq{r^{-3}} decay of the dipole
field. Escape energies of unpolarized photons for both photon splitting
and pair production are shown in Fig.~\ref{fig:escape} as a function of
magnetic colatitude $\theta$ of the photon emission point for different
values of magnetic field strength (see also Harding, Baring and
Gonthier, 1996). The escape energies clearly decline with \teq{\theta}
and are monotonically decreasing functions of \teq{B} for the range of
fields shown. The divergences as \teq{\theta\to 0} are due to the
divergence of the field line radius of curvature at the poles. There
the maximum angle \teq{\theta_{\rm kB}} achieved before the field falls
off and inhibits attenuation is proportional to the colatitude
\teq{\theta}. For photon splitting, since the rate in
equation~(\ref{eq:splitotrate}) is proportional to
\teq{\omega^5\sin^6\theta_{\rm kB}}, and therefore also the attenuation
length \teq{L}, it follows that the escape energy scales as
\teq{\varepsilon_{\rm esc} \propto \theta^{-6/5}} near the poles (see also
Fig.~\ref{fig:icescape}) as is determined by the condition \teq{L\sim
R}. For pair production, the behaviour of the rate (and therefore
\teq{L}) is dominated by the exponential form in
equation~(\ref{eq:ppratlim}), which then quickly yields a dependence
\teq{\varepsilon_{\rm esc} \propto \theta^{-1}} near the poles for
\teq{B_0\lesssim 0.1B_{\rm cr}}. This behaviour extends to higher
surface fields because production then is at threshold, which determines
\teq{\varepsilon_{\rm esc}\sim 2/\theta_{\rm kB} \propto \theta^{-1}}. At high
fields, $B_0 \gtrsim 0.3B_{\rm cr}$, there is a saturation of the photon
splitting attenuation lengths and escape energies, due to the
diminishing dependence of $B$ in the attenuation coefficient. Likewise,
there is a saturation of the pair production escape energy at fields
above which pair production occurs at threshold. The pair production
escape energy curves are bounded below by the pair threshold
$2/\sin\theta_{\rm kB}$ and merge for high $\theta$, at the pair rest
mass limit, $\varepsilon = 2$, blueshifted by the factor $(1-2GM/Rc^2)^{-1/2} \sim
1.3$. Note that photon splitting can attenuate photons well below pair
threshold. For low fields, pair production escape energies are below
those for splitting, but in high fields, splitting escape energies are
lower at all $\theta$. The escape energies are roughly equal for $B_0
\sim 0.3 B_{\rm cr}$.
\placefigure{fig:escape}
The effects of curved spacetime are quite significant when compared to the
attenuation lengths and the escape energies obtained assuming flat spacetime.
A comparison of the escape energies for splitting and pair production, computed
in flat and curved spacetime, is shown in Fig.~\ref{fig:escurvflat}.
The largest effects are due
to the increase of the surface dipole field strength by roughly a factor of
1.4, and the correction for the gravitational redshift of the photon, which
increases the photon energy by roughly a factor 1.2 in the local inertial frame
at the neutron star surface compared to the energy measured by the observer in
flat space (see the Appendix). The combination of these effects decreases
the photon splitting escape energy by a factor of about 2 compared to flat
spacetime. The decrease in escape energy for pair production is also a factor
of about 2, except at the largest values of $\theta$ and $B'$,
where the pair rest mass limit is reached (cf. Fig.~\ref{fig:escape})
The escape energy is then no longer dependent on
field strength, and the ratio of the curved to flat space escape energy
is just the redshift of the photon energy ($\sim 0.8$)
from the conversion point. This is achieved in the upper right hand corner
of the figure; photon splitting has no such strict limit.
The ratios also become insensitive to \teq{\theta} near the poles since there
the photons move almost radially, thus traveling along straight trajectories,
and the curved-space correction to the field is not changing rapidly with
colatitude. The curvature of the photon trajectory in a Schwarzschild
metric does not affect the escape energies, to first order, except in the case
of emission at large colatitudes, where the photon wavevector makes a large
angle to the radial direction.
\placefigure{fig:escurvflat}
High energy emission from curvature radiation, inverse Compton or
synchrotron by relativistic particles with Lorentz factor $\Gamma$ will
not beam the photons precisely along the magnetic field, but within some
angle $\sim 1/\Gamma$ to the field. Fig.~\ref{fig:icescape} illustrates
the effect on the escape energies of a non-zero angle of emission of the
photons, for the case where the photons are emitted at angles toward the
dipole axis. We have chosen the angle $\theta_{\rm kB,0} = .01$ rad $(=
0.57^\circ$) because it would be the angle at which photons with $\epsilon
\sim 100$ would be emitted through the cyclotron upscattering process,
$\theta_{\rm kB,0} \simeq B'/\epsilon$ (Dermer 1990).
For emission angles $\theta_{\rm kB,0} = 0$ in
Fig.~\ref{fig:icescape}a, which plots \teq{\varepsilon_{\rm esc}} for photon
splitting, $\varepsilon_{\rm esc} \propto B_0^{-6/5}$ for $B_0 \lesssim
0.3B_{\rm cr}$ and $\varepsilon_{\rm esc} \propto \theta^{-6/5}$, for $\theta
\lesssim 20^\circ$, dependences that naturally follow from the form of
equation~(\ref{eq:splitotrate}). Generally, the escape energy is
insensitive to the emission angle for $\theta \gtrsim 10 \theta_{\rm
kB,0}$. For small angles, the escape energy decreases and the
$\theta_{\rm kB,0} = .57^\circ$ curves
flatten below the $\theta_{\rm kB,0} = 0$ curves, converging as
\teq{\theta\to 0} to an energy that is proportional to
\teq{(B_0\sin\theta_{\rm kB,0})^{-6/5}} (see Eq.~[\ref{eq:ergs}]). This
convergence is a consequence of the field along photon trajectories that
originate near the pole being almost uniform and tilted at about angle
\teq{\theta_{\rm kB,0}} to the photon path. In
Fig.~\ref{fig:icescape}b, the same effect is seen for pair creation, but
this time the ``saturation'' is at the redshifted threshold energy
$2(1-2GM/Rc^2)^{1/2}/\sin\theta_{\rm kB,0}$, and is independent of
\teq{B_0}. We note that this behaviour at low colatitudes was observed,
in the case of pair creation in flat spacetime, by Chang, Chen and Ho (1996).
\placefigure{fig:icescape}
An obvious exception to this expected behaviour is seen in
Fig.~\ref{fig:icescape}b for the $B' = 3.1$ curve, where the escape
energy is actually \it larger \rm at small colatitudes
(\teq{1^\circ\lesssim\theta\lesssim 10^\circ}) when the emission angle
$\theta_{\rm kB,0}$ is increased. This counter-intuitive result can be
understood with the aid of Fig.~\ref{fig:wsinthet}, which shows the
increase in $\sin\theta_{\rm kB}$, and $\omega\sin\theta_{\rm kB}$,
determined in the local inertial frame, with path length \teq{s} along the
photon trajectory. Note that (i) the \teq{\theta_{kB,0}=0} curves increase in
proportion to \teq{s} when \teq{s/R\ll 1}, as described in the Appendix,
and (ii) the $\sin\theta_{\rm kB}$ curves increase logarithmically with
$s/R$ when \teq{s/R} is not very small. In this large field, pair
production occurs when the threshold $\omega\sin\theta_{\rm kB} = 2$ is
crossed, at the same path length for both $\theta_{\rm kB,0} = 0$ and
$\theta_{\rm kB,0} = 0.57^\circ$. The differences in the
photon trajectories (which are almost radial) for these two cases are so
small that \teq{s} effectively represents the same height above the stellar
surface for both
\teq{\theta_{\rm kB,0}}. Since \teq{\omega\sin\theta_{\rm kB}\approx 2}
defines the pair creation ``wall'' for both photon paths, the only
difference in escape energies is due to the factor of
\teq{\sin\theta_{\rm kB}} at the front of the pair creation rates in
equations~(\ref{eq:ppasymp})--(\ref{eq:tppperp}). Hence, at the point of
pair creation, the value of $\sin\theta_{\rm kB}$ is smaller for the
$\theta_{\rm kB,0} = 0.57^\circ$ case, and therefore the escape energy
is larger. In flat spacetime, which is not depicted in
Figs.~\ref{fig:icescape} or~\ref{fig:wsinthet}, the crossover point of
the $\sin\theta_{\rm kB}$ curves occurs at the same $s/R$ value as pair
threshold, so that the escape energies are the same at this colatitude
for the two cases (this situation was also observed by Chang, Chen and Ho
1996). Note that as photon splitting does not have the same sudden
onset as pair creation, it takes place over a range of path lengths, mostly
around \teq{0.1\lesssim s/R\lesssim 2}. Over this range,
\teq{\sin\theta_{\rm kB}} in Fig.~\ref{fig:wsinthet} is generally larger
for the \teq{\theta_{\rm kB,0}=0.57^\circ} case so that the splitting
escape energy is correspondingly shorter than for emission parallel to
the field, as is evident in Fig.~\ref{fig:icescape}a.
\placefigure{fig:wsinthet}
\section{CASCADE SPECTRA}
Here we describe briefly our Monte Carlo simulation of photon propagation
and attenuation via splitting and pair creation in neutron star magnetospheres,
together with results for single (Section 3.2) and multiple (Section 3.3)
generations of photon splitting.
\subsection{Monte Carlo Calculation}
We model the spectrum of escaping photons from a cascade above a neutron star
polar cap, including both photon splitting and pair production, by means of a
Monte Carlo simulation. The free parameters specified at the start of the
calculation are the magnetic colatitude $\theta$, the angles $\theta_k$ and
$\phi_k$ (see Fig.~\ref{fig:geometry}), the spectrum,
the height above the surface $z_0 = r-R$
of the photon emission, and the surface magnetic field strength $B_0$ (note
that entities with subscripts `0' designate determination at the stellar
surface). From these quantities, and assuming that $\phi = 0$ without loss of
generality, we compute the four-vectors of the photon position and momentum
that are carried through the computation. Injected photons are sampled from a
power-law distribution,
\begin{equation}
N(\varepsilon) = N_0\varepsilon^{-\alpha}, \quad \varepsilon_{min} < \varepsilon < \varepsilon_{max}
\label{eq:N}
\end{equation}
Polarization is chosen randomly to simulate unpolarized emission; this can
be altered, as desired, for any postulated emission mechanism.
The path of each input photon is traced through the magnetic field, in
curved spacetime, accumulating the survival probabilities for splitting,
$P_{\rm surv}^s$, and for pair production, $P_{\rm surv}^p$, independently:
\begin{equation}
P_{\rm surv}(s) = \exp\Bigl\{-\tau(s)\Bigr\}
\end{equation}
where
\begin{equation}
\tau(s) = \int_0^s T(\theta_{\rm kB}, \omega ) ds'
\end{equation}
is the optical depth along the path. These survival probabilities
implicitly depend on the origin \teq{{\bf r}_0} of the photon and its
energy \teq{\varepsilon} at infinity. In computing the attenuation lengths
(Section 2.3), the photon was assumed to split when the survival
probability reaches $1/e$, i.e. when equation~(\ref{eq:tau}) is
satisfied. In the cascade simulation, the photon may either split or
pair produce. The fate of each cascade photon is determined as follows:
if the combined survival probability, $P_{\rm surv}^sP_{\rm surv}^p >
\Re_1$, where $\Re_1$ is a random number between 0 and 1,
chosen at the emission point,
then the photon escapes; if not, then if the probability that the
photon survives splitting but not pair production, $P_{\rm
surv}^s(1-P_{\rm surv}^p)/(1-P_{\rm surv}^s P_{\rm surv}^p) > \Re_2$,
where $\Re_2$ is a second random number, then the photon pair produces;
otherwise, the photon splits. When the photon splits, the energy of one
of the final photons is sampled from the distribution given in
equation~(\ref{eq:splitdiffrate}) and their polarizations are chosen
from the branching ratios given in equation~(\ref{eq:splitpolrate}).
The energy of the second photon from the splitting is determined simply
from energy conservation, since both final photons are assumed to be
collinear in the direction of the parent photon. Each final photon is
then followed in the same way as the injected photon, with a call to a
recursive procedure that stores photon energies and positions through
many generations of splitting. When the photon pair produces, the code
does not follow the radiation from the pairs but simply returns to the
previous cascade generation. For field strengths typical of gamma-ray
pulsars, the pair radiation, most probably synchrotron or inverse
Compton, will not contribute significantly at the energies near the
escape energy for the cascades where all splitting modes operate. An
exception to this may occur for supercritical surface fields, where
synchrotron photons acquire most of the energy of their primary
electrons. When all splitting modes operate, the number of pair
production events is a small fraction of the number of splitting events
for $B_0 = 0.7B_{\rm cr}$. The cascade photons are followed through
many generations of splitting until all of the photons either escape
or pair produce. The escaping photons are binned in energy and
polarization.
\subsection{Partial Splitting Cascade}
For pulsar applications with subcritical fields, as discussed in
Section 2.1, it is probable that the splitting modes allowed by CP
invariance are further limited by kinematic selection rules to only the
$\perp \rightarrow \parallel\parallel$ mode. This restriction may be
confined to regimes of weak vacuum dispersion and may also depend on
subtleties such as field non-uniformity. Such selection rules would
effectively prevent splitting cascades since $\perp$ photons could split
only into $\parallel$ photons which do not split. Here we compute the
emergent spectra in this type of cascade, a partial cascade, where
$\perp$ mode photons can either pair produce or split into $\parallel$
mode photons, while the $\parallel$ mode photons may only pair produce.
There is a limit of two cascade generations: one splitting and one pair
production. The input spectrum is a power-law (Eq.~[\ref{eq:N}]) with
the parameters: $\varepsilon_{min} = 10^{-3}$, $\varepsilon_{max} = 100$ and $\alpha
= 1.6$. The value of the index $\alpha$ is chosen to match the power-law
fit of the OSSE spectrum of PSR1509-58 (Matz et al. 1994). The maximum
energy of the input spectrum $\varepsilon_{max} = 100$ is chosen to fall above
the 30 MeV maximum possible cutoff or turnover energy of the observed
PSR1509-58 spectrum.
For these runs, injection of 5 - 10 million photons are required
to give adequate statistics. The number of pairs produced relative to
photons in these
partial splitting cascades is obviously higher than in the full
cascades examined in the next section.
Note that in more complete gamma-ray pulsar
models that include the pair radiation, multiple generations of splitting
might still be possible, being interspersed with generations of conventional
synchrotron/pair cascading.
\placefigure{fig:specpar}
Figure~\ref{fig:specpar} shows partial splitting cascade spectra in each
final polarization mode for photons injected parallel to the local
magnetic field at different magnetic colatitudes. The spectra for the
two polarization modes are cutoff at slightly different energies,
reflecting the different escape energies for splitting, which cuts off
the $\perp$ mode, and for pair production, which cuts off the
$\parallel$ mode. There is a slight bump below the cutoff in the $\parallel$
mode spectrum, due to escaping photons from $\perp \rightarrow
\parallel \parallel$ mode splitting, but only an attenuation cutoff in the
$perp$ mode spectrum.
Figure~\ref{fig:icspecpar} illustrates the effect of injecting photons
at an angle (in this case $\theta_{\rm kB,0} = 0.57^\circ = 0.01$ radians)
to the local magnetic field direction, toward the magnetic dipole axis.
The high-energy cutoff decreases, compared to the case of injection
parallel to the field, only in the $\perp$ mode spectrum and
not at all in
the $\parallel$ mode spectrum. This behavior is due to the existence of
a threshold for pair production, but not for splitting and can be seen
from Figs.~\ref{fig:icescape}a and~\ref{fig:icescape}b. For field
strengths well above $B' = 0.1$, where photons pair produce at
threshold, the pair escape energy is much less sensitive to increases in
$\theta_{\rm kB,0}$ than is the splitting escape energy. The partial
cascade spectra therefore become more highly polarized at small
colatitudes when $\theta_{\rm kB,0}$ is increased.
\placefigure{fig:icspecpar}
This effect of strong polarization, both in the energy of the
spectral cutoffs and the spectral shape just below the cutoffs,
all but disappears when photon splitting is omitted from the
calculation, thereby defining a characteristic signature of the action
of \teq{\gamma\to\gamma\gamma}. Pair production has much less distinctive polarization
features. For example, from equations~(\ref{eq:tpppar})
and~(\ref{eq:tppperp}), the ratio of the cutoff energies at pair
creation threshold between the polarization states is
\teq{(1+\sqrt{1+2B'})/2}. For surface fields of \teq{B'=0.7}, threshold
is crossed during photon propagation in regions with much lower fields,
typically \teq{B'\sim 0.1}, so that the spectral cutoff (or escape
energy) differs only by about 5\% between polarizations; such a
difference would be virtually invisible in the emission spectra. Clearly
then, splitting is primarily responsible for polarization features
shown.
\subsection{Full Splitting Cascade}
We now present model cascade spectra for the case where all three
photon splitting modes allowed by CP invariance, $\perp \rightarrow
\parallel \parallel$, $\perp \rightarrow \perp\perp$ and $\parallel
\rightarrow \perp\parallel$, are operating, and multiple generations of
splitting can occur. These cascades also allow for pair production by
photons of either mode. As noted above, for the field strength of $B_0'
= 0.7$ used in the spectral models for PSR1509-58, pair production
occurs in less than $10\%$ of conversions.
Figure~\ref{fig:specpol} shows full splitting cascade spectra in each
final polarization mode for 2 million photons injected parallel to the
local magnetic field (in curved spacetime) at different magnetic
colatitudes. Each cascade spectrum shows a cutoff at roughly
the splitting escape energy for that colatitude (compare to
Fig.~\ref{fig:escape}), and a bump below the cutoff from the escaping
cascade photons. The size of the bump is a function of the number of
photons attenuated above the cutoff, which is dependent on the ratio of
the maximum input energy, $\varepsilon_{max}$, and the escape energy. For
these models, the size of the cascade bump grows with increasing
$\theta$ because $\varepsilon_{max}$ is held constant while the escape energy
is decreasing. The number of splitting generations ranges from 12 when
$\theta = 30^\circ$ to 3 when $\theta = 2^\circ$. The size of the
cascade bump at a particular $\theta$ could of course be larger or
smaller if $\varepsilon_{max}$ were increased or decreased, but the positions
of the cutoffs would not vary. The spectrum of the bump is polarized,
with a well-defined zero in polarization that is a characteristic
signature of the splitting cascade (see Baring 1995). Note that
the polarization modes have reversed their flux dominance in the
cascade bump compared to the partial splitting cascade case.
\placefigure{fig:specpol}
Although we have injected unpolarized photons in these calculations for
simplicity, the relative flux (i.e. spectra integrated over energies)
of the two polarization modes generally has a complicated dependence
on the branching ratios for splitting defined by
equation~(\ref{eq:splitpolrate}), due to the cascading process and the
non-uniformity of the field. Notwithstanding, the polarization at a
given energy does not exceed a limiting value of 3/7
(Baring 1991). The cascade spectra for injection of
polarized photons resemble the spectra in Fig.~\ref{fig:specpol},
though deviations from Fig.~\ref{fig:specpol} are not exactly
proportional to the degree of polarization of the injection spectrum due
to the inherent complexity of the interplay of polarization states in
the cascade.
\placefigure{fig:icspecpol}
As shown in
Figure~\ref{fig:icspecpol}, injecting photons at an angle to the local
field again has a much larger effect at small colatitudes (i.e. for
$\theta \lesssim 100\theta_{\rm kB,0}$). The
high-energy cutoffs in both modes now decrease in energy compared
to the case of injection
parallel to the field, and the sizes of the cascade bumps are larger,
both being consequences of decrease in escape energy (see
Fig.~\ref{fig:icescape}a). This effect is larger at smaller
colatitudes.
\section{PHOTON SPLITTING CASCADE MODELS FOR PSR1509-58}
The multiwavelength spectrum of PSR1509-58, compiled from radio to TeV
energies (Thompson 1996), shows that the peak in the power output from
this pulsar, as is the case for most other $\gamma$-ray pulsars, falls
in the $\gamma$-ray band.
Figures~\ref{fig:datpar}--\ref{fig:daticunpol} show the high energy
portion of this spectrum, near the cutoff, which we compare with our
model spectra at different emission colatitudes. No formal procedure
for fitting the data with the model was followed, since a simple visual
comparison demonstrating the spectral cutoff is sufficient for the
scientific goals of this paper. The $\epsilon^2\,F(\epsilon)$ format
plots equal
energy per logarithmic decade and clearly demonstrates the need for a
cutoff or sharp turnover somewhere between the highest OSSE detected
point at 3 MeV and the lowest EGRET upper limit at 30 MeV. Although
there appears to be a discontinuity between the GINGA data points below
100 keV and the OSSE data points, it is common for separate fits of data
from two different detectors to produce disparate results, even in the
same energy range. Furthermore, the $\epsilon^2\,F(\epsilon)$ format
tends to magnify
the differences. The difference in spectral index of the Ginga and OSSE
fits probably indicates a true break in the power-law spectrum around
100 keV. We have taken the OSSE index for the input spectrum for our
cascade simulation since it most accurately measures the observed
spectrum at the energies of importance for the model. The offset between
the Ginga and OSSE data (or their different spectral indices) does not
impact the conclusions of this paper, since the cascade formation is
determined by the photon population in the upper end of the OSSE range.
Note that while EGRET has obtained upper limits to the pulsed emission
above around 30 MeV, there are earlier reports of a marginal detection
by COS-B (e.g.. Hartmann et al. 1993), with data points lying above the
EGRET limits. This apparent discrepancy remains to be resolved, and we
opt here to consider only the later and superior EGRET observations.
The Comptel point and limits in Figs.~\ref{fig:datpar}--\ref{fig:daticunpol}
are a preliminary analysis of data from VP23 (Hermsen et al. 1996),
showing pulsed flux at 0.75 - 1 MeV and upper limits for the pulsed interval
(50\%) of the light curve.
The cutoffs in the model photon splitting cascade spectra in
Figures~\ref{fig:datpar}--\ref{fig:daticunpol}
do in fact fall in the energy range 3--30
MeV for colatitudes less than around $30^\circ$. At colatitudes greater
than $\sim 30^\circ$ the cutoffs are lower and are in severe conflict
with the OSSE data points. The standard polar cap half-angle in flat
spacetime, $\sin\theta = (\Omega R/c)^{1/2}$, for PSR1509-58 is
$2.14^\circ$. Although curved spacetime corrections to the magnetic
dipole field tend to very slightly decrease the polar cap size (GH94),
the polar cap may be larger than the standard size due to distortion of
the field near the light cylinder by plasma loading (Michel 1991).
The results presented here assume, for simplicity, a single colatitude
of emission for each, i.e. a polar rim rather than an extended cap.
It is easy to envisage that a range of polar cap emission locations will
produce a convolution of the spectra presented here, thereby generating
a spectral turnover corresponding to the maximum colatitude of the cap,
with steeper emission extending up to a cutoff defined by the minimum
colatitude. The EGRET upper limits cannot really discern between a
sharp cutoff or a more modest turnover above the Comptel energy range
and so cast little light on the emission as a function of colatitude
when \teq{\theta\lesssim 2^\circ}.
\placefigure{fig:datpar}
\placefigure{fig:daticpar}
The partial splitting cascade spectra, shown in Figs.~\ref{fig:datpar}
and~\ref{fig:daticpar}, exhibit only modest cascade bumps just below the
cutoff. The limits on colatitude of the model spectra are essentially
determined by the cutoff energy and are restricted by the lowest EGRET
upper limit to $2^\circ \lesssim \theta
\lesssim 25^\circ$ in the $\theta_{\rm kB,0} = 0$ case, and $\theta
\lesssim 25^\circ$ in the $\theta_{\rm kB,0} = 0.57^\circ$ case, where
no lower limit to the colatitude is imposed by the observations (see below).
The model spectra for $\theta = 2^\circ$ and $5^\circ$ are only marginally
consistent with the upper limits. The final revision of the Comptel
data for PSR1509-58 (Bennett et al., in preparation)
may require raising the lower bounds to the
colatitude of emission obtained in this model/data comparison.
The cutoff energies of these polarization-averaged spectra are somewhat
larger than the cutoff energies of the full cascade spectra (see
Figs.~\ref{fig:datunpol} and~\ref{fig:daticunpol}) because the
$\parallel$ mode escape energies are determined solely by pair
production, whose escape energies generally exceed those of splitting at
this field strength (see Fig.~\ref{fig:escape}). This is especially
pronounced in the $\theta_{\rm kB,0}=0.57^\circ$ case, due to the fact
that the pair production escape energy is insensitive to the photon
emission angle for $B \gg 0.1$, as is illustrated in
Fig.~\ref{fig:icescape}b.
The full cascade spectra, shown in Figs.~\ref{fig:datunpol}
and~\ref{fig:daticunpol}, have distinctive bumps below the cutoff due
to the redistribution of photon energies via splitting. The size of
the cascade bump further limits the magnetic colatitudes to $\theta \lesssim
5^\circ$ to avoid conflict with the Comptel upper limits. The lowest EGRET
upper limit restricts the colatitudes to $5^\circ \gtrsim \theta
\gtrsim 2^\circ$ in the
case of emission parallel to the field (Fig.~\ref{fig:datunpol}). In
the case of emission at angle $\theta_{\rm kB,0} = 0.57^\circ$
(Fig.~\ref{fig:daticunpol}), the cutoff energy in the cascade spectra
saturates at small $\theta$ at an energy of 25 MeV (see
Fig.~\ref{fig:icescape}a). Consequently there is no low-energy limit to
$\theta$ in this case. For larger values of $\theta_{\rm kB,0}$, the
spectral cutoffs would saturate at larger values of $\theta$ and at
lower energies. We can estimate the dependence of this saturation
escape energy, $\varepsilon_{\rm esc}^{\rm sat}$, on $\theta_{\rm kB,0}$ and
$B'$ using the expression for the splitting attenuation coefficient in
equation~(\ref{eq:splitotrate}). Assuming that $1/T_{\rm sp} \simeq R$
approximately gives the escape energy:
\begin{equation}
\varepsilon_{\rm esc}^{\rm sat} \simeq 0.077\,(B'\sin\theta_{\rm kB,0})^{-6/5}
\;\; ,\quad B' \lesssim 0.3. \label{eq:ergs}
\end{equation}
This formula quite accurately reproduces the escape energies in
Figs.~\ref{fig:datunpol} and~\ref{fig:daticunpol} since they are only
weakly dependent on \teq{R}, specifically
\teq{ \varepsilon_{\rm esc}^{\rm sat}\propto R^{-1/5}}.
When $\varepsilon_{\rm esc}^{\rm sat} \le 7.8$ (i.e. 4 MeV), cascade spectra at
all colatitudes cutoff below the lowest possible observed cutoff
energy for the PSR1509-58 spectrum. From equation~(\ref{eq:ergs}),
this occurs, for surface emission, at $\theta_{\rm kB,0} \gtrsim 0.03$
for $B_0' = 0.7$. Therefore, splitting cascade spectra from photons
emitted at larger angles to the field will not be compatible with the
spectrum of PSR1509-58. For emission at some distance above the
surface, the limit on $\theta_{\rm kB,0}$ would be higher since it
depends inversely on local field strength.
\placefigure{fig:datunpol}
\placefigure{fig:daticunpol}
All the model spectra in Figures~\ref{fig:datpar}--\ref{fig:daticunpol}
assume emission at the neutron star surface. Emission above the surface
would produce higher cutoff energies at a given colatitude, due to the
decrease in the dipole field strength with $r$. The upper limits on
colatitude
stated above would therefore be less restrictive for non-surface
emission. Furthermore, when the field strength at the emission point is
$B \sim 0.3 B_{\rm cr}$ (at height $30\%$ of the neutron star radius)
the splitting and pair production escape energies are comparable,
reducing the size of the splitting cascade bumps in all cases. At
higher altitudes above the surface, pair production dominates the photon
attenuation and conventional pair cascades (e.g. Daugherty \& Harding
1996) would operate. Synchrotron radiation from the pairs would then
result in a significantly softer emergent spectrum than the input
power-law above the cyclotron energy (\teq{\sqrt{1+2B'}-1\approx
280}keV at the stellar surface, lower at greater radii). Consequently,
in order to match the observations, the input
power-law would have to be harder, and because of the remoteness of the
emission point from the stellar surface, the colatitude \teq{\theta} of
emission would have to be increased substantially.
\section{DISCUSSION}
The results of this paper demonstrate that magnetic photon splitting can
have a significant effect on $\gamma$-ray emission from the higher field
($B_0 \gtrsim 10^{13}$ G) pulsars. It can attenuate the $\gamma$-ray
spectrum at lower energies than magnetic pair production and will do so
without the creation of electron-positron pairs. We have found that in
low fields ($B_0 \lesssim 0.3B_{\rm cr}$) and $\theta_{\rm kB,0} = 0$
initially, photon splitting attenuation lengths are never shorter than
those for pair production. In high fields ($ B_0 \gtrsim 0.3B_{\rm
cr}$), photon splitting lengths fall below those for pair production
below a certain energy which depends on the colatitude $\theta$. Photon
splitting escape energies fall below pair production escape energies for
$B_0 \gtrsim 0.5\,B_{\rm cr}$, so that splitting may produce an observable
signature for $\gamma$-ray pulsars having strong magnetic fields: high
energy spectral cutoffs that are quite polarization-dependent. While
pair creation alone will also generate such cutoffs, their dependence on
photon polarization is far diminished from when splitting is active.
We have modeled the shape of such spectral cutoffs through simulation of
photon splitting cascades near the neutron star surface for the case of
PSR1509-58. Two types of cascades result from different assumptions about
the selection rules governing the photon splitting modes: the ``full
splitting cascades" occur when three modes limited only by CP selection
rules operate and the ``partial splitting cascades" occur when only one
mode permitted by kinematic selection rules operates.
In the full cascades, splitting
dominates the attenuation while in the partial cascades, pair production
ultimately limits the rate at which photon energy degrades. However, the
partial cascades show a distinct polarization signature due to the different
escape energies for splitting and for pair production.
The resulting PSR1509-58 model spectral cutoffs due to splitting and pair
production fall in the required range for virtually all colatitudes
$\lesssim 25^\circ$. However, the shape of the spectrum of full splitting
cascades, due to the large reprocessing bump, is compatible with the
data only for a very small range of colatitudes, $\theta \lesssim 5^\circ$.
From these
results we conclude that, although photon splitting is capable of producing
spectral cutoffs well below EGRET energies regardless of which selection rules
govern the splitting modes, the partial splitting cascades have a much
larger range of phase space in which to operate.
Attenuation through magnetic pair production and photon splitting
near the polar cap will produce $\gamma$-ray spectral cutoffs that should
be roughly a function of surface magnetic field strength, although other
parameters such as polar cap size will come into play. Thus the
$\gamma$-ray pulsar PSR0656+14, having the second highest surface field
of $9.3 \times 10^{12}$ G, should have a cutoff energy between that of
PSR1509-58 and the other $\gamma$-ray pulsars. In fact the unusually large
spectral index of 2.8 measured by EGRET (Ramanamurthy et al. 1996) may
be a pair production/photon splitting cutoff.
It is thus possible to understand why PSR1509-58, with the highest
magnetic field of all the $\gamma$-ray pulsars, has by far the lowest
spectral cutoff energy and is the only $\gamma$-ray pulsar not detected
by EGRET. In the case of Vela (Kanbach et al. 1994), Geminga
(Meyer-Hasselwander et al. 1994) and 1055-52 (Fierro et al. 1993), the
spectral cutoffs observed by EGRET at a few GeV are consistent with
one-photon pair production cascades (Daugherty and Harding 1982, 1996).
Although the escape energies at the neutron star surface for the
spin-down fields of these pulsars ($B_0 \sim 2 - 6 \times 10^{12}$ G) is
below 1 GeV (see Fig.~\ref{fig:icescape}a), curvature radiation from
primary electrons at one to two stellar radii above the surface will
have pair production escape energies of several GeV. However, when the
surface field exceeds $\sim 10^{13}$ G, photon splitting becomes the
dominant attenuation mechanism in the electromagnetic cascades.
In addition, the primary electrons may lose energy to resonant Compton
scattering of thermal X-rays from the neutron star surface (Sturner
1995), rather than to curvature radiation, limiting their acceleration
to much lower energies, typically $\gamma \sim 100$. The resulting
upscattered $\gamma$-ray spectrum is radiated much closer to the surface
and will be cut off by photon splitting well below the EGRET energy range.
It is important to emphasize that pair creation acting alone suffices
to inhibit GeV emission in pulsars with spin-down fields as high as
PSR1509-58, and splitting significantly enhances the attenuation and
pushes spectral cutoffs to lower energies.
If resonant Compton scattering losses limit the polar cap particle
acceleration energies to $\gamma \ll 10^6$ when $B \gtrsim 10^{13}$ G,
then the primary particles will radiate $\gamma$-rays via the cyclotron
upscattering process or CUSP (Dermer 1990). CUSP radiation would then
provide the seed photons for the splitting cascade. The $\gamma$-ray
spectrum for this process for power-law and monoenergetic electrons
scattering thermal blackbody X-ray photons above the neutron star
surface (Daugherty \& Harding 1989) is a power-law with maximum energy
$\varepsilon_{max} \simeq \gamma_c B' = 2 \times 10^3\,B'^2/T_6$ (Dermer 1990),
where $\gamma_c$ is the energy above which the electrons scatter
resonantly and $T_6 \equiv T/10^6$ K is the thermal X-ray temperature.
In the case of PSR1509-58 with $B' = 0.7$, $\varepsilon_{max} \simeq 10^3/T_6$.
Since the thermal surface emission component is not observed due to the
strong non-thermal spectrum seen at X-ray energies (Kawai 1993), $T_6$
is not known. However, PSR1509-58 is young ($\sim 1000$ yr) and
probably has $T_6 \sim 1 - 3$. We would then expect $\varepsilon_{max} \simeq
300 - 10^3$, compatible with our choice of $\varepsilon_{max} = 100$ for the
splitting cascade models.
A dozen or so other radio pulsars have spin-down magnetic fields above
$10^{13}$ G. These pulsars would, like PSR1509-58, have photon splitting
dominated cascades rather than pair cascades, producing lower yields of
electron-positron pairs. It is possible that neutron stars with extremely
high magnetic fields, where splitting is dominant at altitudes up to
several stellar radii, do not produce sufficient pairs for coherent radio
emission, an intriguing possibility. If such neutron stars exist, they
would constitute a new class of radio quiet, low-energy $\gamma$-ray pulsars.
\acknowledgements
We thank Dieter Hartmann and David Thompson for reading the manuscript
and for providing helpful comments, and Wim Hermsen, Alberto Carrami\~nana
and Kevin Bennett for providing preliminary Comptel data for PSR1509-58.
This work was supported through Compton Gamma-Ray Observatory Guest
Investigator Phase 5 and NASA Astrophysics Theory Grants.
MGB thanks the Institute for Theoretical Physics at the University of
California, Santa Barbara for support (under NSF grant PHY94-07194)
during part of the period in which work for this paper was completed.
\clearpage
| proofpile-arXiv_065-499 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |